From 1cb0a71f8f5228a6dda7e9c7339481aa521ac154 Mon Sep 17 00:00:00 2001
From: Jason Christopherson <jchristopherson@hotmail.com>
Date: Wed, 17 Apr 2024 05:59:21 -0500
Subject: [PATCH] Update documentation

---
 doc/index.html                                |   2 +-
 doc/interface/anova.html                      |  10 +-
 .../bootstrap_resampling_routine.html         |   4 +-
 .../bootstrap_statistic_routine.html          |   6 +-
 doc/interface/box_muller_sample.html          |   4 +-
 doc/interface/confidence_interval.html        |  14 +-
 doc/interface/distribution_function.html      |   4 +-
 doc/interface/distribution_property.html      |   2 +-
 doc/interface/iteration_update.html           |   6 +-
 doc/interface/pooled_variance.html            |   8 +-
 doc/interface/regression_function.html        |   6 +-
 doc/lists/files.html                          |   2 +-
 doc/lists/modules.html                        |   2 +-
 doc/lists/procedures.html                     |   2 +-
 doc/lists/types.html                          |   2 +-
 doc/module/fstats.html                        |  14 +-
 doc/module/fstats_allan.html                  |   6 +-
 doc/module/fstats_anova.html                  |  20 +--
 doc/module/fstats_bootstrap.html              |  86 +++++-----
 doc/module/fstats_descriptive_statistics.html |  32 ++--
 doc/module/fstats_distributions.html          |  12 +-
 doc/module/fstats_errors.html                 |  18 +-
 doc/module/fstats_experimental_design.html    |  10 +-
 doc/module/fstats_helper_routines.html        |   6 +-
 doc/module/fstats_hypothesis.html             |  52 +++---
 doc/module/fstats_regression.html             | 130 +++++++--------
 doc/module/fstats_sampling.html               |  10 +-
 doc/module/fstats_smoothing.html              |   8 +-
 doc/module/fstats_special_functions.html      |  28 ++--
 doc/module/fstats_types.html                  |   4 +-
 doc/proc/adjusted_r_squared.html              |  10 +-
 doc/proc/allan_variance.html                  |   8 +-
 doc/proc/bartletts_test.html                  |   6 +-
 doc/proc/beta.html                            |   8 +-
 doc/proc/bootstrap.html                       |  10 +-
 doc/proc/bootstrap_linear_least_squares.html  |  24 +--
 .../bootstrap_nonlinear_least_squares.html    |  34 ++--
 doc/proc/calculate_regression_statistics.html |  14 +-
 doc/proc/correlation.html                     |   8 +-
 doc/proc/covariance.html                      |   8 +-
 doc/proc/covariance_matrix.html               |   8 +-
 doc/proc/design_matrix.html                   |  12 +-
 doc/proc/difference.html                      |   6 +-
 doc/proc/digamma.html                         |   6 +-
 doc/proc/f_test.html                          |   8 +-
 doc/proc/factorial.html                       |   6 +-
 doc/proc/full_factorial.html                  |   4 +-
 doc/proc/get_full_factorial_matrix_size.html  |   8 +-
 doc/proc/incomplete_beta.html                 |  10 +-
 doc/proc/incomplete_gamma_lower.html          |   8 +-
 doc/proc/incomplete_gamma_upper.html          |   8 +-
 doc/proc/jacobian.html                        |   8 +-
 doc/proc/levenes_test.html                    |  13 +-
 doc/proc/linear_least_squares.html            |  22 +--
 doc/proc/lowess.html                          |   4 +-
 doc/proc/mean.html                            |   6 +-
 doc/proc/median.html                          |   6 +-
 doc/proc/nonlinear_least_squares.html         |  32 ++--
 doc/proc/quantile.html                        |   6 +-
 doc/proc/r_squared.html                       |   8 +-
 doc/proc/random_resample.html                 |   4 +-
 doc/proc/regularized_beta.html                |  10 +-
 doc/proc/rejection_sample.html                |  10 +-
 doc/proc/report_array_size_error.html         |   6 +-
 .../report_arrays_not_same_size_error.html    |   4 +-
 doc/proc/report_iteration_count_error.html    |   4 +-
 doc/proc/report_matrix_size_error.html        |   4 +-
 doc/proc/report_memory_error.html             |   4 +-
 doc/proc/report_underdefined_error.html       |   6 +-
 doc/proc/scaled_random_resample.html          |   4 +-
 doc/proc/standard_deviation.html              |   6 +-
 doc/proc/t_test_equal_variance.html           |   6 +-
 doc/proc/t_test_paired.html                   |   8 +-
 doc/proc/t_test_unequal_variance.html         |   6 +-
 doc/proc/trimmed_mean.html                    |   8 +-
 doc/proc/variance.html                        |   6 +-
 doc/search.html                               |   2 +-
 doc/sourcefile/fstats.f90.html                |   2 +-
 doc/sourcefile/fstats_allan.f90.html          |   2 +-
 doc/sourcefile/fstats_anova.f90.html          |   2 +-
 doc/sourcefile/fstats_bootstrap.f90.html      |   2 +-
 .../fstats_descriptive_statistics.f90.html    |   2 +-
 doc/sourcefile/fstats_distributions.f90.html  |   2 +-
 doc/sourcefile/fstats_errors.f90.html         |   2 +-
 .../fstats_experimental_design.f90.html       |   2 +-
 .../fstats_helper_routines.f90.html           |   2 +-
 doc/sourcefile/fstats_hypothesis.f90.html     | 155 +++++++++---------
 doc/sourcefile/fstats_regression.f90.html     |   2 +-
 doc/sourcefile/fstats_sampling.f90.html       |   2 +-
 doc/sourcefile/fstats_smoothing.f90.html      |   2 +-
 .../fstats_special_functions.f90.html         |   2 +-
 doc/sourcefile/fstats_types.f90.html          |   2 +-
 doc/src/fstats_hypothesis.f90                 |   5 +
 doc/tipuesearch/tipuesearch_content.js        |   2 +-
 doc/type/anova_factor.html                    |  10 +-
 doc/type/array_container.html                 |   6 +-
 doc/type/binomial_distribution.html           |   6 +-
 doc/type/bootstrap_regression_statistics.html |  10 +-
 doc/type/bootstrap_statistics.html            |   6 +-
 doc/type/chi_squared_distribution.html        |   2 +-
 doc/type/convergence_info.html                |   2 +-
 doc/type/distribution.html                    |   2 +-
 doc/type/f_distribution.html                  |  10 +-
 doc/type/iteration_controls.html              |   2 +-
 doc/type/lm_solver_options.html               |   6 +-
 doc/type/normal_distribution.html             |   2 +-
 doc/type/regression_statistics.html           |  14 +-
 doc/type/single_factor_anova_table.html       |   2 +-
 doc/type/t_distribution.html                  |   2 +-
 doc/type/two_factor_anova_table.html          |   2 +-
 110 files changed, 608 insertions(+), 593 deletions(-)

diff --git a/doc/index.html b/doc/index.html
index 5ea9ef9..7b74b92 100644
--- a/doc/index.html
+++ b/doc/index.html
@@ -140,7 +140,7 @@ <h3>Derived Types</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/anova.html b/doc/interface/anova.html
index c294646..2c2e24e 100644
--- a/doc/interface/anova.html
+++ b/doc/interface/anova.html
@@ -237,7 +237,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~9"></span>
+              <span class="anchor" id="variable-x~41"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -282,7 +282,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~10"></span>
+              <span class="anchor" id="variable-x~42"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -357,7 +357,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod"></span>
+              <span class="anchor" id="variable-ymod~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -372,7 +372,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err"></span>
+              <span class="anchor" id="variable-err~18"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -416,7 +416,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/bootstrap_resampling_routine.html b/doc/interface/bootstrap_resampling_routine.html
index 9d553de..49c10be 100644
--- a/doc/interface/bootstrap_resampling_routine.html
+++ b/doc/interface/bootstrap_resampling_routine.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~23"></span>
+              <span class="anchor" id="variable-x~34"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -205,7 +205,7 @@ <h3>Description</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/bootstrap_statistic_routine.html b/doc/interface/bootstrap_statistic_routine.html
index 101948e..3d22012 100644
--- a/doc/interface/bootstrap_statistic_routine.html
+++ b/doc/interface/bootstrap_statistic_routine.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~24"></span>
+              <span class="anchor" id="variable-x~35"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -169,7 +169,7 @@ <h3>Arguments</h3>
     </tbody>
   </table>
 
-      <h3>Return Value <span class="anchor" id="variable-rst~18"></span><small>real(kind=real64)</small></h3>
+      <h3>Return Value <span class="anchor" id="variable-rst~29"></span><small>real(kind=real64)</small></h3>
     <p>The resulting statistic.</p>
     <h3>Description</h3>
     <p>Defines the signature of a function for computing the desired
@@ -191,7 +191,7 @@ <h3>Description</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/box_muller_sample.html b/doc/interface/box_muller_sample.html
index 69327d1..6114717 100644
--- a/doc/interface/box_muller_sample.html
+++ b/doc/interface/box_muller_sample.html
@@ -262,7 +262,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~37"></span>
+              <span class="anchor" id="variable-n~12"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -301,7 +301,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/confidence_interval.html b/doc/interface/confidence_interval.html
index 2ae817e..aa24a60 100644
--- a/doc/interface/confidence_interval.html
+++ b/doc/interface/confidence_interval.html
@@ -174,7 +174,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dist"></span>
+              <span class="anchor" id="variable-dist~3"></span>
               class(<a href='..\type\distribution.html'>distribution</a>),
             </td>
 <td>intent(in)</td>
@@ -190,7 +190,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha"></span>
+              <span class="anchor" id="variable-alpha~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -221,7 +221,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~15"></span>
+              <span class="anchor" id="variable-n~14"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -265,7 +265,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dist~2"></span>
+              <span class="anchor" id="variable-dist~4"></span>
               class(<a href='..\type\distribution.html'>distribution</a>),
             </td>
 <td>intent(in)</td>
@@ -281,7 +281,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~2"></span>
+              <span class="anchor" id="variable-alpha~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -297,7 +297,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~16"></span>
+              <span class="anchor" id="variable-x~14"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -336,7 +336,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/distribution_function.html b/doc/interface/distribution_function.html
index 57bac22..3703fe2 100644
--- a/doc/interface/distribution_function.html
+++ b/doc/interface/distribution_function.html
@@ -168,7 +168,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~46"></span>
+              <span class="anchor" id="variable-x~44"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -205,7 +205,7 @@ <h3>Description</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/distribution_property.html b/doc/interface/distribution_property.html
index a0f56a9..e123014 100644
--- a/doc/interface/distribution_property.html
+++ b/doc/interface/distribution_property.html
@@ -190,7 +190,7 @@ <h3>Description</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/iteration_update.html b/doc/interface/iteration_update.html
index a1378a9..1c3db08 100644
--- a/doc/interface/iteration_update.html
+++ b/doc/interface/iteration_update.html
@@ -183,7 +183,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~4"></span>
+              <span class="anchor" id="variable-resid~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -198,7 +198,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~3"></span>
+              <span class="anchor" id="variable-params~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -246,7 +246,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/pooled_variance.html b/doc/interface/pooled_variance.html
index 720c8ce..c57e532 100644
--- a/doc/interface/pooled_variance.html
+++ b/doc/interface/pooled_variance.html
@@ -170,7 +170,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-si"></span>
+              <span class="anchor" id="variable-si~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -186,7 +186,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ni"></span>
+              <span class="anchor" id="variable-ni~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in),</td>
@@ -231,7 +231,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~8"></span>
+              <span class="anchor" id="variable-x~27"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -270,7 +270,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/interface/regression_function.html b/doc/interface/regression_function.html
index 2df513f..c994fd0 100644
--- a/doc/interface/regression_function.html
+++ b/doc/interface/regression_function.html
@@ -168,7 +168,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~2"></span>
+              <span class="anchor" id="variable-params"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -183,7 +183,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~3"></span>
+              <span class="anchor" id="variable-resid"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -231,7 +231,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/lists/files.html b/doc/lists/files.html
index 6933a33..ea8775e 100644
--- a/doc/lists/files.html
+++ b/doc/lists/files.html
@@ -144,7 +144,7 @@ <h1>Source Files</h1>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/lists/modules.html b/doc/lists/modules.html
index b400bab..3fd7cf3 100644
--- a/doc/lists/modules.html
+++ b/doc/lists/modules.html
@@ -161,7 +161,7 @@ <h1>Modules</h1>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/lists/procedures.html b/doc/lists/procedures.html
index 42d26ea..92ef81d 100644
--- a/doc/lists/procedures.html
+++ b/doc/lists/procedures.html
@@ -452,7 +452,7 @@ <h1>Procedures</h1>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/lists/types.html b/doc/lists/types.html
index d538081..82ec6e5 100644
--- a/doc/lists/types.html
+++ b/doc/lists/types.html
@@ -184,7 +184,7 @@ <h1>Derived Types</h1>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats.html b/doc/module/fstats.html
index 17f433b..cf13cd2 100644
--- a/doc/module/fstats.html
+++ b/doc/module/fstats.html
@@ -142,19 +142,19 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
+                  <li class="list-inline-item"><a href='fstats_smoothing.html'>fstats_smoothing</a></li>
+                  <li class="list-inline-item"><a href='fstats_allan.html'>fstats_allan</a></li>
                   <li class="list-inline-item"><a href='fstats_helper_routines.html'>fstats_helper_routines</a></li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_sampling.html'>fstats_sampling</a></li>
                   <li class="list-inline-item"><a href='fstats_regression.html'>fstats_regression</a></li>
-                  <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
-                  <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
-                  <li class="list-inline-item"><a href='fstats_allan.html'>fstats_allan</a></li>
                   <li class="list-inline-item"><a href='fstats_anova.html'>fstats_anova</a></li>
-                  <li class="list-inline-item"><a href='fstats_sampling.html'>fstats_sampling</a></li>
+                  <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
+                  <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
                   <li class="list-inline-item"><a href='fstats_bootstrap.html'>fstats_bootstrap</a></li>
-                  <li class="list-inline-item"><a href='fstats_smoothing.html'>fstats_smoothing</a></li>
                   <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
                   <li class="list-inline-item"><a href='fstats_experimental_design.html'>fstats_experimental_design</a></li>
                   <li class="list-inline-item"><a href='fstats_hypothesis.html'>fstats_hypothesis</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
               </ul>
             </li>
         </ul>
@@ -189,7 +189,7 @@ <h3 class="card-header card-title bg-light">Uses</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_allan.html b/doc/module/fstats_allan.html
index e401804..9db4011 100644
--- a/doc/module/fstats_allan.html
+++ b/doc/module/fstats_allan.html
@@ -189,7 +189,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~22"></span>
+              <span class="anchor" id="variable-x~33"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -220,7 +220,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~12"></span>
+              <span class="anchor" id="variable-err~15"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -273,7 +273,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_anova.html b/doc/module/fstats_anova.html
index 7de8843..9ca107c 100644
--- a/doc/module/fstats_anova.html
+++ b/doc/module/fstats_anova.html
@@ -164,11 +164,11 @@ <h3 class="card-header card-title bg-light">Uses</h3>
             <li class="list-group-item">
               <ul class="list-inline">
                   <li class="list-inline-item">ferror</li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
                   <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/ieee_arithmetic'>ieee_arithmetic</a></li>
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
                   <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/ieee_arithmetic'>ieee_arithmetic</a></li>
               </ul>
             </li>
         </ul>
@@ -279,7 +279,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~9"></span>
+              <span class="anchor" id="variable-x~41"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -323,7 +323,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~10"></span>
+              <span class="anchor" id="variable-x~42"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -397,7 +397,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod"></span>
+              <span class="anchor" id="variable-ymod~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -412,7 +412,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err"></span>
+              <span class="anchor" id="variable-err~18"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -474,7 +474,7 @@ <h4>Components</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof"></span>
+              <span class="anchor" id="variable-dof~7"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -508,7 +508,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-probability"></span>
+              <span class="anchor" id="variable-probability~3"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -890,7 +890,7 @@ <h4>Components</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_bootstrap.html b/doc/module/fstats_bootstrap.html
index 9f8ef8f..e780610 100644
--- a/doc/module/fstats_bootstrap.html
+++ b/doc/module/fstats_bootstrap.html
@@ -188,14 +188,14 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
+                  <li class="list-inline-item">linalg</li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
                   <li class="list-inline-item"><a href='fstats_regression.html'>fstats_regression</a></li>
-                  <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
                   <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
-                  <li class="list-inline-item">linalg</li>
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
                   <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
+                  <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
                   <li class="list-inline-item"><a href='https://www.openmp.org/spec-html/5.1/openmpch3.html#x156-1890003'>omp_lib</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
               </ul>
             </li>
         </ul>
@@ -238,7 +238,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~23"></span>
+              <span class="anchor" id="variable-x~34"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -303,7 +303,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~24"></span>
+              <span class="anchor" id="variable-x~35"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -400,7 +400,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error"></span>
+              <span class="anchor" id="variable-standard_error~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -417,7 +417,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-t_statistic"></span>
+              <span class="anchor" id="variable-t_statistic~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -429,7 +429,7 @@ <h4>Components</h4>
                 <td></td>
                 <td></td>
             <td>
-<p>The T-statistic for the model coefficient.</p><a href="../type/bootstrap_regression_statistics.html#variable-t_statistic" class="pull-right"><emph>Read more&hellip;</emph></a>
+<p>The T-statistic for the model coefficient.</p><a href="../type/bootstrap_regression_statistics.html#variable-t_statistic~2" class="pull-right"><emph>Read more&hellip;</emph></a>
             </td>
         </tr>
         <tr>
@@ -535,7 +535,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error~2"></span>
+              <span class="anchor" id="variable-standard_error~3"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -636,7 +636,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~27"></span>
+              <span class="anchor" id="variable-x~38"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -651,7 +651,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method"></span>
+              <span class="anchor" id="variable-method~2"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -683,7 +683,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~3"></span>
+              <span class="anchor" id="variable-alpha~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -739,7 +739,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order"></span>
+              <span class="anchor" id="variable-order~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -756,7 +756,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept"></span>
+              <span class="anchor" id="variable-intercept~3"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -772,7 +772,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~28"></span>
+              <span class="anchor" id="variable-x~39"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -788,7 +788,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~3"></span>
+              <span class="anchor" id="variable-y~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -804,7 +804,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-coeffs"></span>
+              <span class="anchor" id="variable-coeffs~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -820,7 +820,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~2"></span>
+              <span class="anchor" id="variable-ymod~3"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -835,7 +835,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid"></span>
+              <span class="anchor" id="variable-resid~8"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -866,7 +866,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats"></span>
+              <span class="anchor" id="variable-stats~3"></span>
               type(<a href='..\type\bootstrap_regression_statistics.html'>bootstrap_regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -902,7 +902,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~4"></span>
+              <span class="anchor" id="variable-alpha~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -919,7 +919,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~2"></span>
+              <span class="anchor" id="variable-method~3"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -952,7 +952,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~13"></span>
+              <span class="anchor" id="variable-err~16"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1004,7 +1004,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun"></span>
+              <span class="anchor" id="variable-fun~8"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -1019,7 +1019,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~29"></span>
+              <span class="anchor" id="variable-x~40"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1034,7 +1034,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~4"></span>
+              <span class="anchor" id="variable-y~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1049,7 +1049,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params"></span>
+              <span class="anchor" id="variable-params~7"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -1066,7 +1066,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~3"></span>
+              <span class="anchor" id="variable-ymod~4"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1082,7 +1082,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~2"></span>
+              <span class="anchor" id="variable-resid~9"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1113,7 +1113,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-weights"></span>
+              <span class="anchor" id="variable-weights~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1129,7 +1129,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-maxp"></span>
+              <span class="anchor" id="variable-maxp~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1147,7 +1147,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-minp"></span>
+              <span class="anchor" id="variable-minp~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1166,7 +1166,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~2"></span>
+              <span class="anchor" id="variable-stats~4"></span>
               type(<a href='..\type\bootstrap_regression_statistics.html'>bootstrap_regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -1182,7 +1182,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~5"></span>
+              <span class="anchor" id="variable-alpha~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1199,7 +1199,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-controls"></span>
+              <span class="anchor" id="variable-controls~5"></span>
               type(<a href='..\type\iteration_controls.html'>iteration_controls</a>),
             </td>
 <td>intent(in),</td>
@@ -1214,7 +1214,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-settings"></span>
+              <span class="anchor" id="variable-settings~2"></span>
               type(<a href='..\type\lm_solver_options.html'>lm_solver_options</a>),
             </td>
 <td>intent(in),</td>
@@ -1229,7 +1229,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-info"></span>
+              <span class="anchor" id="variable-info~4"></span>
               type(<a href='..\type\convergence_info.html'>convergence_info</a>),
             </td>
 <td>intent(out),</td>
@@ -1265,7 +1265,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~3"></span>
+              <span class="anchor" id="variable-method~4"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -1298,7 +1298,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~14"></span>
+              <span class="anchor" id="variable-err~17"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1350,7 +1350,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~25"></span>
+              <span class="anchor" id="variable-x~36"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1406,7 +1406,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~26"></span>
+              <span class="anchor" id="variable-x~37"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1461,7 +1461,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_descriptive_statistics.html b/doc/module/fstats_descriptive_statistics.html
index 18b5732..df20234 100644
--- a/doc/module/fstats_descriptive_statistics.html
+++ b/doc/module/fstats_descriptive_statistics.html
@@ -167,11 +167,11 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
-                  <li class="list-inline-item"><a href='fstats_types.html'>fstats_types</a></li>
-                  <li class="list-inline-item">ferror</li>
                   <li class="list-inline-item">linalg</li>
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
+                  <li class="list-inline-item">ferror</li>
                   <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_types.html'>fstats_types</a></li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
               </ul>
             </li>
         </ul>
@@ -216,7 +216,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-si"></span>
+              <span class="anchor" id="variable-si~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -232,7 +232,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ni"></span>
+              <span class="anchor" id="variable-ni~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in),</td>
@@ -276,7 +276,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~8"></span>
+              <span class="anchor" id="variable-x~27"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -329,7 +329,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~7"></span>
+              <span class="anchor" id="variable-x~26"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -344,7 +344,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y"></span>
+              <span class="anchor" id="variable-y~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -389,7 +389,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x"></span>
+              <span class="anchor" id="variable-x~20"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -434,7 +434,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~4"></span>
+              <span class="anchor" id="variable-x~23"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -481,7 +481,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~5"></span>
+              <span class="anchor" id="variable-x~24"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -541,7 +541,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~3"></span>
+              <span class="anchor" id="variable-x~22"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -586,7 +586,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~6"></span>
+              <span class="anchor" id="variable-x~25"></span>
               real(kind=real64),
             </td>
 <td>intent(inout),</td>
@@ -602,7 +602,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p"></span>
+              <span class="anchor" id="variable-p~14"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -649,7 +649,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~2"></span>
+              <span class="anchor" id="variable-x~21"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -695,7 +695,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_distributions.html b/doc/module/fstats_distributions.html
index a3b1405..ee7e1b5 100644
--- a/doc/module/fstats_distributions.html
+++ b/doc/module/fstats_distributions.html
@@ -167,9 +167,9 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
+                  <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
                   <li class="list-inline-item"><a href='fstats_helper_routines.html'>fstats_helper_routines</a></li>
                   <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/ieee_arithmetic'>ieee_arithmetic</a></li>
-                  <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
                   <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
               </ul>
             </li>
@@ -227,7 +227,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~46"></span>
+              <span class="anchor" id="variable-x~44"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -338,7 +338,7 @@ <h4>Components</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~39"></span>
+              <span class="anchor" id="variable-n~37"></span>
               integer(kind=int32),
             </td>
               <td>public</td>
@@ -577,7 +577,7 @@ <h4>Components</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-d1~3"></span>
+              <span class="anchor" id="variable-d1~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -594,7 +594,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-d2~3"></span>
+              <span class="anchor" id="variable-d2~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -865,7 +865,7 @@ <h4>Type-Bound Procedures</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_errors.html b/doc/module/fstats_errors.html
index 224493e..5be8829 100644
--- a/doc/module/fstats_errors.html
+++ b/doc/module/fstats_errors.html
@@ -366,7 +366,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~7"></span>
+              <span class="anchor" id="variable-err~20"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -426,7 +426,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-actual"></span>
+              <span class="anchor" id="variable-actual~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -467,7 +467,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~9"></span>
+              <span class="anchor" id="variable-err~22"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -582,7 +582,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~11"></span>
+              <span class="anchor" id="variable-err~24"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -667,7 +667,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~8"></span>
+              <span class="anchor" id="variable-err~21"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -797,7 +797,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~6"></span>
+              <span class="anchor" id="variable-err~19"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -867,7 +867,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~10"></span>
+              <span class="anchor" id="variable-err~23"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -912,7 +912,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-actual~2"></span>
+              <span class="anchor" id="variable-actual~4"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -952,7 +952,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_experimental_design.html b/doc/module/fstats_experimental_design.html
index e2d2a7d..b331cc2 100644
--- a/doc/module/fstats_experimental_design.html
+++ b/doc/module/fstats_experimental_design.html
@@ -227,7 +227,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~3"></span>
+              <span class="anchor" id="variable-err~14"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -290,7 +290,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-m"></span>
+              <span class="anchor" id="variable-m~6"></span>
               integer(kind=int32),
             </td>
 <td>intent(out)</td>
@@ -305,7 +305,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~13"></span>
+              <span class="anchor" id="variable-n~26"></span>
               integer(kind=int32),
             </td>
 <td>intent(out)</td>
@@ -320,7 +320,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~2"></span>
+              <span class="anchor" id="variable-err~13"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -364,7 +364,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_helper_routines.html b/doc/module/fstats_helper_routines.html
index 7ee0f3d..c2408bb 100644
--- a/doc/module/fstats_helper_routines.html
+++ b/doc/module/fstats_helper_routines.html
@@ -189,7 +189,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~12"></span>
+              <span class="anchor" id="variable-x~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -235,7 +235,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~13"></span>
+              <span class="anchor" id="variable-x~12"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -281,7 +281,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_hypothesis.html b/doc/module/fstats_hypothesis.html
index e2a39ca..8dbd229 100644
--- a/doc/module/fstats_hypothesis.html
+++ b/doc/module/fstats_hypothesis.html
@@ -166,13 +166,13 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
                   <li class="list-inline-item"><a href='fstats_types.html'>fstats_types</a></li>
                   <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
-                  <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/ieee_arithmetic'>ieee_arithmetic</a></li>
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
                   <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/ieee_arithmetic'>ieee_arithmetic</a></li>
               </ul>
             </li>
         </ul>
@@ -221,7 +221,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dist"></span>
+              <span class="anchor" id="variable-dist~3"></span>
               class(<a href='..\type\distribution.html'>distribution</a>),
             </td>
 <td>intent(in)</td>
@@ -237,7 +237,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha"></span>
+              <span class="anchor" id="variable-alpha~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -268,7 +268,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~15"></span>
+              <span class="anchor" id="variable-n~14"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -311,7 +311,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dist~2"></span>
+              <span class="anchor" id="variable-dist~4"></span>
               class(<a href='..\type\distribution.html'>distribution</a>),
             </td>
 <td>intent(in)</td>
@@ -327,7 +327,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~2"></span>
+              <span class="anchor" id="variable-alpha~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -343,7 +343,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~16"></span>
+              <span class="anchor" id="variable-x~14"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -397,7 +397,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~20"></span>
+              <span class="anchor" id="variable-x~18"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -427,7 +427,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~7"></span>
+              <span class="anchor" id="variable-p~12"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -515,7 +515,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~6"></span>
+              <span class="anchor" id="variable-p~11"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -532,7 +532,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof1~2"></span>
+              <span class="anchor" id="variable-dof1"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -547,7 +547,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof2~2"></span>
+              <span class="anchor" id="variable-dof2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -587,7 +587,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~21"></span>
+              <span class="anchor" id="variable-x~19"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -617,7 +617,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~8"></span>
+              <span class="anchor" id="variable-p~13"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -634,7 +634,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~5"></span>
+              <span class="anchor" id="variable-err~12"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -720,7 +720,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~3"></span>
+              <span class="anchor" id="variable-p~8"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -737,7 +737,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~2"></span>
+              <span class="anchor" id="variable-dof~4"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -822,7 +822,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~5"></span>
+              <span class="anchor" id="variable-p~10"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -839,7 +839,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~4"></span>
+              <span class="anchor" id="variable-dof~6"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -854,7 +854,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~4"></span>
+              <span class="anchor" id="variable-err~11"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -944,7 +944,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~4"></span>
+              <span class="anchor" id="variable-p~9"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -961,7 +961,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~3"></span>
+              <span class="anchor" id="variable-dof~5"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1001,7 +1001,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_regression.html b/doc/module/fstats_regression.html
index d8f17b2..1be312b 100644
--- a/doc/module/fstats_regression.html
+++ b/doc/module/fstats_regression.html
@@ -207,15 +207,15 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
+                  <li class="list-inline-item">linalg</li>
+                  <li class="list-inline-item">blas</li>
                   <li class="list-inline-item">ferror</li>
+                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
                   <li class="list-inline-item"><a href='fstats_descriptive_statistics.html'>fstats_descriptive_statistics</a></li>
                   <li class="list-inline-item"><a href='fstats_distributions.html'>fstats_distributions</a></li>
-                  <li class="list-inline-item">linalg</li>
-                  <li class="list-inline-item"><a href='fstats_hypothesis.html'>fstats_hypothesis</a></li>
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
                   <li class="list-inline-item"><a href='fstats_special_functions.html'>fstats_special_functions</a></li>
-                  <li class="list-inline-item">blas</li>
-                  <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
+                  <li class="list-inline-item"><a href='fstats_hypothesis.html'>fstats_hypothesis</a></li>
               </ul>
             </li>
         </ul>
@@ -355,7 +355,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~4"></span>
+              <span class="anchor" id="variable-resid~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -370,7 +370,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~3"></span>
+              <span class="anchor" id="variable-params~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -448,7 +448,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~2"></span>
+              <span class="anchor" id="variable-params"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -463,7 +463,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~3"></span>
+              <span class="anchor" id="variable-resid"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -969,7 +969,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~4"></span>
+              <span class="anchor" id="variable-method"></span>
               integer(kind=int32),
             </td>
               <td>public</td>
@@ -1049,7 +1049,7 @@ <h4>Components</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-probability~3"></span>
+              <span class="anchor" id="variable-probability"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -1066,12 +1066,12 @@ <h4>Components</h4>
 low probability (p-value), typically 0.05 or lower; however, a 
 p-value of up to ~0.2 may be acceptable dependent upon the 
 problem.  Typically any p-value larger than ~0.2 indicates the 
-parameter is not statistically important for the model.</p><a href="../type/regression_statistics.html#variable-probability~3" class="pull-right"><emph>Read more&hellip;</emph></a>
+parameter is not statistically important for the model.</p><a href="../type/regression_statistics.html#variable-probability" class="pull-right"><emph>Read more&hellip;</emph></a>
             </td>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error~3"></span>
+              <span class="anchor" id="variable-standard_error"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -1083,12 +1083,12 @@ <h4>Components</h4>
                 <td></td>
                 <td></td>
             <td>
-<p>The standard error for the model coefficient.</p><a href="../type/regression_statistics.html#variable-standard_error~3" class="pull-right"><emph>Read more&hellip;</emph></a>
+<p>The standard error for the model coefficient.</p><a href="../type/regression_statistics.html#variable-standard_error" class="pull-right"><emph>Read more&hellip;</emph></a>
             </td>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-t_statistic~2"></span>
+              <span class="anchor" id="variable-t_statistic"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -1100,7 +1100,7 @@ <h4>Components</h4>
                 <td></td>
                 <td></td>
             <td>
-<p>The T-statistic for the model coefficient.</p><a href="../type/regression_statistics.html#variable-t_statistic~2" class="pull-right"><emph>Read more&hellip;</emph></a>
+<p>The T-statistic for the model coefficient.</p><a href="../type/regression_statistics.html#variable-t_statistic" class="pull-right"><emph>Read more&hellip;</emph></a>
             </td>
         </tr>
     </tbody>
@@ -1137,7 +1137,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~9"></span>
+              <span class="anchor" id="variable-p"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -1152,7 +1152,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~31"></span>
+              <span class="anchor" id="variable-x~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1184,7 +1184,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~16"></span>
+              <span class="anchor" id="variable-err~2"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1235,7 +1235,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~6"></span>
+              <span class="anchor" id="variable-resid~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1250,7 +1250,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~4"></span>
+              <span class="anchor" id="variable-params~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1265,7 +1265,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~5"></span>
+              <span class="anchor" id="variable-c~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1280,7 +1280,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~8"></span>
+              <span class="anchor" id="variable-alpha~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1297,7 +1297,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~20"></span>
+              <span class="anchor" id="variable-err~6"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1349,7 +1349,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~32"></span>
+              <span class="anchor" id="variable-x~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1364,7 +1364,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~5"></span>
+              <span class="anchor" id="variable-y"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -1409,7 +1409,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~30"></span>
+              <span class="anchor" id="variable-x"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1441,7 +1441,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~15"></span>
+              <span class="anchor" id="variable-err"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1498,7 +1498,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~34"></span>
+              <span class="anchor" id="variable-x~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1514,7 +1514,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~3"></span>
+              <span class="anchor" id="variable-c~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1529,7 +1529,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~18"></span>
+              <span class="anchor" id="variable-err~4"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1579,7 +1579,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order~2"></span>
+              <span class="anchor" id="variable-order"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -1595,7 +1595,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept~3"></span>
+              <span class="anchor" id="variable-intercept"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -1611,7 +1611,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~33"></span>
+              <span class="anchor" id="variable-x~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1627,7 +1627,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~2"></span>
+              <span class="anchor" id="variable-c"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1644,7 +1644,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~17"></span>
+              <span class="anchor" id="variable-err~3"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1688,7 +1688,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun~2"></span>
+              <span class="anchor" id="variable-fun"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -1718,7 +1718,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~5"></span>
+              <span class="anchor" id="variable-params~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1815,7 +1815,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~21"></span>
+              <span class="anchor" id="variable-err~7"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -1861,7 +1861,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order~3"></span>
+              <span class="anchor" id="variable-order~2"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -1878,7 +1878,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept~4"></span>
+              <span class="anchor" id="variable-intercept~2"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -1894,7 +1894,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~35"></span>
+              <span class="anchor" id="variable-x~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1910,7 +1910,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~6"></span>
+              <span class="anchor" id="variable-y~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -1926,7 +1926,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-coeffs~3"></span>
+              <span class="anchor" id="variable-coeffs"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1941,7 +1941,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~4"></span>
+              <span class="anchor" id="variable-ymod"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1956,7 +1956,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~5"></span>
+              <span class="anchor" id="variable-resid~3"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -1972,7 +1972,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~4"></span>
+              <span class="anchor" id="variable-stats"></span>
               type(<a href='..\type\regression_statistics.html'>regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -1989,7 +1989,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~7"></span>
+              <span class="anchor" id="variable-alpha"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -2006,7 +2006,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~19"></span>
+              <span class="anchor" id="variable-err~5"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -2054,7 +2054,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun~3"></span>
+              <span class="anchor" id="variable-fun~2"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -2069,7 +2069,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~36"></span>
+              <span class="anchor" id="variable-x~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -2084,7 +2084,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~7"></span>
+              <span class="anchor" id="variable-y~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -2099,7 +2099,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~6"></span>
+              <span class="anchor" id="variable-params~5"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -2116,7 +2116,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~5"></span>
+              <span class="anchor" id="variable-ymod~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -2132,7 +2132,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~7"></span>
+              <span class="anchor" id="variable-resid~5"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -2148,7 +2148,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-weights~2"></span>
+              <span class="anchor" id="variable-weights"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -2164,7 +2164,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-maxp~2"></span>
+              <span class="anchor" id="variable-maxp"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -2182,7 +2182,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-minp~2"></span>
+              <span class="anchor" id="variable-minp"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -2201,7 +2201,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~5"></span>
+              <span class="anchor" id="variable-stats~2"></span>
               type(<a href='..\type\regression_statistics.html'>regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -2217,7 +2217,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~9"></span>
+              <span class="anchor" id="variable-alpha~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -2234,7 +2234,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-controls~2"></span>
+              <span class="anchor" id="variable-controls"></span>
               type(<a href='..\type\iteration_controls.html'>iteration_controls</a>),
             </td>
 <td>intent(in),</td>
@@ -2249,7 +2249,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-settings~2"></span>
+              <span class="anchor" id="variable-settings"></span>
               type(<a href='..\type\lm_solver_options.html'>lm_solver_options</a>),
             </td>
 <td>intent(in),</td>
@@ -2264,7 +2264,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-info~2"></span>
+              <span class="anchor" id="variable-info"></span>
               type(<a href='..\type\convergence_info.html'>convergence_info</a>),
             </td>
 <td>intent(out),</td>
@@ -2296,7 +2296,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~22"></span>
+              <span class="anchor" id="variable-err~8"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -2348,7 +2348,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_sampling.html b/doc/module/fstats_sampling.html
index 9c61d30..6880121 100644
--- a/doc/module/fstats_sampling.html
+++ b/doc/module/fstats_sampling.html
@@ -299,7 +299,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~37"></span>
+              <span class="anchor" id="variable-n~12"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -367,7 +367,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~38"></span>
+              <span class="anchor" id="variable-n~13"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -382,7 +382,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-xmin~2"></span>
+              <span class="anchor" id="variable-xmin"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -397,7 +397,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-xmax~2"></span>
+              <span class="anchor" id="variable-xmax"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -444,7 +444,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_smoothing.html b/doc/module/fstats_smoothing.html
index 6508334..6efae3d 100644
--- a/doc/module/fstats_smoothing.html
+++ b/doc/module/fstats_smoothing.html
@@ -150,10 +150,10 @@ <h3 class="card-header card-title bg-light">Uses</h3>
         <ul class="list-group list-group-flush">
             <li class="list-group-item">
               <ul class="list-inline">
-                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
+                  <li class="list-inline-item">linalg</li>
                   <li class="list-inline-item">ferror</li>
+                  <li class="list-inline-item"><a href='fstats_errors.html'>fstats_errors</a></li>
                   <li class="list-inline-item"><a href='http://fortranwiki.org/fortran/show/iso_fortran_env'>iso_fortran_env</a></li>
-                  <li class="list-inline-item">linalg</li>
               </ul>
             </li>
         </ul>
@@ -194,7 +194,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~39"></span>
+              <span class="anchor" id="variable-x~55"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -367,7 +367,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_special_functions.html b/doc/module/fstats_special_functions.html
index 4a4e6c9..9520d83 100644
--- a/doc/module/fstats_special_functions.html
+++ b/doc/module/fstats_special_functions.html
@@ -194,7 +194,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~11"></span>
+              <span class="anchor" id="variable-a~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -209,7 +209,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~7"></span>
+              <span class="anchor" id="variable-b~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -254,7 +254,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~45"></span>
+              <span class="anchor" id="variable-x~32"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -299,7 +299,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~13"></span>
+              <span class="anchor" id="variable-a~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -314,7 +314,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~9"></span>
+              <span class="anchor" id="variable-b~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -329,7 +329,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~42"></span>
+              <span class="anchor" id="variable-x~29"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -374,7 +374,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~15"></span>
+              <span class="anchor" id="variable-a~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -389,7 +389,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~44"></span>
+              <span class="anchor" id="variable-x~31"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -434,7 +434,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~14"></span>
+              <span class="anchor" id="variable-a~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -449,7 +449,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~43"></span>
+              <span class="anchor" id="variable-x~30"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -494,7 +494,7 @@ <h4>Arguments</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~12"></span>
+              <span class="anchor" id="variable-a~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -509,7 +509,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~8"></span>
+              <span class="anchor" id="variable-b~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -524,7 +524,7 @@ <h4>Arguments</h4>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~41"></span>
+              <span class="anchor" id="variable-x~28"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -570,7 +570,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/module/fstats_types.html b/doc/module/fstats_types.html
index 7ad8097..04c6313 100644
--- a/doc/module/fstats_types.html
+++ b/doc/module/fstats_types.html
@@ -193,7 +193,7 @@ <h4>Components</h4>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~14"></span>
+              <span class="anchor" id="variable-x~10"></span>
               real(kind=real64),
             </td>
               <td>public,</td>
@@ -239,7 +239,7 @@ <h4>Components</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/adjusted_r_squared.html b/doc/proc/adjusted_r_squared.html
index eeb5aeb..deba474 100644
--- a/doc/proc/adjusted_r_squared.html
+++ b/doc/proc/adjusted_r_squared.html
@@ -162,7 +162,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~9"></span>
+              <span class="anchor" id="variable-p"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -177,7 +177,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~31"></span>
+              <span class="anchor" id="variable-x~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -209,7 +209,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~16"></span>
+              <span class="anchor" id="variable-err~2"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -232,7 +232,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~21"></span>
+          <span class="anchor" id="variable-rst~2"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -265,7 +265,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/allan_variance.html b/doc/proc/allan_variance.html
index 3d8dc19..7955050 100644
--- a/doc/proc/allan_variance.html
+++ b/doc/proc/allan_variance.html
@@ -161,7 +161,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~22"></span>
+              <span class="anchor" id="variable-x~33"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -192,7 +192,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~12"></span>
+              <span class="anchor" id="variable-err~15"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -214,7 +214,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~17"></span>
+          <span class="anchor" id="variable-rst~28"></span>
           real(kind=real64), allocatable, dimension(:,:)
         </small>
       </h3>
@@ -250,7 +250,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/bartletts_test.html b/doc/proc/bartletts_test.html
index a0725eb..6128d2e 100644
--- a/doc/proc/bartletts_test.html
+++ b/doc/proc/bartletts_test.html
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~20"></span>
+              <span class="anchor" id="variable-x~18"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -201,7 +201,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~7"></span>
+              <span class="anchor" id="variable-p~12"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -247,7 +247,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/beta.html b/doc/proc/beta.html
index 42f3a93..1d956d0 100644
--- a/doc/proc/beta.html
+++ b/doc/proc/beta.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~11"></span>
+              <span class="anchor" id="variable-a~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -175,7 +175,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~7"></span>
+              <span class="anchor" id="variable-b~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -193,7 +193,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~25"></span>
+          <span class="anchor" id="variable-rst~22"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -226,7 +226,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/bootstrap.html b/doc/proc/bootstrap.html
index 127f57e..d2e893a 100644
--- a/doc/proc/bootstrap.html
+++ b/doc/proc/bootstrap.html
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~27"></span>
+              <span class="anchor" id="variable-x~38"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -186,7 +186,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method"></span>
+              <span class="anchor" id="variable-method~2"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -218,7 +218,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~3"></span>
+              <span class="anchor" id="variable-alpha~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -237,7 +237,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~19"></span>
+          <span class="anchor" id="variable-rst~30"></span>
           type(<a href='..\type\bootstrap_statistics.html'>bootstrap_statistics</a>)
         </small>
       </h3>
@@ -271,7 +271,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/bootstrap_linear_least_squares.html b/doc/proc/bootstrap_linear_least_squares.html
index dccaec3..6df8e85 100644
--- a/doc/proc/bootstrap_linear_least_squares.html
+++ b/doc/proc/bootstrap_linear_least_squares.html
@@ -157,7 +157,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order"></span>
+              <span class="anchor" id="variable-order~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -174,7 +174,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept"></span>
+              <span class="anchor" id="variable-intercept~3"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -190,7 +190,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~28"></span>
+              <span class="anchor" id="variable-x~39"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -206,7 +206,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~3"></span>
+              <span class="anchor" id="variable-y~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -222,7 +222,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-coeffs"></span>
+              <span class="anchor" id="variable-coeffs~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -238,7 +238,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~2"></span>
+              <span class="anchor" id="variable-ymod~3"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -253,7 +253,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid"></span>
+              <span class="anchor" id="variable-resid~8"></span>
               real(kind=real64),
             </td>
 <td>intent(out),</td>
@@ -284,7 +284,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats"></span>
+              <span class="anchor" id="variable-stats~3"></span>
               type(<a href='..\type\bootstrap_regression_statistics.html'>bootstrap_regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -320,7 +320,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~4"></span>
+              <span class="anchor" id="variable-alpha~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -337,7 +337,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~2"></span>
+              <span class="anchor" id="variable-method~3"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -370,7 +370,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~13"></span>
+              <span class="anchor" id="variable-err~16"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -421,7 +421,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/bootstrap_nonlinear_least_squares.html b/doc/proc/bootstrap_nonlinear_least_squares.html
index 26cd88e..bcf317f 100644
--- a/doc/proc/bootstrap_nonlinear_least_squares.html
+++ b/doc/proc/bootstrap_nonlinear_least_squares.html
@@ -158,7 +158,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun"></span>
+              <span class="anchor" id="variable-fun~8"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -173,7 +173,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~29"></span>
+              <span class="anchor" id="variable-x~40"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -188,7 +188,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~4"></span>
+              <span class="anchor" id="variable-y~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -203,7 +203,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params"></span>
+              <span class="anchor" id="variable-params~7"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -220,7 +220,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~3"></span>
+              <span class="anchor" id="variable-ymod~4"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -236,7 +236,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~2"></span>
+              <span class="anchor" id="variable-resid~9"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -267,7 +267,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-weights"></span>
+              <span class="anchor" id="variable-weights~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -283,7 +283,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-maxp"></span>
+              <span class="anchor" id="variable-maxp~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -301,7 +301,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-minp"></span>
+              <span class="anchor" id="variable-minp~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -320,7 +320,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~2"></span>
+              <span class="anchor" id="variable-stats~4"></span>
               type(<a href='..\type\bootstrap_regression_statistics.html'>bootstrap_regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -336,7 +336,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~5"></span>
+              <span class="anchor" id="variable-alpha~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -353,7 +353,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-controls"></span>
+              <span class="anchor" id="variable-controls~5"></span>
               type(<a href='..\type\iteration_controls.html'>iteration_controls</a>),
             </td>
 <td>intent(in),</td>
@@ -368,7 +368,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-settings"></span>
+              <span class="anchor" id="variable-settings~2"></span>
               type(<a href='..\type\lm_solver_options.html'>lm_solver_options</a>),
             </td>
 <td>intent(in),</td>
@@ -383,7 +383,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-info"></span>
+              <span class="anchor" id="variable-info~4"></span>
               type(<a href='..\type\convergence_info.html'>convergence_info</a>),
             </td>
 <td>intent(out),</td>
@@ -419,7 +419,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~3"></span>
+              <span class="anchor" id="variable-method~4"></span>
               procedure(<a href='..\interface\bootstrap_resampling_routine.html'>bootstrap_resampling_routine</a>),
             </td>
 <td>intent(in),</td>
@@ -452,7 +452,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~14"></span>
+              <span class="anchor" id="variable-err~17"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -508,7 +508,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/calculate_regression_statistics.html b/doc/proc/calculate_regression_statistics.html
index 97da9bd..9f5f857 100644
--- a/doc/proc/calculate_regression_statistics.html
+++ b/doc/proc/calculate_regression_statistics.html
@@ -154,7 +154,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~6"></span>
+              <span class="anchor" id="variable-resid~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -169,7 +169,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~4"></span>
+              <span class="anchor" id="variable-params~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -184,7 +184,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~5"></span>
+              <span class="anchor" id="variable-c~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -199,7 +199,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~8"></span>
+              <span class="anchor" id="variable-alpha~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -216,7 +216,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~20"></span>
+              <span class="anchor" id="variable-err~6"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -240,7 +240,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~23"></span>
+          <span class="anchor" id="variable-rst~4"></span>
           type(<a href='..\type\regression_statistics.html'>regression_statistics</a>), allocatable, (:)
         </small>
       </h3>
@@ -273,7 +273,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/correlation.html b/doc/proc/correlation.html
index 2e18103..8d439f5 100644
--- a/doc/proc/correlation.html
+++ b/doc/proc/correlation.html
@@ -158,7 +158,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~32"></span>
+              <span class="anchor" id="variable-x~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -173,7 +173,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~5"></span>
+              <span class="anchor" id="variable-y"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -191,7 +191,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~22"></span>
+          <span class="anchor" id="variable-rst~3"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -224,7 +224,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/covariance.html b/doc/proc/covariance.html
index 3ff3e61..b4c7b15 100644
--- a/doc/proc/covariance.html
+++ b/doc/proc/covariance.html
@@ -156,7 +156,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~7"></span>
+              <span class="anchor" id="variable-x~26"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y"></span>
+              <span class="anchor" id="variable-y~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -189,7 +189,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~7"></span>
+          <span class="anchor" id="variable-rst~19"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -222,7 +222,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/covariance_matrix.html b/doc/proc/covariance_matrix.html
index dd65463..9a4436a 100644
--- a/doc/proc/covariance_matrix.html
+++ b/doc/proc/covariance_matrix.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~34"></span>
+              <span class="anchor" id="variable-x~5"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -176,7 +176,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~3"></span>
+              <span class="anchor" id="variable-c~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -191,7 +191,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~18"></span>
+              <span class="anchor" id="variable-err~4"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -241,7 +241,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/design_matrix.html b/doc/proc/design_matrix.html
index 45d770b..01639d2 100644
--- a/doc/proc/design_matrix.html
+++ b/doc/proc/design_matrix.html
@@ -163,7 +163,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order~2"></span>
+              <span class="anchor" id="variable-order"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -179,7 +179,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept~3"></span>
+              <span class="anchor" id="variable-intercept"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -195,7 +195,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~33"></span>
+              <span class="anchor" id="variable-x~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -211,7 +211,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-c~2"></span>
+              <span class="anchor" id="variable-c"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -228,7 +228,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~17"></span>
+              <span class="anchor" id="variable-err~3"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -276,7 +276,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/difference.html b/doc/proc/difference.html
index 90f6061..8557950 100644
--- a/doc/proc/difference.html
+++ b/doc/proc/difference.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~12"></span>
+              <span class="anchor" id="variable-x~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~13"></span>
+          <span class="anchor" id="variable-rst~6"></span>
           real(kind=real64), allocatable, dimension(:)
         </small>
       </h3>
@@ -205,7 +205,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/digamma.html b/doc/proc/digamma.html
index 9723b41..2b936f1 100644
--- a/doc/proc/digamma.html
+++ b/doc/proc/digamma.html
@@ -162,7 +162,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~45"></span>
+              <span class="anchor" id="variable-x~32"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -180,7 +180,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~30"></span>
+          <span class="anchor" id="variable-rst~27"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -213,7 +213,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/f_test.html b/doc/proc/f_test.html
index 237cfdb..f345e04 100644
--- a/doc/proc/f_test.html
+++ b/doc/proc/f_test.html
@@ -203,7 +203,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~6"></span>
+              <span class="anchor" id="variable-p~11"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -220,7 +220,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof1~2"></span>
+              <span class="anchor" id="variable-dof1"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -235,7 +235,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof2~2"></span>
+              <span class="anchor" id="variable-dof2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -279,7 +279,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/factorial.html b/doc/proc/factorial.html
index e7b49de..ff2fa27 100644
--- a/doc/proc/factorial.html
+++ b/doc/proc/factorial.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~13"></span>
+              <span class="anchor" id="variable-x~12"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~14"></span>
+          <span class="anchor" id="variable-rst~7"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -204,7 +204,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/full_factorial.html b/doc/proc/full_factorial.html
index 66ed438..d4e934b 100644
--- a/doc/proc/full_factorial.html
+++ b/doc/proc/full_factorial.html
@@ -236,7 +236,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~3"></span>
+              <span class="anchor" id="variable-err~14"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -285,7 +285,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/get_full_factorial_matrix_size.html b/doc/proc/get_full_factorial_matrix_size.html
index 2e312a2..36b7973 100644
--- a/doc/proc/get_full_factorial_matrix_size.html
+++ b/doc/proc/get_full_factorial_matrix_size.html
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-m"></span>
+              <span class="anchor" id="variable-m~6"></span>
               integer(kind=int32),
             </td>
 <td>intent(out)</td>
@@ -186,7 +186,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~13"></span>
+              <span class="anchor" id="variable-n~26"></span>
               integer(kind=int32),
             </td>
 <td>intent(out)</td>
@@ -201,7 +201,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~2"></span>
+              <span class="anchor" id="variable-err~13"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -249,7 +249,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/incomplete_beta.html b/doc/proc/incomplete_beta.html
index 8f6fb42..362293d 100644
--- a/doc/proc/incomplete_beta.html
+++ b/doc/proc/incomplete_beta.html
@@ -159,7 +159,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~13"></span>
+              <span class="anchor" id="variable-a~9"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -174,7 +174,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~9"></span>
+              <span class="anchor" id="variable-b~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -189,7 +189,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~42"></span>
+              <span class="anchor" id="variable-x~29"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -207,7 +207,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~27"></span>
+          <span class="anchor" id="variable-rst~24"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -240,7 +240,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/incomplete_gamma_lower.html b/doc/proc/incomplete_gamma_lower.html
index a366b97..b3ed4b9 100644
--- a/doc/proc/incomplete_gamma_lower.html
+++ b/doc/proc/incomplete_gamma_lower.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~15"></span>
+              <span class="anchor" id="variable-a~11"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -175,7 +175,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~44"></span>
+              <span class="anchor" id="variable-x~31"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -193,7 +193,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~29"></span>
+          <span class="anchor" id="variable-rst~26"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -226,7 +226,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/incomplete_gamma_upper.html b/doc/proc/incomplete_gamma_upper.html
index c1317a4..8edd927 100644
--- a/doc/proc/incomplete_gamma_upper.html
+++ b/doc/proc/incomplete_gamma_upper.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~14"></span>
+              <span class="anchor" id="variable-a~10"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -175,7 +175,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~43"></span>
+              <span class="anchor" id="variable-x~30"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -193,7 +193,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~28"></span>
+          <span class="anchor" id="variable-rst~25"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -226,7 +226,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/jacobian.html b/doc/proc/jacobian.html
index 36a75c4..ceed164 100644
--- a/doc/proc/jacobian.html
+++ b/doc/proc/jacobian.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun~2"></span>
+              <span class="anchor" id="variable-fun"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -183,7 +183,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~5"></span>
+              <span class="anchor" id="variable-params~4"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -280,7 +280,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~21"></span>
+              <span class="anchor" id="variable-err~7"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -330,7 +330,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/levenes_test.html b/doc/proc/levenes_test.html
index 9f0f9f4..c35f008 100644
--- a/doc/proc/levenes_test.html
+++ b/doc/proc/levenes_test.html
@@ -143,6 +143,11 @@ <h2>public  subroutine levenes_test(x, stat, p, err)
 Z{..} \right)^{2}}{ \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} \left( Z_{ij} -
 Z_{i.} \right)^{2} } </script>
 </p>
+<p>Where:
+<script type="math/tex; mode=display"> Z_{ij} = |X_{ij} - \overline{X_{i.}}| </script>
+<script type="math/tex; mode=display"> Z_{i.} = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} Z_{ij} </script>
+<script type="math/tex; mode=display"> Z_{..} = \frac{1}{N} \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} Z_{ij} </script>
+</p>
 <p>As the test statistic is approximately F-distributed, the F-distribution
 is used to calculate the probability term.</p>
 <p>See Also</p>
@@ -164,7 +169,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~21"></span>
+              <span class="anchor" id="variable-x~19"></span>
               type(<a href='..\type\array_container.html'>array_container</a>),
             </td>
 <td>intent(in),</td>
@@ -194,7 +199,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~8"></span>
+              <span class="anchor" id="variable-p~13"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -211,7 +216,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~5"></span>
+              <span class="anchor" id="variable-err~12"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -255,7 +260,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/linear_least_squares.html b/doc/proc/linear_least_squares.html
index 28d9713..aab17b7 100644
--- a/doc/proc/linear_least_squares.html
+++ b/doc/proc/linear_least_squares.html
@@ -158,7 +158,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-order~3"></span>
+              <span class="anchor" id="variable-order~2"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -175,7 +175,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-intercept~4"></span>
+              <span class="anchor" id="variable-intercept~2"></span>
               logical,
             </td>
 <td>intent(in)</td>
@@ -191,7 +191,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~35"></span>
+              <span class="anchor" id="variable-x~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -207,7 +207,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~6"></span>
+              <span class="anchor" id="variable-y~2"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -223,7 +223,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-coeffs~3"></span>
+              <span class="anchor" id="variable-coeffs"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -238,7 +238,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~4"></span>
+              <span class="anchor" id="variable-ymod"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -253,7 +253,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~5"></span>
+              <span class="anchor" id="variable-resid~3"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -269,7 +269,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~4"></span>
+              <span class="anchor" id="variable-stats"></span>
               type(<a href='..\type\regression_statistics.html'>regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -286,7 +286,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~7"></span>
+              <span class="anchor" id="variable-alpha"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -303,7 +303,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~19"></span>
+              <span class="anchor" id="variable-err~5"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -354,7 +354,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/lowess.html b/doc/proc/lowess.html
index b4bd313..5d73aa2 100644
--- a/doc/proc/lowess.html
+++ b/doc/proc/lowess.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~39"></span>
+              <span class="anchor" id="variable-x~55"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -337,7 +337,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/mean.html b/doc/proc/mean.html
index cd585b7..a508914 100644
--- a/doc/proc/mean.html
+++ b/doc/proc/mean.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x"></span>
+              <span class="anchor" id="variable-x~20"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -171,7 +171,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst"></span>
+          <span class="anchor" id="variable-rst~13"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -204,7 +204,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/median.html b/doc/proc/median.html
index 9517277..52f2ffc 100644
--- a/doc/proc/median.html
+++ b/doc/proc/median.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~4"></span>
+              <span class="anchor" id="variable-x~23"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -172,7 +172,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~4"></span>
+          <span class="anchor" id="variable-rst~16"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -205,7 +205,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/nonlinear_least_squares.html b/doc/proc/nonlinear_least_squares.html
index 051ad53..0365df8 100644
--- a/doc/proc/nonlinear_least_squares.html
+++ b/doc/proc/nonlinear_least_squares.html
@@ -154,7 +154,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-fun~3"></span>
+              <span class="anchor" id="variable-fun~2"></span>
               procedure(<a href='..\interface\regression_function.html'>regression_function</a>),
             </td>
 <td>intent(in),</td>
@@ -169,7 +169,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~36"></span>
+              <span class="anchor" id="variable-x~7"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -184,7 +184,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-y~7"></span>
+              <span class="anchor" id="variable-y~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -199,7 +199,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-params~6"></span>
+              <span class="anchor" id="variable-params~5"></span>
               real(kind=real64),
             </td>
 <td>intent(inout)</td>
@@ -216,7 +216,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-ymod~5"></span>
+              <span class="anchor" id="variable-ymod~2"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -232,7 +232,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-resid~7"></span>
+              <span class="anchor" id="variable-resid~5"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -248,7 +248,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-weights~2"></span>
+              <span class="anchor" id="variable-weights"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -264,7 +264,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-maxp~2"></span>
+              <span class="anchor" id="variable-maxp"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -282,7 +282,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-minp~2"></span>
+              <span class="anchor" id="variable-minp"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -301,7 +301,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-stats~5"></span>
+              <span class="anchor" id="variable-stats~2"></span>
               type(<a href='..\type\regression_statistics.html'>regression_statistics</a>),
             </td>
 <td>intent(out),</td>
@@ -317,7 +317,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-alpha~9"></span>
+              <span class="anchor" id="variable-alpha~3"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -334,7 +334,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-controls~2"></span>
+              <span class="anchor" id="variable-controls"></span>
               type(<a href='..\type\iteration_controls.html'>iteration_controls</a>),
             </td>
 <td>intent(in),</td>
@@ -349,7 +349,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-settings~2"></span>
+              <span class="anchor" id="variable-settings"></span>
               type(<a href='..\type\lm_solver_options.html'>lm_solver_options</a>),
             </td>
 <td>intent(in),</td>
@@ -364,7 +364,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-info~2"></span>
+              <span class="anchor" id="variable-info"></span>
               type(<a href='..\type\convergence_info.html'>convergence_info</a>),
             </td>
 <td>intent(out),</td>
@@ -396,7 +396,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~22"></span>
+              <span class="anchor" id="variable-err~8"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -452,7 +452,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/quantile.html b/doc/proc/quantile.html
index 215c4e2..481e281 100644
--- a/doc/proc/quantile.html
+++ b/doc/proc/quantile.html
@@ -158,7 +158,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~5"></span>
+              <span class="anchor" id="variable-x~24"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -191,7 +191,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~5"></span>
+          <span class="anchor" id="variable-rst~17"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -224,7 +224,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/r_squared.html b/doc/proc/r_squared.html
index d3e6455..77de9e5 100644
--- a/doc/proc/r_squared.html
+++ b/doc/proc/r_squared.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~30"></span>
+              <span class="anchor" id="variable-x"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -196,7 +196,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~15"></span>
+              <span class="anchor" id="variable-err"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -219,7 +219,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~20"></span>
+          <span class="anchor" id="variable-rst"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -252,7 +252,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/random_resample.html b/doc/proc/random_resample.html
index 82bf39d..89c7b62 100644
--- a/doc/proc/random_resample.html
+++ b/doc/proc/random_resample.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~25"></span>
+              <span class="anchor" id="variable-x~36"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -212,7 +212,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/regularized_beta.html b/doc/proc/regularized_beta.html
index d3caf06..334664c 100644
--- a/doc/proc/regularized_beta.html
+++ b/doc/proc/regularized_beta.html
@@ -160,7 +160,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-a~12"></span>
+              <span class="anchor" id="variable-a~8"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -175,7 +175,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-b~8"></span>
+              <span class="anchor" id="variable-b~6"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -190,7 +190,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~41"></span>
+              <span class="anchor" id="variable-x~28"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -208,7 +208,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~26"></span>
+          <span class="anchor" id="variable-rst~23"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -241,7 +241,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/rejection_sample.html b/doc/proc/rejection_sample.html
index c8e2016..370655a 100644
--- a/doc/proc/rejection_sample.html
+++ b/doc/proc/rejection_sample.html
@@ -168,7 +168,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~38"></span>
+              <span class="anchor" id="variable-n~13"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -183,7 +183,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-xmin~2"></span>
+              <span class="anchor" id="variable-xmin"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -198,7 +198,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-xmax~2"></span>
+              <span class="anchor" id="variable-xmax"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -216,7 +216,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~33"></span>
+          <span class="anchor" id="variable-rst~10"></span>
           real(kind=real64), allocatable, dimension(:)
         </small>
       </h3>
@@ -250,7 +250,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_array_size_error.html b/doc/proc/report_array_size_error.html
index 3b256d8..8e14335 100644
--- a/doc/proc/report_array_size_error.html
+++ b/doc/proc/report_array_size_error.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~7"></span>
+              <span class="anchor" id="variable-err~20"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -224,7 +224,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-actual"></span>
+              <span class="anchor" id="variable-actual~3"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -302,7 +302,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_arrays_not_same_size_error.html b/doc/proc/report_arrays_not_same_size_error.html
index f9b75bb..e467674 100644
--- a/doc/proc/report_arrays_not_same_size_error.html
+++ b/doc/proc/report_arrays_not_same_size_error.html
@@ -165,7 +165,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~9"></span>
+              <span class="anchor" id="variable-err~22"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -318,7 +318,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_iteration_count_error.html b/doc/proc/report_iteration_count_error.html
index e07ca2f..72e5208 100644
--- a/doc/proc/report_iteration_count_error.html
+++ b/doc/proc/report_iteration_count_error.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~11"></span>
+              <span class="anchor" id="variable-err~24"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -287,7 +287,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_matrix_size_error.html b/doc/proc/report_matrix_size_error.html
index 6ba6e0d..4603362 100644
--- a/doc/proc/report_matrix_size_error.html
+++ b/doc/proc/report_matrix_size_error.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~8"></span>
+              <span class="anchor" id="variable-err~21"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -332,7 +332,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_memory_error.html b/doc/proc/report_memory_error.html
index 7ccb8e1..50af783 100644
--- a/doc/proc/report_memory_error.html
+++ b/doc/proc/report_memory_error.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~6"></span>
+              <span class="anchor" id="variable-err~19"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -272,7 +272,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/report_underdefined_error.html b/doc/proc/report_underdefined_error.html
index ac00237..e77c6a4 100644
--- a/doc/proc/report_underdefined_error.html
+++ b/doc/proc/report_underdefined_error.html
@@ -164,7 +164,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~10"></span>
+              <span class="anchor" id="variable-err~23"></span>
               class(errors),
             </td>
 <td>intent(inout)</td>
@@ -209,7 +209,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-actual~2"></span>
+              <span class="anchor" id="variable-actual~4"></span>
               integer(kind=int32),
             </td>
 <td>intent(in)</td>
@@ -287,7 +287,7 @@ <h2>Variables</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/scaled_random_resample.html b/doc/proc/scaled_random_resample.html
index 6acd7a9..489af7f 100644
--- a/doc/proc/scaled_random_resample.html
+++ b/doc/proc/scaled_random_resample.html
@@ -154,7 +154,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~26"></span>
+              <span class="anchor" id="variable-x~37"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -213,7 +213,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/standard_deviation.html b/doc/proc/standard_deviation.html
index 342a94a..77d2d1b 100644
--- a/doc/proc/standard_deviation.html
+++ b/doc/proc/standard_deviation.html
@@ -156,7 +156,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~3"></span>
+              <span class="anchor" id="variable-x~22"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -174,7 +174,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~3"></span>
+          <span class="anchor" id="variable-rst~15"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -207,7 +207,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/t_test_equal_variance.html b/doc/proc/t_test_equal_variance.html
index 134b9fd..a6f3128 100644
--- a/doc/proc/t_test_equal_variance.html
+++ b/doc/proc/t_test_equal_variance.html
@@ -203,7 +203,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~3"></span>
+              <span class="anchor" id="variable-p~8"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -220,7 +220,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~2"></span>
+              <span class="anchor" id="variable-dof~4"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -264,7 +264,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/t_test_paired.html b/doc/proc/t_test_paired.html
index 540f211..9a8a53d 100644
--- a/doc/proc/t_test_paired.html
+++ b/doc/proc/t_test_paired.html
@@ -202,7 +202,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~5"></span>
+              <span class="anchor" id="variable-p~10"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -219,7 +219,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~4"></span>
+              <span class="anchor" id="variable-dof~6"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -234,7 +234,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-err~4"></span>
+              <span class="anchor" id="variable-err~11"></span>
               class(errors),
             </td>
 <td>intent(inout),</td>
@@ -282,7 +282,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/t_test_unequal_variance.html b/doc/proc/t_test_unequal_variance.html
index 2216789..dd916f3 100644
--- a/doc/proc/t_test_unequal_variance.html
+++ b/doc/proc/t_test_unequal_variance.html
@@ -203,7 +203,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p~4"></span>
+              <span class="anchor" id="variable-p~9"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -220,7 +220,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof~3"></span>
+              <span class="anchor" id="variable-dof~5"></span>
               real(kind=real64),
             </td>
 <td>intent(out)</td>
@@ -264,7 +264,7 @@ <h3>Arguments</h3>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/trimmed_mean.html b/doc/proc/trimmed_mean.html
index c2db54d..eec696a 100644
--- a/doc/proc/trimmed_mean.html
+++ b/doc/proc/trimmed_mean.html
@@ -153,7 +153,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~6"></span>
+              <span class="anchor" id="variable-x~25"></span>
               real(kind=real64),
             </td>
 <td>intent(inout),</td>
@@ -169,7 +169,7 @@ <h3>Arguments</h3>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-p"></span>
+              <span class="anchor" id="variable-p~14"></span>
               real(kind=real64),
             </td>
 <td>intent(in),</td>
@@ -189,7 +189,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~6"></span>
+          <span class="anchor" id="variable-rst~18"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -222,7 +222,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/proc/variance.html b/doc/proc/variance.html
index 7f97555..03c6211 100644
--- a/doc/proc/variance.html
+++ b/doc/proc/variance.html
@@ -155,7 +155,7 @@ <h3>Arguments</h3>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~2"></span>
+              <span class="anchor" id="variable-x~21"></span>
               real(kind=real64),
             </td>
 <td>intent(in)</td>
@@ -173,7 +173,7 @@ <h3>Arguments</h3>
 
       <h3>Return Value
         <small>
-          <span class="anchor" id="variable-rst~2"></span>
+          <span class="anchor" id="variable-rst~14"></span>
           real(kind=real64)
         </small>
       </h3>
@@ -206,7 +206,7 @@ <h3>Return Value
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/search.html b/doc/search.html
index e540210..bc815a6 100644
--- a/doc/search.html
+++ b/doc/search.html
@@ -95,7 +95,7 @@ <h1>Search Results</h1>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats.f90.html b/doc/sourcefile/fstats.f90.html
index 9141d6f..157910f 100644
--- a/doc/sourcefile/fstats.f90.html
+++ b/doc/sourcefile/fstats.f90.html
@@ -257,7 +257,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_allan.f90.html b/doc/sourcefile/fstats_allan.f90.html
index eaba49f..fe019a0 100644
--- a/doc/sourcefile/fstats_allan.f90.html
+++ b/doc/sourcefile/fstats_allan.f90.html
@@ -269,7 +269,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_anova.f90.html b/doc/sourcefile/fstats_anova.f90.html
index ab35bae..d2c0cb7 100644
--- a/doc/sourcefile/fstats_anova.f90.html
+++ b/doc/sourcefile/fstats_anova.f90.html
@@ -617,7 +617,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_bootstrap.f90.html b/doc/sourcefile/fstats_bootstrap.f90.html
index b2f5f26..b2124c4 100644
--- a/doc/sourcefile/fstats_bootstrap.f90.html
+++ b/doc/sourcefile/fstats_bootstrap.f90.html
@@ -869,7 +869,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_descriptive_statistics.f90.html b/doc/sourcefile/fstats_descriptive_statistics.f90.html
index 8b1662f..5e16def 100644
--- a/doc/sourcefile/fstats_descriptive_statistics.f90.html
+++ b/doc/sourcefile/fstats_descriptive_statistics.f90.html
@@ -465,7 +465,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_distributions.f90.html b/doc/sourcefile/fstats_distributions.f90.html
index ce24946..6e51145 100644
--- a/doc/sourcefile/fstats_distributions.f90.html
+++ b/doc/sourcefile/fstats_distributions.f90.html
@@ -797,7 +797,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_errors.f90.html b/doc/sourcefile/fstats_errors.f90.html
index 791470f..56178d4 100644
--- a/doc/sourcefile/fstats_errors.f90.html
+++ b/doc/sourcefile/fstats_errors.f90.html
@@ -353,7 +353,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_experimental_design.f90.html b/doc/sourcefile/fstats_experimental_design.f90.html
index 11e4776..b9a3781 100644
--- a/doc/sourcefile/fstats_experimental_design.f90.html
+++ b/doc/sourcefile/fstats_experimental_design.f90.html
@@ -338,7 +338,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_helper_routines.f90.html b/doc/sourcefile/fstats_helper_routines.f90.html
index b4d8f99..af77d17 100644
--- a/doc/sourcefile/fstats_helper_routines.f90.html
+++ b/doc/sourcefile/fstats_helper_routines.f90.html
@@ -209,7 +209,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_hypothesis.f90.html b/doc/sourcefile/fstats_hypothesis.f90.html
index 0b31f3e..39b4010 100644
--- a/doc/sourcefile/fstats_hypothesis.f90.html
+++ b/doc/sourcefile/fstats_hypothesis.f90.html
@@ -542,81 +542,86 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
 <a id="ln-389" name="ln-389" href="#ln-389"></a><span class="w">    </span><span class="c">!! Z{..} \right)^{2}}{ \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} \left( Z_{ij} -</span>
 <a id="ln-390" name="ln-390" href="#ln-390"></a><span class="w">    </span><span class="c">!! Z_{i.} \right)^{2} } $$</span>
 <a id="ln-391" name="ln-391" href="#ln-391"></a><span class="w">    </span><span class="c">!!</span>
-<a id="ln-392" name="ln-392" href="#ln-392"></a><span class="w">    </span><span class="c">!! As the test statistic is approximately F-distributed, the F-distribution</span>
-<a id="ln-393" name="ln-393" href="#ln-393"></a><span class="w">    </span><span class="c">!! is used to calculate the probability term.</span>
-<a id="ln-394" name="ln-394" href="#ln-394"></a><span class="w">    </span><span class="c">!!</span>
-<a id="ln-395" name="ln-395" href="#ln-395"></a><span class="w">    </span><span class="c">!! See Also</span>
+<a id="ln-392" name="ln-392" href="#ln-392"></a><span class="w">    </span><span class="c">!! Where:</span>
+<a id="ln-393" name="ln-393" href="#ln-393"></a><span class="w">    </span><span class="c">!! $$ Z_{ij} = |X_{ij} - \overline{X_{i.}}| $$</span>
+<a id="ln-394" name="ln-394" href="#ln-394"></a><span class="w">    </span><span class="c">!! $$ Z_{i.} = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} Z_{ij} $$ </span>
+<a id="ln-395" name="ln-395" href="#ln-395"></a><span class="w">    </span><span class="c">!! $$ Z_{..} = \frac{1}{N} \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} Z_{ij} $$</span>
 <a id="ln-396" name="ln-396" href="#ln-396"></a><span class="w">    </span><span class="c">!!</span>
-<a id="ln-397" name="ln-397" href="#ln-397"></a><span class="w">    </span><span class="c">!! - [Wikipedia](https://en.wikipedia.org/wiki/Levene%27s_test)</span>
-<a id="ln-398" name="ln-398" href="#ln-398"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">array_container</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">in</span><span class="p">),</span><span class="w"> </span><span class="k">dimension</span><span class="p">(:)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">x</span>
-<a id="ln-399" name="ln-399" href="#ln-399"></a><span class="w">        </span><span class="c">!! The arrays of data to analyze.</span>
-<a id="ln-400" name="ln-400" href="#ln-400"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">out</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="nb">stat</span>
-<a id="ln-401" name="ln-401" href="#ln-401"></a><span class="w">        </span><span class="c">!! The Bartlett&#39;s test statistic.</span>
-<a id="ln-402" name="ln-402" href="#ln-402"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">out</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">p</span>
-<a id="ln-403" name="ln-403" href="#ln-403"></a><span class="w">        </span><span class="c">!! The probability value that the variances of each data set are</span>
-<a id="ln-404" name="ln-404" href="#ln-404"></a><span class="w">        </span><span class="c">!! equivalent.  A low p-value, less than some significance level,</span>
-<a id="ln-405" name="ln-405" href="#ln-405"></a><span class="w">        </span><span class="c">!! indicates a non-equivalance of variances.</span>
-<a id="ln-406" name="ln-406" href="#ln-406"></a><span class="w">    </span><span class="k">class</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">inout</span><span class="p">),</span><span class="w"> </span><span class="k">optional</span><span class="p">,</span><span class="w"> </span><span class="k">target</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">err</span>
-<a id="ln-407" name="ln-407" href="#ln-407"></a>
-<a id="ln-408" name="ln-408" href="#ln-408"></a><span class="w">    </span><span class="c">! Local Variables</span>
-<a id="ln-409" name="ln-409" href="#ln-409"></a><span class="w">    </span><span class="kt">integer</span><span class="p">(</span><span class="n">int32</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">ni</span><span class="p">,</span><span class="w"> </span><span class="n">flag</span>
-<a id="ln-410" name="ln-410" href="#ln-410"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">numer</span><span class="p">,</span><span class="w"> </span><span class="n">denom</span><span class="p">,</span><span class="w"> </span><span class="n">inner</span><span class="p">,</span><span class="w"> </span><span class="n">yi</span><span class="p">,</span><span class="w"> </span><span class="n">z</span><span class="p">,</span><span class="w"> </span><span class="n">zij</span>
-<a id="ln-411" name="ln-411" href="#ln-411"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">allocatable</span><span class="p">,</span><span class="w"> </span><span class="k">dimension</span><span class="p">(:)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="w"> </span><span class="n">zt</span><span class="p">,</span><span class="w"> </span><span class="n">zi</span>
-<a id="ln-412" name="ln-412" href="#ln-412"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">f_distribution</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">dist</span>
-<a id="ln-413" name="ln-413" href="#ln-413"></a><span class="w">    </span><span class="k">class</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">pointer</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">errmgr</span>
-<a id="ln-414" name="ln-414" href="#ln-414"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">target</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">deferr</span>
-<a id="ln-415" name="ln-415" href="#ln-415"></a><span class="w">    </span>
-<a id="ln-416" name="ln-416" href="#ln-416"></a><span class="w">    </span><span class="c">! Initialization</span>
-<a id="ln-417" name="ln-417" href="#ln-417"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nb">present</span><span class="p">(</span><span class="n">err</span><span class="p">))</span><span class="w"> </span><span class="k">then</span>
-<a id="ln-418" name="ln-418" href="#ln-418"></a><span class="k">        </span><span class="n">errmgr</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">err</span>
-<a id="ln-419" name="ln-419" href="#ln-419"></a><span class="w">    </span><span class="k">else</span>
-<a id="ln-420" name="ln-420" href="#ln-420"></a><span class="k">        </span><span class="n">errmgr</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">deferr</span>
-<a id="ln-421" name="ln-421" href="#ln-421"></a><span class="w">    </span><span class="k">end if</span>
-<a id="ln-422" name="ln-422" href="#ln-422"></a><span class="k">    </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<a id="ln-423" name="ln-423" href="#ln-423"></a>
-<a id="ln-424" name="ln-424" href="#ln-424"></a><span class="w">    </span><span class="c">! Local Memory Allocations</span>
-<a id="ln-425" name="ln-425" href="#ln-425"></a><span class="w">    </span><span class="k">allocate</span><span class="p">(</span><span class="n">y</span><span class="p">(</span><span class="n">k</span><span class="p">),</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">k</span><span class="p">),</span><span class="w"> </span><span class="nb">stat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">flag</span><span class="p">)</span>
-<a id="ln-426" name="ln-426" href="#ln-426"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">flag</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">then</span>
-<a id="ln-427" name="ln-427" href="#ln-427"></a><span class="k">        call </span><span class="n">report_memory_error</span><span class="p">(</span><span class="n">errmgr</span><span class="p">,</span><span class="w"> </span><span class="s2">&quot;levenes_test&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">flag</span><span class="p">)</span>
-<a id="ln-428" name="ln-428" href="#ln-428"></a><span class="w">        </span><span class="k">return</span>
-<a id="ln-429" name="ln-429" href="#ln-429"></a><span class="k">    end if</span>
-<a id="ln-430" name="ln-430" href="#ln-430"></a>
-<a id="ln-431" name="ln-431" href="#ln-431"></a><span class="w">    </span><span class="c">! Compute the total mean</span>
-<a id="ln-432" name="ln-432" href="#ln-432"></a><span class="w">    </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
-<a id="ln-433" name="ln-433" href="#ln-433"></a><span class="w">    </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="ln-434" name="ln-434" href="#ln-434"></a><span class="w">    </span><span class="k">do </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span>
-<a id="ln-435" name="ln-435" href="#ln-435"></a><span class="w">        </span><span class="n">ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
-<a id="ln-436" name="ln-436" href="#ln-436"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">ni</span>
-<a id="ln-437" name="ln-437" href="#ln-437"></a><span class="w">        </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
-<a id="ln-438" name="ln-438" href="#ln-438"></a><span class="w">        </span><span class="n">zt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-<a id="ln-439" name="ln-439" href="#ln-439"></a><span class="w">        </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mean</span><span class="p">(</span><span class="n">zt</span><span class="p">)</span>
-<a id="ln-440" name="ln-440" href="#ln-440"></a><span class="w">        </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">ni</span>
-<a id="ln-441" name="ln-441" href="#ln-441"></a><span class="w">    </span><span class="k">end do</span>
-<a id="ln-442" name="ln-442" href="#ln-442"></a><span class="k">    </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span>
-<a id="ln-443" name="ln-443" href="#ln-443"></a>
-<a id="ln-444" name="ln-444" href="#ln-444"></a><span class="w">    </span><span class="c">! Process</span>
-<a id="ln-445" name="ln-445" href="#ln-445"></a><span class="w">    </span><span class="n">numer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
-<a id="ln-446" name="ln-446" href="#ln-446"></a><span class="w">    </span><span class="n">denom</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
-<a id="ln-447" name="ln-447" href="#ln-447"></a><span class="w">    </span><span class="k">do </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span>
-<a id="ln-448" name="ln-448" href="#ln-448"></a><span class="w">        </span><span class="n">ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
-<a id="ln-449" name="ln-449" href="#ln-449"></a><span class="w">        </span><span class="n">yi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="ln-450" name="ln-450" href="#ln-450"></a><span class="w">        </span><span class="n">numer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numer</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">ni</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<a id="ln-451" name="ln-451" href="#ln-451"></a><span class="w">        </span>
-<a id="ln-452" name="ln-452" href="#ln-452"></a><span class="w">        </span><span class="n">inner</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
-<a id="ln-453" name="ln-453" href="#ln-453"></a><span class="w">        </span><span class="k">do </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">ni</span>
-<a id="ln-454" name="ln-454" href="#ln-454"></a><span class="w">            </span><span class="n">zij</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">(</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">yi</span><span class="p">)</span>
-<a id="ln-455" name="ln-455" href="#ln-455"></a><span class="w">            </span><span class="n">inner</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inner</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">zij</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
-<a id="ln-456" name="ln-456" href="#ln-456"></a><span class="w">        </span><span class="k">end do</span>
-<a id="ln-457" name="ln-457" href="#ln-457"></a><span class="k">        </span><span class="n">denom</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">denom</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">inner</span>
-<a id="ln-458" name="ln-458" href="#ln-458"></a><span class="w">    </span><span class="k">end do</span>
-<a id="ln-459" name="ln-459" href="#ln-459"></a><span class="k">    </span><span class="nb">stat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">real</span><span class="p">((</span><span class="n">N</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="n">real64</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">numer</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">denom</span><span class="p">)</span>
-<a id="ln-460" name="ln-460" href="#ln-460"></a><span class="w">    </span><span class="n">dist</span><span class="p">%</span><span class="n">d1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1.0d0</span>
-<a id="ln-461" name="ln-461" href="#ln-461"></a><span class="w">    </span><span class="n">dist</span><span class="p">%</span><span class="n">d2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">real</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="n">real64</span><span class="p">)</span>
-<a id="ln-462" name="ln-462" href="#ln-462"></a><span class="w">    </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1.0d0</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">dist</span><span class="p">%</span><span class="n">cdf</span><span class="p">(</span><span class="nb">stat</span><span class="p">)</span>
-<a id="ln-463" name="ln-463" href="#ln-463"></a><span class="k">end subroutine</span>
-<a id="ln-464" name="ln-464" href="#ln-464"></a>
-<a id="ln-465" name="ln-465" href="#ln-465"></a><span class="c">! ------------------------------------------------------------------------------</span>
-<a id="ln-466" name="ln-466" href="#ln-466"></a><span class="k">end module</span>
+<a id="ln-397" name="ln-397" href="#ln-397"></a><span class="w">    </span><span class="c">!! As the test statistic is approximately F-distributed, the F-distribution</span>
+<a id="ln-398" name="ln-398" href="#ln-398"></a><span class="w">    </span><span class="c">!! is used to calculate the probability term.</span>
+<a id="ln-399" name="ln-399" href="#ln-399"></a><span class="w">    </span><span class="c">!!</span>
+<a id="ln-400" name="ln-400" href="#ln-400"></a><span class="w">    </span><span class="c">!! See Also</span>
+<a id="ln-401" name="ln-401" href="#ln-401"></a><span class="w">    </span><span class="c">!!</span>
+<a id="ln-402" name="ln-402" href="#ln-402"></a><span class="w">    </span><span class="c">!! - [Wikipedia](https://en.wikipedia.org/wiki/Levene%27s_test)</span>
+<a id="ln-403" name="ln-403" href="#ln-403"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">array_container</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">in</span><span class="p">),</span><span class="w"> </span><span class="k">dimension</span><span class="p">(:)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">x</span>
+<a id="ln-404" name="ln-404" href="#ln-404"></a><span class="w">        </span><span class="c">!! The arrays of data to analyze.</span>
+<a id="ln-405" name="ln-405" href="#ln-405"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">out</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="nb">stat</span>
+<a id="ln-406" name="ln-406" href="#ln-406"></a><span class="w">        </span><span class="c">!! The Bartlett&#39;s test statistic.</span>
+<a id="ln-407" name="ln-407" href="#ln-407"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">out</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">p</span>
+<a id="ln-408" name="ln-408" href="#ln-408"></a><span class="w">        </span><span class="c">!! The probability value that the variances of each data set are</span>
+<a id="ln-409" name="ln-409" href="#ln-409"></a><span class="w">        </span><span class="c">!! equivalent.  A low p-value, less than some significance level,</span>
+<a id="ln-410" name="ln-410" href="#ln-410"></a><span class="w">        </span><span class="c">!! indicates a non-equivalance of variances.</span>
+<a id="ln-411" name="ln-411" href="#ln-411"></a><span class="w">    </span><span class="k">class</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">intent</span><span class="p">(</span><span class="n">inout</span><span class="p">),</span><span class="w"> </span><span class="k">optional</span><span class="p">,</span><span class="w"> </span><span class="k">target</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">err</span>
+<a id="ln-412" name="ln-412" href="#ln-412"></a>
+<a id="ln-413" name="ln-413" href="#ln-413"></a><span class="w">    </span><span class="c">! Local Variables</span>
+<a id="ln-414" name="ln-414" href="#ln-414"></a><span class="w">    </span><span class="kt">integer</span><span class="p">(</span><span class="n">int32</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">ni</span><span class="p">,</span><span class="w"> </span><span class="n">flag</span>
+<a id="ln-415" name="ln-415" href="#ln-415"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">numer</span><span class="p">,</span><span class="w"> </span><span class="n">denom</span><span class="p">,</span><span class="w"> </span><span class="n">inner</span><span class="p">,</span><span class="w"> </span><span class="n">yi</span><span class="p">,</span><span class="w"> </span><span class="n">z</span><span class="p">,</span><span class="w"> </span><span class="n">zij</span>
+<a id="ln-416" name="ln-416" href="#ln-416"></a><span class="w">    </span><span class="kt">real</span><span class="p">(</span><span class="n">real64</span><span class="p">),</span><span class="w"> </span><span class="k">allocatable</span><span class="p">,</span><span class="w"> </span><span class="k">dimension</span><span class="p">(:)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">y</span><span class="p">,</span><span class="w"> </span><span class="n">zt</span><span class="p">,</span><span class="w"> </span><span class="n">zi</span>
+<a id="ln-417" name="ln-417" href="#ln-417"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">f_distribution</span><span class="p">)</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">dist</span>
+<a id="ln-418" name="ln-418" href="#ln-418"></a><span class="w">    </span><span class="k">class</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">pointer</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">errmgr</span>
+<a id="ln-419" name="ln-419" href="#ln-419"></a><span class="w">    </span><span class="k">type</span><span class="p">(</span><span class="n">errors</span><span class="p">),</span><span class="w"> </span><span class="k">target</span><span class="w"> </span><span class="kd">::</span><span class="w"> </span><span class="n">deferr</span>
+<a id="ln-420" name="ln-420" href="#ln-420"></a><span class="w">    </span>
+<a id="ln-421" name="ln-421" href="#ln-421"></a><span class="w">    </span><span class="c">! Initialization</span>
+<a id="ln-422" name="ln-422" href="#ln-422"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nb">present</span><span class="p">(</span><span class="n">err</span><span class="p">))</span><span class="w"> </span><span class="k">then</span>
+<a id="ln-423" name="ln-423" href="#ln-423"></a><span class="k">        </span><span class="n">errmgr</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">err</span>
+<a id="ln-424" name="ln-424" href="#ln-424"></a><span class="w">    </span><span class="k">else</span>
+<a id="ln-425" name="ln-425" href="#ln-425"></a><span class="k">        </span><span class="n">errmgr</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">deferr</span>
+<a id="ln-426" name="ln-426" href="#ln-426"></a><span class="w">    </span><span class="k">end if</span>
+<a id="ln-427" name="ln-427" href="#ln-427"></a><span class="k">    </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+<a id="ln-428" name="ln-428" href="#ln-428"></a>
+<a id="ln-429" name="ln-429" href="#ln-429"></a><span class="w">    </span><span class="c">! Local Memory Allocations</span>
+<a id="ln-430" name="ln-430" href="#ln-430"></a><span class="w">    </span><span class="k">allocate</span><span class="p">(</span><span class="n">y</span><span class="p">(</span><span class="n">k</span><span class="p">),</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">k</span><span class="p">),</span><span class="w"> </span><span class="nb">stat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">flag</span><span class="p">)</span>
+<a id="ln-431" name="ln-431" href="#ln-431"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">flag</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">then</span>
+<a id="ln-432" name="ln-432" href="#ln-432"></a><span class="k">        call </span><span class="n">report_memory_error</span><span class="p">(</span><span class="n">errmgr</span><span class="p">,</span><span class="w"> </span><span class="s2">&quot;levenes_test&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">flag</span><span class="p">)</span>
+<a id="ln-433" name="ln-433" href="#ln-433"></a><span class="w">        </span><span class="k">return</span>
+<a id="ln-434" name="ln-434" href="#ln-434"></a><span class="k">    end if</span>
+<a id="ln-435" name="ln-435" href="#ln-435"></a>
+<a id="ln-436" name="ln-436" href="#ln-436"></a><span class="w">    </span><span class="c">! Compute the total mean</span>
+<a id="ln-437" name="ln-437" href="#ln-437"></a><span class="w">    </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
+<a id="ln-438" name="ln-438" href="#ln-438"></a><span class="w">    </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
+<a id="ln-439" name="ln-439" href="#ln-439"></a><span class="w">    </span><span class="k">do </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span>
+<a id="ln-440" name="ln-440" href="#ln-440"></a><span class="w">        </span><span class="n">ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
+<a id="ln-441" name="ln-441" href="#ln-441"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">ni</span>
+<a id="ln-442" name="ln-442" href="#ln-442"></a><span class="w">        </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
+<a id="ln-443" name="ln-443" href="#ln-443"></a><span class="w">        </span><span class="n">zt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
+<a id="ln-444" name="ln-444" href="#ln-444"></a><span class="w">        </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mean</span><span class="p">(</span><span class="n">zt</span><span class="p">)</span>
+<a id="ln-445" name="ln-445" href="#ln-445"></a><span class="w">        </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">ni</span>
+<a id="ln-446" name="ln-446" href="#ln-446"></a><span class="w">    </span><span class="k">end do</span>
+<a id="ln-447" name="ln-447" href="#ln-447"></a><span class="k">    </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span>
+<a id="ln-448" name="ln-448" href="#ln-448"></a>
+<a id="ln-449" name="ln-449" href="#ln-449"></a><span class="w">    </span><span class="c">! Process</span>
+<a id="ln-450" name="ln-450" href="#ln-450"></a><span class="w">    </span><span class="n">numer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
+<a id="ln-451" name="ln-451" href="#ln-451"></a><span class="w">    </span><span class="n">denom</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
+<a id="ln-452" name="ln-452" href="#ln-452"></a><span class="w">    </span><span class="k">do </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span>
+<a id="ln-453" name="ln-453" href="#ln-453"></a><span class="w">        </span><span class="n">ni</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">)</span>
+<a id="ln-454" name="ln-454" href="#ln-454"></a><span class="w">        </span><span class="n">yi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
+<a id="ln-455" name="ln-455" href="#ln-455"></a><span class="w">        </span><span class="n">numer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numer</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">ni</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
+<a id="ln-456" name="ln-456" href="#ln-456"></a><span class="w">        </span>
+<a id="ln-457" name="ln-457" href="#ln-457"></a><span class="w">        </span><span class="n">inner</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0d0</span>
+<a id="ln-458" name="ln-458" href="#ln-458"></a><span class="w">        </span><span class="k">do </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">ni</span>
+<a id="ln-459" name="ln-459" href="#ln-459"></a><span class="w">            </span><span class="n">zij</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="n">i</span><span class="p">)%</span><span class="n">x</span><span class="p">(</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">yi</span><span class="p">)</span>
+<a id="ln-460" name="ln-460" href="#ln-460"></a><span class="w">            </span><span class="n">inner</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inner</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">zij</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">zi</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
+<a id="ln-461" name="ln-461" href="#ln-461"></a><span class="w">        </span><span class="k">end do</span>
+<a id="ln-462" name="ln-462" href="#ln-462"></a><span class="k">        </span><span class="n">denom</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">denom</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">inner</span>
+<a id="ln-463" name="ln-463" href="#ln-463"></a><span class="w">    </span><span class="k">end do</span>
+<a id="ln-464" name="ln-464" href="#ln-464"></a><span class="k">    </span><span class="nb">stat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">real</span><span class="p">((</span><span class="n">N</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="n">real64</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">numer</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">denom</span><span class="p">)</span>
+<a id="ln-465" name="ln-465" href="#ln-465"></a><span class="w">    </span><span class="n">dist</span><span class="p">%</span><span class="n">d1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1.0d0</span>
+<a id="ln-466" name="ln-466" href="#ln-466"></a><span class="w">    </span><span class="n">dist</span><span class="p">%</span><span class="n">d2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">real</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="n">real64</span><span class="p">)</span>
+<a id="ln-467" name="ln-467" href="#ln-467"></a><span class="w">    </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1.0d0</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">dist</span><span class="p">%</span><span class="n">cdf</span><span class="p">(</span><span class="nb">stat</span><span class="p">)</span>
+<a id="ln-468" name="ln-468" href="#ln-468"></a><span class="k">end subroutine</span>
+<a id="ln-469" name="ln-469" href="#ln-469"></a>
+<a id="ln-470" name="ln-470" href="#ln-470"></a><span class="c">! ------------------------------------------------------------------------------</span>
+<a id="ln-471" name="ln-471" href="#ln-471"></a><span class="k">end module</span>
 </pre></div>
 
       </section>
@@ -635,7 +640,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_regression.f90.html b/doc/sourcefile/fstats_regression.f90.html
index 9986965..811f4d4 100644
--- a/doc/sourcefile/fstats_regression.f90.html
+++ b/doc/sourcefile/fstats_regression.f90.html
@@ -1828,7 +1828,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_sampling.f90.html b/doc/sourcefile/fstats_sampling.f90.html
index 4f57231..86fbcda 100644
--- a/doc/sourcefile/fstats_sampling.f90.html
+++ b/doc/sourcefile/fstats_sampling.f90.html
@@ -307,7 +307,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_smoothing.f90.html b/doc/sourcefile/fstats_smoothing.f90.html
index 6312ab1..abdffa1 100644
--- a/doc/sourcefile/fstats_smoothing.f90.html
+++ b/doc/sourcefile/fstats_smoothing.f90.html
@@ -458,7 +458,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_special_functions.f90.html b/doc/sourcefile/fstats_special_functions.f90.html
index 2b04601..53c4750 100644
--- a/doc/sourcefile/fstats_special_functions.f90.html
+++ b/doc/sourcefile/fstats_special_functions.f90.html
@@ -476,7 +476,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/sourcefile/fstats_types.f90.html b/doc/sourcefile/fstats_types.f90.html
index ad14859..17c3b0d 100644
--- a/doc/sourcefile/fstats_types.f90.html
+++ b/doc/sourcefile/fstats_types.f90.html
@@ -180,7 +180,7 @@ <h2><span class="anchor" id="src"></span>Source Code</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/src/fstats_hypothesis.f90 b/doc/src/fstats_hypothesis.f90
index 9a1b407..7d1d7fc 100644
--- a/doc/src/fstats_hypothesis.f90
+++ b/doc/src/fstats_hypothesis.f90
@@ -389,6 +389,11 @@ subroutine levenes_test(x, stat, p, err)
     !! Z{..} \right)^{2}}{ \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} \left( Z_{ij} -
     !! Z_{i.} \right)^{2} } $$
     !!
+    !! Where:
+    !! $$ Z_{ij} = |X_{ij} - \overline{X_{i.}}| $$
+    !! $$ Z_{i.} = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} Z_{ij} $$ 
+    !! $$ Z_{..} = \frac{1}{N} \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} Z_{ij} $$
+    !!
     !! As the test statistic is approximately F-distributed, the F-distribution
     !! is used to calculate the probability term.
     !!
diff --git a/doc/tipuesearch/tipuesearch_content.js b/doc/tipuesearch/tipuesearch_content.js
index 91594d5..39e9797 100644
--- a/doc/tipuesearch/tipuesearch_content.js
+++ b/doc/tipuesearch/tipuesearch_content.js
@@ -1 +1 @@
-var tipuesearch = {"pages":[{"title":" FSTATS ","text":"FSTATS Developer Info Jason Christopherson","tags":"home","loc":"index.html"},{"title":"anova_factor – FSTATS ","text":"type, public :: anova_factor Defines an ANOVA factor result. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedome. real(kind=real64), public :: f_statistic The F-statistic. real(kind=real64), public :: probability The variance probability term. real(kind=real64), public :: sum_of_squares The sum of the squares. real(kind=real64), public :: variance The estimate of variance.","tags":"","loc":"type\\anova_factor.html"},{"title":"single_factor_anova_table – FSTATS ","text":"type, public :: single_factor_anova_table Defines a single-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: main_factor The main, or main factor, results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within-treatement (error) results.","tags":"","loc":"type\\single_factor_anova_table.html"},{"title":"two_factor_anova_table – FSTATS ","text":"type, public :: two_factor_anova_table Defines a two-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: interaction The interaction effects. type( anova_factor ), public :: main_factor_1 The first main-factor results. type( anova_factor ), public :: main_factor_2 The second main-factor results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within (error) factor results.","tags":"","loc":"type\\two_factor_anova_table.html"},{"title":"array_container – FSTATS ","text":"type, public :: array_container Provides a container for a real-valued array.  A practical use of\nthis construct is in the construction of jagged arrays. Components Type Visibility Attributes Name Initial real(kind=real64), public, allocatable, dimension(:) :: x The array.","tags":"","loc":"type\\array_container.html"},{"title":"bootstrap_regression_statistics – FSTATS ","text":"type, public :: bootstrap_regression_statistics A container for regression-related statistical information as \ncomputed in a bootstrap, or equivalent, calculation. Components Type Visibility Attributes Name Initial real(kind=real64), public :: lower_confidence_interval The lower limit of the confidence interval for the parameter. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. real(kind=real64), public :: upper_confidence_interval The upper limit of the confidence interval for the parameter.","tags":"","loc":"type\\bootstrap_regression_statistics.html"},{"title":"bootstrap_statistics – FSTATS ","text":"type, public :: bootstrap_statistics A collection of statistics resulting from the bootstrap process. Components Type Visibility Attributes Name Initial real(kind=real64), public :: bias The bias in the statistic. real(kind=real64), public :: lower_confidence_interval The lower confidence limit on the statistic. real(kind=real64), public, allocatable, dimension(:) :: population An array of the population values generated by the bootstrap\nprocess. real(kind=real64), public :: standard_error The standard error of the statistic. real(kind=real64), public :: statistic_value The value of the statistic of interest. real(kind=real64), public :: upper_confidence_interval The upper confidence limit on the statistic.","tags":"","loc":"type\\bootstrap_statistics.html"},{"title":"convergence_info – FSTATS ","text":"type, public :: convergence_info Provides information regarding convergence status. Components Type Visibility Attributes Name Initial logical, public :: converge_on_gradient True if convergence on the gradient was achieved; else, false. logical, public :: converge_on_residual_parameter True if convergence on the residual error parameter was achieved; \nelse, false. logical, public :: converge_on_solution_change True if convergence on the change in solution was achieved; else,\nfalse. integer(kind=int32), public :: function_evaluation_count The function evaluation count. real(kind=real64), public :: gradient_value The value of the gradient test parameter. integer(kind=int32), public :: iteration_count The iteration count. logical, public :: reach_function_evaluation_limit True if the solution did not converge in the allowed number of\nfunction evaluations. logical, public :: reach_iteration_limit True if the solution did not converge in the allowed number of \niterations. real(kind=real64), public :: residual_value The value of the residual error parameter. real(kind=real64), public :: solution_change_value The value of the change in solution parameter. logical, public :: user_requested_stop True if the user requested the stop; else, false.","tags":"","loc":"type\\convergence_info.html"},{"title":"iteration_controls – FSTATS ","text":"type, public :: iteration_controls Provides a collection of iteration control parameters. Components Type Visibility Attributes Name Initial real(kind=real64), public :: change_in_solution_tolerance Defines a tolerance on the change in parameter values. real(kind=real64), public :: gradient_tolerance Defines a tolerance on the gradient of the fitted function. real(kind=real64), public :: iteration_improvement_tolerance Defines a tolerance to ensure adequate improvement on each \niteration. integer(kind=int32), public :: max_function_evaluations Defines the maximum number of function evaluations allowed. integer(kind=int32), public :: max_iteration_between_updates Defines how many iterations can pass before a re-evaluation of \nthe Jacobian matrix is forced. integer(kind=int32), public :: max_iteration_count Defines the maximum number of iterations allowed. real(kind=real64), public :: residual_tolerance Defines a tolerance on the metric associated with the residual \nerror. Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_tolerances private  subroutine lm_set_default_tolerances(x) Arguments Type Intent Optional Attributes Name class( iteration_controls ), intent(inout) :: x","tags":"","loc":"type\\iteration_controls.html"},{"title":"lm_solver_options – FSTATS ","text":"type, public :: lm_solver_options Options to control the Levenberg-Marquardt solver. Components Type Visibility Attributes Name Initial real(kind=real64), public :: damping_decrease_factor The factor to use when decreasing the damping parameter. real(kind=real64), public :: damping_increase_factor The factor to use when increasing the damping parameter. real(kind=real64), public :: finite_difference_step_size The step size used for the finite difference calculations of the\nJacobian matrix. integer(kind=int32), public :: method The solver method to utilize.\n- FS_LEVENBERG_MARQUARDT_UPDATE:\n- FS_QUADRATIC_UPDATE:\n- FS_NIELSEN_UDPATE: Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_settings private  subroutine lm_set_default_settings(x) Arguments Type Intent Optional Attributes Name class( lm_solver_options ), intent(inout) :: x","tags":"","loc":"type\\lm_solver_options.html"},{"title":"regression_statistics – FSTATS ","text":"type, public :: regression_statistics A container for regression-related statistical information. Components Type Visibility Attributes Name Initial real(kind=real64), public :: confidence_interval The confidence interval for the parameter at the level \ndetermined by the regression process. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient.","tags":"","loc":"type\\regression_statistics.html"},{"title":"binomial_distribution – FSTATS ","text":"type, public, extends( distribution ) :: binomial_distribution Defines a binomial distribution.  The binomial distribution describes\nthe probability p of getting k successes in n independent trials. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: n The number of independent trials. real(kind=real64), public :: p The success probability for each trial.  This parameter must\nexist on the set [0, 1]. Type-Bound Procedures procedure, public :: cdf => bd_cdf private pure elemental function bd_cdf(this, x) result(rst) Computes the cumulative distribution funtion. The CDF for a binomial distribution is given as , which is simply\nthe regularized incomplete beta function. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function.  This parameter\nis the number k successes in the n independent trials.  As\nsuch, this parameter must exist on the set [0, n]. Return Value real(kind=real64) The value of the function. procedure, public :: mean => bd_mean private pure function bd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => bd_median private pure function bd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => bd_mode private pure function bd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => bd_pdf private pure elemental function bd_pdf(this, x) result(rst) Computes the probability mass function. The PMF for a binomial distribution is given as . Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function.  This parameter\nis the number k successes in the n independent trials.  As\nsuch, this parameter must exist on the set [0, n]. Return Value real(kind=real64) The value of the function. procedure, public :: variance => bd_variance private pure function bd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\binomial_distribution.html"},{"title":"chi_squared_distribution – FSTATS ","text":"type, public, extends( distribution ) :: chi_squared_distribution Defines a Chi-squared distribution. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => cs_cdf private pure elemental function cs_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a Chi-squared distribution is given as . Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => cs_mean private pure function cs_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => cs_median private pure function cs_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => cs_mode private pure function cs_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => cs_pdf private pure elemental function cs_pdf(this, x) result(rst) Computes the probability density function. The PDF for a Chi-squared distribution is given as . Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => cs_variance private pure function cs_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\chi_squared_distribution.html"},{"title":"distribution – FSTATS ","text":"type, public :: distribution Defines a probability distribution. Type-Bound Procedures procedure( distribution_function ), public, deferred, pass :: cdf Computes the cumulative distribution function. pure elemental function distribution_function(this, x) result(rst) Prototype Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure( distribution_property ), public, deferred, pass :: mean Computes the mean of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_property ), public, deferred, pass :: median Computes the median of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_property ), public, deferred, pass :: mode Computes the mode of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_function ), public, deferred, pass :: pdf Computes the probability density function. pure elemental function distribution_function(this, x) result(rst) Prototype Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure( distribution_property ), public, deferred, pass :: variance Computes the variance of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value.","tags":"","loc":"type\\distribution.html"},{"title":"f_distribution – FSTATS ","text":"type, public, extends( distribution ) :: f_distribution Defines an F-distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: d1 The measure of degrees of freedom for the first data set. real(kind=real64), public :: d2 The measure of degrees of freedom for the second data set. Type-Bound Procedures procedure, public :: cdf => fd_cdf private pure elemental function fd_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a F distribution is given as . Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => fd_mean private pure function fd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => fd_median private pure function fd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => fd_mode private pure function fd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => fd_pdf private pure elemental function fd_pdf(this, x) result(rst) Computes the probability density function. The PDF for a F distribution is given as . Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => fd_variance private pure function fd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\f_distribution.html"},{"title":"normal_distribution – FSTATS ","text":"type, public, extends( distribution ) :: normal_distribution Defines a normal distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: mean_value The mean value of the distribution. real(kind=real64), public :: standard_deviation The standard deviation of the distribution. Type-Bound Procedures procedure, public :: cdf => nd_cdf private pure elemental function nd_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a normal distribution is given as . Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => nd_mean private pure function nd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The mean procedure, public :: median => nd_median private pure function nd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => nd_mode private pure function nd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => nd_pdf private pure elemental function nd_pdf(this, x) result(rst) Computes the probability density function. The PDF for a normal distribution is given as . Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: standardize => nd_standardize private  subroutine nd_standardize(this) Standardizes the normal distribution to a mean of 0 and a \nstandard deviation of 1. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(inout) :: this The normal_distribution object. procedure, public :: variance => nd_variance private pure function nd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\normal_distribution.html"},{"title":"t_distribution – FSTATS ","text":"type, public, extends( distribution ) :: t_distribution Defines Student's T-Distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => td_cdf private pure elemental function td_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for Student's T-Distribution is given as where . Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => td_mean private pure function td_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => td_median private pure function td_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) procedure, public :: mode => td_mode private pure function td_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => td_pdf private pure elemental function td_pdf(this, x) result(rst) Computes the probability density function. The PDF for Student's T-Distribution is given as . Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => td_variance private pure function td_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\t_distribution.html"},{"title":"covariance – FSTATS","text":"public pure function covariance(x, y) result(rst) Computes the sample covariance of two data sets. The covariance computed is the sample covariance such that . Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The covariance.","tags":"","loc":"proc\\covariance.html"},{"title":"mean – FSTATS","text":"public pure function mean(x) result(rst) Computes the mean of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\mean.html"},{"title":"median – FSTATS","text":"public  function median(x) result(rst) Computes the median of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout) :: x (:) The array of values to analyze.  On output, this array is sorted into\nascending order. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\median.html"},{"title":"quantile – FSTATS","text":"public pure function quantile(x, q) result(rst) Computes the specified quantile of a data set using the SAS \nMethod 4. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the data. real(kind=real64), intent(in) :: q The quantile to compute (e.g. 0.25 computes the 25% quantile). Return Value real(kind=real64) The result.","tags":"","loc":"proc\\quantile.html"},{"title":"standard_deviation – FSTATS","text":"public pure function standard_deviation(x) result(rst) Computes the sample standard deviation of the values in an array. The value computed is the sample standard deviation. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\standard_deviation.html"},{"title":"trimmed_mean – FSTATS","text":"public  function trimmed_mean(x, p) result(rst) Computes the trimmed mean of a data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout), dimension(:) :: x An N-element array containing the data.  On output, the\narray is sorted into ascending order. real(kind=real64), intent(in), optional :: p An optional parameter specifying the percentage of values\nfrom either end of the distribution to remove.  The default\nis 0.05 such that the bottom 5% and top 5% are removed. Return Value real(kind=real64) The trimmed mean.","tags":"","loc":"proc\\trimmed_mean.html"},{"title":"variance – FSTATS","text":"public pure function variance(x) result(rst) Computes the sample variance of the values in an array. The variance computed is the sample variance such that . Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64)","tags":"","loc":"proc\\variance.html"},{"title":"pooled_variance – FSTATS","text":"public interface pooled_variance Computes the pooled estimate of variance. Module Procedures private pure function pooled_variance_1(si, ni) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: si An N-element array containing the estimates for each of the N\nvariances. integer(kind=int32), intent(in), dimension(size(si)) :: ni An N-element array containing the number of data points in each\nof the data sets used to compute the variances in si. Return Value real(kind=real64) The pooled variance. private pure function pooled_variance_2(x) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x An array of arrays of data. Return Value real(kind=real64) The pooled variance.","tags":"","loc":"interface\\pooled_variance.html"},{"title":"anova – FSTATS","text":"public interface anova Performs an analysis of variance (ANOVA) on the supplied data \nset. The following example illustrates a single-factor ANOVA on a \ndata set. program example use iso_fortran_env use fstats implicit none ! Local Variables character , parameter :: tab = achar ( 9 ) real ( real64 ) :: x ( 10 , 2 ) type ( single_factor_anova_table ) :: tbl ! Define the data x = reshape ( & [ & 3.086d3 , 3.082d3 , 3.069d3 , 3.072d3 , 3.045d3 , 3.070d3 , 3.079d3 , & 3.050d3 , 3.062d3 , 3.062d3 , 3.075d3 , 3.061d3 , 3.063d3 , 3.038d3 , & 3.070d3 , 3.062d3 , 3.070d3 , 3.049d3 , 3.042d3 , 3.063d3 & ], & [ 10 , 2 ] & ) ! Perform the ANOVA tbl = anova ( x ) ! Print out the table print '(A)' , \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" print '(AF2.0AF5.1AF5.1AF5.3AF5.3)' , \"Main Factor: \" // tab , & tbl % main_factor % dof , tab , & tbl % main_factor % sum_of_squares , tab // tab , & tbl % main_factor % variance , tab // tab , & tbl % main_factor % f_statistic , tab , & tbl % main_factor % probability print '(AF3.0AF6.1AF5.1)' , \"Within: \" // tab , & tbl % within_factor % dof , tab , & tbl % within_factor % sum_of_squares , tab // tab , & tbl % within_factor % variance print '(AF3.0AF6.1AF5.1)' , \"Total: \" // tab // tab , & tbl % total_dof , tab , & tbl % total_sum_of_squares , tab // tab , & tbl % total_variance print '(AF6.1)' , \"Overall Mean: \" , tbl % overall_mean end program The above program produces the following output. Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value\nMain Factor:    1.      352.8           352.8           2.147   0.160\nWithin:         18.     2958.2          164.3\nTotal:          19.     3311.0          174.3\nOverall Mean: 3063.5 See Also Wikipedia SPC Excel Single Factor ANOVA SPC Excel Gage R&R SPC Excel Understanding Regression Statistics NIST - Two Way ANOVA Module Procedures private  function anova_1_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the M replications of the N test \npoints of interest. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. private  function anova_2_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:,:) An M-by-N-by-K array containing the M replications of the\nN first factor results, and the K second factor results. Return Value type( two_factor_anova_table ) A two_factor_anova_table instance containing the ANOVA results. private  function anova_model_fit(nmodelparams, ymeas, ymod, err) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: nmodelparams The number of model parameters. real(kind=real64), intent(in) :: ymeas (:) An N-element array containing the measured dependent variable data. real(kind=real64), intent(in) :: ymod (:) An N-element array containing the modeled dependent variable data. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the \n  same length.\n- FS_MEMORY_ERROR: Occurs if a memory error is encountered. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results.","tags":"","loc":"interface\\anova.html"},{"title":"difference – FSTATS","text":"public pure function difference(x) result(rst) Computes the difference between elements in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array on which to operate. Return Value real(kind=real64), allocatable, dimension(:) The (N-1)-element array containing the differences between adjacent\nelements.","tags":"","loc":"proc\\difference.html"},{"title":"factorial – FSTATS","text":"public pure elemental function factorial(x) result(rst) Computes the factorial of X. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value whose factorial is to be computed. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\factorial.html"},{"title":"full_factorial – FSTATS","text":"public  subroutine full_factorial(vars, tbl, err) Computes a table with values scaled from 1 to N describing a \nfull-factorial design. program example use iso_fortran_env use fstats implicit none ! Local Variables integer ( int32 ) :: i , vars ( 3 ), tbl ( 24 , 3 ) ! Define the number of design points for each of the 3 factors to study vars = [ 2 , 4 , 3 ] ! Determine the design table call full_factorial ( vars , tbl ) ! Display the table do i = 1 , size ( tbl , 1 ) print * , tbl ( i ,:) end do end program The above program produces the following output. 1           1           1\n1           1           2\n1           1           3\n1           2           1\n1           2           2\n1           2           3\n1           3           1\n1           3           2\n1           3           3\n1           4           1\n1           4           2\n1           4           3\n2           1           1\n2           1           2\n2           1           3\n2           2           1\n2           2           2\n2           2           3\n2           3           1\n2           3           2\n2           3           3\n2           4           1\n2           4           2\n2           4           3 Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain \nthe number of options for that particular factor to explore. \nThis value must be greater than or equal to 1. integer(kind=int32), intent(out) :: tbl (:,:) A table where the design will be written.  Use \nget_full_factorial_matrix_size to determine the appropriate \ntable size. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.\n- FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized.","tags":"","loc":"proc\\full_factorial.html"},{"title":"get_full_factorial_matrix_size – FSTATS","text":"public  subroutine get_full_factorial_matrix_size(vars, m, n, err) Computes the appropriate size for a full-factorial design table. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study.  Each \nof the M entries to the array is expected to contain the \nnumber of options for that particular factor to explore. This value must be greater than or equal to 1. integer(kind=int32), intent(out) :: m The number of rows for the table. integer(kind=int32), intent(out) :: n The number of columns for the table. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.","tags":"","loc":"proc\\get_full_factorial_matrix_size.html"},{"title":"bartletts_test – FSTATS","text":"public  subroutine bartletts_test(x, stat, p) Computes Bartlett's test statistic and associated probability. The statistic is calculated as follows. Where and is the pooled\nvariance. The probability is calculated as the right-tail probability of the\nchi-squared distribution. Bartlett's test is most relevant for distributions showing strong \nnormality.  For distributions lacking strong normality, consider \nLevene's test instead. See Also Wikipedia Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances.","tags":"","loc":"proc\\bartletts_test.html"},{"title":"f_test – FSTATS","text":"public  subroutine f_test(x1, x2, stat, p, dof1, dof2) Computes the F-test and returns the probability (two-tailed) that\nthe variances of two data sets are not significantly different. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The F-statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from the two underlying populations that \nhave the same variance. real(kind=real64), intent(out) :: dof1 A measure of the degrees of freedom. real(kind=real64), intent(out) :: dof2 A measure of the degrees of freedom.","tags":"","loc":"proc\\f_test.html"},{"title":"levenes_test – FSTATS","text":"public  subroutine levenes_test(x, stat, p, err) Computes Levene's test statistic and associated probability. The statistic is calculated as follows. As the test statistic is approximately F-distributed, the F-distribution\nis used to calculate the probability term. See Also Wikipedia Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. class(errors), intent(inout), optional, target :: err","tags":"","loc":"proc\\levenes_test.html"},{"title":"t_test_equal_variance – FSTATS","text":"public  subroutine t_test_equal_variance(x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed equivalent variances. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"proc\\t_test_equal_variance.html"},{"title":"t_test_paired – FSTATS","text":"public  subroutine t_test_paired(x1, x2, stat, p, dof, err) Computes the 2-tailed Student's T-Test for two paired data sets. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An N-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from  two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same \n  length.","tags":"","loc":"proc\\t_test_paired.html"},{"title":"t_test_unequal_variance – FSTATS","text":"public  subroutine t_test_unequal_variance(x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed non-equivalent variances. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"proc\\t_test_unequal_variance.html"},{"title":"confidence_interval – FSTATS","text":"public interface confidence_interval Computes the confidence interval for the specified distribution. See Also Wikipedia Module Procedures private pure function confidence_interval_scalar(dist, alpha, s, n) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: s The sample standard deviation. integer(kind=int32), intent(in) :: n The number of samples in the data set. Return Value real(kind=real64) The result. private pure function confidence_interval_array(dist, alpha, x) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: x (:) An N-element array containing the data to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"interface\\confidence_interval.html"},{"title":"report_array_size_error – FSTATS","text":"public  subroutine report_array_size_error(err, fname, name, expect, actual) Reports an array size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the array. integer(kind=int32), intent(in) :: expect The expected size of the array. integer(kind=int32), intent(in) :: actual The actual size of the array. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_array_size_error.html"},{"title":"report_arrays_not_same_size_error – FSTATS","text":"public  subroutine report_arrays_not_same_size_error(err, fname, name1, name2, size1, size2) Reports an error relating to two arrays not being the same size\nwhen they should be the same size. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name1 The name of the first array. character(len=*), intent(in) :: name2 The name of the second array. integer(kind=int32), intent(in) :: size1 The size of the first array. integer(kind=int32), intent(in) :: size2 The size of the second array. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_arrays_not_same_size_error.html"},{"title":"report_iteration_count_error – FSTATS","text":"public  subroutine report_iteration_count_error(err, fname, msg, mincount) Reports an iteration count error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*) :: fname The name of the routine in which the error occurred. character(len=*) :: msg The error message. integer(kind=int32), intent(in) :: mincount The minimum iteration count expected. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: emsg","tags":"","loc":"proc\\report_iteration_count_error.html"},{"title":"report_matrix_size_error – FSTATS","text":"public  subroutine report_matrix_size_error(err, fname, name, expect_rows, expect_cols, actual_rows, actual_cols) Reports a matrix size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the matrix. integer(kind=int32), intent(in) :: expect_rows The expected number of rows. integer(kind=int32), intent(in) :: expect_cols The expected number of columns. integer(kind=int32), intent(in) :: actual_rows The actual number of rows. integer(kind=int32), intent(in) :: actual_cols The actual number of columns. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_matrix_size_error.html"},{"title":"report_memory_error – FSTATS","text":"public  subroutine report_memory_error(err, fname, code) Reports a memory allocation related error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: code The error code returned by the allocation routine. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_memory_error.html"},{"title":"report_underdefined_error – FSTATS","text":"public  subroutine report_underdefined_error(err, fname, expect, actual) Reports an underdefined problem error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: expect The expected minimum number of equations. integer(kind=int32), intent(in) :: actual The actual number of equations. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_underdefined_error.html"},{"title":"allan_variance – FSTATS","text":"public  function allan_variance(x, dt, err) result(rst) Computes the Allan variance of a data set. Remarks This implementation computes the fully overlapped Allan variance \nusing the method presented by Yadav et. al. Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, \nViraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm \nfor Fully Overlapped Allan Variance and Total Variance for Analysis and \nModeling of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. \n10.1109/LSENS.2018.2829799. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element data set to analyze. real(kind=real64), intent(in), optional :: dt An optional input specifying the time increment between \nsamples in x.  If not specified, this value is set to 1. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value real(kind=real64), allocatable, dimension(:,:) An M-by-2 array containing the results where M is N / 2 - 1\nif N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The \nfirst column contains the averaging times associated with \nthe M results stored in the second column.","tags":"","loc":"proc\\allan_variance.html"},{"title":"bootstrap – FSTATS","text":"public  function bootstrap(stat, x, method, nsamples, alpha) result(rst) Performs a bootstrap calculation on the supplied data set for the given\nstatistic.  The default implementation utlizes a random resampling with \nreplacement.  Other resampling methods may be defined by specifying an \nappropriate routine by means of the method input. Arguments Type Intent Optional Attributes Name procedure( bootstrap_statistic_routine ), intent(in), pointer :: stat The routine used to compute the desired statistic. real(kind=real64), intent(in), dimension(:) :: x The N-element data set. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. integer(kind=int32), intent(in), optional :: nsamples An optional input, that if supplied, specifies the number of \nresampling runs to perform.  The default is 10 000. real(kind=real64), intent(in), optional :: alpha An optional input, that if supplied, defines the significance level\nto use for the analysis.  The default is 0.05. Return Value type( bootstrap_statistics ) The resulting bootstrap_statistics type containing the confidence\nintervals, bias, standard error, etc. for the analyzed statistic.","tags":"","loc":"proc\\bootstrap.html"},{"title":"bootstrap_linear_least_squares – FSTATS","text":"public  subroutine bootstrap_linear_least_squares(order, intercept, x, y, coeffs, ymod, resid, nsamples, stats, bias, alpha, method, bscoeffs, err) Computes a linear least-squares regression to fit a set of data.\nBootstrapping is utilized to gain insight into the quality of \nthe fit.  Resampling for the bootstrap process is a random resampling \nwith replacement process with the range of values limited by the \nstandard deviation of the original data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out), dimension(:) :: coeffs An ORDER+1 element array where the coefficients will\nbe written. real(kind=real64), intent(out), dimension(:) :: ymod An N-element array where the modeled data will be written. real(kind=real64), intent(out), dimension(:) :: resid An N-element array where the residual error data will be \nwritten (modeled - actual). integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. type( bootstrap_regression_statistics ), intent(out), optional, dimension(:) :: stats An M-element array of bootstrap_regression_statistics items \nwhere M = ORDER + 1 when intercept is set to true; however, \nif intercept is set to false, M = ORDER. real(kind=real64), intent(out), optional, dimension(:) :: bias An ORDER+1 element array where an estimate of the bias of\neach coefficient is returned based upon the results of the\nbootstrapping analysis.  The bias is computed as the difference \nbetween the mean of the boostrap population results for the given \nparameter and the original estimate of the given parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\bootstrap_linear_least_squares.html"},{"title":"bootstrap_nonlinear_least_squares – FSTATS","text":"public  subroutine bootstrap_nonlinear_least_squares(fun, x, y, params, ymod, resid, nsamples, weights, maxp, minp, stats, alpha, controls, settings, info, bias, method, bscoeffs, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain \ninsight into the quality of the fit.  Resampling for the bootstrap \nprocess is a random resampling with replacement process with the \nrange of values limited by the standard deviation of the original \ndata set. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( bootstrap_regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. real(kind=real64), intent(out), optional, dimension(:) :: bias An optional N-element array that, if supplied, will be used to \nprovide an estimate of the bias of each model parameter based upon\nthe results of the bootstrapping analysis.  The bias is computed as \nthe difference between the mean of the boostrap population results \nfor the given parameter and the original estimate of the given \nparameter. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"proc\\bootstrap_nonlinear_least_squares.html"},{"title":"random_resample – FSTATS","text":"public  subroutine random_resample(x, xn) Random resampling, with replacement, based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"proc\\random_resample.html"},{"title":"scaled_random_resample – FSTATS","text":"public  subroutine scaled_random_resample(x, xn) A random resampling, scaled by the standard deviation of the original\ndata, but based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"proc\\scaled_random_resample.html"},{"title":"bootstrap_resampling_routine – FSTATS","text":"interface public  subroutine bootstrap_resampling_routine(x, xn) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be \nwritten. Description Defines the signature of a subroutine used to compute a \nresampling of data for bootstrapping purposes.","tags":"","loc":"interface\\bootstrap_resampling_routine.html"},{"title":"bootstrap_statistic_routine – FSTATS","text":"interface public  function bootstrap_statistic_routine(x) result(rst) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The array of data to analyze. Return Value real(kind=real64) The resulting statistic. Description Defines the signature of a function for computing the desired\nbootstrap statistic.","tags":"","loc":"interface\\bootstrap_statistic_routine.html"},{"title":"adjusted_r_squared – FSTATS","text":"public  function adjusted_r_squared(p, x, xm, err) result(rst) Computes the adjusted R-squared value for a data set. The adjusted R-squared provides a mechanism for tempering the effects\nof extra explanatory variables on the traditional R-squared \ncalculation.  It is computed by noting the sample size and \nthe number of variables . . See Also: Wikipedia Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: p The number of variables. real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\adjusted_r_squared.html"},{"title":"calculate_regression_statistics – FSTATS","text":"public  function calculate_regression_statistics(resid, params, c, alpha, err) result(rst) Computes statistics for the quality of fit for a regression \nmodel. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: resid (:) An M-element array containing the model residual errors. real(kind=real64), intent(in) :: params (:) An N-element array containing the model parameters. real(kind=real64), intent(in) :: c (:,:) The N-by-N covariance matrix. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value type( regression_statistics ), allocatable, (:) A regression_statistics object containing the analysis results.","tags":"","loc":"proc\\calculate_regression_statistics.html"},{"title":"correlation – FSTATS","text":"public pure function correlation(x, y) result(rst) Computes the sample correlation coefficient (an estimate to the \npopulation Pearson correlation) as follows. . Where, & are the sample standard deviations of\nx and y respectively. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The correlation coefficient.","tags":"","loc":"proc\\correlation.html"},{"title":"r_squared – FSTATS","text":"public  function r_squared(x, xm, err) result(rst) Computes the R-squared value for a data set. The R-squared value is computed by determining the sum of the squares\nof the residuals: The total sum of the squares: . \nThe R-squared value is then: . See Also: Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\r_squared.html"},{"title":"covariance_matrix – FSTATS","text":"public  subroutine covariance_matrix(x, c, err) Computes the covariance matrix where and is computed\nby design_matrix. See Also Wikipedia Wikipedia - Regression Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the formatted independent data\n matrix as computed by design_matrix. real(kind=real64), intent(out) :: c (:,:) The N-by-N covariance matrix. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not \n     sized correctly.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\covariance_matrix.html"},{"title":"design_matrix – FSTATS","text":"public  subroutine design_matrix(order, intercept, x, c, err) Computes the design matrix for the linear \nleast-squares regression problem of , where is the matrix computed here, is \nthe vector of coefficients to be determined, and is the \nvector of measured dependent variables. See Also Wikipedia Wikipedia Wikipedia Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be\nat least one (linear equation), but can be higher as desired. logical, intent(in) :: intercept Set to true if the intercept is being computed\nas part of the regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(out) :: c (:,:) An N-by-K matrix where the results will be written.  K\nmust equal order + 1 in the event intercept is true; \nhowever, if intercept is false, K must equal order. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.","tags":"","loc":"proc\\design_matrix.html"},{"title":"jacobian – FSTATS","text":"public  subroutine jacobian(fun, xdata, params, jac, stop, f0, f1, step, err) Computes the Jacobian matrix for a nonlinear regression problem. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: xdata (:) The M-element array containing x-coordinate data. real(kind=real64), intent(in) :: params (:) The N-element array containing the model parameters. real(kind=real64), intent(out) :: jac (:,:) The M-by-N matrix where the Jacobian will be written. logical, intent(out) :: stop A value that the user can set in fun forcing the\nevaluation process to stop prior to completion. real(kind=real64), intent(in), optional, target :: f0 (:) An optional M-element array containing the model values\n using the current parameters as defined in m.  This input \ncan be used to prevent the routine from performing a \nfunction evaluation at the model parameter state defined in \nparams. real(kind=real64), intent(out), optional, target :: f1 (:) An optional M-element workspace array used for function\nevaluations. real(kind=real64), intent(in), optional :: step The differentiation step size.  The default is the square \nroot of machine precision. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\jacobian.html"},{"title":"linear_least_squares – FSTATS","text":"public  subroutine linear_least_squares(order, intercept, x, y, coeffs, ymod, resid, stats, alpha, err) Computes a linear least-squares regression to fit a set of data. See Also Wikipedia SPC Excel Understanding Regression Statistics Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in) :: y (:) An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out) :: coeffs (:) An ORDER+1 element array where the coefficients will be written. real(kind=real64), intent(out) :: ymod (:) An N-element array where the modeled data will be written. real(kind=real64), intent(out) :: resid (:) An N-element array where the residual error data will be \nwritten (modeled - actual). type( regression_statistics ), intent(out), optional :: stats (:) An M-element array of regression_statistics items where \nM = ORDER + 1 when intercept is set to true; however, if \nintercept is set to false, M = ORDER. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\linear_least_squares.html"},{"title":"nonlinear_least_squares – FSTATS","text":"public  subroutine nonlinear_least_squares(fun, x, y, params, ymod, resid, weights, maxp, minp, stats, alpha, controls, settings, info, status, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. procedure( iteration_update ), intent(in), optional, pointer :: status An optional pointer to a routine that can be used to extract\niteration information. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"proc\\nonlinear_least_squares.html"},{"title":"iteration_update – FSTATS","text":"interface public  subroutine iteration_update(iter, funvals, resid, params, step) Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: iter real(kind=real64), intent(in) :: funvals (:) real(kind=real64), intent(in) :: resid (:) real(kind=real64), intent(in) :: params (:) real(kind=real64), intent(in) :: step (:)","tags":"","loc":"interface\\iteration_update.html"},{"title":"regression_function – FSTATS","text":"interface public  subroutine regression_function(xdata, params, resid, stop) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: xdata real(kind=real64), intent(in), dimension(:) :: params real(kind=real64), intent(out), dimension(:) :: resid logical, intent(out) :: stop","tags":"","loc":"interface\\regression_function.html"},{"title":"lowess – FSTATS","text":"public  subroutine lowess(x, y, ys, fsmooth, nstps, del, rweights, resid, err) Computes the smoothing of a data set using a robust locally weighted\nscatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at\neach of the supplied x values. Remarks The code is a reimplementation of the LOWESS library.  For a detailed\nunderstanding, see [this]\n(http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf) \npaper by William Cleveland. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable data.  This\narray must be monotonically increasing. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable data. real(kind=real64), intent(out), dimension(:) :: ys An N-element array where the smoothed results will be written. real(kind=real64), intent(in), optional :: fsmooth An optional input that specifies the amount of smoothing. Specifically, this value is the fraction of points used to compute\neach value.  As this value increases, the output becomes smoother.\nChoosing a value in the range of 0.2 to 0.8 typically results in a\ngood fit.  The default value is 0.2. integer(kind=int32), intent(in), optional :: nstps An optional input that specifies the numb of iterations.  If set to\nzero, a non-robust fit is returned.  The default value is set to 2. real(kind=real64), intent(in), optional :: del real(kind=real64), intent(out), optional, dimension(:), target :: rweights An optional N-element array, that if supplied, will be used to\nreturn the weights given to each data point. real(kind=real64), intent(out), optional, dimension(:), target :: resid An optional N-element array, that if supplied, will be used to \nreturn the residual. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation error.","tags":"","loc":"proc\\lowess.html"},{"title":"beta – FSTATS","text":"public pure elemental function beta(a, b) result(rst) Computes the beta function. The beta function is related to the gamma function\nby the following relationship. . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. Return Value real(kind=real64) The value of the beta function at and .","tags":"","loc":"proc\\beta.html"},{"title":"digamma – FSTATS","text":"public pure elemental function digamma(x) result(rst) Computes the digamma function. The digamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\digamma.html"},{"title":"incomplete_beta – FSTATS","text":"public pure elemental function incomplete_beta(a, b, x) result(rst) Computes the incomplete beta function. The incomplete beta function is defind as: . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the incomplete beta function.","tags":"","loc":"proc\\incomplete_beta.html"},{"title":"incomplete_gamma_lower – FSTATS","text":"public pure elemental function incomplete_gamma_lower(a, x) result(rst) Computes the lower incomplete gamma function. The lower incomplete gamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\incomplete_gamma_lower.html"},{"title":"incomplete_gamma_upper – FSTATS","text":"public pure elemental function incomplete_gamma_upper(a, x) result(rst) Computes the upper incomplete gamma function. The upper incomplete gamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\incomplete_gamma_upper.html"},{"title":"regularized_beta – FSTATS","text":"public pure elemental function regularized_beta(a, b, x) result(rst) Computes the regularized beta function. The regularized beta function is defined as the ratio between\nthe incomplete beta function and the beta function. . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the regularized beta function.","tags":"","loc":"proc\\regularized_beta.html"},{"title":"rejection_sample – FSTATS","text":"public  function rejection_sample(tdist, n, xmin, xmax) result(rst) Uses rejection sampling to randomly sample a target distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: tdist The distribution to sample integer(kind=int32), intent(in) :: n The number of samples to make. real(kind=real64), intent(in) :: xmin The minimum range to explore. real(kind=real64), intent(in) :: xmax The maximum range to explore. Return Value real(kind=real64), allocatable, dimension(:) An N-element array containing the N samples from the \ndistribution.","tags":"","loc":"proc\\rejection_sample.html"},{"title":"box_muller_sample – FSTATS","text":"public interface box_muller_sample Generates random, normally distributed values via the Box-Muller \ntransform. Module Procedures private  function box_muller_sample_scalar(mu, sigma) result(rst) Generates a pair of independent, standard, normally distributed\nrandom values using the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. Return Value real(kind=real64), (2) The pair of random values. private  function box_muller_array(mu, sigma, n) result(rst) Generates an array of normally distributed random values sampled\nby the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. integer(kind=int32), intent(in) :: n The number of Box-Muller pairs to generate. Return Value real(kind=real64), allocatable, dimension(:) A 2N-element array containing the N Box-Muller pairs.","tags":"","loc":"interface\\box_muller_sample.html"},{"title":"distribution_function – FSTATS","text":"interface public pure elemental function distribution_function(this, x) result(rst) Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. Description Defines the interface for a probability distribution function.","tags":"","loc":"interface\\distribution_function.html"},{"title":"distribution_property – FSTATS","text":"interface public pure function distribution_property(this) result(rst) Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. Description Computes the value of a distribution property.","tags":"","loc":"interface\\distribution_property.html"},{"title":"fstats – FSTATS","text":"FSTATS is a modern Fortran statistical library containing routines for \ncomputing basic statistical properties, hypothesis testing, regression, \nspecial functions, and experimental design. Uses fstats_helper_routines fstats_regression fstats_distributions fstats_descriptive_statistics fstats_allan fstats_anova fstats_sampling fstats_bootstrap fstats_smoothing fstats_special_functions fstats_experimental_design fstats_hypothesis iso_fortran_env","tags":"","loc":"module\\fstats.html"},{"title":"fstats_descriptive_statistics – FSTATS","text":"Uses fstats_types ferror linalg fstats_errors iso_fortran_env Interfaces public        interface pooled_variance Computes the pooled estimate of variance. private pure function pooled_variance_1(si, ni) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: si An N-element array containing the estimates for each of the N\nvariances. integer(kind=int32), intent(in), dimension(size(si)) :: ni An N-element array containing the number of data points in each\nof the data sets used to compute the variances in si. Return Value real(kind=real64) The pooled variance. private pure function pooled_variance_2(x) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x An array of arrays of data. Return Value real(kind=real64) The pooled variance. Functions public pure function covariance (x, y) result(rst) Computes the sample covariance of two data sets. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The covariance. public pure function mean (x) result(rst) Computes the mean of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result. public  function median (x) result(rst) Computes the median of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout) :: x (:) The array of values to analyze.  On output, this array is sorted into\nascending order. Return Value real(kind=real64) The result. public pure function quantile (x, q) result(rst) Computes the specified quantile of a data set using the SAS \nMethod 4. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the data. real(kind=real64), intent(in) :: q The quantile to compute (e.g. 0.25 computes the 25% quantile). Return Value real(kind=real64) The result. public pure function standard_deviation (x) result(rst) Computes the sample standard deviation of the values in an array. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result. public  function trimmed_mean (x, p) result(rst) Computes the trimmed mean of a data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout), dimension(:) :: x An N-element array containing the data.  On output, the\narray is sorted into ascending order. real(kind=real64), intent(in), optional :: p An optional parameter specifying the percentage of values\nfrom either end of the distribution to remove.  The default\nis 0.05 such that the bottom 5% and top 5% are removed. Return Value real(kind=real64) The trimmed mean. public pure function variance (x) result(rst) Computes the sample variance of the values in an array. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64)","tags":"","loc":"module\\fstats_descriptive_statistics.html"},{"title":"fstats_anova – FSTATS","text":"Uses ferror fstats_descriptive_statistics ieee_arithmetic fstats_errors fstats_special_functions iso_fortran_env Interfaces public        interface anova Performs an analysis of variance (ANOVA) on the supplied data \nset. The following example illustrates a single-factor ANOVA on a \ndata set. program example use iso_fortran_env use fstats implicit none ! Local Variables character , parameter :: tab = achar ( 9 ) real ( real64 ) :: x ( 10 , 2 ) type ( single_factor_anova_table ) :: tbl ! Define the data x = reshape ( & [ & 3.086d3 , 3.082d3 , 3.069d3 , 3.072d3 , 3.045d3 , 3.070d3 , 3.079d3 , & 3.050d3 , 3.062d3 , 3.062d3 , 3.075d3 , 3.061d3 , 3.063d3 , 3.038d3 , & 3.070d3 , 3.062d3 , 3.070d3 , 3.049d3 , 3.042d3 , 3.063d3 & ], & [ 10 , 2 ] & ) ! Perform the ANOVA tbl = anova ( x ) ! Print out the table print '(A)' , \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" print '(AF2.0AF5.1AF5.1AF5.3AF5.3)' , \"Main Factor: \" // tab , & tbl % main_factor % dof , tab , & tbl % main_factor % sum_of_squares , tab // tab , & tbl % main_factor % variance , tab // tab , & tbl % main_factor % f_statistic , tab , & tbl % main_factor % probability print '(AF3.0AF6.1AF5.1)' , \"Within: \" // tab , & tbl % within_factor % dof , tab , & tbl % within_factor % sum_of_squares , tab // tab , & tbl % within_factor % variance print '(AF3.0AF6.1AF5.1)' , \"Total: \" // tab // tab , & tbl % total_dof , tab , & tbl % total_sum_of_squares , tab // tab , & tbl % total_variance print '(AF6.1)' , \"Overall Mean: \" , tbl % overall_mean end program The above program produces the following output. Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value\nMain Factor:    1.      352.8           352.8           2.147   0.160\nWithin:         18.     2958.2          164.3\nTotal:          19.     3311.0          174.3\nOverall Mean: 3063.5 See Also Wikipedia SPC Excel Single Factor ANOVA SPC Excel Gage R&R SPC Excel Understanding Regression Statistics NIST - Two Way ANOVA private  function anova_1_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the M replications of the N test \npoints of interest. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. private  function anova_2_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:,:) An M-by-N-by-K array containing the M replications of the\nN first factor results, and the K second factor results. Return Value type( two_factor_anova_table ) A two_factor_anova_table instance containing the ANOVA results. private  function anova_model_fit(nmodelparams, ymeas, ymod, err) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: nmodelparams The number of model parameters. real(kind=real64), intent(in) :: ymeas (:) An N-element array containing the measured dependent variable data. real(kind=real64), intent(in) :: ymod (:) An N-element array containing the modeled dependent variable data. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the \n  same length.\n- FS_MEMORY_ERROR: Occurs if a memory error is encountered. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. Derived Types type, public :: anova_factor Defines an ANOVA factor result. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedome. real(kind=real64), public :: f_statistic The F-statistic. real(kind=real64), public :: probability The variance probability term. real(kind=real64), public :: sum_of_squares The sum of the squares. real(kind=real64), public :: variance The estimate of variance. type, public :: single_factor_anova_table Defines a single-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: main_factor The main, or main factor, results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within-treatement (error) results. type, public :: two_factor_anova_table Defines a two-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: interaction The interaction effects. type( anova_factor ), public :: main_factor_1 The first main-factor results. type( anova_factor ), public :: main_factor_2 The second main-factor results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within (error) factor results.","tags":"","loc":"module\\fstats_anova.html"},{"title":"fstats_helper_routines – FSTATS","text":"Uses iso_fortran_env Functions public pure function difference (x) result(rst) Computes the difference between elements in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array on which to operate. Return Value real(kind=real64), allocatable, dimension(:) The (N-1)-element array containing the differences between adjacent\nelements. public pure elemental function factorial (x) result(rst) Computes the factorial of X. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value whose factorial is to be computed. Return Value real(kind=real64) The result.","tags":"","loc":"module\\fstats_helper_routines.html"},{"title":"fstats_types – FSTATS","text":"Uses iso_fortran_env Derived Types type, public :: array_container Provides a container for a real-valued array.  A practical use of\nthis construct is in the construction of jagged arrays. Components Type Visibility Attributes Name Initial real(kind=real64), public, allocatable, dimension(:) :: x The array.","tags":"","loc":"module\\fstats_types.html"},{"title":"fstats_experimental_design – FSTATS","text":"Uses fstats_errors iso_fortran_env Subroutines public  subroutine full_factorial (vars, tbl, err) Computes a table with values scaled from 1 to N describing a \nfull-factorial design. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain \nthe number of options for that particular factor to explore. \nThis value must be greater than or equal to 1. integer(kind=int32), intent(out) :: tbl (:,:) A table where the design will be written.  Use \nget_full_factorial_matrix_size to determine the appropriate \ntable size. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.\n- FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized. public  subroutine get_full_factorial_matrix_size (vars, m, n, err) Computes the appropriate size for a full-factorial design table. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study.  Each \nof the M entries to the array is expected to contain the \nnumber of options for that particular factor to explore. This value must be greater than or equal to 1. integer(kind=int32), intent(out) :: m The number of rows for the table. integer(kind=int32), intent(out) :: n The number of columns for the table. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.","tags":"","loc":"module\\fstats_experimental_design.html"},{"title":"fstats_hypothesis – FSTATS","text":"Uses fstats_types fstats_distributions fstats_descriptive_statistics ieee_arithmetic fstats_errors fstats_special_functions iso_fortran_env Interfaces public        interface confidence_interval Computes the confidence interval for the specified distribution. See Also Wikipedia private pure function confidence_interval_scalar(dist, alpha, s, n) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: s The sample standard deviation. integer(kind=int32), intent(in) :: n The number of samples in the data set. Return Value real(kind=real64) The result. private pure function confidence_interval_array(dist, alpha, x) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: x (:) An N-element array containing the data to analyze. Return Value real(kind=real64) The result. Subroutines public  subroutine bartletts_test (x, stat, p) Computes Bartlett's test statistic and associated probability. Read more… Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. public  subroutine f_test (x1, x2, stat, p, dof1, dof2) Computes the F-test and returns the probability (two-tailed) that\nthe variances of two data sets are not significantly different. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The F-statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from the two underlying populations that \nhave the same variance. real(kind=real64), intent(out) :: dof1 A measure of the degrees of freedom. real(kind=real64), intent(out) :: dof2 A measure of the degrees of freedom. public  subroutine levenes_test (x, stat, p, err) Computes Levene's test statistic and associated probability. Read more… Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. class(errors), intent(inout), optional, target :: err public  subroutine t_test_equal_variance (x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed equivalent variances. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. public  subroutine t_test_paired (x1, x2, stat, p, dof, err) Computes the 2-tailed Student's T-Test for two paired data sets. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An N-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from  two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same \n  length. public  subroutine t_test_unequal_variance (x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed non-equivalent variances. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"module\\fstats_hypothesis.html"},{"title":"fstats_errors – FSTATS","text":"Uses ferror iso_fortran_env Variables Type Visibility Attributes Name Initial integer(kind=int32), public, parameter :: FS_ARRAY_SIZE_ERROR = 10000 integer(kind=int32), public, parameter :: FS_INVALID_INPUT_ERROR = 10002 integer(kind=int32), public, parameter :: FS_MATRIX_SIZE_ERROR = 10001 integer(kind=int32), public, parameter :: FS_MEMORY_ERROR = 10003 integer(kind=int32), public, parameter :: FS_NO_ERROR = 0 integer(kind=int32), public, parameter :: FS_TOLERANCE_TOO_SMALL_ERROR = 10005 integer(kind=int32), public, parameter :: FS_TOO_FEW_ITERATION_ERROR = 10006 integer(kind=int32), public, parameter :: FS_UNDERDEFINED_PROBLEM_ERROR = 10004 Subroutines public  subroutine report_array_size_error (err, fname, name, expect, actual) Reports an array size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the array. integer(kind=int32), intent(in) :: expect The expected size of the array. integer(kind=int32), intent(in) :: actual The actual size of the array. public  subroutine report_arrays_not_same_size_error (err, fname, name1, name2, size1, size2) Reports an error relating to two arrays not being the same size\nwhen they should be the same size. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name1 The name of the first array. character(len=*), intent(in) :: name2 The name of the second array. integer(kind=int32), intent(in) :: size1 The size of the first array. integer(kind=int32), intent(in) :: size2 The size of the second array. public  subroutine report_iteration_count_error (err, fname, msg, mincount) Reports an iteration count error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*) :: fname The name of the routine in which the error occurred. character(len=*) :: msg The error message. integer(kind=int32), intent(in) :: mincount The minimum iteration count expected. public  subroutine report_matrix_size_error (err, fname, name, expect_rows, expect_cols, actual_rows, actual_cols) Reports a matrix size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the matrix. integer(kind=int32), intent(in) :: expect_rows The expected number of rows. integer(kind=int32), intent(in) :: expect_cols The expected number of columns. integer(kind=int32), intent(in) :: actual_rows The actual number of rows. integer(kind=int32), intent(in) :: actual_cols The actual number of columns. public  subroutine report_memory_error (err, fname, code) Reports a memory allocation related error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: code The error code returned by the allocation routine. public  subroutine report_underdefined_error (err, fname, expect, actual) Reports an underdefined problem error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: expect The expected minimum number of equations. integer(kind=int32), intent(in) :: actual The actual number of equations.","tags":"","loc":"module\\fstats_errors.html"},{"title":"fstats_allan – FSTATS","text":"Uses fstats_errors iso_fortran_env Functions public  function allan_variance (x, dt, err) result(rst) Computes the Allan variance of a data set. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element data set to analyze. real(kind=real64), intent(in), optional :: dt An optional input specifying the time increment between \nsamples in x.  If not specified, this value is set to 1. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value real(kind=real64), allocatable, dimension(:,:) An M-by-2 array containing the results where M is N / 2 - 1\nif N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The \nfirst column contains the averaging times associated with \nthe M results stored in the second column.","tags":"","loc":"module\\fstats_allan.html"},{"title":"fstats_bootstrap – FSTATS","text":"Uses fstats_regression fstats_distributions fstats_descriptive_statistics linalg fstats_errors fstats_special_functions omp_lib iso_fortran_env Interfaces interface public  subroutine bootstrap_resampling_routine(x, xn) Defines the signature of a subroutine used to compute a \nresampling of data for bootstrapping purposes. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be \nwritten. interface public  function bootstrap_statistic_routine(x) result(rst) Defines the signature of a function for computing the desired\nbootstrap statistic. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The array of data to analyze. Return Value real(kind=real64) The resulting statistic. Derived Types type, public :: bootstrap_regression_statistics A container for regression-related statistical information as \ncomputed in a bootstrap, or equivalent, calculation. Components Type Visibility Attributes Name Initial real(kind=real64), public :: lower_confidence_interval The lower limit of the confidence interval for the parameter. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. Read more… real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. Read more… real(kind=real64), public :: upper_confidence_interval The upper limit of the confidence interval for the parameter. type, public :: bootstrap_statistics A collection of statistics resulting from the bootstrap process. Components Type Visibility Attributes Name Initial real(kind=real64), public :: bias The bias in the statistic. real(kind=real64), public :: lower_confidence_interval The lower confidence limit on the statistic. real(kind=real64), public, allocatable, dimension(:) :: population An array of the population values generated by the bootstrap\nprocess. real(kind=real64), public :: standard_error The standard error of the statistic. real(kind=real64), public :: statistic_value The value of the statistic of interest. real(kind=real64), public :: upper_confidence_interval The upper confidence limit on the statistic. Functions public  function bootstrap (stat, x, method, nsamples, alpha) result(rst) Performs a bootstrap calculation on the supplied data set for the given\nstatistic.  The default implementation utlizes a random resampling with \nreplacement.  Other resampling methods may be defined by specifying an \nappropriate routine by means of the method input. Arguments Type Intent Optional Attributes Name procedure( bootstrap_statistic_routine ), intent(in), pointer :: stat The routine used to compute the desired statistic. real(kind=real64), intent(in), dimension(:) :: x The N-element data set. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. integer(kind=int32), intent(in), optional :: nsamples An optional input, that if supplied, specifies the number of \nresampling runs to perform.  The default is 10 000. real(kind=real64), intent(in), optional :: alpha An optional input, that if supplied, defines the significance level\nto use for the analysis.  The default is 0.05. Return Value type( bootstrap_statistics ) The resulting bootstrap_statistics type containing the confidence\nintervals, bias, standard error, etc. for the analyzed statistic. Subroutines public  subroutine bootstrap_linear_least_squares (order, intercept, x, y, coeffs, ymod, resid, nsamples, stats, bias, alpha, method, bscoeffs, err) Computes a linear least-squares regression to fit a set of data.\nBootstrapping is utilized to gain insight into the quality of \nthe fit.  Resampling for the bootstrap process is a random resampling \nwith replacement process with the range of values limited by the \nstandard deviation of the original data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out), dimension(:) :: coeffs An ORDER+1 element array where the coefficients will\nbe written. real(kind=real64), intent(out), dimension(:) :: ymod An N-element array where the modeled data will be written. real(kind=real64), intent(out), dimension(:) :: resid An N-element array where the residual error data will be \nwritten (modeled - actual). integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. type( bootstrap_regression_statistics ), intent(out), optional, dimension(:) :: stats An M-element array of bootstrap_regression_statistics items \nwhere M = ORDER + 1 when intercept is set to true; however, \nif intercept is set to false, M = ORDER. real(kind=real64), intent(out), optional, dimension(:) :: bias An ORDER+1 element array where an estimate of the bias of\neach coefficient is returned based upon the results of the\nbootstrapping analysis.  The bias is computed as the difference \nbetween the mean of the boostrap population results for the given \nparameter and the original estimate of the given parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine bootstrap_nonlinear_least_squares (fun, x, y, params, ymod, resid, nsamples, weights, maxp, minp, stats, alpha, controls, settings, info, bias, method, bscoeffs, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain \ninsight into the quality of the fit.  Resampling for the bootstrap \nprocess is a random resampling with replacement process with the \nrange of values limited by the standard deviation of the original \ndata set. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( bootstrap_regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. real(kind=real64), intent(out), optional, dimension(:) :: bias An optional N-element array that, if supplied, will be used to \nprovide an estimate of the bias of each model parameter based upon\nthe results of the bootstrapping analysis.  The bias is computed as \nthe difference between the mean of the boostrap population results \nfor the given parameter and the original estimate of the given \nparameter. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed. public  subroutine random_resample (x, xn) Random resampling, with replacement, based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written. public  subroutine scaled_random_resample (x, xn) A random resampling, scaled by the standard deviation of the original\ndata, but based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"module\\fstats_bootstrap.html"},{"title":"fstats_regression – FSTATS","text":"Uses ferror fstats_descriptive_statistics fstats_distributions linalg fstats_hypothesis fstats_errors fstats_special_functions blas iso_fortran_env Variables Type Visibility Attributes Name Initial integer(kind=int32), public, parameter :: FS_LEVENBERG_MARQUARDT_UPDATE = 1 integer(kind=int32), public, parameter :: FS_NIELSEN_UPDATE = 3 integer(kind=int32), public, parameter :: FS_QUADRATIC_UPDATE = 2 Interfaces interface public  subroutine iteration_update(iter, funvals, resid, params, step) Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: iter real(kind=real64), intent(in) :: funvals (:) real(kind=real64), intent(in) :: resid (:) real(kind=real64), intent(in) :: params (:) real(kind=real64), intent(in) :: step (:) interface public  subroutine regression_function(xdata, params, resid, stop) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: xdata real(kind=real64), intent(in), dimension(:) :: params real(kind=real64), intent(out), dimension(:) :: resid logical, intent(out) :: stop Derived Types type, public :: convergence_info Provides information regarding convergence status. Components Type Visibility Attributes Name Initial logical, public :: converge_on_gradient True if convergence on the gradient was achieved; else, false. logical, public :: converge_on_residual_parameter True if convergence on the residual error parameter was achieved; \nelse, false. logical, public :: converge_on_solution_change True if convergence on the change in solution was achieved; else,\nfalse. integer(kind=int32), public :: function_evaluation_count The function evaluation count. real(kind=real64), public :: gradient_value The value of the gradient test parameter. integer(kind=int32), public :: iteration_count The iteration count. logical, public :: reach_function_evaluation_limit True if the solution did not converge in the allowed number of\nfunction evaluations. logical, public :: reach_iteration_limit True if the solution did not converge in the allowed number of \niterations. real(kind=real64), public :: residual_value The value of the residual error parameter. real(kind=real64), public :: solution_change_value The value of the change in solution parameter. logical, public :: user_requested_stop True if the user requested the stop; else, false. type, public :: iteration_controls Provides a collection of iteration control parameters. Components Type Visibility Attributes Name Initial real(kind=real64), public :: change_in_solution_tolerance Defines a tolerance on the change in parameter values. real(kind=real64), public :: gradient_tolerance Defines a tolerance on the gradient of the fitted function. real(kind=real64), public :: iteration_improvement_tolerance Defines a tolerance to ensure adequate improvement on each \niteration. integer(kind=int32), public :: max_function_evaluations Defines the maximum number of function evaluations allowed. integer(kind=int32), public :: max_iteration_between_updates Defines how many iterations can pass before a re-evaluation of \nthe Jacobian matrix is forced. integer(kind=int32), public :: max_iteration_count Defines the maximum number of iterations allowed. real(kind=real64), public :: residual_tolerance Defines a tolerance on the metric associated with the residual \nerror. Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_tolerances type, public :: lm_solver_options Options to control the Levenberg-Marquardt solver. Components Type Visibility Attributes Name Initial real(kind=real64), public :: damping_decrease_factor The factor to use when decreasing the damping parameter. real(kind=real64), public :: damping_increase_factor The factor to use when increasing the damping parameter. real(kind=real64), public :: finite_difference_step_size The step size used for the finite difference calculations of the\nJacobian matrix. integer(kind=int32), public :: method The solver method to utilize.\n- FS_LEVENBERG_MARQUARDT_UPDATE:\n- FS_QUADRATIC_UPDATE:\n- FS_NIELSEN_UDPATE: Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_settings type, public :: regression_statistics A container for regression-related statistical information. Components Type Visibility Attributes Name Initial real(kind=real64), public :: confidence_interval The confidence interval for the parameter at the level \ndetermined by the regression process. Read more… real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. Read more… real(kind=real64), public :: standard_error The standard error for the model coefficient. Read more… real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. Read more… Functions public  function adjusted_r_squared (p, x, xm, err) result(rst) Computes the adjusted R-squared value for a data set. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: p The number of variables. real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result. public  function calculate_regression_statistics (resid, params, c, alpha, err) result(rst) Computes statistics for the quality of fit for a regression \nmodel. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: resid (:) An M-element array containing the model residual errors. real(kind=real64), intent(in) :: params (:) An N-element array containing the model parameters. real(kind=real64), intent(in) :: c (:,:) The N-by-N covariance matrix. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value type( regression_statistics ), allocatable, (:) A regression_statistics object containing the analysis results. public pure function correlation (x, y) result(rst) Computes the sample correlation coefficient (an estimate to the \npopulation Pearson correlation) as follows. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The correlation coefficient. public  function r_squared (x, xm, err) result(rst) Computes the R-squared value for a data set. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result. Subroutines public  subroutine covariance_matrix (x, c, err) Computes the covariance matrix where and is computed\nby design_matrix. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the formatted independent data\n matrix as computed by design_matrix. real(kind=real64), intent(out) :: c (:,:) The N-by-N covariance matrix. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not \n     sized correctly.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine design_matrix (order, intercept, x, c, err) Computes the design matrix for the linear \nleast-squares regression problem of , where is the matrix computed here, is \nthe vector of coefficients to be determined, and is the \nvector of measured dependent variables. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be\nat least one (linear equation), but can be higher as desired. logical, intent(in) :: intercept Set to true if the intercept is being computed\nas part of the regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(out) :: c (:,:) An N-by-K matrix where the results will be written.  K\nmust equal order + 1 in the event intercept is true; \nhowever, if intercept is false, K must equal order. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. public  subroutine jacobian (fun, xdata, params, jac, stop, f0, f1, step, err) Computes the Jacobian matrix for a nonlinear regression problem. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: xdata (:) The M-element array containing x-coordinate data. real(kind=real64), intent(in) :: params (:) The N-element array containing the model parameters. real(kind=real64), intent(out) :: jac (:,:) The M-by-N matrix where the Jacobian will be written. logical, intent(out) :: stop A value that the user can set in fun forcing the\nevaluation process to stop prior to completion. real(kind=real64), intent(in), optional, target :: f0 (:) An optional M-element array containing the model values\n using the current parameters as defined in m.  This input \ncan be used to prevent the routine from performing a \nfunction evaluation at the model parameter state defined in \nparams. real(kind=real64), intent(out), optional, target :: f1 (:) An optional M-element workspace array used for function\nevaluations. real(kind=real64), intent(in), optional :: step The differentiation step size.  The default is the square \nroot of machine precision. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine linear_least_squares (order, intercept, x, y, coeffs, ymod, resid, stats, alpha, err) Computes a linear least-squares regression to fit a set of data. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in) :: y (:) An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out) :: coeffs (:) An ORDER+1 element array where the coefficients will be written. real(kind=real64), intent(out) :: ymod (:) An N-element array where the modeled data will be written. real(kind=real64), intent(out) :: resid (:) An N-element array where the residual error data will be \nwritten (modeled - actual). type( regression_statistics ), intent(out), optional :: stats (:) An M-element array of regression_statistics items where \nM = ORDER + 1 when intercept is set to true; however, if \nintercept is set to false, M = ORDER. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine nonlinear_least_squares (fun, x, y, params, ymod, resid, weights, maxp, minp, stats, alpha, controls, settings, info, status, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. procedure( iteration_update ), intent(in), optional, pointer :: status An optional pointer to a routine that can be used to extract\niteration information. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"module\\fstats_regression.html"},{"title":"fstats_smoothing – FSTATS","text":"Uses fstats_errors ferror iso_fortran_env linalg Subroutines public  subroutine lowess (x, y, ys, fsmooth, nstps, del, rweights, resid, err) Computes the smoothing of a data set using a robust locally weighted\nscatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at\neach of the supplied x values. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable data.  This\narray must be monotonically increasing. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable data. real(kind=real64), intent(out), dimension(:) :: ys An N-element array where the smoothed results will be written. real(kind=real64), intent(in), optional :: fsmooth An optional input that specifies the amount of smoothing. Specifically, this value is the fraction of points used to compute\neach value.  As this value increases, the output becomes smoother.\nChoosing a value in the range of 0.2 to 0.8 typically results in a\ngood fit.  The default value is 0.2. integer(kind=int32), intent(in), optional :: nstps An optional input that specifies the numb of iterations.  If set to\nzero, a non-robust fit is returned.  The default value is set to 2. real(kind=real64), intent(in), optional :: del real(kind=real64), intent(out), optional, dimension(:), target :: rweights An optional N-element array, that if supplied, will be used to\nreturn the weights given to each data point. real(kind=real64), intent(out), optional, dimension(:), target :: resid An optional N-element array, that if supplied, will be used to \nreturn the residual. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation error.","tags":"","loc":"module\\fstats_smoothing.html"},{"title":"fstats_special_functions – FSTATS","text":"Uses ieee_arithmetic iso_fortran_env Functions public pure elemental function beta (a, b) result(rst) Computes the beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. Return Value real(kind=real64) The value of the beta function at and . public pure elemental function digamma (x) result(rst) Computes the digamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function incomplete_beta (a, b, x) result(rst) Computes the incomplete beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the incomplete beta function. public pure elemental function incomplete_gamma_lower (a, x) result(rst) Computes the lower incomplete gamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function incomplete_gamma_upper (a, x) result(rst) Computes the upper incomplete gamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function regularized_beta (a, b, x) result(rst) Computes the regularized beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the regularized beta function.","tags":"","loc":"module\\fstats_special_functions.html"},{"title":"fstats_sampling – FSTATS","text":"Uses fstats_distributions linalg iso_fortran_env Interfaces public        interface box_muller_sample Generates random, normally distributed values via the Box-Muller \ntransform. private  function box_muller_sample_scalar(mu, sigma) result(rst) Generates a pair of independent, standard, normally distributed\nrandom values using the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. Return Value real(kind=real64), (2) The pair of random values. private  function box_muller_array(mu, sigma, n) result(rst) Generates an array of normally distributed random values sampled\nby the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. integer(kind=int32), intent(in) :: n The number of Box-Muller pairs to generate. Return Value real(kind=real64), allocatable, dimension(:) A 2N-element array containing the N Box-Muller pairs. Functions public  function rejection_sample (tdist, n, xmin, xmax) result(rst) Uses rejection sampling to randomly sample a target distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: tdist The distribution to sample integer(kind=int32), intent(in) :: n The number of samples to make. real(kind=real64), intent(in) :: xmin The minimum range to explore. real(kind=real64), intent(in) :: xmax The maximum range to explore. Return Value real(kind=real64), allocatable, dimension(:) An N-element array containing the N samples from the \ndistribution.","tags":"","loc":"module\\fstats_sampling.html"},{"title":"fstats_distributions – FSTATS","text":"Uses fstats_helper_routines ieee_arithmetic fstats_special_functions iso_fortran_env Interfaces interface public pure elemental function distribution_function(this, x) result(rst) Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. interface public pure function distribution_property(this) result(rst) Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. Derived Types type, public, extends( distribution ) :: binomial_distribution Defines a binomial distribution.  The binomial distribution describes\nthe probability p of getting k successes in n independent trials. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: n The number of independent trials. real(kind=real64), public :: p The success probability for each trial.  This parameter must\nexist on the set [0, 1]. Type-Bound Procedures procedure, public :: cdf => bd_cdf procedure, public :: mean => bd_mean procedure, public :: median => bd_median procedure, public :: mode => bd_mode procedure, public :: pdf => bd_pdf procedure, public :: variance => bd_variance type, public, extends( distribution ) :: chi_squared_distribution Defines a Chi-squared distribution. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => cs_cdf procedure, public :: mean => cs_mean procedure, public :: median => cs_median procedure, public :: mode => cs_mode procedure, public :: pdf => cs_pdf procedure, public :: variance => cs_variance type, public :: distribution Defines a probability distribution. Type-Bound Procedures procedure( distribution_function ), public, deferred, pass :: cdf ..\\..\\ Computes the cumulative distribution function.<\\p> procedure( distribution_property ), public, deferred, pass :: mean ..\\..\\ Computes the mean of the distribution.<\\p> procedure( distribution_property ), public, deferred, pass :: median ..\\..\\ Computes the median of the distribution.<\\p> procedure( distribution_property ), public, deferred, pass :: mode ..\\..\\ Computes the mode of the distribution.<\\p> procedure( distribution_function ), public, deferred, pass :: pdf ..\\..\\ Computes the probability density function.<\\p> procedure( distribution_property ), public, deferred, pass :: variance ..\\..\\ Computes the variance of the distribution.<\\p> type, public, extends( distribution ) :: f_distribution Defines an F-distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: d1 The measure of degrees of freedom for the first data set. real(kind=real64), public :: d2 The measure of degrees of freedom for the second data set. Type-Bound Procedures procedure, public :: cdf => fd_cdf procedure, public :: mean => fd_mean procedure, public :: median => fd_median procedure, public :: mode => fd_mode procedure, public :: pdf => fd_pdf procedure, public :: variance => fd_variance type, public, extends( distribution ) :: normal_distribution Defines a normal distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: mean_value The mean value of the distribution. real(kind=real64), public :: standard_deviation The standard deviation of the distribution. Type-Bound Procedures procedure, public :: cdf => nd_cdf procedure, public :: mean => nd_mean procedure, public :: median => nd_median procedure, public :: mode => nd_mode procedure, public :: pdf => nd_pdf procedure, public :: standardize => nd_standardize procedure, public :: variance => nd_variance type, public, extends( distribution ) :: t_distribution Defines Student's T-Distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => td_cdf procedure, public :: mean => td_mean procedure, public :: median => td_median procedure, public :: mode => td_mode procedure, public :: pdf => td_pdf procedure, public :: variance => td_variance","tags":"","loc":"module\\fstats_distributions.html"},{"title":"fstats.f90 – FSTATS","text":"Source Code module fstats !! FSTATS is a modern Fortran statistical library containing routines for !! computing basic statistical properties, hypothesis testing, regression, !! special functions, and experimental design. use iso_fortran_env use fstats_special_functions use fstats_descriptive_statistics use fstats_hypothesis use fstats_distributions use fstats_anova use fstats_helper_routines use fstats_regression use fstats_experimental_design use fstats_allan use fstats_bootstrap use fstats_sampling use fstats_smoothing implicit none private public :: distribution public :: distribution_function public :: distribution_property public :: t_distribution public :: normal_distribution public :: f_distribution public :: chi_squared_distribution public :: binomial_distribution public :: mean public :: variance public :: standard_deviation public :: median public :: covariance public :: r_squared public :: adjusted_r_squared public :: correlation public :: quantile public :: t_test_equal_variance public :: t_test_unequal_variance public :: t_test_paired public :: f_test public :: anova public :: anova_factor public :: single_factor_anova_table public :: two_factor_anova_table public :: confidence_interval public :: beta public :: regularized_beta public :: incomplete_beta public :: digamma public :: incomplete_gamma_upper public :: incomplete_gamma_lower public :: design_matrix public :: covariance_matrix public :: linear_least_squares public :: regression_statistics public :: get_full_factorial_matrix_size public :: full_factorial public :: iteration_controls public :: lm_solver_options public :: convergence_info public :: regression_function public :: iteration_update public :: jacobian public :: nonlinear_least_squares public :: allan_variance public :: trimmed_mean public :: difference public :: factorial public :: bootstrap_resampling_routine public :: bootstrap_statistic_routine public :: random_resample public :: scaled_random_resample public :: bootstrap_statistics public :: bootstrap public :: bootstrap_regression_statistics public :: bootstrap_linear_least_squares public :: bootstrap_nonlinear_least_squares public :: box_muller_sample public :: rejection_sample public :: lowess public :: pooled_variance public :: bartletts_test public :: levenes_test public :: FS_LEVENBERG_MARQUARDT_UPDATE public :: FS_QUADRATIC_UPDATE public :: FS_NIELSEN_UPDATE end module","tags":"","loc":"sourcefile\\fstats.f90.html"},{"title":"fstats_descriptive_statistics.f90 – FSTATS","text":"Source Code module fstats_descriptive_statistics use iso_fortran_env use linalg , only : sort use ferror use fstats_errors use fstats_types implicit none private public :: mean public :: variance public :: standard_deviation public :: median public :: quantile public :: trimmed_mean public :: covariance public :: pooled_variance interface pooled_variance !! Computes the pooled estimate of variance. module procedure :: pooled_variance_1 module procedure :: pooled_variance_2 end interface contains ! ------------------------------------------------------------------------------ pure function mean ( x ) result ( rst ) !! Computes the mean of the values in an array. real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 ! Local Variables integer ( int32 ) :: i , n ! Process n = size ( x ) if ( n == 0 ) then rst = zero else rst = x ( 1 ) do i = 2 , n rst = rst + ( x ( i ) - rst ) / i end do end if end function ! ------------------------------------------------------------------------------ pure function variance ( x ) result ( rst ) !! Computes the sample variance of the values in an array. !! !! The variance computed is the sample variance such that !!  s&#94;2 = \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right)&#94;2}{n - 1} . real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: oldMean , newMean ! Process n = size ( x ) if ( n <= 1 ) then rst = zero else oldMean = x ( 1 ) rst = zero do i = 2 , n newMean = oldMean + ( x ( i ) - oldMean ) / i rst = rst + ( x ( i ) - oldMean ) * ( x ( i ) - newMean ) oldMean = newMean end do rst = rst / ( n - one ) end if end function ! ------------------------------------------------------------------------------ pure function standard_deviation ( x ) result ( rst ) !! Computes the sample standard deviation of the values in an array. !! !! The value computed is the sample standard deviation. !!  s = \\sqrt{ \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right)&#94;2}{n - 1} }  real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst !! The result. ! Process rst = sqrt ( variance ( x )) end function ! ------------------------------------------------------------------------------ function median ( x ) result ( rst ) !! Computes the median of the values in an array. real ( real64 ), intent ( inout ) :: x (:) !! The array of values to analyze.  On output, this array is sorted into !! ascending order. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: n , nmid , nmidp1 , flag , iflag ! Initialization n = size ( x ) nmid = n / 2 nmidp1 = nmid + 1 iflag = n - 2 * nmid ! Sort the array in ascending order call sort ( x , . true .) ! Find the median if ( iflag == 0 ) then rst = half * ( x ( nmid ) + x ( nmidp1 )) else rst = x ( nmidp1 ) end if end function ! ------------------------------------------------------------------------------ ! REF: https://fortranwiki.org/fortran/show/Quartiles ! ! This is the method used by Minitab pure function quantile ( x , q ) result ( rst ) !! Computes the specified quantile of a data set using the SAS !! Method 4. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Quantile) real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the data. real ( real64 ), intent ( in ) :: q !! The quantile to compute (e.g. 0.25 computes the 25% quantile). real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Local Variables real ( real64 ) :: a , b , c , tol integer ( int32 ) :: n , ib ! Initialization tol = sqrt ( epsilon ( tol )) n = size ( x ) ! Process a = ( n + one ) * q b = mod ( a , one ) c = a - b ib = int ( c , int32 ) if (( ib + 1 ) > n ) then rst = ( one - b ) * x ( ib ) + b * x ( n ) else rst = ( one - b ) * x ( ib ) + b * x ( ib + 1 ) end if end function ! ------------------------------------------------------------------------------ function trimmed_mean ( x , p ) result ( rst ) !! Computes the trimmed mean of a data set. real ( real64 ), intent ( inout ), dimension (:) :: x !! An N-element array containing the data.  On output, the !! array is sorted into ascending order. real ( real64 ), intent ( in ), optional :: p !! An optional parameter specifying the percentage of values !! from either end of the distribution to remove.  The default !! is 0.05 such that the bottom 5% and top 5% are removed. real ( real64 ) :: rst !! The trimmed mean. ! Local Variables integer ( int32 ) :: i1 , i2 , n real ( real64 ) :: pv ! Initialization if ( present ( p )) then pv = abs ( p ) else pv = 0.05d0 end if ! Sort the array into ascending order call sort ( x , . true .) ! Find the limiting indices n = size ( x ) i1 = max ( floor ( n * pv , int32 ), 1 ) i2 = min ( n , n - i1 + 1 ) rst = mean ( x ( i1 : i2 )) end function ! ------------------------------------------------------------------------------ pure function covariance ( x , y ) result ( rst ) !! Computes the sample covariance of two data sets. !! !! The covariance computed is the sample covariance such that !!  q_{jk} = \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right) !! \\left( y_{i} - \\bar{y} \\right)}{n - 1} . real ( real64 ), intent ( in ), dimension (:) :: x !! The first N-element data set. real ( real64 ), intent ( in ), dimension ( size ( x )) :: y !! The second N-element data set. real ( real64 ) :: rst !! The covariance. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: meanX , meanY ! Process n = size ( x ) if ( n <= 1 ) then rst = zero else ! Compute the means meanX = x ( 1 ) meanY = y ( 1 ) do i = 2 , n meanX = meanX + ( x ( i ) - meanX ) / i meanY = meanY + ( y ( i ) - meanY ) / i end do ! Compute the covariance rst = sum (( x - meanX ) * ( y - meanY )) / ( n - one ) end if end function ! ------------------------------------------------------------------------------ pure function pooled_variance_1 ( si , ni ) result ( rst ) !! Computes the pooled estimate of variance. real ( real64 ), intent ( in ), dimension (:) :: si !! An N-element array containing the estimates for each of the N !! variances. integer ( int32 ), intent ( in ), dimension ( size ( si )) :: ni !! An N-element array containing the number of data points in each !! of the data sets used to compute the variances in si. real ( real64 ) :: rst !! The pooled variance. ! Local Variables integer ( int32 ) :: i , k , n ! Process k = size ( si ) rst = 0.0d0 n = 0 do i = 1 , k n = n + ni ( i ) rst = rst + ( ni ( i ) - 1.0d0 ) * si ( i ) end do rst = rst / real ( n - k , real64 ) end function pure function pooled_variance_2 ( x ) result ( rst ) !! Computes the pooled estimate of variance. type ( array_container ), intent ( in ), dimension (:) :: x !! An array of arrays of data. real ( real64 ) :: rst !! The pooled variance. ! Local Variables integer ( int32 ) :: i , k , n , ni ! Process k = size ( x ) n = 0 rst = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni rst = rst + variance ( x ( i )% x ) * ( ni - 1.0 ) end do rst = rst / real ( n - k , real64 ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_descriptive_statistics.f90.html"},{"title":"fstats_anova.f90 – FSTATS","text":"Source Code module fstats_anova use iso_fortran_env use ieee_arithmetic use fstats_special_functions use fstats_descriptive_statistics use ferror use fstats_errors implicit none private public :: anova_factor public :: single_factor_anova_table public :: two_factor_anova_table public :: anova type anova_factor !! Defines an ANOVA factor result. real ( real64 ) :: dof !! The number of degrees of freedome. real ( real64 ) :: variance !! The estimate of variance. real ( real64 ) :: sum_of_squares !! The sum of the squares. real ( real64 ) :: f_statistic !! The F-statistic. real ( real64 ) :: probability !! The variance probability term. end type type single_factor_anova_table !! Defines a single-factor ANOVA results table. type ( anova_factor ) :: main_factor !! The main, or main factor, results. type ( anova_factor ) :: within_factor !! The within-treatement (error) results. real ( real64 ) :: total_dof !! The total number of degrees of freedom. real ( real64 ) :: total_sum_of_squares !! The total sum of squares. real ( real64 ) :: total_variance !! The total variance estimate. real ( real64 ) :: overall_mean !! The overall mean value. end type type two_factor_anova_table !! Defines a two-factor ANOVA results table. type ( anova_factor ) :: main_factor_1 !! The first main-factor results. type ( anova_factor ) :: main_factor_2 !! The second main-factor results. type ( anova_factor ) :: interaction !! The interaction effects. type ( anova_factor ) :: within_factor !! The within (error) factor results. real ( real64 ) :: total_dof !! The total number of degrees of freedom. real ( real64 ) :: total_sum_of_squares !! The total sum of squares. real ( real64 ) :: total_variance !! The total variance estimate. real ( real64 ) :: overall_mean !! The overall mean value. end type interface anova !! Performs an analysis of variance (ANOVA) on the supplied data !! set. !! !! The following example illustrates a single-factor ANOVA on a !! data set. !! ```fortran !! program example !!     use iso_fortran_env !!     use fstats !!     implicit none !! !!     ! Local Variables !!     character, parameter :: tab = achar(9) !!     real(real64) :: x(10, 2) !!     type(single_factor_anova_table) :: tbl !! !!     ! Define the data !!     x = reshape( & !!         [ & !!             3.086d3, 3.082d3, 3.069d3, 3.072d3, 3.045d3, 3.070d3, 3.079d3, & !!             3.050d3, 3.062d3, 3.062d3, 3.075d3, 3.061d3, 3.063d3, 3.038d3, & !!             3.070d3, 3.062d3, 3.070d3, 3.049d3, 3.042d3, 3.063d3 & !!         ], & !!         [10, 2] & !!     ) !! !!     ! Perform the ANOVA !!     tbl = anova(x) !! !!     ! Print out the table !!     print '(A)', \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & !!         tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" !!     print '(AF2.0AF5.1AF5.1AF5.3AF5.3)', \"Main Factor: \" // tab, & !!         tbl%main_factor%dof, tab, & !!         tbl%main_factor%sum_of_squares, tab // tab, & !!         tbl%main_factor%variance, tab // tab, & !!         tbl%main_factor%f_statistic, tab, & !!         tbl%main_factor%probability !! !!     print '(AF3.0AF6.1AF5.1)', \"Within: \" // tab, & !!         tbl%within_factor%dof, tab, & !!         tbl%within_factor%sum_of_squares, tab // tab, & !!         tbl%within_factor%variance !! !!     print '(AF3.0AF6.1AF5.1)', \"Total: \" // tab // tab, & !!         tbl%total_dof, tab, & !!         tbl%total_sum_of_squares, tab // tab, & !!         tbl%total_variance !! !!     print '(AF6.1)', \"Overall Mean: \", tbl%overall_mean !! end program !! ``` !! The above program produces the following output. !! ```text !! Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value !! Main Factor:    1.      352.8           352.8           2.147   0.160 !! Within:         18.     2958.2          164.3 !! Total:          19.     3311.0          174.3 !! Overall Mean: 3063.5 !! ``` !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Analysis_of_variance) !! - [SPC Excel Single Factor ANOVA](https://www.spcforexcel.com/knowledge/root-cause-analysis/single-factor-anova) !! - [SPC Excel Gage R&R](https://www.spcforexcel.com/knowledge/measurement-systems-analysis/anova-gage-rr-part-1) !! - [SPC Excel Understanding Regression Statistics](https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1) !! - [NIST - Two Way ANOVA](https://www.itl.nist.gov/div898/handbook/prc/section4/prc427.htm) module procedure :: anova_1_factor module procedure :: anova_2_factor module procedure :: anova_model_fit end interface contains ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/root-cause-analysis/single-factor-anova function anova_1_factor ( x ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. real ( real64 ), intent ( in ) :: x (:,:) !! An M-by-N matrix containing the M replications of the N test !! points of interest. type ( single_factor_anova_table ) :: rst !! A single_factor_anova_table instance containing the ANOVA results. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 ! Local Variables integer ( int32 ) :: j , a , n , nt real ( real64 ) :: sum_all , tssq , essq , bssq ! Initialization a = size ( x , 2 ) nt = size ( x , 1 ) n = nt * a rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Determine the degrees of freedom rst % main_factor % dof = a - 1 rst % within_factor % dof = n - a rst % total_dof = n - 1 ! Quick Return if ( a == 1 . or . nt == 1 ) then rst % main_factor % sum_of_squares = zero rst % main_factor % variance = zero rst % main_factor % f_statistic = zero rst % main_factor % probability = zero rst % within_factor % sum_of_squares = zero rst % within_factor % variance = zero rst % total_variance = variance ( pack ( x , . true .)) rst % total_sum_of_squares = rst % total_variance * rst % total_dof rst % overall_mean = mean ( pack ( x , . true .)) return end if ! Compute the sum of squares for all factors sum_all = sum ( x ) tssq = sum ( x ** 2 ) - ( sum_all ** 2 / n ) bssq = zero do j = 1 , a bssq = bssq + sum ( x (:, j )) ** 2 end do bssq = ( bssq / nt ) - ( sum_all ** 2 / n ) essq = tssq - bssq rst % main_factor % sum_of_squares = bssq rst % within_factor % sum_of_squares = essq rst % total_sum_of_squares = tssq ! Compute the variance terms rst % main_factor % variance = bssq / rst % main_factor % dof rst % within_factor % variance = essq / rst % within_factor % dof rst % total_variance = tssq / rst % total_dof ! Compute the overall mean rst % overall_mean = mean ( pack ( x , . true .)) ! Compute the F-statistic and probability term call anova_probability ( & rst % main_factor % variance , & rst % within_factor % variance , & rst % main_factor % dof , & rst % within_factor % dof , & rst % main_factor % f_statistic , & rst % main_factor % probability & ) end function ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/measurement-systems-analysis/anova-gage-rr-part-1 ! REF: https://www.itl.nist.gov/div898/handbook/prc/section4/prc427.htm ! Data set is expected as a 3D array with each of the K pages containing the R !   treatments of N tests such that the array size is N-by-R-by-K function anova_2_factor ( x ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. real ( real64 ), intent ( in ) :: x (:,:,:) !! An M-by-N-by-K array containing the M replications of the !! N first factor results, and the K second factor results. type ( two_factor_anova_table ) :: rst !! A two_factor_anova_table instance containing the ANOVA results. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , j , jj , k , r , n real ( real64 ) :: factorMean , sum_all real ( real64 ), allocatable :: xpack (:) ! Initialization n = size ( x , 3 ) k = size ( x , 2 ) r = size ( x , 1 ) rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Quick Return if ( k == 1 ) then ! This is a one-factor anova end if ! Determine the number of DOF rst % main_factor_1 % dof = k - one rst % main_factor_2 % dof = n - 1 rst % interaction % dof = ( k - 1 ) * ( n - 1 ) rst % within_factor % dof = n * k * ( r - 1 ) rst % total_dof = n * k * r - 1 ! Compute the overall mean, sum of squares, and variance xpack = pack ( x , . true .) rst % overall_mean = mean ( xpack ) rst % total_sum_of_squares = sum (( xpack - rst % overall_mean ) ** 2 ) rst % total_variance = rst % total_sum_of_squares / rst % total_dof ! Compute factor 1 results rst % main_factor_1 % sum_of_squares = zero do i = 1 , k factorMean = mean ( pack ( x (:, i ,:), . true .)) rst % main_factor_1 % sum_of_squares = rst % main_factor_1 % sum_of_squares + & ( factorMean - rst % overall_mean ) ** 2 end do rst % main_factor_1 % sum_of_squares = n * r * rst % main_factor_1 % sum_of_squares rst % main_factor_1 % variance = rst % main_factor_1 % sum_of_squares / & rst % main_factor_1 % dof ! Compute factor 2 results rst % main_factor_2 % sum_of_squares = zero do i = 1 , n factorMean = mean ( pack ( x (:,:, i ), . true .)) rst % main_factor_2 % sum_of_squares = rst % main_factor_2 % sum_of_squares + & ( factorMean - rst % overall_mean ) ** 2 end do rst % main_factor_2 % sum_of_squares = k * r * rst % main_factor_2 % sum_of_squares rst % main_factor_2 % variance = rst % main_factor_2 % sum_of_squares / & rst % main_factor_2 % dof ! Compute the within (error) term rst % within_factor % sum_of_squares = zero do j = 1 , k do i = 1 , n factorMean = mean ( x (:, j , i )) do jj = 1 , r rst % within_factor % sum_of_squares = & rst % within_factor % sum_of_squares + & ( x ( jj , j , i ) - factorMean ) ** 2 end do end do end do rst % within_factor % variance = rst % within_factor % sum_of_squares / & rst % within_factor % dof ! Compute the interaction term rst % interaction % sum_of_squares = rst % total_sum_of_squares - ( & rst % main_factor_1 % sum_of_squares + & rst % main_factor_2 % sum_of_squares + & rst % within_factor % sum_of_squares & ) rst % interaction % variance = rst % interaction % sum_of_squares / & rst % interaction % dof ! Compute the F-statistics call anova_probability ( & rst % main_factor_1 % variance , & rst % within_factor % variance , & rst % main_factor_1 % dof , & rst % within_factor % dof , & rst % main_factor_1 % f_statistic , & rst % main_factor_1 % probability & ) call anova_probability ( & rst % main_factor_2 % variance , & rst % within_factor % variance , & rst % main_factor_2 % dof , & rst % within_factor % dof , & rst % main_factor_2 % f_statistic , & rst % main_factor_2 % probability & ) call anova_probability ( & rst % interaction % variance , & rst % within_factor % variance , & rst % interaction % dof , & rst % within_factor % dof , & rst % interaction % f_statistic , & rst % interaction % probability & ) end function ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1 function anova_model_fit ( nmodelparams , ymeas , ymod , err ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. integer ( int32 ), intent ( in ) :: nmodelparams !! The number of model parameters. real ( real64 ), intent ( in ) :: ymeas (:) !! An N-element array containing the measured dependent variable data. real ( real64 ), intent ( in ) :: ymod (:) !! An N-element array containing the modeled dependent variable data. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the !!   same length. !! - FS_MEMORY_ERROR: Occurs if a memory error is encountered. type ( single_factor_anova_table ) :: rst !! A single_factor_anova_table instance containing the ANOVA results. ! Local Variables integer ( int32 ) :: n , flag real ( real64 ), allocatable :: ypack (:) real ( real64 ) :: sum_all class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization n = size ( ymeas ) if ( present ( err )) then errmgr => err else errmgr => deferr end if rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Input Checking if ( size ( ymod ) /= n ) then call report_arrays_not_same_size_error ( errmgr , \"anova_model_fit\" , & \"YMEAS\" , \"YMOD\" , n , size ( ymod )) return end if ! Memory Allocation allocate ( ypack ( 2 * n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"anova_model_fit\" , flag ) return end if ! Determine the number of DOF rst % main_factor % dof = nmodelparams - 1 rst % within_factor % dof = n - rst % main_factor % dof - 1 rst % total_dof = n - 1 ! Process ypack ( 1 : n ) = ymeas ypack ( n + 1 : 2 * n ) = ymod rst % overall_mean = mean ( ypack ) rst % total_sum_of_squares = sum (( ymeas - rst % overall_mean ) ** 2 ) rst % main_factor % sum_of_squares = sum (( ymod - rst % overall_mean ) ** 2 ) rst % within_factor % sum_of_squares = sum (( ymeas - ymod ) ** 2 ) rst % total_variance = rst % total_sum_of_squares / rst % total_dof rst % main_factor % variance = rst % main_factor % sum_of_squares / & rst % main_factor % dof rst % within_factor % variance = rst % within_factor % sum_of_squares / & rst % within_factor % dof ! Compute the F-statistic and probability term call anova_probability ( & rst % main_factor % variance , & rst % within_factor % variance , & rst % main_factor % dof , & rst % within_factor % dof , & rst % main_factor % f_statistic , & rst % main_factor % probability & ) ! Formatting 100 format ( A , I0 , A , I0 , A ) 101 format ( A , I0 , A ) end function ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ subroutine anova_probability ( v1 , v2 , dof1 , dof2 , f , p ) ! Arguments real ( real64 ), intent ( in ) :: v1 , v2 , dof1 , dof2 real ( real64 ), intent ( out ) :: f , p ! Local Variables real ( real64 ) :: d1 , d2 , a , b , x ! Process f = v1 / v2 d1 = dof1 d2 = dof2 a = 0.5d0 * d2 b = 0.5d0 * d1 x = d2 / ( d2 + d1 * f ) p = regularized_beta ( a , b , x ) if ( p > 1.0d0 ) then p = 2.0d0 - p end if end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_anova.f90.html"},{"title":"fstats_helper_routines.f90 – FSTATS","text":"Source Code module fstats_helper_routines use iso_fortran_env implicit none private public :: difference public :: factorial contains ! ------------------------------------------------------------------------------ pure function difference ( x ) result ( rst ) !! Computes the difference between elements in an array. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array on which to operate. real ( real64 ), allocatable , dimension (:) :: rst !! The (N-1)-element array containing the differences between adjacent !! elements. ! Local Variables integer ( int32 ) :: i , n ! Process n = size ( x ) allocate ( rst ( n - 1 )) do i = 1 , n - 1 rst ( i ) = x ( i + 1 ) - x ( i ) end do end function ! ------------------------------------------------------------------------------ pure elemental function factorial ( x ) result ( rst ) !! Computes the factorial of X. real ( real64 ), intent ( in ) :: x !! The value whose factorial is to be computed. real ( real64 ) :: rst !! The result. rst = gamma ( x + 1.0d0 ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_helper_routines.f90.html"},{"title":"fstats_types.f90 – FSTATS","text":"Source Code module fstats_types use iso_fortran_env implicit none type array_container !! Provides a container for a real-valued array.  A practical use of !! this construct is in the construction of jagged arrays. real ( real64 ), allocatable , dimension (:) :: x !! The array. end type end module","tags":"","loc":"sourcefile\\fstats_types.f90.html"},{"title":"fstats_experimental_design.f90 – FSTATS","text":"Source Code module fstats_experimental_design use iso_fortran_env use fstats_errors implicit none private public :: get_full_factorial_matrix_size public :: full_factorial contains ! ------------------------------------------------------------------------------ subroutine get_full_factorial_matrix_size ( vars , m , n , err ) !! Computes the appropriate size for a full-factorial design table. integer ( int32 ), intent ( in ) :: vars (:) !! An M-element array containing the M factors to study.  Each !! of the M entries to the array is expected to contain the !! number of options for that particular factor to explore. !! This value must be greater than or equal to 1. integer ( int32 ), intent ( out ) :: m !! The number of rows for the table. integer ( int32 ), intent ( out ) :: n !! The number of columns for the table. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_INVALID_INPUT_ERROR: Occurs if any items in vars are !!      less than 1. ! Local Variables integer ( int32 ) :: i class ( errors ), pointer :: errmgr type ( errors ), target :: deferr character ( len = 256 ) :: errmsg ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if m = 0 n = 0 ! Ensure every value is greater than 1 do i = 1 , size ( vars ) if ( vars ( i ) < 1 ) then write ( errmsg , 100 ) \"A value less than 1 was found at index \" , & i , \" of the input array.  All values must be greater \" // & \"than or equal to 1.\" call errmgr % report_error ( \"get_full_factorial_matrix_size\" , & trim ( errmsg ), FS_INVALID_INPUT_ERROR ) return end if end do ! Process m = product ( vars ) n = size ( vars ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine full_factorial ( vars , tbl , err ) !! Computes a table with values scaled from 1 to N describing a !! full-factorial design. !! !! ```fortran !! program example !!     use iso_fortran_env !!     use fstats !!     implicit none !! !!     ! Local Variables !!     integer(int32) :: i, vars(3), tbl(24, 3) !! !!     ! Define the number of design points for each of the 3 factors to study !!     vars = [2, 4, 3] !! !!     ! Determine the design table !!     call full_factorial(vars, tbl) !! !!     ! Display the table !!     do i = 1, size(tbl, 1) !!         print *, tbl(i,:) !!     end do !! end program !! ``` !! The above program produces the following output. !! ```text !! 1           1           1 !! 1           1           2 !! 1           1           3 !! 1           2           1 !! 1           2           2 !! 1           2           3 !! 1           3           1 !! 1           3           2 !! 1           3           3 !! 1           4           1 !! 1           4           2 !! 1           4           3 !! 2           1           1 !! 2           1           2 !! 2           1           3 !! 2           2           1 !! 2           2           2 !! 2           2           3 !! 2           3           1 !! 2           3           2 !! 2           3           3 !! 2           4           1 !! 2           4           2 !! 2           4           3 !! ``` integer ( int32 ), intent ( in ) :: vars (:) !! An M-element array containing the M factors to study. !! Each of the M entries to the array is expected to contain !! the number of options for that particular factor to explore. !! This value must be greater than or equal to 1. integer ( int32 ), intent ( out ) :: tbl (:,:) !! A table where the design will be written.  Use !! get_full_factorial_matrix_size to determine the appropriate !! table size. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_INVALID_INPUT_ERROR: Occurs if any items in vars are !!      less than 1. !! - FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized. ! Local Variables integer ( int32 ) :: i , col , stride , last , val , m , n class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Verify the size of the input table call get_full_factorial_matrix_size ( vars , m , n , errmgr ) if ( errmgr % has_error_occurred ()) return if ( size ( tbl , 1 ) /= m . or . size ( tbl , 2 ) /= n ) then call report_matrix_size_error ( errmgr , \"full_factorial\" , & \"tbl\" , m , n , size ( tbl , 1 ), size ( tbl , 2 )) return end if ! Process do col = 1 , n stride = 1 if ( col /= n ) stride = product ( vars ( col + 1 : n )) val = 1 do i = 1 , m , stride last = i + stride - 1 tbl ( i : last , col ) = val val = val + 1 if ( val > vars ( col )) val = 1 end do end do end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_experimental_design.f90.html"},{"title":"fstats_hypothesis.f90 – FSTATS","text":"Source Code module fstats_hypothesis use iso_fortran_env use ieee_arithmetic use fstats_errors use fstats_special_functions use fstats_distributions use fstats_descriptive_statistics use fstats_types private public :: confidence_interval public :: t_test_equal_variance public :: t_test_unequal_variance public :: t_test_paired public :: f_test public :: bartletts_test public :: levenes_test interface confidence_interval !! Computes the confidence interval for the specified distribution. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Confidence_interval) module procedure :: confidence_interval_scalar module procedure :: confidence_interval_array end interface contains ! ------------------------------------------------------------------------------ pure function confidence_interval_scalar ( dist , alpha , s , n ) result ( rst ) !! Computes the confidence interval for the specified distribution. class ( distribution ), intent ( in ) :: dist !! The distribution object defining the probability distribution !! to establish the confidence level. real ( real64 ), intent ( in ) :: alpha !! The probability value of interest.  For instance, use a value of 0.05 !! for a confidence level of 95%. real ( real64 ), intent ( in ) :: s !! The sample standard deviation. integer ( int32 ), intent ( in ) :: n !! The number of samples in the data set. real ( real64 ) :: rst !! The result. ! Local Variables integer ( int32 ), parameter :: maxiter = 100 real ( real64 ), parameter :: tol = 1.0d-6 integer ( int32 ) :: i real ( real64 ) :: x , f , df , h , twoh , dy ! Process ! ! We use a simplified Newton's method to solve for the independent variable ! of the CDF function where it equals 1 - alpha / 2. h = 1.0d-6 twoh = 2.0d0 * h x = 1.0d0 - alpha / 2.0d0 rst = 0.5d0 do i = 1 , maxiter ! Compute the CDF and its derivative at y f = dist % cdf ( rst ) - x df = ( dist % cdf ( rst + h ) - dist % cdf ( rst - h )) / twoh dy = f / df rst = rst - dy if ( abs ( dy ) < tol ) exit end do ! Determine the actual interval rst = rst * s / sqrt ( real ( n , real64 )) end function ! ------------------------------------------------------------------------------ pure function confidence_interval_array ( dist , alpha , x ) result ( rst ) !! Computes the confidence interval for the specified distribution. class ( distribution ), intent ( in ) :: dist !! The distribution object defining the probability distribution !! to establish the confidence level. real ( real64 ), intent ( in ) :: alpha !! The probability value of interest.  For instance, use a value of 0.05 !! for a confidence level of 95%. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the data to analyze. real ( real64 ) :: rst !! The result. ! Process rst = confidence_interval ( dist , alpha , standard_deviation ( x ), size ( x )) end function ! ------------------------------------------------------------------------------ subroutine t_test_equal_variance ( x1 , x2 , stat , p , dof ) !! Computes the 2-tailed Student's T-Test for two data sets of !! assumed equivalent variances. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables real ( real64 ) :: v1 , v2 , m1 , m2 , sv , a , b , x integer ( int32 ) :: n1 , n2 ! Compute the T-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = n1 + n2 - two sv = (( n1 - one ) * v1 + ( n2 - one ) * v2 ) / dof stat = abs ( m1 - m2 ) / sqrt ( sv * ( one / real ( n1 ) + one / real ( n2 ))) ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine t_test_unequal_variance ( x1 , x2 , stat , p , dof ) !! Computes the 2-tailed Student's T-Test for two data sets of !! assumed non-equivalent variances. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables real ( real64 ) :: v1 , v2 , m1 , m2 , sv , a , b , x integer ( int32 ) :: n1 , n2 ! Compute the T-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = ( v1 / real ( n1 ) + v2 / real ( n2 )) ** 2 / (( v1 / n1 ) ** 2 / ( n1 - one ) + & ( v2 / n2 ) ** 2 / ( n2 - one )) sv = sqrt ( v1 / n1 + v2 / n2 ) stat = ( m1 - m2 ) / sv ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine t_test_paired ( x1 , x2 , stat , p , dof , err ) !! Computes the 2-tailed Student's T-Test for two paired data sets. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An N-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from  two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same !!   length. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr real ( real64 ) :: v1 , v2 , m1 , m2 , sd , cov , a , b , x integer ( int32 ) :: i , n1 , n2 , n ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n1 = size ( x1 ) n2 = size ( x2 ) n = min ( n1 , n2 ) ! Input Checking if ( n1 /= n2 ) then call report_arrays_not_same_size_error ( errmgr , \"t_test_paired_real64\" , & \"X1\" , \"X2\" , n1 , n2 ) return end if ! Compute the T-statistic m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = real ( n1 ) - one cov = zero do i = 1 , n cov = cov + ( x1 ( i ) - m1 ) * ( x2 ( i ) - m2 ) end do cov = cov / dof sd = sqrt (( v1 + v2 - two * cov ) / n ) stat = ( m1 - m2 ) / sd ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine f_test ( x1 , x2 , stat , p , dof1 , dof2 ) !! Computes the F-test and returns the probability (two-tailed) that !! the variances of two data sets are not significantly different. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/F-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The F-statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from the two underlying populations that !! have the same variance. real ( real64 ), intent ( out ) :: dof1 !! A measure of the degrees of freedom. real ( real64 ), intent ( out ) :: dof2 !! A measure of the degrees of freedom. ! Parameters real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables integer ( int32 ) :: n1 , n2 real ( real64 ) :: v1 , v2 , m1 , m2 type ( f_distribution ) :: dist ! Compute the F-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) if ( v1 > v2 ) then stat = v1 / v2 dof1 = n1 - one dof2 = n2 - one else stat = v2 / v1 dof1 = n2 - one dof2 = n1 - one end if dist % d1 = dof1 dist % d2 = dof2 p = two * ( one - dist % cdf ( stat )) end subroutine ! ------------------------------------------------------------------------------ subroutine bartletts_test ( x , stat , p ) !! Computes Bartlett's test statistic and associated probability. !! !! The statistic is calculated as follows. !! !!  \\chi&#94;{2} = \\frac{(N - k) \\ln(S_{p}&#94;{2}) \\sum_{i = 1}&#94;{k} !! \\left(n_{i} - 1 \\right) \\ln(S_{i}&#94;{2})}{1 + !! \\frac{1}{3 \\left( k - 1 \\right)} \\left( \\sum_{i = 1}&#94;{k} !! \\left( \\frac{1}{n_{i} - 1} \\right) - \\frac{1}{N - k} \\right)}  !! !! Where  N = \\sum_{i = 1}&#94;{k} n_{i}  and  S_{p}&#94;{2}  is the pooled !! variance. !! !! The probability is calculated as the right-tail probability of the !! chi-squared distribution. !! !! Bartlett's test is most relevant for distributions showing strong !! normality.  For distributions lacking strong normality, consider !! Levene's test instead. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Bartlett%27s_test) type ( array_container ), intent ( in ), dimension (:) :: x !! The arrays of data to analyze. real ( real64 ), intent ( out ) :: stat !! The Bartlett's test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the variances of each data set are !! equivalent.  A low p-value, less than some significance level, !! indicates a non-equivalance of variances. ! Local Variables integer ( int32 ) :: i , n , k , ni real ( real64 ) :: si , sp , numer , denom type ( chi_squared_distribution ) :: dist ! Initialization k = size ( x ) n = 0 do i = 1 , k n = n + size ( x ( i )% x ) end do ! Compute the statistic n = 0 sp = 0.0d0 numer = 0.0d0 denom = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni si = variance ( x ( i )% x ) sp = sp + ( ni - 1.0d0 ) * si numer = numer + ( ni - 1.0d0 ) * log ( variance ( x ( i )% x )) denom = denom + 1.0d0 / ( ni - 1.0d0 ) end do sp = sp / real ( n - k , real64 ) stat = (( n - k ) * log ( sp ) - numer ) / & ( 1.0d0 + ( 1.0d0 / ( 3.0d0 * k - 3.0d0 )) * & ( denom - 1.0d0 / real ( n - k , real64 ))) ! Compute the p-value dist % dof = k - 1 p = 1.0d0 - dist % cdf ( stat ) end subroutine ! ------------------------------------------------------------------------------ subroutine levenes_test ( x , stat , p , err ) !! Computes Levene's test statistic and associated probability. !! !! The statistic is calculated as follows. !!  W = \\frac{N - k}{k - 1} \\frac{ \\sum_{i = 1}&#94;{k} N_{i} \\left( Z_{i.} - !! Z{..} \\right)&#94;{2}}{ \\sum_{i = 1}&#94;{k} \\sum_{j = 1}&#94;{n_{i}} \\left( Z_{ij} - !! Z_{i.} \\right)&#94;{2} }  !! !! As the test statistic is approximately F-distributed, the F-distribution !! is used to calculate the probability term. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Levene%27s_test) type ( array_container ), intent ( in ), dimension (:) :: x !! The arrays of data to analyze. real ( real64 ), intent ( out ) :: stat !! The Bartlett's test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the variances of each data set are !! equivalent.  A low p-value, less than some significance level, !! indicates a non-equivalance of variances. class ( errors ), intent ( inout ), optional , target :: err ! Local Variables integer ( int32 ) :: i , j , k , n , ni , flag real ( real64 ) :: numer , denom , inner , yi , z , zij real ( real64 ), allocatable , dimension (:) :: y , zt , zi type ( f_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if k = size ( x ) ! Local Memory Allocations allocate ( y ( k ), zi ( k ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"levenes_test\" , flag ) return end if ! Compute the total mean z = 0.0d0 n = 0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni y ( i ) = mean ( x ( i )% x ) zt = abs ( x ( i )% x - y ( i )) zi ( i ) = mean ( zt ) z = z + zi ( i ) * ni end do z = z / n ! Process numer = 0.0d0 denom = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) yi = y ( i ) numer = numer + ni * ( zi ( i ) - z ) ** 2 inner = 0.0d0 do j = 1 , ni zij = abs ( x ( i )% x ( j ) - yi ) inner = inner + ( zij - zi ( i )) ** 2 end do denom = denom + inner end do stat = real (( N - k ) / ( k - 1 ), real64 ) * ( numer / denom ) dist % d1 = k - 1.0d0 dist % d2 = real ( n - k , real64 ) p = 1.0d0 - dist % cdf ( stat ) end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_hypothesis.f90.html"},{"title":"fstats_errors.f90 – FSTATS","text":"Source Code ! A module providing a set of routines to handle errors for the FSTATS library. module fstats_errors use ferror use iso_fortran_env , only : int32 implicit none ! ****************************************************************************** ! ERROR CODES ! ------------------------------------------------------------------------------ integer ( int32 ), parameter :: FS_NO_ERROR = 0 integer ( int32 ), parameter :: FS_ARRAY_SIZE_ERROR = 10000 integer ( int32 ), parameter :: FS_MATRIX_SIZE_ERROR = 10001 integer ( int32 ), parameter :: FS_INVALID_INPUT_ERROR = 10002 integer ( int32 ), parameter :: FS_MEMORY_ERROR = 10003 integer ( int32 ), parameter :: FS_UNDERDEFINED_PROBLEM_ERROR = 10004 integer ( int32 ), parameter :: FS_TOLERANCE_TOO_SMALL_ERROR = 10005 integer ( int32 ), parameter :: FS_TOO_FEW_ITERATION_ERROR = 10006 ! ------------------------------------------------------------------------------ integer ( int32 ), private , parameter :: MESSAGE_SIZE = 1024 contains ! ------------------------------------------------------------------------------ subroutine report_memory_error ( err , fname , code ) !! Reports a memory allocation related error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. integer ( int32 ), intent ( in ) :: code !! The error code returned by the allocation routine. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) & \"A memory allocation error occurred with code \" , code , \".\" call err % report_error ( fname , trim ( msg ), FS_MEMORY_ERROR ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_array_size_error ( err , fname , name , expect , actual ) !! Reports an array size error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name !! The name of the array. integer ( int32 ), intent ( in ) :: expect !! The expected size of the array. integer ( int32 ), intent ( in ) :: actual !! The actual size of the array. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Expected array \" // name // \" to be of length \" , & expect , \", but found it to be of length \" , actual , \".\" call err % report_error ( fname , trim ( msg ), FS_ARRAY_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_matrix_size_error ( err , fname , name , expect_rows , & expect_cols , actual_rows , actual_cols ) !! Reports a matrix size error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name !! The name of the matrix. integer ( int32 ), intent ( in ) :: expect_rows !! The expected number of rows. integer ( int32 ), intent ( in ) :: expect_cols !! The expected number of columns. integer ( int32 ), intent ( in ) :: actual_rows !! The actual number of rows. integer ( int32 ), intent ( in ) :: actual_cols !! The actual number of columns. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Expected matrix \" // name // \" to be of size (\" , & expect_rows , \", \" , expect_cols , \"), but found it to be of size (\" , & actual_rows , \", \" , actual_cols , \").\" call err % report_error ( fname , trim ( msg ), FS_MATRIX_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_arrays_not_same_size_error ( err , fname , name1 , name2 , & size1 , size2 ) !! Reports an error relating to two arrays not being the same size !! when they should be the same size. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name1 !! The name of the first array. character ( len = * ), intent ( in ) :: name2 !! The name of the second array. integer ( int32 ), intent ( in ) :: size1 !! The size of the first array. integer ( int32 ), intent ( in ) :: size2 !! The size of the second array. ! Local Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Array \" // name1 // \" and array \" // name2 // & \"were expected to be the same size, but instead were found \" // & \"to be sized \" , size1 , \" and \" , size2 , \" respectively.\" call err % report_error ( fname , trim ( msg ), FS_ARRAY_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_underdefined_error ( err , fname , expect , actual ) !! Reports an underdefined problem error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. integer ( int32 ), intent ( in ) :: expect !! The expected minimum number of equations. integer ( int32 ), intent ( in ) :: actual !! The actual number of equations. ! Local Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"The problem is underdefined.  The number of \" // & \"equations was found to be \" , actual , & \", but must be at least equal to the number of unknowns \" , & expect , \".\" call err % report_error ( fname , trim ( msg ), FS_UNDERDEFINED_PROBLEM_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_iteration_count_error ( err , fname , msg , mincount ) !! Reports an iteration count error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ) :: fname !! The name of the routine in which the error occurred. character ( len = * ) :: msg !! The error message. integer ( int32 ), intent ( in ) :: mincount !! The minimum iteration count expected. ! Local Variables character ( len = MESSAGE_SIZE ) :: emsg ! Process write ( emsg , 100 ) msg // \"  A minimum of \" , mincount , \" is expected.\" call err % report_error ( fname , trim ( emsg ), FS_TOO_FEW_ITERATION_ERROR ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_errors.f90.html"},{"title":"fstats_allan.f90 – FSTATS","text":"Source Code module fstats_allan use iso_fortran_env use fstats_errors implicit none private public :: allan_variance contains ! ------------------------------------------------------------------------------ ! REF: Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, ! Viraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm for ! Fully Overlapped Allan Variance and Total Variance for Analysis and Modeling ! of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. ! 10.1109/LSENS.2018.2829799. ! ! https://www.researchgate.net/publication/324738301_A_Fast_Parallel_Algorithm_for_Fully_Overlapped_Allan_Variance_and_Total_Variance_for_Analysis_and_Modeling_of_Noise_in_Inertial_Sensors ! https://github.com/shrikanth95/Fast-Parallel-Fully-Overlapped-Allan-Variance-and-Total-Variance/blob/master/fast_FOAV.m function allan_variance ( x , dt , err ) result ( rst ) !! Computes the Allan variance of a data set. !! !! Remarks !! !! This implementation computes the fully overlapped Allan variance !! using the method presented by Yadav et. al. !! !! Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, !! Viraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm !! for Fully Overlapped Allan Variance and Total Variance for Analysis and !! Modeling of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. !! 10.1109/LSENS.2018.2829799. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element data set to analyze. real ( real64 ), intent ( in ), optional :: dt !! An optional input specifying the time increment between !! samples in x.  If not specified, this value is set to 1. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. real ( real64 ), allocatable , dimension (:,:) :: rst !! An M-by-2 array containing the results where M is N / 2 - 1 !! if N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The !! first column contains the averaging times associated with !! the M results stored in the second column. ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr integer ( int32 ) :: flag , j , m , n , limit , nr real ( real64 ), allocatable , dimension (:) :: tall1 , tall2 real ( real64 ) :: temp , deltaT ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( dt )) then deltaT = dt else deltaT = 1.0d0 end if ! Initialization n = size ( x ) limit = n nr = floor ( 0.5 * n ) - 1 allocate ( tall1 ( n - 1 ), source = x (: n - 1 ), stat = flag ) if ( flag == 0 ) allocate ( tall2 ( n - 1 ), source = x ( 2 : n )) if ( flag == 0 ) allocate ( rst ( nr , 2 ), source = 0.0d0 ) if ( flag /= 0 ) go to 10 ! Process do m = 1 , nr temp = 0.0d0 do j = 1 , limit - 1 temp = temp + ( tall2 ( j ) - tall1 ( j )) ** 2 tall1 ( j ) = tall1 ( j ) + x ( min ( n , m + j )) tall2 ( j ) = tall2 ( min ( n - 1 , j + 1 )) + x ( min ( n , 2 * m + j + 1 )) end do limit = limit - 2 rst ( m , 1 ) = dt * m rst ( m , 2 ) = temp / ( 2.0d0 * ( n - 2 * m + 1 ) * m ** 2 ) end do ! End return ! Memory Error Handling 10 continue call report_memory_error ( errmgr , \"allan_variance\" , flag ) return end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_allan.f90.html"},{"title":"fstats_bootstrap.f90 – FSTATS","text":"Source Code module fstats_bootstrap use iso_fortran_env use fstats_errors use omp_lib use fstats_distributions use fstats_descriptive_statistics use fstats_special_functions use fstats_regression use linalg , only : sort implicit none private public :: bootstrap_resampling_routine public :: bootstrap_statistic_routine public :: random_resample public :: scaled_random_resample public :: bootstrap_statistics public :: bootstrap public :: bootstrap_regression_statistics public :: bootstrap_linear_least_squares public :: bootstrap_nonlinear_least_squares ! REFERENCES: ! - https://medium.com/@m21413108/bootstrapping-maximum-entropy-non-parametric-boot-python-3b1e23ea589d ! - https://cran.r-project.org/web/packages/meboot/vignettes/meboot.pdf ! - https://gist.github.com/christianjauregui/314456688a3c2fead43a48be3a47dad6 type bootstrap_regression_statistics !! A container for regression-related statistical information as !! computed in a bootstrap, or equivalent, calculation. real ( real64 ) :: standard_error !! The standard error for the model coefficient. real ( real64 ) :: t_statistic !! The T-statistic for the model coefficient. !! !!  t_o = \\frac{ \\beta_{i} }{E_{s}(\\beta_{i})}  real ( real64 ) :: probability !! The probability that the coefficient is not statistically !! important.  A statistically important coefficient will have a !! low probability (p-value), typically 0.05 or lower; however, a !! p-value of up to ~0.2 may be acceptable dependent upon the !! problem.  Typically any p-value larger than ~0.2 indicates the !! parameter is not statistically important for the model. !! !!  p = t_{|t_o|, df_{residual}}  real ( real64 ) :: upper_confidence_interval !! The upper limit of the confidence interval for the parameter. real ( real64 ) :: lower_confidence_interval !! The lower limit of the confidence interval for the parameter. end type type bootstrap_statistics !! A collection of statistics resulting from the bootstrap process. real ( real64 ) :: statistic_value !! The value of the statistic of interest. real ( real64 ) :: upper_confidence_interval !! The upper confidence limit on the statistic. real ( real64 ) :: lower_confidence_interval !! The lower confidence limit on the statistic. real ( real64 ) :: bias !! The bias in the statistic. real ( real64 ) :: standard_error !! The standard error of the statistic. real ( real64 ), allocatable , dimension (:) :: population !! An array of the population values generated by the bootstrap !! process. end type interface subroutine bootstrap_resampling_routine ( x , xn ) !! Defines the signature of a subroutine used to compute a !! resampling of data for bootstrapping purposes. use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be !! written. end subroutine function bootstrap_statistic_routine ( x ) result ( rst ) !! Defines the signature of a function for computing the desired !! bootstrap statistic. use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: x !! The array of data to analyze. real ( real64 ) :: rst !! The resulting statistic. end function end interface contains ! ****************************************************************************** ! RESAMPLING ! ------------------------------------------------------------------------------ subroutine random_resample ( x , xn ) !! Random resampling, with replacement, based upon a normal distribution. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be written. ! Parameters real ( real64 ), parameter :: scale = 1.25d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: xmin , xmax , rng ! Process n = size ( x ) xmin = x ( 1 ) xmax = x ( 1 ) do i = 2 , n xmin = min ( xmin , x ( i )) xmax = max ( xmax , x ( i )) end do rng = ( xmax - xmin ) call random_number ( xn ) xn = xn * rng + xmin end subroutine ! ------------------------------------------------------------------------------ subroutine scaled_random_resample ( x , xn ) !! A random resampling, scaled by the standard deviation of the original !! data, but based upon a normal distribution. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be written. ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: n real ( real64 ) :: eps ! Process n = size ( x ) eps = standard_deviation ( x ) / sqrt ( real ( n , real64 )) call random_number ( xn ) xn = eps * ( xn - half ) + x end subroutine ! ****************************************************************************** ! BOOTSTRAPPING ! ------------------------------------------------------------------------------ function bootstrap ( stat , x , method , nsamples , alpha ) result ( rst ) !! Performs a bootstrap calculation on the supplied data set for the given !! statistic.  The default implementation utlizes a random resampling with !! replacement.  Other resampling methods may be defined by specifying an !! appropriate routine by means of the method input. procedure ( bootstrap_statistic_routine ), pointer , intent ( in ) :: stat !! The routine used to compute the desired statistic. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element data set. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. integer ( int32 ), intent ( in ), optional :: nsamples !! An optional input, that if supplied, specifies the number of !! resampling runs to perform.  The default is 10 000. real ( real64 ), intent ( in ), optional :: alpha !! An optional input, that if supplied, defines the significance level !! to use for the analysis.  The default is 0.05. type ( bootstrap_statistics ) :: rst !! The resulting bootstrap_statistics type containing the confidence !! intervals, bias, standard error, etc. for the analyzed statistic. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , i1 , i2 , n , ns real ( real64 ) :: a real ( real64 ), allocatable , dimension (:) :: xn procedure ( bootstrap_resampling_routine ), pointer :: resample ! Initialization n = size ( x ) if ( present ( method )) then resample => method else resample => random_resample end if if ( present ( nsamples )) then ns = nsamples else ns = 10000 end if if ( present ( alpha )) then a = alpha else a = p05 end if allocate ( rst % population ( ns )) i1 = floor ( half * a * ns , int32 ) i2 = ns - i1 + 1 ! Analyze the basic data set rst % statistic_value = stat ( x ) rst % population ( 1 ) = rst % statistic_value ! Resampling Process #ifdef USEOPENMP ! Use OpenMP to run operations in parallel !$OMP PARALLEL DO PRIVATE(xn) SHARED(rst) do i = 2 , ns ! Per-thread memory allocation if (. not . allocated ( xn )) allocate ( xn ( n )) ! Resample the data call resample ( x , xn ) ! Compute the statistic rst % population ( i ) = stat ( xn ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( xn ( n )) do i = 2 , ns ! Resample the data call resample ( x , xn ) ! Compute the statistic for the resampled data rst % population ( i ) = stat ( xn ) end do #endif ! Compute the relevant quantities on the resampled statistic call sort ( rst % population , . true .) rst % upper_confidence_interval = rst % population ( i2 ) rst % lower_confidence_interval = rst % population ( i1 ) rst % bias = mean ( rst % population ) - rst % statistic_value rst % standard_error = standard_deviation ( rst % population ) end function ! ****************************************************************************** ! LINEAR REGRESSION ! ------------------------------------------------------------------------------ subroutine bootstrap_linear_least_squares ( order , intercept , x , y , & coeffs , ymod , resid , nsamples , stats , bias , alpha , method , bscoeffs , err ) !! Computes a linear least-squares regression to fit a set of data. !! Bootstrapping is utilized to gain insight into the quality of !! the fit.  Resampling for the bootstrap process is a random resampling !! with replacement process with the range of values limited by the !! standard deviation of the original data set. integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be at !! least one (linear equation), but can be higher as desired, !! as long as there is sufficient data. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed as part of !! the regression; else, false. real ( real64 ), intent ( in ), dimension (:) :: x !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( in ), dimension (:) :: y !! An N-element array containing the dependent variable !! measurement points. real ( real64 ), intent ( out ), dimension (:) :: coeffs !! An ORDER+1 element array where the coefficients will !! be written. real ( real64 ), intent ( out ), dimension (:) :: ymod !! An N-element array where the modeled data will be written. real ( real64 ), intent ( out ), dimension (:) :: resid !! An N-element array where the residual error data will be !! written (modeled - actual). integer ( int32 ), intent ( in ), optional :: nsamples !! The number of bootstrapping samples to utilize. type ( bootstrap_regression_statistics ), intent ( out ), optional , & dimension (:) :: stats !! An M-element array of bootstrap_regression_statistics items !! where M = ORDER + 1 when intercept is set to true; however, !! if intercept is set to false, M = ORDER. real ( real64 ), intent ( out ), optional , dimension (:) :: bias !! An ORDER+1 element array where an estimate of the bias of !! each coefficient is returned based upon the results of the !! bootstrapping analysis.  The bias is computed as the difference !! between the mean of the boostrap population results for the given !! parameter and the original estimate of the given parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. real ( real64 ), intent ( out ), optional , allocatable , target , dimension (:,:) :: bscoeffs !! An optional, allocatable matrix, containing the bootstrap !! distributions for each parameter stored in each row of the matrix !! such that the resulting matrix is NCOEFFS -by- NSAMPLES. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , j , n , ns , nc , ncoeffs , flag , nthreads , thread real ( real64 ) :: alph real ( real64 ), allocatable , dimension (:) :: fLocal , yLocal , rLocal real ( real64 ), allocatable , target , dimension (:,:) :: coeffstorage real ( real64 ), pointer , dimension (:,:) :: allcoeffs class ( errors ), pointer :: errmgr type ( errors ), target :: deferr procedure ( bootstrap_resampling_routine ), pointer :: resample ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( nsamples )) then ns = nsamples else ns = 1000 end if if ( present ( alpha )) then alph = alpha else alph = p05 end if if ( present ( method )) then resample => method else resample => scaled_random_resample end if n = size ( x ) ncoeffs = order + 1 nc = order if ( intercept ) nc = nc + 1 nthreads = omp_get_num_threads () ! Compute the fit call linear_least_squares ( order , intercept , x , y , coeffs , & ymod , resid , alpha = alph , err = errmgr ) if ( errmgr % has_error_occurred ()) return ! Memory Allocations if ( present ( bscoeffs )) then allocate ( bscoeffs ( ncoeffs , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"bootstrap_linear_least_squares\" , & flag ) return end if allcoeffs => bscoeffs else allocate ( coeffstorage ( ncoeffs , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"bootstrap_linear_least_squares\" , & flag ) return end if allcoeffs => coeffstorage end if allcoeffs (:, 1 ) = coeffs ! Cycle over each data set and perform the fit #ifdef USEOPENMP !$OMP PARALLEL DO PRIVATE(fLocal, yLocal, rLocal) SHARED(allcoeffs) do i = 2 , ns ! Get the current thread number ! The +1 is because OpenMP is zero-based for thread numbering thread = omp_get_thread_num () + 1 ! Allocate local arrays on a per-thread basis if (. not . allocated ( fLocal )) allocate ( fLocal ( n )) if (. not . allocated ( yLocal )) allocate ( yLocal ( n )) if (. not . allocated ( rLocal )) allocate ( rLocal ( n )) ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call linear_least_squares ( order , intercept , x , yLocal , & allcoeffs (:, i ), fLocal , rLocal , alpha = alph ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( fLocal ( n ), yLocal ( n ), rLocal ( n )) do i = 2 , ns ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call linear_least_squares ( order , intercept , x , yLocal , & allcoeffs (:, i ), fLocal , rLocal , alpha = alph ) end do #endif ! Perform statistics calculations, if needed if ( present ( stats )) then call compute_stats ( coeffs , allcoeffs , alph , intercept , stats ) end if ! Compute the bias for each parameter, if needed if ( present ( bias )) then ! Verify the size of the array if ( size ( bias ) /= ncoeffs ) then call report_array_size_error ( errmgr , & \"bootstrap_linear_least_squares\" , \"bias\" , ncoeffs , size ( bias )) return end if ! Perform the calculations do i = 1 , ncoeffs bias ( i ) = mean ( allcoeffs ( i ,:)) - coeffs ( i ) end do end if end subroutine ! ****************************************************************************** ! NONLINEAR REGRESSION ! ------------------------------------------------------------------------------ subroutine bootstrap_nonlinear_least_squares ( fun , x , y , params , ymod , resid , & nsamples , weights , maxp , minp , stats , alpha , controls , settings , info , & bias , method , bscoeffs , err ) !! Performs a nonlinear regression to fit a model using a version !! of the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain !! insight into the quality of the fit.  Resampling for the bootstrap !! process is a random resampling with replacement process with the !! range of values limited by the standard deviation of the original !! data set. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: x (:) !! The M-element array containing independent data. real ( real64 ), intent ( in ) :: y (:) !! The M-element array containing dependent data. real ( real64 ), intent ( inout ) :: params (:) !! On input, the N-element array containing the initial estimate !! of the model parameters.  On output, the computed model !! parameters. real ( real64 ), intent ( out ) :: ymod (:) !! An M-element array where the modeled dependent data will !! be written. real ( real64 ), intent ( out ) :: resid (:) !! An M-element array where the model residuals will be !! written. integer ( int32 ), intent ( in ), optional :: nsamples !! The number of bootstrapping samples to utilize. real ( real64 ), intent ( in ), optional , target :: weights (:) !! An optional M-element array allowing the weighting of !! individual points. real ( real64 ), intent ( in ), optional , target :: maxp (:) !! An optional N-element array that can be used as upper limits !! on the parameter values.  If no upper limit is requested for !! a particular parameter, utilize a very large value.  The !! internal default is to utilize huge() as a value. real ( real64 ), intent ( in ), optional , target :: minp (:) !! An optional N-element array that can be used as lower limits !! on the parameter values.  If no lower limit is requested for !! a particalar parameter, utilize a very large magnitude, but !! negative, value.  The internal default is to utilize -huge() !! as a value. type ( bootstrap_regression_statistics ), intent ( out ), optional :: stats (:) !! An optional N-element array that, if supplied, will be used !! to return statistics about the fit for each parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. type ( iteration_controls ), intent ( in ), optional :: controls !! An optional input providing custom iteration controls. type ( lm_solver_options ), intent ( in ), optional :: settings !! An optional input providing custom settings for the solver. type ( convergence_info ), intent ( out ), optional , target :: info !! An optional output that can be used to gain information about !! the iterative solution and the nature of the convergence. real ( real64 ), intent ( out ), optional , dimension (:) :: bias !! An optional N-element array that, if supplied, will be used to !! provide an estimate of the bias of each model parameter based upon !! the results of the bootstrapping analysis.  The bias is computed as !! the difference between the mean of the boostrap population results !! for the given parameter and the original estimate of the given !! parameter. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. real ( real64 ), intent ( out ), optional , allocatable , target , dimension (:,:) :: bscoeffs !! An optional, allocatable matrix, containing the bootstrap !! distributions for each parameter stored in each row of the matrix !! such that the resulting matrix is NCOEFFS -by- NSAMPLES. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. !! - FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed !!      is underdetetermined (M < N). !! - FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied !!      tolerances are too small to be practical. !! - FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations !!      are allowed. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , n , ns , nparams , flag real ( real64 ) :: alph real ( real64 ), allocatable , dimension (:) :: fLocal , yLocal , rLocal real ( real64 ), allocatable , target , dimension (:,:) :: coeffstorage real ( real64 ), pointer , dimension (:,:) :: allcoeffs procedure ( bootstrap_resampling_routine ), pointer :: resample class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( nsamples )) then ns = nsamples else ns = 1000 end if if ( present ( alpha )) then alph = alpha else alph = p05 end if if ( present ( method )) then resample => method else resample => scaled_random_resample end if n = size ( x ) nparams = size ( params ) ! Compute the fit call nonlinear_least_squares ( fun , x , y , params , ymod , resid , & weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info , err = err ) ! Memory Allocations if ( present ( bscoeffs )) then allocate ( bscoeffs ( nparams , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , flag ) return end if allcoeffs => bscoeffs else allocate ( coeffstorage ( nparams , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , flag ) return end if allcoeffs => coeffstorage end if allcoeffs (:, 1 ) = params ! Define initial guesses for each step.  Base upon the results of the ! initial analysis as this should provide a strong starting point for ! subsequent analysis do i = 1 , nparams allcoeffs ( i ,:) = params ( i ) end do ! Cycle over each data set and perform the fit #ifdef USEOPENMP !$OMP PARALLEL DO PRIVATE(fLocal, yLocal, rLocal) do i = 2 , ns ! Allocate local arrays on a per-thread basis if (. not . allocated ( fLocal )) allocate ( fLocal ( n )) if (. not . allocated ( yLocal )) allocate ( yLocal ( n )) if (. not . allocated ( rLocal )) allocate ( rLocal ( n )) ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call nonlinear_least_squares ( fun , x , yLocal , allcoeffs (:, i ), fLocal , & rLocal , weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( fLocal ( n ), yLocal ( n ), rLocal ( n )) do i = 2 , ns ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call nonlinear_least_squares ( fun , x , yLocal , allcoeffs (:, i ), fLocal , & rLocal , weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info ) end do #endif ! Perform the statistics calculations, if needed if ( present ( stats )) then ! Verify the size of stats if ( size ( stats ) /= nparams ) then call report_array_size_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , \"stats\" , & nparams , size ( stats )) return end if ! Perform the calculations call compute_stats ( params , allcoeffs , alph , . true ., stats ) end if ! Compute the bias for each parameter, if needed if ( present ( bias )) then ! Verify the size of the array if ( size ( bias ) /= nparams ) then call report_array_size_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , \"bias\" , & nparams , size ( bias )) return end if ! Perform the calculations do i = 1 , nparams bias ( i ) = mean ( allcoeffs ( i ,:)) - params ( i ) end do end if end subroutine ! ****************************************************************************** ! PRIVATE HELPER ROUTINES ! ------------------------------------------------------------------------------ subroutine compute_stats ( mdl , coeffs , alpha , intercept , stats ) ! Arguments real ( real64 ), intent ( in ), dimension (:) :: mdl real ( real64 ), intent ( inout ), dimension (:,:) :: coeffs real ( real64 ), intent ( in ) :: alpha logical , intent ( in ) :: intercept type ( bootstrap_regression_statistics ), intent ( out ), dimension (:) :: stats ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: i , j , i1 , i2 , ncoeffs , nc , nsamples real ( real64 ) :: ms type ( t_distribution ) :: dist ! Initialization ncoeffs = size ( coeffs , 1 ) nsamples = size ( coeffs , 2 ) nc = ncoeffs if (. not . intercept ) nc = ncoeffs - 1 i1 = floor ( half * alpha * nsamples , int32 ) i2 = nsamples - i1 + 1 dist % dof = real ( nsamples - nc ) ! Process j = 1 if ( intercept ) j = 0 do i = 1 , nc j = j + 1 ms = trimmed_mean ( coeffs ( j ,:), p = half * alpha ) ! As we have a distribution of mean values, the standard deviation ! of this population yields the standard error estimate for the ! overall problem stats ( i )% standard_error = standard_deviation ( coeffs ( j ,:)) ! As before, this is a distribution of mean values.  The CI can ! be directly estimated by considering the values of the bottom ! alpha/2 and top alpha/2 terms. stats ( i )% upper_confidence_interval = coeffs ( j , i2 ) stats ( i )% lower_confidence_interval = coeffs ( j , i1 ) ! Compute the remaining parameters stats ( i )% t_statistic = mdl ( j ) / stats ( i )% standard_error stats ( i )% probability = regularized_beta ( half * dist % dof , half , & dist % dof / ( dist % dof + ( stats ( i )% t_statistic ) ** 2 )) end do end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_bootstrap.f90.html"},{"title":"fstats_regression.f90 – FSTATS","text":"Source Code module fstats_regression use iso_fortran_env use linalg use fstats_errors use blas use ferror use fstats_descriptive_statistics use fstats_distributions use fstats_special_functions use fstats_hypothesis implicit none private public :: iteration_controls public :: convergence_info public :: lm_solver_options public :: regression_function public :: iteration_update public :: regression_statistics public :: r_squared public :: adjusted_r_squared public :: correlation public :: design_matrix public :: covariance_matrix public :: linear_least_squares public :: calculate_regression_statistics public :: jacobian public :: nonlinear_least_squares public :: FS_LEVENBERG_MARQUARDT_UPDATE public :: FS_QUADRATIC_UPDATE public :: FS_NIELSEN_UPDATE ! ****************************************************************************** ! CONSTANTS ! ------------------------------------------------------------------------------ integer ( int32 ), parameter :: FS_LEVENBERG_MARQUARDT_UPDATE = 1 integer ( int32 ), parameter :: FS_QUADRATIC_UPDATE = 2 integer ( int32 ), parameter :: FS_NIELSEN_UPDATE = 3 ! ****************************************************************************** ! TYPES ! ------------------------------------------------------------------------------ type regression_statistics !! A container for regression-related statistical information. real ( real64 ) :: standard_error !! The standard error for the model coefficient. !! !!  E_{s}(\\beta_{i}) = \\sqrt{\\sigma&#94;{2} C_{ii}}  real ( real64 ) :: t_statistic !! The T-statistic for the model coefficient. !! !!  t_o = \\frac{ \\beta_{i} }{E_{s}(\\beta_{i})}  real ( real64 ) :: probability !! The probability that the coefficient is not statistically !! important.  A statistically important coefficient will have a !! low probability (p-value), typically 0.05 or lower; however, a !! p-value of up to ~0.2 may be acceptable dependent upon the !! problem.  Typically any p-value larger than ~0.2 indicates the !! parameter is not statistically important for the model. !! !!  p = t_{|t_o|, df_{residual}}  real ( real64 ) :: confidence_interval !! The confidence interval for the parameter at the level !! determined by the regression process. !! !!  c = t_{\\alpha, df} E_{s}(\\beta_{i})  end type type iteration_controls !! Provides a collection of iteration control parameters. integer ( int32 ) :: max_iteration_count !! Defines the maximum number of iterations allowed. integer ( int32 ) :: max_function_evaluations !! Defines the maximum number of function evaluations allowed. real ( real64 ) :: gradient_tolerance !! Defines a tolerance on the gradient of the fitted function. real ( real64 ) :: change_in_solution_tolerance !! Defines a tolerance on the change in parameter values. real ( real64 ) :: residual_tolerance !! Defines a tolerance on the metric associated with the residual !! error. real ( real64 ) :: iteration_improvement_tolerance !! Defines a tolerance to ensure adequate improvement on each !! iteration. integer ( int32 ) :: max_iteration_between_updates !! Defines how many iterations can pass before a re-evaluation of !! the Jacobian matrix is forced. contains procedure , public :: set_to_default => lm_set_default_tolerances end type type convergence_info !! Provides information regarding convergence status. logical :: converge_on_gradient !! True if convergence on the gradient was achieved; else, false. real ( real64 ) :: gradient_value !! The value of the gradient test parameter. logical :: converge_on_solution_change !! True if convergence on the change in solution was achieved; else, !! false. real ( real64 ) :: solution_change_value !! The value of the change in solution parameter. logical :: converge_on_residual_parameter !! True if convergence on the residual error parameter was achieved; !! else, false. real ( real64 ) :: residual_value !! The value of the residual error parameter. logical :: reach_iteration_limit !! True if the solution did not converge in the allowed number of !! iterations. integer ( int32 ) :: iteration_count !! The iteration count. logical :: reach_function_evaluation_limit !! True if the solution did not converge in the allowed number of !! function evaluations. integer ( int32 ) :: function_evaluation_count !! The function evaluation count. logical :: user_requested_stop !! True if the user requested the stop; else, false. end type type lm_solver_options !! Options to control the Levenberg-Marquardt solver. integer ( int32 ) :: method !! The solver method to utilize. !! - FS_LEVENBERG_MARQUARDT_UPDATE: !! - FS_QUADRATIC_UPDATE: !! - FS_NIELSEN_UDPATE: real ( real64 ) :: finite_difference_step_size !! The step size used for the finite difference calculations of the !! Jacobian matrix. real ( real64 ) :: damping_increase_factor !! The factor to use when increasing the damping parameter. real ( real64 ) :: damping_decrease_factor !! The factor to use when decreasing the damping parameter. contains procedure , public :: set_to_default => lm_set_default_settings end type interface subroutine regression_function ( xdata , params , resid , stop ) use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: xdata , params real ( real64 ), intent ( out ), dimension (:) :: resid logical , intent ( out ) :: stop end subroutine subroutine iteration_update ( iter , funvals , resid , params , step ) use iso_fortran_env , only : int32 , real64 integer ( int32 ), intent ( in ) :: iter real ( real64 ), intent ( in ) :: funvals (:), resid (:), params (:), step (:) end subroutine end interface contains ! ------------------------------------------------------------------------------ function r_squared ( x , xm , err ) result ( rst ) !! Computes the R-squared value for a data set. !! !! The R-squared value is computed by determining the sum of the squares !! of the residuals: !!  SS_{res} = \\Sigma \\left( y_i - f_i \\right)&#94;2  !! The total sum of the squares: !!  SS_{tot} = \\Sigma \\left( y_i - \\bar{y} \\right)&#94;2 . !! The R-squared value is then: !!  R&#94;2 = 1 - \\frac{SS_{res}}{SS_{tot}} . !! !! See Also: !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Coefficient_of_determination) real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the dependent variables from !! the data set. real ( real64 ), intent ( in ) :: xm (:) !! An N-element array containing the corresponding modeled !! values. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings !! to the caller.  Possible warning and error codes are as !! follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the !!   same size. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: esum , vt class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Input Check n = size ( x ) if ( size ( xm ) /= n ) then call report_array_size_error ( errmgr , \"r_squared_real64\" , \"XM\" , n , & size ( xm )) return end if ! Process esum = zero do i = 1 , n esum = esum + ( x ( i ) - xm ( i )) ** 2 end do vt = variance ( x ) * ( n - one ) rst = one - esum / vt end function ! ------------------------------------------------------------------------------ function adjusted_r_squared ( p , x , xm , err ) result ( rst ) !! Computes the adjusted R-squared value for a data set. !! !! The adjusted R-squared provides a mechanism for tempering the effects !! of extra explanatory variables on the traditional R-squared !! calculation.  It is computed by noting the sample size  n  and !! the number of variables  p . !!  \\bar{R}&#94;2 = 1 - \\left( 1 - R&#94;2 \\right) \\frac{n - 1}{n - p} . !! !! See Also: !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2) integer ( int32 ), intent ( in ) :: p !! The number of variables. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the dependent variables from !! the data set. real ( real64 ), intent ( in ) :: xm (:) !! An N-element array containing the corresponding modeled !! values. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings !! to the caller.  Possible warning and error codes are as !! follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the !!   same size. real ( real64 ) :: rst !! The result. ! Local Variables integer ( int32 ) :: n real ( real64 ) :: r2 class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n = size ( x ) ! Process r2 = r_squared ( x , xm , errmgr ) if ( errmgr % has_error_occurred ()) return rst = one - ( one - r2 ) * ( n - one ) / ( n - p - one ) end function ! ------------------------------------------------------------------------------ pure function correlation ( x , y ) result ( rst ) !! Computes the sample correlation coefficient (an estimate to the !! population Pearson correlation) as follows. !! !!  r_{xy} = \\frac{cov(x, y)}{s_{x} s_{y}} . !! !! Where,  s_{x}  &  s_{y}  are the sample standard deviations of !! x and y respectively. real ( real64 ), intent ( in ), dimension (:) :: x !! The first N-element data set. real ( real64 ), intent ( in ), dimension ( size ( x )) :: y !! The second N-element data set. real ( real64 ) :: rst !! The correlation coefficient. ! Process rst = covariance ( x , y ) / ( standard_deviation ( x ) * standard_deviation ( y )) end function ! ------------------------------------------------------------------------------ subroutine design_matrix ( order , intercept , x , c , err ) !! Computes the design matrix  X  for the linear !! least-squares regression problem of  X \\beta = y , where !!  X  is the matrix computed here,  \\beta  is !! the vector of coefficients to be determined, and  y  is the !! vector of measured dependent variables. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Linear_regression) !! - [Wikipedia](https://en.wikipedia.org/wiki/Vandermonde_matrix) !! - [Wikipedia](https://en.wikipedia.org/wiki/Design_matrix) integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be !! at least one (linear equation), but can be higher as desired. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed !! as part of the regression; else, false. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( out ) :: c (:,:) !! An N-by-K matrix where the results will be written.  K !! must equal order + 1 in the event intercept is true; !! however, if intercept is false, K must equal order. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , start , npts , ncols class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x ) ncols = order if ( intercept ) ncols = ncols + 1 ! Input Check if ( order < 1 ) then call errmgr % report_error ( \"design_matrix\" , & \"The model order must be at least one.\" , FS_INVALID_INPUT_ERROR ) return end if if ( size ( c , 1 ) /= npts . or . size ( c , 2 ) /= ncols ) then call report_matrix_size_error ( errmgr , \"design_matrix\" , & \"c\" , npts , ncols , size ( c , 1 ), size ( c , 2 )) return end if ! Process if ( intercept ) then c (:, 1 ) = one c (:, 2 ) = x start = 3 else c (:, 1 ) = x start = 2 end if if ( start >= ncols ) return do i = start , ncols c (:, i ) = c (:, i - 1 ) * x end do end subroutine ! ------------------------------------------------------------------------------ subroutine covariance_matrix ( x , c , err ) !! Computes the covariance matrix  C  where !!  C = \\left( X&#94;{T} X \\right)&#94;{-1}  and  X  is computed !! by design_matrix. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Covariance_matrix) !! - [Wikipedia - Regression](https://en.wikipedia.org/wiki/Linear_regression) real ( real64 ), intent ( in ) :: x (:,:) !! An M-by-N matrix containing the formatted independent data !!  matrix  X  as computed by design_matrix. real ( real64 ), intent ( out ) :: c (:,:) !! The N-by-N covariance matrix. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not !!      sized correctly. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr integer ( int32 ) :: npts , ncoeffs , flag real ( real64 ), allocatable :: xtx (:,:) ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x , 1 ) ncoeffs = size ( x , 2 ) ! Input Checking if ( size ( c , 1 ) /= ncoeffs . or . size ( c , 2 ) /= ncoeffs ) then call report_matrix_size_error ( errmgr , \"covariance_matrix\" , & \"c\" , ncoeffs , ncoeffs , size ( c , 1 ), size ( c , 2 )) return end if ! Local Memory Allocation allocate ( xtx ( ncoeffs , ncoeffs ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"covariance_matrix\" , flag ) return end if ! Compute X**T * X call DGEMM ( \"T\" , \"N\" , ncoeffs , ncoeffs , npts , one , x , npts , x , npts , & zero , xtx , ncoeffs ) ! Compute the inverse of X**T * X to obtain the covariance matrix call mtx_pinverse ( xtx , c , err = errmgr ) if ( errmgr % has_error_occurred ()) return end subroutine ! ------------------------------------------------------------------------------ subroutine linear_least_squares ( order , intercept , x , y , coeffs , & ymod , resid , stats , alpha , err ) !! Computes a linear least-squares regression to fit a set of data. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Linear_regression) !! - [SPC Excel Understanding Regression Statistics](https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1) integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be at !! least one (linear equation), but can be higher as desired, !! as long as there is sufficient data. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed as part of !! the regression; else, false. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( in ) :: y (:) !! An N-element array containing the dependent variable !! measurement points. real ( real64 ), intent ( out ) :: coeffs (:) !! An ORDER+1 element array where the coefficients will be written. real ( real64 ), intent ( out ) :: ymod (:) !! An N-element array where the modeled data will be written. real ( real64 ), intent ( out ) :: resid (:) !! An N-element array where the residual error data will be !! written (modeled - actual). type ( regression_statistics ), intent ( out ), optional :: stats (:) !! An M-element array of regression_statistics items where !! M = ORDER + 1 when intercept is set to true; however, if !! intercept is set to false, M = ORDER. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , npts , ncols , ncoeffs , flag real ( real64 ) :: alph , var , df , ssr , talpha real ( real64 ), allocatable :: a (:,:), c (:,:), cxt (:,:) type ( t_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x ) ncoeffs = order + 1 ncols = order if ( intercept ) ncols = ncols + 1 alph = 0.05d0 if ( present ( alpha )) alph = alpha ! Input Check if ( order < 1 ) then call errmgr % report_error ( \"linear_least_squares\" , & \"The model order must be at least one.\" , FS_INVALID_INPUT_ERROR ) return end if if ( size ( y ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"y\" , npts , size ( y )) return end if if ( size ( coeffs ) /= ncoeffs ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"coeffs\" , ncoeffs , size ( coeffs )) return end if if ( size ( ymod ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"ymod\" , npts , size ( ymod )) return end if if ( size ( resid ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"resid\" , npts , size ( resid )) return end if if ( present ( stats )) then if ( size ( stats ) /= ncols ) then call report_array_size_error ( errmgr , & \"linear_least_squares\" , \"stats\" , ncols , size ( stats )) return end if end if ! Memory Allocation allocate ( a ( npts , ncols ), stat = flag ) if ( flag == 0 ) allocate ( c ( ncols , ncols ), stat = flag ) if ( flag == 0 ) allocate ( cxt ( ncols , npts ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"linear_least_squares\" , flag ) return end if ! Compute the coefficient matrix call design_matrix ( order , intercept , x , a , errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the covariance matrix call covariance_matrix ( a , c , errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the coefficients (NCOLS-by-1) call DGEMM ( \"N\" , \"T\" , ncols , npts , ncols , one , c , ncols , a , npts , zero , & cxt , ncols ) ! C * X**T i = 2 coeffs ( 1 ) = zero if ( intercept ) i = 1 call DGEMM ( \"N\" , \"N\" , ncols , 1 , npts , one , cxt , ncols , y , npts , zero , & coeffs ( i :), ncols ) ! (C * X**T) * Y ! Evaluate the model and compute the residuals call DGEMM ( \"N\" , \"N\" , npts , 1 , ncols , one , a , npts , coeffs ( i :), & ncols , zero , ymod , npts ) resid = ymod - y ! If the user doesn't want the statistics calculations we can stop now if (. not . present ( stats )) return ! Start the process of computing statistics stats = calculate_regression_statistics ( resid , coeffs ( i :), c , alph , & errmgr ) end subroutine ! ------------------------------------------------------------------------------ function calculate_regression_statistics ( resid , params , c , alpha , err ) & result ( rst ) !! Computes statistics for the quality of fit for a regression !! model. real ( real64 ), intent ( in ) :: resid (:) !! An M-element array containing the model residual errors. real ( real64 ), intent ( in ) :: params (:) !! An N-element array containing the model parameters. real ( real64 ), intent ( in ) :: c (:,:) !! The N-by-N covariance matrix. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. type ( regression_statistics ), allocatable :: rst (:) !! A regression_statistics object containing the analysis results. ! Parameters real ( real64 ), parameter :: p05 = 0.05d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , m , n , dof , flag real ( real64 ) :: a , ssr , var , talpha type ( t_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Initialization m = size ( resid ) n = size ( params ) dof = m - n if ( present ( alpha )) then a = alpha else a = p05 end if allocate ( rst ( n ), stat = flag ) if ( flag /= 0 ) then end if ! Input Checking if ( size ( c , 1 ) /= n . or . size ( c , 2 ) /= n ) then end if ! Process ssr = norm2 ( resid ) ** 2 ! sum of the squares of the residual var = ssr / dof dist % dof = real ( dof , real64 ) talpha = confidence_interval ( dist , a , one , 1 ) do i = 1 , n rst ( i )% standard_error = sqrt ( var * c ( i , i )) rst ( i )% t_statistic = params ( i ) / rst ( i )% standard_error rst ( i )% probability = regularized_beta ( & half * dof , & half , & real ( dof , real64 ) / ( dof + ( rst ( i )% t_statistic ) ** 2 ) & ) rst ( i )% confidence_interval = talpha * rst ( i )% standard_error end do end function ! ------------------------------------------------------------------------------ subroutine jacobian ( fun , xdata , params , & jac , stop , f0 , f1 , step , err ) !! Computes the Jacobian matrix for a nonlinear regression problem. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: xdata (:) !! The M-element array containing x-coordinate data. real ( real64 ), intent ( in ) :: params (:) !! The N-element array containing the model parameters. real ( real64 ), intent ( out ) :: jac (:,:) !! The M-by-N matrix where the Jacobian will be written. logical , intent ( out ) :: stop !! A value that the user can set in fun forcing the !! evaluation process to stop prior to completion. real ( real64 ), intent ( in ), optional , target :: f0 (:) !! An optional M-element array containing the model values !!  using the current parameters as defined in m.  This input !! can be used to prevent the routine from performing a !! function evaluation at the model parameter state defined in !! params. real ( real64 ), intent ( out ), optional , target :: f1 (:) !! An optional M-element workspace array used for function !! evaluations. real ( real64 ), intent ( in ), optional :: step !! The differentiation step size.  The default is the square !! root of machine precision. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Local Variables real ( real64 ) :: h integer ( int32 ) :: m , n , flag , expected , actual real ( real64 ), pointer :: f1p (:), f0p (:) real ( real64 ), allocatable , target :: f1a (:), f0a (:), work (:) class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( step )) then h = step else h = sqrt ( epsilon ( h )) end if m = size ( xdata ) n = size ( params ) ! Input Size Checking if ( size ( jac , 1 ) /= m . or . size ( jac , 2 ) /= n ) then call report_matrix_size_error ( errmgr , \"jacobian\" , & \"JAC\" , m , n , size ( jac , 1 ), size ( jac , 2 )) return end if if ( present ( f0 )) then ! Check Size if ( size ( f0 ) /= m ) then call report_array_size_error ( errmgr , \"jacobian\" , & \"F0\" , m , size ( f0 )) return end if f0p ( 1 : m ) => f0 else ! Allocate space, and fill the array with the current function ! results allocate ( f0a ( m ), stat = flag ) if ( flag /= 0 ) go to 20 f0p ( 1 : m ) => f0a call fun ( xdata , params , f0p , stop ) if ( stop ) return end if if ( present ( f1 )) then ! Check Size if ( size ( f1 ) /= m ) then call report_array_size_error ( errmgr , \"jacobian\" , & \"F1\" , m , size ( f1 )) return end if f1p ( 1 : m ) => f1 else ! Allocate space allocate ( f1a ( m ), stat = flag ) if ( flag /= 0 ) go to 20 f1p ( 1 : m ) => f1a end if ! Allocate a workspace array the same size as params allocate ( work ( n ), stat = flag ) if ( flag /= 0 ) go to 20 ! Compute the Jacobian call jacobian_finite_diff ( fun , xdata , params , f0p , jac , f1p , & stop , h , work ) ! End return ! Memroy Allocation Error Handling 20 continue call report_memory_error ( errmgr , \"jacobian\" , flag ) return end subroutine ! ------------------------------------------------------------------------------ subroutine nonlinear_least_squares ( fun , x , y , params , ymod , & resid , weights , maxp , minp , stats , alpha , controls , settings , info , & status , err ) !! Performs a nonlinear regression to fit a model using a version !! of the Levenberg-Marquardt algorithm. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: x (:) !! The M-element array containing independent data. real ( real64 ), intent ( in ) :: y (:) !! The M-element array containing dependent data. real ( real64 ), intent ( inout ) :: params (:) !! On input, the N-element array containing the initial estimate !! of the model parameters.  On output, the computed model !! parameters. real ( real64 ), intent ( out ) :: ymod (:) !! An M-element array where the modeled dependent data will !! be written. real ( real64 ), intent ( out ) :: resid (:) !! An M-element array where the model residuals will be !! written. real ( real64 ), intent ( in ), optional , target :: weights (:) !! An optional M-element array allowing the weighting of !! individual points. real ( real64 ), intent ( in ), optional , target :: maxp (:) !! An optional N-element array that can be used as upper limits !! on the parameter values.  If no upper limit is requested for !! a particular parameter, utilize a very large value.  The !! internal default is to utilize huge() as a value. real ( real64 ), intent ( in ), optional , target :: minp (:) !! An optional N-element array that can be used as lower limits !! on the parameter values.  If no lower limit is requested for !! a particalar parameter, utilize a very large magnitude, but !! negative, value.  The internal default is to utilize -huge() !! as a value. type ( regression_statistics ), intent ( out ), optional :: stats (:) !! An optional N-element array that, if supplied, will be used !! to return statistics about the fit for each parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. type ( iteration_controls ), intent ( in ), optional :: controls !! An optional input providing custom iteration controls. type ( lm_solver_options ), intent ( in ), optional :: settings !! An optional input providing custom settings for the solver. type ( convergence_info ), intent ( out ), optional , target :: info !! An optional output that can be used to gain information about !! the iterative solution and the nature of the convergence. procedure ( iteration_update ), intent ( in ), pointer , optional :: status !! An optional pointer to a routine that can be used to extract !! iteration information. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. !! - FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed !!      is underdetetermined (M < N). !! - FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied !!      tolerances are too small to be practical. !! - FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations !!      are allowed. ! Parameters real ( real64 ), parameter :: too_small = 1.0d-14 integer ( int32 ), parameter :: min_iter_count = 2 integer ( int32 ), parameter :: min_fun_count = 10 integer ( int32 ), parameter :: min_update_count = 1 ! Local Variables logical :: stop integer ( int32 ) :: m , n , actual , expected , flag real ( real64 ), pointer :: w (:), pmax (:), pmin (:) real ( real64 ), allocatable , target :: defaultWeights (:), maxparam (:), & minparam (:), JtWJ (:,:) type ( iteration_controls ) :: tol type ( lm_solver_options ) :: opt type ( convergence_info ) :: cInfo class ( errors ), pointer :: errmgr type ( errors ), target :: deferr type ( convergence_info ), target :: defaultinfo type ( convergence_info ), pointer :: inf ! Initialization stop = . false . m = size ( x ) n = size ( params ) if ( present ( info )) then inf => info else inf => defaultinfo end if if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( controls )) then tol = controls else call tol % set_to_default () end if if ( present ( settings )) then opt = settings else call opt % set_to_default () end if ! Input Checking if ( size ( y ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"y\" , m , size ( y )) return end if if ( size ( ymod ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"ymod\" , m , size ( ymod )) return end if if ( size ( resid ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"resid\" , m , size ( resid )) return end if if ( m < n ) then call report_underdefined_error ( errmgr , & \"nonlinear_least_squares\" , n , m ) return end if ! Tolerance Checking if ( tol % gradient_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The gradient tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % change_in_solution_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The change in solution tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % residual_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The residual error tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % iteration_improvement_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The iteration improvement tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if ! Iteration Count Checking if ( tol % max_iteration_count < min_iter_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few iterations were specified.\" , & min_iter_count ) return end if if ( tol % max_function_evaluations < min_fun_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few function evaluations were specified.\" , & min_fun_count ) return end if if ( tol % max_iteration_between_updates < min_update_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few iterations between updates were specified.\" , & min_update_count ) return end if ! Optional Array Arguments (weights, parameter limits, etc.) if ( present ( weights )) then if ( size ( weights ) < m ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"weights\" , m , size ( weights )) return end if w ( 1 : m ) => weights ( 1 : m ) else allocate ( defaultWeights ( m ), source = 1.0d0 , stat = flag ) if ( flag /= 0 ) go to 50 w ( 1 : m ) => defaultWeights ( 1 : m ) end if if ( present ( maxp )) then if ( size ( maxp ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"maxp\" , n , size ( maxp )) return end if pmax ( 1 : n ) => maxp ( 1 : n ) else allocate ( maxparam ( n ), source = huge ( 1.0d0 ), stat = flag ) if ( flag /= 0 ) go to 50 pmax ( 1 : n ) => maxparam ( 1 : n ) end if if ( present ( minp )) then if ( size ( minp ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"minp\" , n , size ( minp )) return end if pmin ( 1 : n ) => minp ( 1 : n ) else allocate ( minparam ( n ), source = - huge ( 1.0d0 ), stat = flag ) if ( flag /= 0 ) go to 50 pmin ( 1 : n ) => minparam ( 1 : n ) end if ! Local Memory Allocations allocate ( JtWJ ( n , n ), stat = flag ) if ( flag /= 0 ) go to 50 ! Process call lm_solve ( fun , x , y , params , w , pmax , pmin , tol , opt , ymod , & resid , JtWJ , inf , stop , errmgr , status ) ! Statistical Parameters if ( present ( stats )) then if ( size ( stats ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"stats\" , n , size ( stats )) return end if ! Compute the covariance matrix call mtx_inverse ( JtWJ , err = errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the statistics stats = calculate_regression_statistics ( resid , params , JtWJ , & alpha , errmgr ) end if ! End return ! Memory Error Handler 50 continue call report_memory_error ( errmgr , \"nonlinear_least_squares\" , flag ) return end subroutine ! ****************************************************************************** ! SETTINGS DEFAULTS ! ------------------------------------------------------------------------------ ! Sets up default tolerances. subroutine lm_set_default_tolerances ( x ) ! Arguments class ( iteration_controls ), intent ( inout ) :: x ! Set defaults x % max_iteration_count = 500 x % max_function_evaluations = 5000 x % max_iteration_between_updates = 10 x % gradient_tolerance = 1.0d-8 x % residual_tolerance = 0.5d-2 x % change_in_solution_tolerance = 1.0d-6 x % iteration_improvement_tolerance = 1.0d-1 end subroutine ! ------------------------------------------------------------------------------ ! Sets up default solver settings. subroutine lm_set_default_settings ( x ) ! Arguments class ( lm_solver_options ), intent ( inout ) :: x ! Set defaults x % method = FS_LEVENBERG_MARQUARDT_UPDATE x % finite_difference_step_size = sqrt ( epsilon ( 1.0d0 )) x % damping_increase_factor = 1 1.0d0 x % damping_decrease_factor = 9.0d0 end subroutine ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ ! Computes the Jacobian matrix via a forward difference. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - params: The model parameters (N-by-1) ! - f0: The current model estimate (M-by-1) ! - step: The differentiation step size ! ! Outputs: ! - jac: The Jacobian matrix (M-by-N) ! - f1: A workspace array for the model output (M-by-1) ! - stop: A flag allowing the user to terminate model execution ! - work: A workspace array for the model parameters (N-by-1) subroutine jacobian_finite_diff ( fun , xdata , params , f0 , jac , f1 , & stop , step , work ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), params (:) real ( real64 ), intent ( in ) :: f0 (:) real ( real64 ), intent ( out ) :: jac (:,:) real ( real64 ), intent ( out ) :: f1 (:), work (:) logical , intent ( out ) :: stop real ( real64 ), intent ( in ) :: step ! Local Variables integer ( int32 ) :: i , n ! Initialization n = size ( params ) ! Cycle over each column of the Jacobian and calculate the derivative ! via a forward difference scheme ! ! J(i,j) = df(i) / dx(j) work = params do i = 1 , n work ( i ) = work ( i ) + step call fun ( xdata , work , f1 , stop ) if ( stop ) return jac (:, i ) = ( f1 - f0 ) / step work ( i ) = params ( i ) end do end subroutine ! ------------------------------------------------------------------------------ ! Computes a rank-1 update to the Jacobian matrix ! ! Inputs: ! - pOld: previous set of parameters (N-by-1) ! - yOld: model evaluation at previous set of parameters (M-by-1) ! - jac: current Jacobian estimate (M-by-N) ! - p: current set of parameters (N-by-1) ! - y: model evaluation at current set of parameters (M-by-1) ! ! Outputs: ! - jac: updated Jacobian matrix (M-by-N) (dy * dp**T + J) ! - dp: p - pOld (N-by-1) ! - dy: (y - yOld - J * dp) / (dp' * dp) (M-by-1) subroutine broyden_update ( pOld , yOld , jac , p , y , dp , dy ) ! Arguments real ( real64 ), intent ( in ) :: pOld (:), yOld (:), p (:), y (:) real ( real64 ), intent ( inout ) :: jac (:,:) real ( real64 ), intent ( out ) :: dp (:), dy (:) ! Local Variables real ( real64 ) :: h2 ! Process dp = p - pOld h2 = dot_product ( dp , dp ) dy = y - yOld - matmul ( jac , dp ) dy = dy / h2 call rank1_update ( 1.0d0 , dy , dp , jac ) end subroutine ! ------------------------------------------------------------------------------ ! Updates the Levenberg-Marquardt matrix by either computing a new Jacobian ! matrix or performing a rank-1 update to the existing Jacobian matrix. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - pOld: previous set of parameters (N-by-1) ! - yOld: model evaluation at previous set of parameters (M-by-1) ! - dX2: The previous change in the Chi-squared criteria ! - jac: current Jacobian estimate (M-by-N) ! - p: current set of parameters (N-by-1) ! - weights: A weighting vector (M-by-1) ! - neval: Current number of function evaluations ! - update: Set to true to force an update of the Jacobian; else, set to !       false to let the program choose based upon the change in the !       Chi-squared parameter. ! - step: The differentiation step size ! ! Outputs: ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - JtWdy: linearized fitting vector (N-by-1) ! - X2: Updated Chi-squared criteria ! - yNew: model evaluated with parameters of p (M-by-1) ! - jac: updated Jacobian matrix (M-by-N) ! - neval: updated count of function evaluations ! - stop: A flag allowing the user to terminate model execution ! - work: A workspace array (N+M-by-1) ! - mwork: A workspace matrix (N-by-M) ! - update: Reset to false if a Jacobian evaluation was performed. subroutine lm_matrix ( fun , xdata , ydata , pOld , yOld , dX2 , jac , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , yNew , stop , work , mwork ) ! Arguments procedure ( regression_function ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), pOld (:), yOld (:), & p (:), weights (:) real ( real64 ), intent ( in ) :: dX2 , step real ( real64 ), intent ( inout ) :: jac (:,:) integer ( int32 ), intent ( inout ) :: neval logical , intent ( inout ) :: update real ( real64 ), intent ( out ) :: JtWJ (:,:), JtWdy (:) real ( real64 ), intent ( out ) :: X2 , mwork (:,:), yNew (:) logical , intent ( out ) :: stop real ( real64 ), intent ( out ), target :: work (:) ! Local Variables integer ( int32 ) :: m , n real ( real64 ), pointer :: w1 (:), w2 (:) ! Initialization m = size ( xdata ) n = size ( p ) w1 ( 1 : m ) => work ( 1 : m ) w2 ( 1 : n ) => work ( m + 1 : n + m ) ! Perform the next function evaluation call fun ( xdata , p , yNew , stop ) neval = neval + 1 if ( stop ) return ! Update or recompute the Jacobian matrix if ( dX2 > 0 . or . update ) then ! Recompute the Jacobian call jacobian_finite_diff ( fun , xdata , p , yNew , jac , w1 , & stop , step , w2 ) neval = neval + n if ( stop ) return update = . false . else ! Simply perform a rank-1 update to the Jacobian call broyden_update ( pOld , yOld , jac , p , yNew , w2 , w1 ) end if ! Update the Chi-squared estimate w1 = ydata - yNew X2 = dot_product ( w1 , w1 * weights ) ! Compute J**T * (W .* dY) w1 = w1 * weights call mtx_mult (. true ., 1.0d0 , jac , w1 , 0.0d0 , JtWdy ) ! Update the Hessian ! First: J**T * W = MWORK ! Second: (J**T * W) * J call diag_mtx_mult (. false ., . true ., 1.0d0 , weights , jac , 0.0d0 , mwork ) call mtx_mult (. false ., . false ., 1.0d0 , mwork , jac , 0.0d0 , JtWJ ) end subroutine ! ------------------------------------------------------------------------------ ! Performs a single iteration of the Levenberg-Marquardt algorithm. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - p: current set of parameters (N-by-1) ! - neval: current number of function evaluations ! - niter: current iteration number ! - update: set to 1 to use Marquardt's modification; else, ! - step: the differentiation step size ! - lambda: LM damping parameter ! - maxP: maximum limits on the parameters.  Use huge() or larger for no constraints (N-by-1) ! - minP: minimum limits on the parameters.  Use -huge() or smaller for no constraints (N-by-1) ! - weights: a weighting vector (M-by-1) ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - JtWdy: linearized fitting vector (N-by-1) ! ! Outputs: ! - JtWJ: overwritten LU factorization of the original matrix (N-by-N) ! - h: The new estimate of the change in parameter (N-by-1) ! - pNew: The new parameter estimates (N-by-1) ! - deltaY: The new difference between data and model (M-by-1) ! - yNew: model evaluated with parameters of pNew (M-by-1) ! - neval: updated count of function evaluations ! - niter: updated current iteration number ! - X2: updated Chi-squared criteria ! - stop: A flag allowing the user to terminate model execution ! - iwork: A workspace array (N-by-1) ! - err: An error handling mechanism subroutine lm_iter ( fun , xdata , ydata , p , neval , niter , update , lambda , & maxP , minP , weights , JtWJ , JtWdy , h , pNew , deltaY , yNew , X2 , X2Old , & alpha , stop , iwork , err , status ) ! Arguments procedure ( regression_function ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), p (:), maxP (:), & minP (:), weights (:), JtWdy (:) real ( real64 ), intent ( in ) :: lambda , X2Old integer ( int32 ), intent ( inout ) :: neval , niter integer ( int32 ), intent ( in ) :: update real ( real64 ), intent ( inout ) :: JtWJ (:,:) real ( real64 ), intent ( out ) :: h (:), pNew (:), deltaY (:), yNew (:) real ( real64 ), intent ( out ) :: X2 , alpha logical , intent ( out ) :: stop integer ( int32 ), intent ( out ) :: iwork (:) class ( errors ), intent ( inout ) :: err procedure ( iteration_update ), intent ( in ), pointer , optional :: status ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: dpJh ! Initialization n = size ( p ) ! Increment the iteration counter niter = niter + 1 ! Solve the linear system to determine the change in parameters ! A is N-by-N and is stored in JtWJ ! b is N-by-1 if ( update == FS_LEVENBERG_MARQUARDT_UPDATE ) then ! Compute: h = A \\ b ! A = J**T * W * J + lambda * diag(J**T * W * J) ! b = J**T * W * dy do i = 1 , n JtWJ ( i , i ) = JtWJ ( i , i ) * ( 1.0d0 + lambda ) h ( i ) = JtWdy ( i ) end do else ! Compute: h = A \\ b ! A = J**T * W * J + lambda * I ! b = J**T * W * dy do i = 1 , n JtWJ ( i , i ) = JtWJ ( i , i ) + lambda h ( i ) = JtWdy ( i ) end do end if call lu_factor ( JtWJ , iwork , err ) ! overwrites JtWJ with [L\\U] if ( err % has_error_occurred ()) return ! if JtWJ is singular call solve_lu ( JtWJ , iwork , h ) ! solution stored in h ! Compute the new attempted solution, and apply any constraints do i = 1 , n pNew ( i ) = min ( max ( minP ( i ), h ( i ) + p ( i )), maxP ( i )) end do ! Update the residual error call fun ( xdata , pNew , yNew , stop ) neval = neval + 1 deltaY = ydata - yNew if ( stop ) return ! Update the Chi-squared estimate X2 = dot_product ( deltaY , deltaY * weights ) ! Perform a quadratic line update in the H direction, if necessary if ( update == FS_QUADRATIC_UPDATE ) then dpJh = dot_product ( JtWdy , h ) alpha = abs ( dpJh / ( 0.5d0 * ( X2 - X2Old ) + 2.0d0 * dpJh )) h = alpha * h do i = 1 , n pNew ( i ) = min ( max ( minP ( i ), p ( i ) + h ( i )), maxP ( i )) end do call fun ( xdata , pNew , yNew , stop ) if ( stop ) return neval = neval + 1 deltaY = ydata - yNew X2 = dot_product ( deltaY , deltaY * weights ) end if ! Update the status of the iteration, if needed if ( present ( status )) then call status ( niter , yNew , deltaY , pNew , h ) end if end subroutine ! ------------------------------------------------------------------------------ ! A Levenberg-Marquardt solver. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - p: current set of parameters (N-by-1) ! - weights: a weighting vector (M-by-1) ! - maxP: maximum limits on the parameters.  Use huge() or larger for no constraints (N-by-1) ! - minP: minimum limits on the parameters.  Use -huge() or smaller for no constraints (N-by-1) ! - controls: an iteration_controls instance containing solution tolerances ! ! Outputs: ! - p: solution (N-by-1) ! - y: model results at p (M-by-1) ! - resid: residual (ydata - y) (M-by-1) ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - opt: a convergence_info object containing information regarding !       convergence of the iteration ! - stop: A flag allowing the user to terminate model execution ! - err: An error handling object subroutine lm_solve ( fun , xdata , ydata , p , weights , maxP , minP , controls , & opt , y , resid , JtWJ , info , stop , err , status ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), weights (:), maxP (:), & minP (:) real ( real64 ), intent ( inout ) :: p (:) class ( iteration_controls ), intent ( in ) :: controls class ( lm_solver_options ), intent ( in ) :: opt real ( real64 ), intent ( out ) :: y (:), resid (:), JtWJ (:,:) class ( convergence_info ), intent ( out ) :: info logical , intent ( out ) :: stop class ( errors ), intent ( inout ) :: err procedure ( iteration_update ), intent ( in ), pointer , optional :: status ! Local Variables logical :: update integer ( int32 ) :: i , m , n , dof , flag , neval , niter , nupdate real ( real64 ) :: dX2 , X2 , X2Old , X2Try , lambda , alpha , nu , step real ( real64 ), allocatable :: pOld (:), yOld (:), J (:,:), JtWdy (:), & work (:), mwork (:,:), pTry (:), yTemp (:), JtWJc (:,:), h (:) integer ( int32 ), allocatable :: iwork (:) character ( len = :), allocatable :: errmsg ! Initialization update = . true . m = size ( xdata ) n = size ( p ) dof = m - n niter = 0 step = opt % finite_difference_step_size stop = . false . info % user_requested_stop = . false . nupdate = 0 ! Local Memory Allocation allocate ( pOld ( n ), source = 0.0d0 , stat = flag ) if ( flag == 0 ) allocate ( yOld ( m ), source = 0.0d0 , stat = flag ) if ( flag == 0 ) allocate ( J ( m , n ), stat = flag ) if ( flag == 0 ) allocate ( JtWdy ( n ), stat = flag ) if ( flag == 0 ) allocate ( work ( m + n ), stat = flag ) if ( flag == 0 ) allocate ( mwork ( n , m ), stat = flag ) if ( flag == 0 ) allocate ( pTry ( n ), stat = flag ) if ( flag == 0 ) allocate ( h ( n ), stat = flag ) if ( flag == 0 ) allocate ( yTemp ( m ), stat = flag ) if ( flag == 0 ) allocate ( JtWJc ( n , n ), stat = flag ) if ( flag == 0 ) allocate ( iwork ( n ), stat = flag ) if ( flag /= 0 ) go to 10 ! Perform an initial function evaluation call fun ( xdata , p , y , stop ) neval = 1 ! Evaluate the problem matrices call lm_matrix ( fun , xdata , ydata , pOld , yOld , 1.0d0 , J , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , y , stop , work , mwork ) if ( stop ) go to 5 X2Old = X2 JtWJc = JtWJ ! Determine an initial value for lambda if ( opt % method == FS_LEVENBERG_MARQUARDT_UPDATE ) then lambda = 1.0d-2 else call extract_diagonal ( JtWJ , work ( 1 : n )) lambda = 1.0d-2 * maxval ( work ( 1 : n )) nu = 2.0d0 end if ! Main Loop main : do while ( niter < controls % max_iteration_count ) ! Compute the linear solution at the current solution estimate and ! update the new parameter estimates call lm_iter ( fun , xdata , ydata , p , neval , niter , opt % method , & lambda , maxP , minP , weights , JtWJc , JtWdy , h , pTry , resid , & yTemp , X2Try , X2Old , alpha , stop , iwork , err , status ) if ( stop ) go to 5 if ( err % has_error_occurred ()) return ! Update the Chi-squared estimate, update the damping parameter ! lambda, and, if necessary, update the matrices call lm_update ( fun , xdata , ydata , pOld , p , pTry , yOld , y , h , dX2 , & X2Old , X2 , X2Try , lambda , alpha , nu , JtWdy , JtWJ , J , weights , & niter , neval , update , step , work , mwork , controls , opt , stop ) if ( stop ) go to 5 JtWJc = JtWJ ! Determine the matrix update scheme nupdate = nupdate + 1 if ( opt % method == FS_QUADRATIC_UPDATE ) then update = mod ( niter , 2 * n ) > 0 else if ( nupdate >= controls % max_iteration_between_updates ) then update = . true . nupdate = 0 end if ! Test for convergence if ( lm_check_convergence ( controls , dof , resid , niter , neval , & JtWdy , h , p , X2 , info )) & then exit main end if end do main ! End return ! User Requested End 5 continue info % user_requested_stop = . true . return ! Memory Error Handling 10 continue allocate ( character ( len = 512 ) :: errmsg ) write ( errmsg , 100 ) \"Memory allocation error code \" , flag , \".\" call err % report_error ( \"lm_solve\" , & trim ( errmsg ), FS_MEMORY_ERROR ) return ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ ! subroutine lm_update ( fun , xdata , ydata , pOld , p , pTry , yOld , y , h , dX2 , & X2old , X2 , X2try , lambda , alpha , nu , JtWdy , JtWJ , J , weights , niter , & neval , update , step , work , mwork , controls , opt , stop ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), X2try , h (:), step , & pTry (:), weights (:), alpha real ( real64 ), intent ( inout ) :: pOld (:), p (:), yOld (:), y (:), lambda , & JtWdy (:), dX2 , X2 , X2old , JtWJ (:,:), J (:,:), nu real ( real64 ), intent ( out ) :: work (:), mwork (:,:) integer ( int32 ), intent ( in ) :: niter integer ( int32 ), intent ( inout ) :: neval logical , intent ( inout ) :: update class ( iteration_controls ), intent ( in ) :: controls class ( lm_solver_options ), intent ( in ) :: opt logical , intent ( out ) :: stop ! Local Variables integer ( int32 ) :: n real ( real64 ) :: rho ! Initialization n = size ( p ) ! Process if ( opt % method == FS_LEVENBERG_MARQUARDT_UPDATE ) then call extract_diagonal ( JtWJ , work ( 1 : n )) work ( 1 : n ) = lambda * work ( 1 : n ) * h + JtWdy else work ( 1 : n ) = lambda * h + JtWdy end if rho = ( X2 - X2try ) / abs ( dot_product ( h , work ( 1 : n ))) if ( rho > controls % iteration_improvement_tolerance ) then ! Things are getting better at an acceptable rate dX2 = X2 - X2old X2old = X2 pOld = p yOld = y p = pTry ! Recompute the matrices call lm_matrix ( fun , xdata , ydata , pOld , yOld , dX2 , J , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , y , stop , work , mwork ) if ( stop ) return ! Decrease lambda select case ( opt % method ) case ( FS_LEVENBERG_MARQUARDT_UPDATE ) lambda = max ( lambda / opt % damping_decrease_factor , 1.0d-7 ) case ( FS_QUADRATIC_UPDATE ) lambda = max ( lambda / ( 1.0d0 + alpha ), 1.0d-7 ) case ( FS_NIELSEN_UPDATE ) lambda = lambda * max ( 1.0d0 / 3.0d0 , & 1.0d0 - ( 2.0d0 * rho - 1.0d0 ** 3 )) nu = 2.0d0 end select else ! The iteration is not improving in a satisfactory manner X2 = X2old if ( mod ( niter , 2 * n ) /= 0 ) then call lm_matrix ( fun , xdata , ydata , pOld , yOld , - 1.0d0 , J , p , & weights , neval , update , step , JtWJ , JtWdy , dX2 , y , stop , & work , mwork ) if ( stop ) return end if ! Increase lambda select case ( opt % method ) case ( FS_LEVENBERG_MARQUARDT_UPDATE ) lambda = min ( lambda * opt % damping_increase_factor , 1.0d7 ) case ( FS_QUADRATIC_UPDATE ) lambda = lambda + abs (( X2try - X2 ) / 2.0d0 / alpha ) case ( FS_NIELSEN_UPDATE ) lambda = lambda * nu nu = 2.0d0 * nu end select end if end subroutine ! ------------------------------------------------------------------------------ ! Checks the Levenberg-Marquardt solution against the convergence criteria. ! ! Inputs: ! - controls: the solution controls and convergence criteria ! - dof: the statistical degrees of freedom of the system (M - N) ! - resid: the residual error (M-by-1) ! - niter: the number of iterations ! - neval: the number of function evaluations ! - JtWdy: linearized fitting vector (N-by-1) ! - h: the change in parameter (solution) values (N-by-1) ! - p: the parameter (solution) values (N-by-1) ! - X2: the Chi-squared estimate ! ! Outputs: ! - info: The convergence information. ! - rst: True if convergence was achieved; else, false. function lm_check_convergence ( controls , dof , resid , niter , neval , & JtWdy , h , p , X2 , info ) result ( rst ) ! Arguments class ( iteration_controls ), intent ( in ) :: controls real ( real64 ), intent ( in ) :: resid (:), JtWdy (:), h (:), p (:), X2 integer ( int32 ), intent ( in ) :: dof , niter , neval class ( convergence_info ), intent ( out ) :: info logical :: rst ! Initialization rst = . false . ! Iteration Checks info % iteration_count = niter if ( niter >= controls % max_iteration_count ) then info % reach_iteration_limit = . true . rst = . true . else info % reach_iteration_limit = . false . end if info % function_evaluation_count = neval if ( neval >= controls % max_function_evaluations ) then info % reach_function_evaluation_limit = . true . rst = . true . else info % reach_function_evaluation_limit = . false . end if info % gradient_value = maxval ( abs ( JtWdy )) if ( info % gradient_value < controls % gradient_tolerance . and . niter > 2 ) & then info % converge_on_gradient = . true . rst = . true . else info % converge_on_gradient = . false . end if info % solution_change_value = maxval ( abs ( h ) / ( abs ( p ) + 1.0d-12 )) if ( info % solution_change_value < & controls % change_in_solution_tolerance . and . niter > 2 ) & then info % converge_on_solution_change = . true . rst = . true . else info % converge_on_solution_change = . false . end if info % residual_value = X2 / dof if ( info % residual_value < controls % residual_tolerance . and . niter > 2 ) & then info % converge_on_residual_parameter = . true . rst = . true . else info % converge_on_residual_parameter = . false . end if end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_regression.f90.html"},{"title":"fstats_smoothing.f90 – FSTATS","text":"Source Code module fstats_smoothing use iso_fortran_env use ferror use fstats_errors use linalg , only : sort implicit none private public :: lowess contains ! ------------------------------------------------------------------------------ subroutine lowess ( x , y , ys , fsmooth , nstps , del , rweights , resid , err ) !! Computes the smoothing of a data set using a robust locally weighted !! scatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at !! each of the supplied x values. !! !! Remarks !! !! The code is a reimplementation of the LOWESS library.  For a detailed !! understanding, see [this] !! (http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf) !! paper by William Cleveland. real ( real64 ), intent ( in ), dimension (:) :: x !! An N-element array containing the independent variable data.  This !! array must be monotonically increasing. real ( real64 ), intent ( in ), dimension (:) :: y !! An N-element array containing the dependent variable data. real ( real64 ), intent ( out ), dimension (:) :: ys !! An N-element array where the smoothed results will be written. real ( real64 ), intent ( in ), optional :: fsmooth !! An optional input that specifies the amount of smoothing. !! Specifically, this value is the fraction of points used to compute !! each value.  As this value increases, the output becomes smoother. !! Choosing a value in the range of 0.2 to 0.8 typically results in a !! good fit.  The default value is 0.2. integer ( int32 ), intent ( in ), optional :: nstps !! An optional input that specifies the numb of iterations.  If set to !! zero, a non-robust fit is returned.  The default value is set to 2. real ( real64 ), intent ( in ), optional :: del !! real ( real64 ), intent ( out ), optional , dimension (:), target :: rweights !! An optional N-element array, that if supplied, will be used to !! return the weights given to each data point. real ( real64 ), intent ( out ), optional , dimension (:), target :: resid !! An optional N-element array, that if supplied, will be used to !! return the residual. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p2 = 2.0d-1 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: three = 3.0d0 real ( real64 ), parameter :: p001 = 1.0d-3 real ( real64 ), parameter :: p999 = 0.999d0 ! Local Variables logical :: ok integer ( int32 ) :: iter , i , j , nleft , nright , ns , last , m1 , m2 , n , nsteps , flag real ( real64 ) :: f , delta , d1 , d2 , denom , alpha , cut , eps , cmad , c1 , c9 , r real ( real64 ), allocatable , target , dimension (:) :: rwdef , rsdef real ( real64 ), pointer , dimension (:) :: rw , res class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n = size ( x ) if ( present ( fsmooth )) then f = fsmooth else f = p2 end if if ( present ( nstps )) then nsteps = nstps else nsteps = 2 end if if ( present ( del )) then delta = del else delta = 0.0d0 end if if ( present ( rweights )) then if ( size ( rweights ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"rweights\" , n , & size ( rweights )) return end if rw => rweights else allocate ( rwdef ( n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"lowess\" , flag ) return end if rw => rwdef end if if ( present ( resid )) then if ( size ( resid ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"resid\" , n , & size ( resid )) return end if res => resid else allocate ( rsdef ( n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"lowess\" , flag ) return end if res => rsdef end if ns = max ( min ( int ( f * real ( n ), int32 ), n ), 2 ) eps = epsilon ( eps ) ! Input Checking if ( size ( y ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"y\" , n , size ( y )) return end if if ( size ( ys ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"ys\" , n , size ( ys )) return end if ! Quick Return if ( n < 2 ) then ys = y return end if ! Process do iter = 1 , nsteps + 1 nleft = 1 nright = ns last = 0 i = 1 do do while ( nright < n ) d1 = x ( i ) - x ( nleft ) d2 = x ( nright + 1 ) - x ( i ) if ( d1 <= d2 ) exit nleft = nleft + 1 nright = nright + 1 end do call lowest ( x , y , x ( i ), ys ( i ), nleft , nright , res , iter > 1 , & rw , ok ) if (. not . ok ) ys ( i ) = y ( i ) if ( last < i - 1 ) then denom = x ( i ) - x ( last ) do j = last + 1 , i - 1 alpha = ( x ( j ) - x ( last )) / denom ys ( j ) = alpha * ys ( i ) + ( one - alpha ) * ys ( last ) end do end if last = i cut = x ( last ) + delta do i = last + 1 , n if ( x ( i ) > cut ) exit if ( abs ( x ( i ) - x ( last )) < eps ) then ys ( i ) = ys ( last ) last = i end if end do i = max ( last + 1 , i - 1 ) if ( last >= n ) exit end do res = y - ys if ( iter > nsteps ) exit rw = abs ( res ) call sort ( rw , . true .) m1 = 1 + n / 2 m2 = n - m1 + 1 cmad = three * ( rw ( m1 ) + rw ( m2 )) c9 = p999 * cmad c1 = p001 * cmad do i = 1 , n r = abs ( res ( i )) if ( r <= c1 ) then rw ( i ) = one else if ( r > c9 ) then rw ( i ) = zero else rw ( i ) = ( one - ( r / cmad ) ** 2 ) ** 2 end if end do end do end subroutine ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ ! REF: ! - https://en.wikipedia.org/wiki/Local_regression ! - http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf subroutine lowest ( x , y , xs , ys , nleft , nright , w , userw , rw , ok ) ! Arguments real ( real64 ), intent ( in ), dimension (:) :: x , y , rw ! N ELEMENT real ( real64 ), intent ( in ) :: xs real ( real64 ), intent ( out ) :: ys integer ( int32 ), intent ( in ) :: nleft , nright real ( real64 ), intent ( out ), dimension (:) :: w ! N ELEMENT logical , intent ( in ) :: userw logical , intent ( out ) :: ok ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: p001 = 1.0d-3 real ( real64 ), parameter :: p999 = 0.999d0 ! Local Variables integer ( int32 ) :: j , n , nrt real ( real64 ) :: range , h , h9 , h1 , a , b , c , r ! Initialization n = size ( x ) range = x ( n ) - x ( 1 ) h = max ( xs - x ( nleft ), x ( nright ) - xs ) h9 = p999 * h h1 = p001 * h a = zero ! Process do j = nleft , n w ( j ) = zero r = abs ( x ( j ) - xs ) if ( r <= h9 ) then if ( r > h1 ) then w ( j ) = ( one - ( r / h ) ** 3 ) ** 3 else w ( j ) = one end if if ( userw ) w ( j ) = rw ( j ) * w ( j ) a = a + w ( j ) else if ( x ( j ) > xs ) then exit end if end do nrt = j - 1 if ( a <= zero ) then ok = . false . else ok = . true . w ( nleft : nrt ) = w ( nleft : nrt ) / a if ( h > zero ) then a = zero do j = nleft , nrt a = a + w ( j ) * x ( j ) end do b = xs - a c = zero do j = nleft , nrt c = c + w ( j ) * ( x ( j ) - a ) ** 2 end do if ( sqrt ( c ) > p001 * range ) then b = b / c do j = nleft , nrt w ( j ) = w ( j ) * ( one + b * ( x ( j ) - a )) end do end if end if ys = zero do j = nleft , nrt ys = ys + w ( j ) * y ( j ) end do end if end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_smoothing.f90.html"},{"title":"fstats_special_functions.f90 – FSTATS","text":"Source Code module fstats_special_functions use iso_fortran_env use ieee_arithmetic implicit none private public :: beta public :: regularized_beta public :: incomplete_beta public :: incomplete_gamma_lower public :: incomplete_gamma_upper public :: digamma contains ! ------------------------------------------------------------------------------ pure elemental function beta ( a , b ) result ( rst ) !! Computes the beta function. !! !! The beta function is related to the gamma function !! by the following relationship. !!  \\beta(a,b) = \\frac{\\Gamma(a) \\Gamma(b)}{\\Gamma(a + b)} . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ) :: rst !! The value of the beta function at  a  and  b . ! Process ! REF: https://en.wikipedia.org/wiki/Beta_function rst = exp ( log_gamma ( a ) + log_gamma ( b ) - log_gamma ( a + b )) end function ! ------------------------------------------------------------------------------ ! source: https://people.math.sc.edu/Burkardt/f_src/special_functions/special_functions.f90 pure elemental function regularized_beta ( a , b , x ) result ( rst ) !! Computes the regularized beta function. !! !! The regularized beta function is defined as the ratio between !! the incomplete beta function and the beta function. !!  I_{x}(a,b) = \\frac{\\beta(x;a,b)}{\\beta(a,b)} . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ), intent ( in ) :: x !! The upper limit of the integration. real ( real64 ) :: rst !! The value of the regularized beta function. ! Local Variables real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 real ( real64 ) :: bt , dk ( 51 ), fk ( 51 ), s0 , t1 , t2 , ta , tb integer ( int32 ) :: k ! Process s0 = ( a + one ) / ( a + b + two ) bt = beta ( a , b ) if ( x <= s0 ) then do k = 1 , 20 dk ( 2 * k ) = k * ( b - k ) * x / ( a + two * k - one ) / ( a + two * k ) end do do k = 0 , 20 dk ( 2 * k + 1 ) = - ( a + k ) * ( a + b + k ) * x / ( a + two * k ) / & ( a + two * k + one ) end do t1 = zero do k = 20 , 1 , - 1 t1 = dk ( k ) / ( one + t1 ) end do ta = one / ( one + t1 ) rst = x ** a * ( one - x ) ** b / ( a * bt ) * ta else do k = 1 , 20 fk ( 2 * k ) = k * ( a - k ) * ( one - x ) / ( b + two * k - one ) / & ( b + two * k ) end do do k = 0 , 20 fk ( 2 * k + 1 ) = - ( b + k ) * ( a + b + k ) * ( one - x ) / ( b + two * k ) / & ( b + two * k + one ) end do t2 = zero do k = 20 , 1 , - 1 t2 = fk ( k ) / ( one + t2 ) end do tb = one / ( one + t2 ) rst = one - x ** a * ( one - x ) ** b / ( b * bt ) * tb end if end function ! ------------------------------------------------------------------------------ pure elemental function incomplete_beta ( a , b , x ) result ( rst ) !! Computes the incomplete beta function. !! !! The incomplete beta function is defind as: !!  \\beta(x;a,b) = \\int_{0}&#94;{x} t&#94;{a-1} (1 - t)&#94;{b-1} dt . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ), intent ( in ) :: x !! The upper limit of the integration. real ( real64 ) :: rst !! The value of the incomplete beta function. ! Process rst = beta ( a , b ) * regularized_beta ( a , b , x ) end function ! ------------------------------------------------------------------------------ ! REF: https://people.math.sc.edu/Burkardt/f_src/special_functions/special_functions.f90 pure elemental function incomplete_gamma_upper ( a , x ) result ( rst ) !! Computes the upper incomplete gamma function. !! !! The upper incomplete gamma function is defined as: !!  \\Gamma(a, x) = \\int_{x}&#94;{\\infty} t&#94;{a-1} e&#94;{-t} \\,dt  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function) real ( real64 ), intent ( in ) :: a !! The coefficient value. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: ten = 1.0d1 ! Local Variables real ( real64 ) :: ga , gin , gip , r , s , t0 , xam , small integer ( int32 ) :: k ! Process small = ten * epsilon ( small ) xam = - x + a * log ( x ) if ( xam > 7.0d2 . or . a > 1.7d2 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) return end if if ( x == zero ) then rst = gamma ( a ) else if ( x <= one + a ) then s = one / a r = s do k = 1 , 60 r = r * x / ( a + k ) s = s + r if ( abs ( r / s ) < small ) then exit end if end do gin = exp ( xam ) * s ga = gamma ( a ) gip = gin / ga rst = ga - gin else if ( one + a < x ) then t0 = zero do k = 60 , 1 , - 1 t0 = ( k - a ) / ( one + k / ( x + t0 )) end do rst = exp ( xam ) / ( x + t0 ) end if end function ! ------------------------------------------------------------------------------ pure elemental function incomplete_gamma_lower ( a , x ) result ( rst ) !! Computes the lower incomplete gamma function. !! !! The lower incomplete gamma function is defined as: !!  \\gamma(a, x) = \\int_{0}&#94;{x} t&#94;{a-1} e&#94;{-t} \\,dt  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function) real ( real64 ), intent ( in ) :: a !! The coefficient value. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: ten = 1.0d1 ! Local Variables real ( real64 ) :: ga , gim , r , s , t0 , xam , small integer ( int32 ) :: k ! Process small = ten * epsilon ( small ) xam = - x + a * log ( x ) if ( xam > 7.0d2 . or . a > 1.7d2 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) return end if if ( x == zero ) then rst = 0.0d0 else if ( x <= one + a ) then s = one / a r = s do k = 1 , 60 r = r * x / ( a + k ) s = s + r if ( abs ( r / s ) < small ) then exit end if end do rst = exp ( xam ) * s else if ( one + a < x ) then t0 = zero do k = 60 , 1 , - 1 t0 = ( k - a ) / ( one + k / ( x + t0 )) end do gim = exp ( xam ) / ( x + t0 ) ga = gamma ( a ) rst = ga - gim end if end function ! ------------------------------------------------------------------------------ pure elemental function digamma ( x ) result ( rst ) !! Computes the digamma function. !! !! The digamma function is defined as: !!  \\psi(x) = !! \\frac{d}{dx}\\left( \\ln \\left( \\Gamma \\left( x \\right) \\right) !! \\right)  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Digamma_function) real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: c = 8.5d0 real ( real64 ), parameter :: euler_mascheroni = 0.57721566490153286060d0 ! Local Variables real ( real64 ) :: r , x2 , nan ! REF: ! - https://people.sc.fsu.edu/~jburkardt/f_src/asa103/asa103.f90 ! If x <= 0.0 if ( x <= 0.0 ) then nan = ieee_value ( nan , IEEE_QUIET_NAN ) rst = nan return end if ! Approximation for a small argument if ( x <= 1.0d-6 ) then rst = - euler_mascheroni - 1.0d0 / x + 1.6449340668482264365d0 * x return end if ! Process rst = 0.0d0 x2 = x do while ( x2 < c ) rst = rst - 1.0d0 / x2 x2 = x2 + 1.0d0 end do r = 1.0d0 / x2 rst = rst + log ( x2 ) - 0.5d0 * r r = r * r rst = rst & - r * ( 1.0d0 / 1 2.0d0 & - r * ( 1.0d0 / 12 0.0d0 & - r * ( 1.0d0 / 25 2.0d0 & - r * ( 1.0d0 / 24 0.0d0 & - r * ( 1.0d0 / 13 2.0d0 ) & )))) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_special_functions.f90.html"},{"title":"fstats_sampling.f90 – FSTATS","text":"Source Code module fstats_sampling use iso_fortran_env use linalg , only : sort use fstats_distributions implicit none private public :: box_muller_sample public :: rejection_sample real ( real64 ), parameter :: pi = 2.0d0 * acos ( 0.0d0 ) real ( real64 ), parameter :: twopi = 2.0d0 * pi real ( real64 ), parameter :: pi_f = 2.0 * acos ( 0.0 ) real ( real64 ), parameter :: twopi_f = 2.0 * pi_f interface box_muller_sample !! Generates random, normally distributed values via the Box-Muller !! transform. module procedure :: box_muller_sample_scalar module procedure :: box_muller_array end interface contains ! ------------------------------------------------------------------------------ function box_muller_sample_scalar ( mu , sigma ) result ( rst ) !! Generates a pair of independent, standard, normally distributed !! random values using the Box-Muller transform. real ( real64 ), intent ( in ) :: mu !! The mean of the distribution. real ( real64 ), intent ( in ) :: sigma !! The standard deviation of the distribution. real ( real64 ) :: rst ( 2 ) !! The pair of random values. ! Parameters complex ( real64 ), parameter :: j = ( 0.0d0 , 1.0d0 ) ! Local Variables real ( real64 ) :: u1 , u2 complex ( real64 ) :: z ! Process call random_number ( u1 ) call random_number ( u2 ) z = sqrt ( - log ( u1 )) * exp ( j * twopi * u2 ) rst = [ real ( z , real64 ), aimag ( z )] end function ! ------------------------------------------------------------------------------ function box_muller_array ( mu , sigma , n ) result ( rst ) !! Generates an array of normally distributed random values sampled !! by the Box-Muller transform. real ( real64 ), intent ( in ) :: mu !! The mean of the distribution. real ( real64 ), intent ( in ) :: sigma !! The standard deviation of the distribution. integer ( int32 ), intent ( in ) :: n !! The number of Box-Muller pairs to generate. real ( real64 ), allocatable , dimension (:) :: rst !! A 2N-element array containing the N Box-Muller pairs. ! Local Variables integer ( int32 ) :: i ! Process if ( n < 1 ) then allocate ( rst ( 0 )) return end if allocate ( rst ( 2 * n )) do i = 1 , n rst ( 2 * i - 1 : 2 * i ) = box_muller_sample ( mu , sigma ) end do end function ! ****************************************************************************** ! REJECTION SAMPLING ! ------------------------------------------------------------------------------ function rejection_sample ( tdist , n , xmin , xmax ) result ( rst ) !! Uses rejection sampling to randomly sample a target distribution. class ( distribution ), intent ( in ) :: tdist !! The distribution to sample integer ( int32 ), intent ( in ) :: n !! The number of samples to make. real ( real64 ), intent ( in ) :: xmin !! The minimum range to explore. real ( real64 ), intent ( in ) :: xmax !! The maximum range to explore. real ( real64 ), allocatable , dimension (:) :: rst !! An N-element array containing the N samples from the !! distribution. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: c_start = 1.01d0 ! Local Variables integer ( int32 ) :: i , j , jmax real ( real64 ) :: u , c , g , f , rng ! Quick Return if ( n < 1 ) then allocate ( rst ( 0 ), source = zero ) end if ! Process i = 0 j = 0 jmax = min ( 1000 * n , huge ( j )) ! Guard against insanity rng = xmax - xmin c = c_start allocate ( rst ( n ), source = zero ) do while ( i <= n ) ! Update the acceptance threshold call random_number ( u ) ! Sample from the proposal distribution call random_number ( g ) g = g * rng + xmin ! Sample the target distribution f = tdist % pdf ( g ) ! Test if ( u <= f / ( c * g )) then i = i + 1 rst ( i ) = g end if ! Update C c = max ( c , f / g ) ! Update the infinite loop guard variable j = j + 1 if ( j == jmax ) exit end do end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_sampling.f90.html"},{"title":"fstats_distributions.f90 – FSTATS","text":"Source Code module fstats_distributions use iso_fortran_env use ieee_arithmetic use fstats_special_functions use fstats_helper_routines implicit none private public :: distribution public :: distribution_function public :: distribution_property public :: t_distribution public :: normal_distribution public :: f_distribution public :: chi_squared_distribution public :: binomial_distribution real ( real64 ), parameter :: pi = 2.0d0 * acos ( 0.0d0 ) type , abstract :: distribution !! Defines a probability distribution. contains procedure ( distribution_function ), deferred , pass :: pdf !! Computes the probability density function. procedure ( distribution_function ), deferred , pass :: cdf !! Computes the cumulative distribution function. procedure ( distribution_property ), deferred , pass :: mean !! Computes the mean of the distribution. procedure ( distribution_property ), deferred , pass :: median !! Computes the median of the distribution. procedure ( distribution_property ), deferred , pass :: mode !! Computes the mode of the distribution. procedure ( distribution_property ), deferred , pass :: variance !! Computes the variance of the distribution. end type interface pure elemental function distribution_function ( this , x ) result ( rst ) !! Defines the interface for a probability distribution function. use iso_fortran_env , only : real64 import distribution class ( distribution ), intent ( in ) :: this !! The distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. end function pure function distribution_property ( this ) result ( rst ) !! Computes the value of a distribution property. use iso_fortran_env , only : real64 import distribution class ( distribution ), intent ( in ) :: this !! The distribution object. real ( real64 ) :: rst !! The property value. end function end interface ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: t_distribution !! Defines Student's T-Distribution. real ( real64 ) :: dof !! The number of degrees of freedom. contains procedure , public :: pdf => td_pdf procedure , public :: cdf => td_cdf procedure , public :: mean => td_mean procedure , public :: median => td_median procedure , public :: mode => td_mode procedure , public :: variance => td_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: normal_distribution !! Defines a normal distribution. real ( real64 ) :: standard_deviation !! The standard deviation of the distribution. real ( real64 ) :: mean_value !! The mean value of the distribution. contains procedure , public :: pdf => nd_pdf procedure , public :: cdf => nd_cdf procedure , public :: mean => nd_mean procedure , public :: median => nd_median procedure , public :: mode => nd_mode procedure , public :: variance => nd_variance procedure , public :: standardize => nd_standardize end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: f_distribution !! Defines an F-distribution. real ( real64 ) :: d1 !! The measure of degrees of freedom for the first data set. real ( real64 ) :: d2 !! The measure of degrees of freedom for the second data set. contains procedure , public :: pdf => fd_pdf procedure , public :: cdf => fd_cdf procedure , public :: mean => fd_mean procedure , public :: median => fd_median procedure , public :: mode => fd_mode procedure , public :: variance => fd_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: chi_squared_distribution !! Defines a Chi-squared distribution. integer ( int32 ) :: dof !! The number of degrees of freedom. contains procedure , public :: pdf => cs_pdf procedure , public :: cdf => cs_cdf procedure , public :: mean => cs_mean procedure , public :: median => cs_median procedure , public :: mode => cs_mode procedure , public :: variance => cs_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: binomial_distribution !! Defines a binomial distribution.  The binomial distribution describes !! the probability p of getting k successes in n independent trials. integer ( int32 ) :: n !! The number of independent trials. real ( real64 ) :: p !! The success probability for each trial.  This parameter must !! exist on the set [0, 1]. contains procedure , public :: pdf => bd_pdf procedure , public :: cdf => bd_cdf procedure , public :: mean => bd_mean procedure , public :: median => bd_median procedure , public :: mode => bd_mode procedure , public :: variance => bd_variance end type ! ------------------------------------------------------------------------------ contains ! ****************************************************************************** ! STUDENT'S T-DISTRIBUTION ! ------------------------------------------------------------------------------ ! REF: https://en.wikipedia.org/wiki/Student%27s_t-distribution pure elemental function td_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for Student's T-Distribution is given as !!  f(t) = \\frac{ \\Gamma \\left( \\frac{\\nu + 1}{2} \\right) } !! { \\sqrt{\\nu \\pi} \\Gamma \\left( \\frac{\\nu}{2} \\right) } !! \\left( 1 + \\frac{t&#94;2}{\\nu} \\right)&#94;{-(\\nu + 1) / 2} . class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process rst = gamma (( this % dof + 1.0d0 ) / 2.0d0 ) / & ( sqrt ( this % dof * pi ) * gamma ( this % dof / 2.0d0 )) * & ( 1.0d0 + x ** 2 / this % dof ) ** ( - 0.5d0 * ( 1.0d0 + this % dof )) end function ! ------------------------------------------------------------------------------ pure elemental function td_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for Student's T-Distribution is given as !!  F(t) = \\int_{-\\infty}&#94;{t} f(u) \\,du = 1 - \\frac{1}{2} I_{x(t)} !! \\left( \\frac{\\nu}{2}, \\frac{1}{2} \\right)  !! where  x(t) = \\frac{\\nu}{\\nu + t&#94;2} . class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: t t = this % dof / ( this % dof + x ** 2 ) rst = 1.0d0 - 0.5d0 * regularized_beta ( 0.5d0 * this % dof , 0.5d0 , t ) if ( x < 0 ) rst = 1.0d0 - rst end function ! ------------------------------------------------------------------------------ pure function td_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The mean. ! Process if ( this % dof < 1.0d0 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) else rst = 0.0d0 end if end function ! ------------------------------------------------------------------------------ pure function td_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst ! Process rst = 0.0d0 end function ! ------------------------------------------------------------------------------ pure function td_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The mode. ! Process rst = 0.0d0 end function ! ------------------------------------------------------------------------------ pure function td_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The variance. ! Process if ( this % dof <= 1.0d0 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) else if ( this % dof > 1.0d0 . and . this % dof <= 2.0d0 ) then rst = ieee_value ( rst , IEEE_POSITIVE_INF ) else rst = this % dof / ( this % dof - 2.0d0 ) end if end function ! ****************************************************************************** ! NORMAL DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function nd_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a normal distribution is given as !!  f(x) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\exp \\left(-\\frac{1}{2} !! \\left( \\frac{x - \\mu}{\\sigma} \\right)&#94;2 \\right) . class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. rst = exp ( - 0.5d0 * (( x - this % mean_value ) / this % standard_deviation ) ** 2 ) / & ( this % standard_deviation * sqrt ( 2.0d0 * pi )) end function ! ------------------------------------------------------------------------------ pure elemental function nd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a normal distribution is given as !!  F(x) = \\frac{1}{2} \\left( 1 + erf \\left( \\frac{x - \\mu} !! {\\sigma \\sqrt{2}} \\right) \\right) . class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. rst = 0.5d0 * ( 1.0d0 + erf (( x - this % mean_value ) / & ( this % standard_deviation * sqrt ( 2.0d0 )))) end function ! ------------------------------------------------------------------------------ pure function nd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The mean rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The median. rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The mode. rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The variance. rst = this % standard_deviation ** 2 end function ! ------------------------------------------------------------------------------ subroutine nd_standardize ( this ) !! Standardizes the normal distribution to a mean of 0 and a !! standard deviation of 1. class ( normal_distribution ), intent ( inout ) :: this !! The normal_distribution object. this % mean_value = 0.0d0 this % standard_deviation = 1.0d0 end subroutine ! ****************************************************************************** ! F DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function fd_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a F distribution is given as !!  f(x) = !! \\sqrt{ \\frac{ (d_1 x)&#94;{d_1} d_{2}&#94;{d_2} }{ (d_1 x + d_2)&#94;{d_1 + d_2} } } !! \\frac{1}{x \\beta \\left( \\frac{d_1}{2}, \\frac{d_2}{2} \\right) } . class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 rst = ( 1.0d0 / beta ( 0.5d0 * d1 , 0.5d0 * d2 )) * ( d1 / d2 ) ** ( 0.5d0 * d1 ) * & x ** ( 0.5d0 * d1 - 1.0d0 ) * ( 1.0d0 + d1 * x / d2 ) ** ( - 0.5d0 * ( d1 + d2 )) end function ! ------------------------------------------------------------------------------ pure elemental function fd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a F distribution is given as !!  F(x) = I_{d_1 x/(d_1 x + d_2)} \\left( \\frac{d_1}{2}, !! \\frac{d_2}{2} \\right) . class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 rst = regularized_beta ( 0.5d0 * d1 , 0.5d0 * d2 , d1 * x / ( d1 * x + d2 )) end function ! ------------------------------------------------------------------------------ pure function fd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The mean. ! Process if ( this % d2 > 2.0d0 ) then rst = this % d2 / ( this % d2 - 2.0d0 ) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ------------------------------------------------------------------------------ pure function fd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The median. rst = ieee_value ( rst , IEEE_QUIET_NAN ) end function ! ------------------------------------------------------------------------------ pure function fd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The mode. ! Process if ( this % d1 > 2.0d0 ) then rst = (( this % d1 - 2.0d0 ) / this % d1 ) * ( this % d2 / ( this % d2 + 2.0d0 )) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ------------------------------------------------------------------------------ pure function fd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The variance. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 if ( d2 > 4.0d0 ) then rst = ( 2.0d0 * d2 ** 2 * ( d1 + d2 - 2.0d0 )) / & ( d1 * ( d2 - 2.0d0 ) ** 2 * ( d2 - 4.0d0 )) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ****************************************************************************** ! CHI-SQUARED DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function cs_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a Chi-squared distribution is given as !!  f(x) = \\frac{x&#94;{k/2 - 1} \\exp{-x / 2}} {2&#94;{k / 2} !! \\Gamma \\left( \\frac{k}{2} \\right)} . class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: arg ! Process arg = 0.5d0 * this % dof rst = 1.0d0 / ( 2.0d0 ** arg * gamma ( arg )) * x ** ( arg - 1.0d0 ) * exp ( - 0.5d0 * x ) end function ! ------------------------------------------------------------------------------ pure elemental function cs_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a Chi-squared distribution is given as !!  F(x) = \\frac{ \\gamma \\left( \\frac{k}{2}, \\frac{x}{2} \\right) } !! { \\Gamma \\left( \\frac{k}{2} \\right)} . class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: arg ! Process arg = 0.5d0 * this % dof rst = incomplete_gamma_lower ( arg , 0.5d0 * x ) / gamma ( arg ) end function ! ------------------------------------------------------------------------------ pure function cs_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The mean. ! Process rst = real ( this % dof , real64 ) end function ! ------------------------------------------------------------------------------ pure function cs_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The median. ! Process rst = this % dof * ( 1.0d0 - 2.0d0 / ( 9.0d0 * this % dof )) ** 3 end function ! ------------------------------------------------------------------------------ pure function cs_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The mode. ! Process rst = max ( this % dof - 2.0d0 , 0.0d0 ) end function ! ------------------------------------------------------------------------------ pure function cs_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The variance. ! Process rst = 2.0d0 * this % dof end function ! ****************************************************************************** ! BINOMIAL DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function bd_pdf ( this , x ) result ( rst ) !! Computes the probability mass function. !! !! The PMF for a binomial distribution is given as !!  f(k,n,p) = \\frac{n!}{k! \\left( n - k! \\right)} p&#94;k !! \\left( 1 - p \\right)&#94;{n-k} . class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function.  This parameter !! is the number k successes in the n independent trials.  As !! such, this parameter must exist on the set [0, n]. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: dn ! Process dn = real ( this % n , real64 ) rst = ( factorial ( dn ) / ( factorial ( x ) * factorial ( dn - x ))) * ( this % p ** x ) * ( 1.0d0 - this % p ) ** ( dn - x ) end function ! ------------------------------------------------------------------------------ pure elemental function bd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution funtion. !! !! The CDF for a binomial distribution is given as !!  F(k,n,p) = I_{1-p} \\left( n - k, 1 + k \\right) , which is simply !! the regularized incomplete beta function. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function.  This parameter !! is the number k successes in the n independent trials.  As !! such, this parameter must exist on the set [0, n]. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: dn ! Process dn = real ( this % n , real64 ) rst = regularized_beta ( dn - x , x + 1.0d0 , 1.0d0 - this % p ) end function ! ------------------------------------------------------------------------------ pure function bd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The mean. rst = real ( this % n * this % p , real64 ) end function ! ------------------------------------------------------------------------------ pure function bd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The median. rst = real ( this % n * this % p , real64 ) end function ! ------------------------------------------------------------------------------ pure function bd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The mode. rst = ( this % n + 1.0d0 ) * this % p end function ! ------------------------------------------------------------------------------ pure function bd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The variance. rst = this % n * this % p * ( 1.0d0 - this % p ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_distributions.f90.html"}]}
\ No newline at end of file
+var tipuesearch = {"pages":[{"title":" FSTATS ","text":"FSTATS Developer Info Jason Christopherson","tags":"home","loc":"index.html"},{"title":"convergence_info – FSTATS ","text":"type, public :: convergence_info Provides information regarding convergence status. Components Type Visibility Attributes Name Initial logical, public :: converge_on_gradient True if convergence on the gradient was achieved; else, false. logical, public :: converge_on_residual_parameter True if convergence on the residual error parameter was achieved; \nelse, false. logical, public :: converge_on_solution_change True if convergence on the change in solution was achieved; else,\nfalse. integer(kind=int32), public :: function_evaluation_count The function evaluation count. real(kind=real64), public :: gradient_value The value of the gradient test parameter. integer(kind=int32), public :: iteration_count The iteration count. logical, public :: reach_function_evaluation_limit True if the solution did not converge in the allowed number of\nfunction evaluations. logical, public :: reach_iteration_limit True if the solution did not converge in the allowed number of \niterations. real(kind=real64), public :: residual_value The value of the residual error parameter. real(kind=real64), public :: solution_change_value The value of the change in solution parameter. logical, public :: user_requested_stop True if the user requested the stop; else, false.","tags":"","loc":"type\\convergence_info.html"},{"title":"iteration_controls – FSTATS ","text":"type, public :: iteration_controls Provides a collection of iteration control parameters. Components Type Visibility Attributes Name Initial real(kind=real64), public :: change_in_solution_tolerance Defines a tolerance on the change in parameter values. real(kind=real64), public :: gradient_tolerance Defines a tolerance on the gradient of the fitted function. real(kind=real64), public :: iteration_improvement_tolerance Defines a tolerance to ensure adequate improvement on each \niteration. integer(kind=int32), public :: max_function_evaluations Defines the maximum number of function evaluations allowed. integer(kind=int32), public :: max_iteration_between_updates Defines how many iterations can pass before a re-evaluation of \nthe Jacobian matrix is forced. integer(kind=int32), public :: max_iteration_count Defines the maximum number of iterations allowed. real(kind=real64), public :: residual_tolerance Defines a tolerance on the metric associated with the residual \nerror. Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_tolerances private  subroutine lm_set_default_tolerances(x) Arguments Type Intent Optional Attributes Name class( iteration_controls ), intent(inout) :: x","tags":"","loc":"type\\iteration_controls.html"},{"title":"lm_solver_options – FSTATS ","text":"type, public :: lm_solver_options Options to control the Levenberg-Marquardt solver. Components Type Visibility Attributes Name Initial real(kind=real64), public :: damping_decrease_factor The factor to use when decreasing the damping parameter. real(kind=real64), public :: damping_increase_factor The factor to use when increasing the damping parameter. real(kind=real64), public :: finite_difference_step_size The step size used for the finite difference calculations of the\nJacobian matrix. integer(kind=int32), public :: method The solver method to utilize.\n- FS_LEVENBERG_MARQUARDT_UPDATE:\n- FS_QUADRATIC_UPDATE:\n- FS_NIELSEN_UDPATE: Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_settings private  subroutine lm_set_default_settings(x) Arguments Type Intent Optional Attributes Name class( lm_solver_options ), intent(inout) :: x","tags":"","loc":"type\\lm_solver_options.html"},{"title":"regression_statistics – FSTATS ","text":"type, public :: regression_statistics A container for regression-related statistical information. Components Type Visibility Attributes Name Initial real(kind=real64), public :: confidence_interval The confidence interval for the parameter at the level \ndetermined by the regression process. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient.","tags":"","loc":"type\\regression_statistics.html"},{"title":"array_container – FSTATS ","text":"type, public :: array_container Provides a container for a real-valued array.  A practical use of\nthis construct is in the construction of jagged arrays. Components Type Visibility Attributes Name Initial real(kind=real64), public, allocatable, dimension(:) :: x The array.","tags":"","loc":"type\\array_container.html"},{"title":"bootstrap_regression_statistics – FSTATS ","text":"type, public :: bootstrap_regression_statistics A container for regression-related statistical information as \ncomputed in a bootstrap, or equivalent, calculation. Components Type Visibility Attributes Name Initial real(kind=real64), public :: lower_confidence_interval The lower limit of the confidence interval for the parameter. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. real(kind=real64), public :: upper_confidence_interval The upper limit of the confidence interval for the parameter.","tags":"","loc":"type\\bootstrap_regression_statistics.html"},{"title":"bootstrap_statistics – FSTATS ","text":"type, public :: bootstrap_statistics A collection of statistics resulting from the bootstrap process. Components Type Visibility Attributes Name Initial real(kind=real64), public :: bias The bias in the statistic. real(kind=real64), public :: lower_confidence_interval The lower confidence limit on the statistic. real(kind=real64), public, allocatable, dimension(:) :: population An array of the population values generated by the bootstrap\nprocess. real(kind=real64), public :: standard_error The standard error of the statistic. real(kind=real64), public :: statistic_value The value of the statistic of interest. real(kind=real64), public :: upper_confidence_interval The upper confidence limit on the statistic.","tags":"","loc":"type\\bootstrap_statistics.html"},{"title":"anova_factor – FSTATS ","text":"type, public :: anova_factor Defines an ANOVA factor result. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedome. real(kind=real64), public :: f_statistic The F-statistic. real(kind=real64), public :: probability The variance probability term. real(kind=real64), public :: sum_of_squares The sum of the squares. real(kind=real64), public :: variance The estimate of variance.","tags":"","loc":"type\\anova_factor.html"},{"title":"single_factor_anova_table – FSTATS ","text":"type, public :: single_factor_anova_table Defines a single-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: main_factor The main, or main factor, results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within-treatement (error) results.","tags":"","loc":"type\\single_factor_anova_table.html"},{"title":"two_factor_anova_table – FSTATS ","text":"type, public :: two_factor_anova_table Defines a two-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: interaction The interaction effects. type( anova_factor ), public :: main_factor_1 The first main-factor results. type( anova_factor ), public :: main_factor_2 The second main-factor results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within (error) factor results.","tags":"","loc":"type\\two_factor_anova_table.html"},{"title":"binomial_distribution – FSTATS ","text":"type, public, extends( distribution ) :: binomial_distribution Defines a binomial distribution.  The binomial distribution describes\nthe probability p of getting k successes in n independent trials. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: n The number of independent trials. real(kind=real64), public :: p The success probability for each trial.  This parameter must\nexist on the set [0, 1]. Type-Bound Procedures procedure, public :: cdf => bd_cdf private pure elemental function bd_cdf(this, x) result(rst) Computes the cumulative distribution funtion. The CDF for a binomial distribution is given as , which is simply\nthe regularized incomplete beta function. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function.  This parameter\nis the number k successes in the n independent trials.  As\nsuch, this parameter must exist on the set [0, n]. Return Value real(kind=real64) The value of the function. procedure, public :: mean => bd_mean private pure function bd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => bd_median private pure function bd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => bd_mode private pure function bd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => bd_pdf private pure elemental function bd_pdf(this, x) result(rst) Computes the probability mass function. The PMF for a binomial distribution is given as . Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function.  This parameter\nis the number k successes in the n independent trials.  As\nsuch, this parameter must exist on the set [0, n]. Return Value real(kind=real64) The value of the function. procedure, public :: variance => bd_variance private pure function bd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( binomial_distribution ), intent(in) :: this The binomial_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\binomial_distribution.html"},{"title":"chi_squared_distribution – FSTATS ","text":"type, public, extends( distribution ) :: chi_squared_distribution Defines a Chi-squared distribution. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => cs_cdf private pure elemental function cs_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a Chi-squared distribution is given as . Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => cs_mean private pure function cs_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => cs_median private pure function cs_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => cs_mode private pure function cs_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => cs_pdf private pure elemental function cs_pdf(this, x) result(rst) Computes the probability density function. The PDF for a Chi-squared distribution is given as . Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => cs_variance private pure function cs_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( chi_squared_distribution ), intent(in) :: this The chi_squared_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\chi_squared_distribution.html"},{"title":"distribution – FSTATS ","text":"type, public :: distribution Defines a probability distribution. Type-Bound Procedures procedure( distribution_function ), public, deferred, pass :: cdf Computes the cumulative distribution function. pure elemental function distribution_function(this, x) result(rst) Prototype Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure( distribution_property ), public, deferred, pass :: mean Computes the mean of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_property ), public, deferred, pass :: median Computes the median of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_property ), public, deferred, pass :: mode Computes the mode of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. procedure( distribution_function ), public, deferred, pass :: pdf Computes the probability density function. pure elemental function distribution_function(this, x) result(rst) Prototype Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure( distribution_property ), public, deferred, pass :: variance Computes the variance of the distribution. pure function distribution_property(this) result(rst) Prototype Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value.","tags":"","loc":"type\\distribution.html"},{"title":"f_distribution – FSTATS ","text":"type, public, extends( distribution ) :: f_distribution Defines an F-distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: d1 The measure of degrees of freedom for the first data set. real(kind=real64), public :: d2 The measure of degrees of freedom for the second data set. Type-Bound Procedures procedure, public :: cdf => fd_cdf private pure elemental function fd_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a F distribution is given as . Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => fd_mean private pure function fd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => fd_median private pure function fd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => fd_mode private pure function fd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => fd_pdf private pure elemental function fd_pdf(this, x) result(rst) Computes the probability density function. The PDF for a F distribution is given as . Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => fd_variance private pure function fd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( f_distribution ), intent(in) :: this The f_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\f_distribution.html"},{"title":"normal_distribution – FSTATS ","text":"type, public, extends( distribution ) :: normal_distribution Defines a normal distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: mean_value The mean value of the distribution. real(kind=real64), public :: standard_deviation The standard deviation of the distribution. Type-Bound Procedures procedure, public :: cdf => nd_cdf private pure elemental function nd_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for a normal distribution is given as . Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => nd_mean private pure function nd_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The mean procedure, public :: median => nd_median private pure function nd_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The median. procedure, public :: mode => nd_mode private pure function nd_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => nd_pdf private pure elemental function nd_pdf(this, x) result(rst) Computes the probability density function. The PDF for a normal distribution is given as . Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: standardize => nd_standardize private  subroutine nd_standardize(this) Standardizes the normal distribution to a mean of 0 and a \nstandard deviation of 1. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(inout) :: this The normal_distribution object. procedure, public :: variance => nd_variance private pure function nd_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( normal_distribution ), intent(in) :: this The normal_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\normal_distribution.html"},{"title":"t_distribution – FSTATS ","text":"type, public, extends( distribution ) :: t_distribution Defines Student's T-Distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => td_cdf private pure elemental function td_cdf(this, x) result(rst) Computes the cumulative distribution function. The CDF for Student's T-Distribution is given as where . Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: mean => td_mean private pure function td_mean(this) result(rst) Computes the mean of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The mean. procedure, public :: median => td_median private pure function td_median(this) result(rst) Computes the median of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) procedure, public :: mode => td_mode private pure function td_mode(this) result(rst) Computes the mode of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The mode. procedure, public :: pdf => td_pdf private pure elemental function td_pdf(this, x) result(rst) Computes the probability density function. The PDF for Student's T-Distribution is given as . Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. procedure, public :: variance => td_variance private pure function td_variance(this) result(rst) Computes the variance of the distribution. Arguments Type Intent Optional Attributes Name class( t_distribution ), intent(in) :: this The t_distribution object. Return Value real(kind=real64) The variance.","tags":"","loc":"type\\t_distribution.html"},{"title":"adjusted_r_squared – FSTATS","text":"public  function adjusted_r_squared(p, x, xm, err) result(rst) Computes the adjusted R-squared value for a data set. The adjusted R-squared provides a mechanism for tempering the effects\nof extra explanatory variables on the traditional R-squared \ncalculation.  It is computed by noting the sample size and \nthe number of variables . . See Also: Wikipedia Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: p The number of variables. real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\adjusted_r_squared.html"},{"title":"calculate_regression_statistics – FSTATS","text":"public  function calculate_regression_statistics(resid, params, c, alpha, err) result(rst) Computes statistics for the quality of fit for a regression \nmodel. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: resid (:) An M-element array containing the model residual errors. real(kind=real64), intent(in) :: params (:) An N-element array containing the model parameters. real(kind=real64), intent(in) :: c (:,:) The N-by-N covariance matrix. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value type( regression_statistics ), allocatable, (:) A regression_statistics object containing the analysis results.","tags":"","loc":"proc\\calculate_regression_statistics.html"},{"title":"correlation – FSTATS","text":"public pure function correlation(x, y) result(rst) Computes the sample correlation coefficient (an estimate to the \npopulation Pearson correlation) as follows. . Where, & are the sample standard deviations of\nx and y respectively. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The correlation coefficient.","tags":"","loc":"proc\\correlation.html"},{"title":"r_squared – FSTATS","text":"public  function r_squared(x, xm, err) result(rst) Computes the R-squared value for a data set. The R-squared value is computed by determining the sum of the squares\nof the residuals: The total sum of the squares: . \nThe R-squared value is then: . See Also: Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\r_squared.html"},{"title":"covariance_matrix – FSTATS","text":"public  subroutine covariance_matrix(x, c, err) Computes the covariance matrix where and is computed\nby design_matrix. See Also Wikipedia Wikipedia - Regression Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the formatted independent data\n matrix as computed by design_matrix. real(kind=real64), intent(out) :: c (:,:) The N-by-N covariance matrix. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not \n     sized correctly.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\covariance_matrix.html"},{"title":"design_matrix – FSTATS","text":"public  subroutine design_matrix(order, intercept, x, c, err) Computes the design matrix for the linear \nleast-squares regression problem of , where is the matrix computed here, is \nthe vector of coefficients to be determined, and is the \nvector of measured dependent variables. See Also Wikipedia Wikipedia Wikipedia Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be\nat least one (linear equation), but can be higher as desired. logical, intent(in) :: intercept Set to true if the intercept is being computed\nas part of the regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(out) :: c (:,:) An N-by-K matrix where the results will be written.  K\nmust equal order + 1 in the event intercept is true; \nhowever, if intercept is false, K must equal order. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.","tags":"","loc":"proc\\design_matrix.html"},{"title":"jacobian – FSTATS","text":"public  subroutine jacobian(fun, xdata, params, jac, stop, f0, f1, step, err) Computes the Jacobian matrix for a nonlinear regression problem. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: xdata (:) The M-element array containing x-coordinate data. real(kind=real64), intent(in) :: params (:) The N-element array containing the model parameters. real(kind=real64), intent(out) :: jac (:,:) The M-by-N matrix where the Jacobian will be written. logical, intent(out) :: stop A value that the user can set in fun forcing the\nevaluation process to stop prior to completion. real(kind=real64), intent(in), optional, target :: f0 (:) An optional M-element array containing the model values\n using the current parameters as defined in m.  This input \ncan be used to prevent the routine from performing a \nfunction evaluation at the model parameter state defined in \nparams. real(kind=real64), intent(out), optional, target :: f1 (:) An optional M-element workspace array used for function\nevaluations. real(kind=real64), intent(in), optional :: step The differentiation step size.  The default is the square \nroot of machine precision. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\jacobian.html"},{"title":"linear_least_squares – FSTATS","text":"public  subroutine linear_least_squares(order, intercept, x, y, coeffs, ymod, resid, stats, alpha, err) Computes a linear least-squares regression to fit a set of data. See Also Wikipedia SPC Excel Understanding Regression Statistics Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in) :: y (:) An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out) :: coeffs (:) An ORDER+1 element array where the coefficients will be written. real(kind=real64), intent(out) :: ymod (:) An N-element array where the modeled data will be written. real(kind=real64), intent(out) :: resid (:) An N-element array where the residual error data will be \nwritten (modeled - actual). type( regression_statistics ), intent(out), optional :: stats (:) An M-element array of regression_statistics items where \nM = ORDER + 1 when intercept is set to true; however, if \nintercept is set to false, M = ORDER. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\linear_least_squares.html"},{"title":"nonlinear_least_squares – FSTATS","text":"public  subroutine nonlinear_least_squares(fun, x, y, params, ymod, resid, weights, maxp, minp, stats, alpha, controls, settings, info, status, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. procedure( iteration_update ), intent(in), optional, pointer :: status An optional pointer to a routine that can be used to extract\niteration information. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"proc\\nonlinear_least_squares.html"},{"title":"iteration_update – FSTATS","text":"interface public  subroutine iteration_update(iter, funvals, resid, params, step) Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: iter real(kind=real64), intent(in) :: funvals (:) real(kind=real64), intent(in) :: resid (:) real(kind=real64), intent(in) :: params (:) real(kind=real64), intent(in) :: step (:)","tags":"","loc":"interface\\iteration_update.html"},{"title":"regression_function – FSTATS","text":"interface public  subroutine regression_function(xdata, params, resid, stop) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: xdata real(kind=real64), intent(in), dimension(:) :: params real(kind=real64), intent(out), dimension(:) :: resid logical, intent(out) :: stop","tags":"","loc":"interface\\regression_function.html"},{"title":"difference – FSTATS","text":"public pure function difference(x) result(rst) Computes the difference between elements in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array on which to operate. Return Value real(kind=real64), allocatable, dimension(:) The (N-1)-element array containing the differences between adjacent\nelements.","tags":"","loc":"proc\\difference.html"},{"title":"factorial – FSTATS","text":"public pure elemental function factorial(x) result(rst) Computes the factorial of X. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value whose factorial is to be computed. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\factorial.html"},{"title":"rejection_sample – FSTATS","text":"public  function rejection_sample(tdist, n, xmin, xmax) result(rst) Uses rejection sampling to randomly sample a target distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: tdist The distribution to sample integer(kind=int32), intent(in) :: n The number of samples to make. real(kind=real64), intent(in) :: xmin The minimum range to explore. real(kind=real64), intent(in) :: xmax The maximum range to explore. Return Value real(kind=real64), allocatable, dimension(:) An N-element array containing the N samples from the \ndistribution.","tags":"","loc":"proc\\rejection_sample.html"},{"title":"box_muller_sample – FSTATS","text":"public interface box_muller_sample Generates random, normally distributed values via the Box-Muller \ntransform. Module Procedures private  function box_muller_sample_scalar(mu, sigma) result(rst) Generates a pair of independent, standard, normally distributed\nrandom values using the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. Return Value real(kind=real64), (2) The pair of random values. private  function box_muller_array(mu, sigma, n) result(rst) Generates an array of normally distributed random values sampled\nby the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. integer(kind=int32), intent(in) :: n The number of Box-Muller pairs to generate. Return Value real(kind=real64), allocatable, dimension(:) A 2N-element array containing the N Box-Muller pairs.","tags":"","loc":"interface\\box_muller_sample.html"},{"title":"bartletts_test – FSTATS","text":"public  subroutine bartletts_test(x, stat, p) Computes Bartlett's test statistic and associated probability. The statistic is calculated as follows. Where and is the pooled\nvariance. The probability is calculated as the right-tail probability of the\nchi-squared distribution. Bartlett's test is most relevant for distributions showing strong \nnormality.  For distributions lacking strong normality, consider \nLevene's test instead. See Also Wikipedia Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances.","tags":"","loc":"proc\\bartletts_test.html"},{"title":"f_test – FSTATS","text":"public  subroutine f_test(x1, x2, stat, p, dof1, dof2) Computes the F-test and returns the probability (two-tailed) that\nthe variances of two data sets are not significantly different. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The F-statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from the two underlying populations that \nhave the same variance. real(kind=real64), intent(out) :: dof1 A measure of the degrees of freedom. real(kind=real64), intent(out) :: dof2 A measure of the degrees of freedom.","tags":"","loc":"proc\\f_test.html"},{"title":"levenes_test – FSTATS","text":"public  subroutine levenes_test(x, stat, p, err) Computes Levene's test statistic and associated probability. The statistic is calculated as follows. Where: As the test statistic is approximately F-distributed, the F-distribution\nis used to calculate the probability term. See Also Wikipedia Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. class(errors), intent(inout), optional, target :: err","tags":"","loc":"proc\\levenes_test.html"},{"title":"t_test_equal_variance – FSTATS","text":"public  subroutine t_test_equal_variance(x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed equivalent variances. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"proc\\t_test_equal_variance.html"},{"title":"t_test_paired – FSTATS","text":"public  subroutine t_test_paired(x1, x2, stat, p, dof, err) Computes the 2-tailed Student's T-Test for two paired data sets. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An N-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from  two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same \n  length.","tags":"","loc":"proc\\t_test_paired.html"},{"title":"t_test_unequal_variance – FSTATS","text":"public  subroutine t_test_unequal_variance(x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed non-equivalent variances. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"proc\\t_test_unequal_variance.html"},{"title":"confidence_interval – FSTATS","text":"public interface confidence_interval Computes the confidence interval for the specified distribution. See Also Wikipedia Module Procedures private pure function confidence_interval_scalar(dist, alpha, s, n) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: s The sample standard deviation. integer(kind=int32), intent(in) :: n The number of samples in the data set. Return Value real(kind=real64) The result. private pure function confidence_interval_array(dist, alpha, x) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: x (:) An N-element array containing the data to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"interface\\confidence_interval.html"},{"title":"covariance – FSTATS","text":"public pure function covariance(x, y) result(rst) Computes the sample covariance of two data sets. The covariance computed is the sample covariance such that . Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The covariance.","tags":"","loc":"proc\\covariance.html"},{"title":"mean – FSTATS","text":"public pure function mean(x) result(rst) Computes the mean of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\mean.html"},{"title":"median – FSTATS","text":"public  function median(x) result(rst) Computes the median of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout) :: x (:) The array of values to analyze.  On output, this array is sorted into\nascending order. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\median.html"},{"title":"quantile – FSTATS","text":"public pure function quantile(x, q) result(rst) Computes the specified quantile of a data set using the SAS \nMethod 4. See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the data. real(kind=real64), intent(in) :: q The quantile to compute (e.g. 0.25 computes the 25% quantile). Return Value real(kind=real64) The result.","tags":"","loc":"proc\\quantile.html"},{"title":"standard_deviation – FSTATS","text":"public pure function standard_deviation(x) result(rst) Computes the sample standard deviation of the values in an array. The value computed is the sample standard deviation. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result.","tags":"","loc":"proc\\standard_deviation.html"},{"title":"trimmed_mean – FSTATS","text":"public  function trimmed_mean(x, p) result(rst) Computes the trimmed mean of a data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout), dimension(:) :: x An N-element array containing the data.  On output, the\narray is sorted into ascending order. real(kind=real64), intent(in), optional :: p An optional parameter specifying the percentage of values\nfrom either end of the distribution to remove.  The default\nis 0.05 such that the bottom 5% and top 5% are removed. Return Value real(kind=real64) The trimmed mean.","tags":"","loc":"proc\\trimmed_mean.html"},{"title":"variance – FSTATS","text":"public pure function variance(x) result(rst) Computes the sample variance of the values in an array. The variance computed is the sample variance such that . Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64)","tags":"","loc":"proc\\variance.html"},{"title":"pooled_variance – FSTATS","text":"public interface pooled_variance Computes the pooled estimate of variance. Module Procedures private pure function pooled_variance_1(si, ni) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: si An N-element array containing the estimates for each of the N\nvariances. integer(kind=int32), intent(in), dimension(size(si)) :: ni An N-element array containing the number of data points in each\nof the data sets used to compute the variances in si. Return Value real(kind=real64) The pooled variance. private pure function pooled_variance_2(x) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x An array of arrays of data. Return Value real(kind=real64) The pooled variance.","tags":"","loc":"interface\\pooled_variance.html"},{"title":"beta – FSTATS","text":"public pure elemental function beta(a, b) result(rst) Computes the beta function. The beta function is related to the gamma function\nby the following relationship. . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. Return Value real(kind=real64) The value of the beta function at and .","tags":"","loc":"proc\\beta.html"},{"title":"digamma – FSTATS","text":"public pure elemental function digamma(x) result(rst) Computes the digamma function. The digamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\digamma.html"},{"title":"incomplete_beta – FSTATS","text":"public pure elemental function incomplete_beta(a, b, x) result(rst) Computes the incomplete beta function. The incomplete beta function is defind as: . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the incomplete beta function.","tags":"","loc":"proc\\incomplete_beta.html"},{"title":"incomplete_gamma_lower – FSTATS","text":"public pure elemental function incomplete_gamma_lower(a, x) result(rst) Computes the lower incomplete gamma function. The lower incomplete gamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\incomplete_gamma_lower.html"},{"title":"incomplete_gamma_upper – FSTATS","text":"public pure elemental function incomplete_gamma_upper(a, x) result(rst) Computes the upper incomplete gamma function. The upper incomplete gamma function is defined as: See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value.","tags":"","loc":"proc\\incomplete_gamma_upper.html"},{"title":"regularized_beta – FSTATS","text":"public pure elemental function regularized_beta(a, b, x) result(rst) Computes the regularized beta function. The regularized beta function is defined as the ratio between\nthe incomplete beta function and the beta function. . See Also Wikipedia Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the regularized beta function.","tags":"","loc":"proc\\regularized_beta.html"},{"title":"full_factorial – FSTATS","text":"public  subroutine full_factorial(vars, tbl, err) Computes a table with values scaled from 1 to N describing a \nfull-factorial design. program example use iso_fortran_env use fstats implicit none ! Local Variables integer ( int32 ) :: i , vars ( 3 ), tbl ( 24 , 3 ) ! Define the number of design points for each of the 3 factors to study vars = [ 2 , 4 , 3 ] ! Determine the design table call full_factorial ( vars , tbl ) ! Display the table do i = 1 , size ( tbl , 1 ) print * , tbl ( i ,:) end do end program The above program produces the following output. 1           1           1\n1           1           2\n1           1           3\n1           2           1\n1           2           2\n1           2           3\n1           3           1\n1           3           2\n1           3           3\n1           4           1\n1           4           2\n1           4           3\n2           1           1\n2           1           2\n2           1           3\n2           2           1\n2           2           2\n2           2           3\n2           3           1\n2           3           2\n2           3           3\n2           4           1\n2           4           2\n2           4           3 Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain \nthe number of options for that particular factor to explore. \nThis value must be greater than or equal to 1. integer(kind=int32), intent(out) :: tbl (:,:) A table where the design will be written.  Use \nget_full_factorial_matrix_size to determine the appropriate \ntable size. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.\n- FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized.","tags":"","loc":"proc\\full_factorial.html"},{"title":"get_full_factorial_matrix_size – FSTATS","text":"public  subroutine get_full_factorial_matrix_size(vars, m, n, err) Computes the appropriate size for a full-factorial design table. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study.  Each \nof the M entries to the array is expected to contain the \nnumber of options for that particular factor to explore. This value must be greater than or equal to 1. integer(kind=int32), intent(out) :: m The number of rows for the table. integer(kind=int32), intent(out) :: n The number of columns for the table. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.","tags":"","loc":"proc\\get_full_factorial_matrix_size.html"},{"title":"allan_variance – FSTATS","text":"public  function allan_variance(x, dt, err) result(rst) Computes the Allan variance of a data set. Remarks This implementation computes the fully overlapped Allan variance \nusing the method presented by Yadav et. al. Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, \nViraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm \nfor Fully Overlapped Allan Variance and Total Variance for Analysis and \nModeling of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. \n10.1109/LSENS.2018.2829799. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element data set to analyze. real(kind=real64), intent(in), optional :: dt An optional input specifying the time increment between \nsamples in x.  If not specified, this value is set to 1. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value real(kind=real64), allocatable, dimension(:,:) An M-by-2 array containing the results where M is N / 2 - 1\nif N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The \nfirst column contains the averaging times associated with \nthe M results stored in the second column.","tags":"","loc":"proc\\allan_variance.html"},{"title":"bootstrap – FSTATS","text":"public  function bootstrap(stat, x, method, nsamples, alpha) result(rst) Performs a bootstrap calculation on the supplied data set for the given\nstatistic.  The default implementation utlizes a random resampling with \nreplacement.  Other resampling methods may be defined by specifying an \nappropriate routine by means of the method input. Arguments Type Intent Optional Attributes Name procedure( bootstrap_statistic_routine ), intent(in), pointer :: stat The routine used to compute the desired statistic. real(kind=real64), intent(in), dimension(:) :: x The N-element data set. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. integer(kind=int32), intent(in), optional :: nsamples An optional input, that if supplied, specifies the number of \nresampling runs to perform.  The default is 10 000. real(kind=real64), intent(in), optional :: alpha An optional input, that if supplied, defines the significance level\nto use for the analysis.  The default is 0.05. Return Value type( bootstrap_statistics ) The resulting bootstrap_statistics type containing the confidence\nintervals, bias, standard error, etc. for the analyzed statistic.","tags":"","loc":"proc\\bootstrap.html"},{"title":"bootstrap_linear_least_squares – FSTATS","text":"public  subroutine bootstrap_linear_least_squares(order, intercept, x, y, coeffs, ymod, resid, nsamples, stats, bias, alpha, method, bscoeffs, err) Computes a linear least-squares regression to fit a set of data.\nBootstrapping is utilized to gain insight into the quality of \nthe fit.  Resampling for the bootstrap process is a random resampling \nwith replacement process with the range of values limited by the \nstandard deviation of the original data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out), dimension(:) :: coeffs An ORDER+1 element array where the coefficients will\nbe written. real(kind=real64), intent(out), dimension(:) :: ymod An N-element array where the modeled data will be written. real(kind=real64), intent(out), dimension(:) :: resid An N-element array where the residual error data will be \nwritten (modeled - actual). integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. type( bootstrap_regression_statistics ), intent(out), optional, dimension(:) :: stats An M-element array of bootstrap_regression_statistics items \nwhere M = ORDER + 1 when intercept is set to true; however, \nif intercept is set to false, M = ORDER. real(kind=real64), intent(out), optional, dimension(:) :: bias An ORDER+1 element array where an estimate of the bias of\neach coefficient is returned based upon the results of the\nbootstrapping analysis.  The bias is computed as the difference \nbetween the mean of the boostrap population results for the given \nparameter and the original estimate of the given parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.","tags":"","loc":"proc\\bootstrap_linear_least_squares.html"},{"title":"bootstrap_nonlinear_least_squares – FSTATS","text":"public  subroutine bootstrap_nonlinear_least_squares(fun, x, y, params, ymod, resid, nsamples, weights, maxp, minp, stats, alpha, controls, settings, info, bias, method, bscoeffs, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain \ninsight into the quality of the fit.  Resampling for the bootstrap \nprocess is a random resampling with replacement process with the \nrange of values limited by the standard deviation of the original \ndata set. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( bootstrap_regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. real(kind=real64), intent(out), optional, dimension(:) :: bias An optional N-element array that, if supplied, will be used to \nprovide an estimate of the bias of each model parameter based upon\nthe results of the bootstrapping analysis.  The bias is computed as \nthe difference between the mean of the boostrap population results \nfor the given parameter and the original estimate of the given \nparameter. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"proc\\bootstrap_nonlinear_least_squares.html"},{"title":"random_resample – FSTATS","text":"public  subroutine random_resample(x, xn) Random resampling, with replacement, based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"proc\\random_resample.html"},{"title":"scaled_random_resample – FSTATS","text":"public  subroutine scaled_random_resample(x, xn) A random resampling, scaled by the standard deviation of the original\ndata, but based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"proc\\scaled_random_resample.html"},{"title":"bootstrap_resampling_routine – FSTATS","text":"interface public  subroutine bootstrap_resampling_routine(x, xn) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be \nwritten. Description Defines the signature of a subroutine used to compute a \nresampling of data for bootstrapping purposes.","tags":"","loc":"interface\\bootstrap_resampling_routine.html"},{"title":"bootstrap_statistic_routine – FSTATS","text":"interface public  function bootstrap_statistic_routine(x) result(rst) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The array of data to analyze. Return Value real(kind=real64) The resulting statistic. Description Defines the signature of a function for computing the desired\nbootstrap statistic.","tags":"","loc":"interface\\bootstrap_statistic_routine.html"},{"title":"anova – FSTATS","text":"public interface anova Performs an analysis of variance (ANOVA) on the supplied data \nset. The following example illustrates a single-factor ANOVA on a \ndata set. program example use iso_fortran_env use fstats implicit none ! Local Variables character , parameter :: tab = achar ( 9 ) real ( real64 ) :: x ( 10 , 2 ) type ( single_factor_anova_table ) :: tbl ! Define the data x = reshape ( & [ & 3.086d3 , 3.082d3 , 3.069d3 , 3.072d3 , 3.045d3 , 3.070d3 , 3.079d3 , & 3.050d3 , 3.062d3 , 3.062d3 , 3.075d3 , 3.061d3 , 3.063d3 , 3.038d3 , & 3.070d3 , 3.062d3 , 3.070d3 , 3.049d3 , 3.042d3 , 3.063d3 & ], & [ 10 , 2 ] & ) ! Perform the ANOVA tbl = anova ( x ) ! Print out the table print '(A)' , \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" print '(AF2.0AF5.1AF5.1AF5.3AF5.3)' , \"Main Factor: \" // tab , & tbl % main_factor % dof , tab , & tbl % main_factor % sum_of_squares , tab // tab , & tbl % main_factor % variance , tab // tab , & tbl % main_factor % f_statistic , tab , & tbl % main_factor % probability print '(AF3.0AF6.1AF5.1)' , \"Within: \" // tab , & tbl % within_factor % dof , tab , & tbl % within_factor % sum_of_squares , tab // tab , & tbl % within_factor % variance print '(AF3.0AF6.1AF5.1)' , \"Total: \" // tab // tab , & tbl % total_dof , tab , & tbl % total_sum_of_squares , tab // tab , & tbl % total_variance print '(AF6.1)' , \"Overall Mean: \" , tbl % overall_mean end program The above program produces the following output. Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value\nMain Factor:    1.      352.8           352.8           2.147   0.160\nWithin:         18.     2958.2          164.3\nTotal:          19.     3311.0          174.3\nOverall Mean: 3063.5 See Also Wikipedia SPC Excel Single Factor ANOVA SPC Excel Gage R&R SPC Excel Understanding Regression Statistics NIST - Two Way ANOVA Module Procedures private  function anova_1_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the M replications of the N test \npoints of interest. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. private  function anova_2_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:,:) An M-by-N-by-K array containing the M replications of the\nN first factor results, and the K second factor results. Return Value type( two_factor_anova_table ) A two_factor_anova_table instance containing the ANOVA results. private  function anova_model_fit(nmodelparams, ymeas, ymod, err) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: nmodelparams The number of model parameters. real(kind=real64), intent(in) :: ymeas (:) An N-element array containing the measured dependent variable data. real(kind=real64), intent(in) :: ymod (:) An N-element array containing the modeled dependent variable data. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the \n  same length.\n- FS_MEMORY_ERROR: Occurs if a memory error is encountered. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results.","tags":"","loc":"interface\\anova.html"},{"title":"report_array_size_error – FSTATS","text":"public  subroutine report_array_size_error(err, fname, name, expect, actual) Reports an array size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the array. integer(kind=int32), intent(in) :: expect The expected size of the array. integer(kind=int32), intent(in) :: actual The actual size of the array. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_array_size_error.html"},{"title":"report_arrays_not_same_size_error – FSTATS","text":"public  subroutine report_arrays_not_same_size_error(err, fname, name1, name2, size1, size2) Reports an error relating to two arrays not being the same size\nwhen they should be the same size. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name1 The name of the first array. character(len=*), intent(in) :: name2 The name of the second array. integer(kind=int32), intent(in) :: size1 The size of the first array. integer(kind=int32), intent(in) :: size2 The size of the second array. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_arrays_not_same_size_error.html"},{"title":"report_iteration_count_error – FSTATS","text":"public  subroutine report_iteration_count_error(err, fname, msg, mincount) Reports an iteration count error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*) :: fname The name of the routine in which the error occurred. character(len=*) :: msg The error message. integer(kind=int32), intent(in) :: mincount The minimum iteration count expected. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: emsg","tags":"","loc":"proc\\report_iteration_count_error.html"},{"title":"report_matrix_size_error – FSTATS","text":"public  subroutine report_matrix_size_error(err, fname, name, expect_rows, expect_cols, actual_rows, actual_cols) Reports a matrix size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the matrix. integer(kind=int32), intent(in) :: expect_rows The expected number of rows. integer(kind=int32), intent(in) :: expect_cols The expected number of columns. integer(kind=int32), intent(in) :: actual_rows The actual number of rows. integer(kind=int32), intent(in) :: actual_cols The actual number of columns. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_matrix_size_error.html"},{"title":"report_memory_error – FSTATS","text":"public  subroutine report_memory_error(err, fname, code) Reports a memory allocation related error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: code The error code returned by the allocation routine. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_memory_error.html"},{"title":"report_underdefined_error – FSTATS","text":"public  subroutine report_underdefined_error(err, fname, expect, actual) Reports an underdefined problem error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: expect The expected minimum number of equations. integer(kind=int32), intent(in) :: actual The actual number of equations. Variables Type Visibility Attributes Name Initial character(len=MESSAGE_SIZE), public :: msg","tags":"","loc":"proc\\report_underdefined_error.html"},{"title":"distribution_function – FSTATS","text":"interface public pure elemental function distribution_function(this, x) result(rst) Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. Description Defines the interface for a probability distribution function.","tags":"","loc":"interface\\distribution_function.html"},{"title":"distribution_property – FSTATS","text":"interface public pure function distribution_property(this) result(rst) Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. Description Computes the value of a distribution property.","tags":"","loc":"interface\\distribution_property.html"},{"title":"lowess – FSTATS","text":"public  subroutine lowess(x, y, ys, fsmooth, nstps, del, rweights, resid, err) Computes the smoothing of a data set using a robust locally weighted\nscatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at\neach of the supplied x values. Remarks The code is a reimplementation of the LOWESS library.  For a detailed\nunderstanding, see [this]\n(http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf) \npaper by William Cleveland. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable data.  This\narray must be monotonically increasing. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable data. real(kind=real64), intent(out), dimension(:) :: ys An N-element array where the smoothed results will be written. real(kind=real64), intent(in), optional :: fsmooth An optional input that specifies the amount of smoothing. Specifically, this value is the fraction of points used to compute\neach value.  As this value increases, the output becomes smoother.\nChoosing a value in the range of 0.2 to 0.8 typically results in a\ngood fit.  The default value is 0.2. integer(kind=int32), intent(in), optional :: nstps An optional input that specifies the numb of iterations.  If set to\nzero, a non-robust fit is returned.  The default value is set to 2. real(kind=real64), intent(in), optional :: del real(kind=real64), intent(out), optional, dimension(:), target :: rweights An optional N-element array, that if supplied, will be used to\nreturn the weights given to each data point. real(kind=real64), intent(out), optional, dimension(:), target :: resid An optional N-element array, that if supplied, will be used to \nreturn the residual. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation error.","tags":"","loc":"proc\\lowess.html"},{"title":"fstats_regression – FSTATS","text":"Uses linalg blas ferror iso_fortran_env fstats_descriptive_statistics fstats_distributions fstats_special_functions fstats_errors fstats_hypothesis Variables Type Visibility Attributes Name Initial integer(kind=int32), public, parameter :: FS_LEVENBERG_MARQUARDT_UPDATE = 1 integer(kind=int32), public, parameter :: FS_NIELSEN_UPDATE = 3 integer(kind=int32), public, parameter :: FS_QUADRATIC_UPDATE = 2 Interfaces interface public  subroutine iteration_update(iter, funvals, resid, params, step) Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: iter real(kind=real64), intent(in) :: funvals (:) real(kind=real64), intent(in) :: resid (:) real(kind=real64), intent(in) :: params (:) real(kind=real64), intent(in) :: step (:) interface public  subroutine regression_function(xdata, params, resid, stop) Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: xdata real(kind=real64), intent(in), dimension(:) :: params real(kind=real64), intent(out), dimension(:) :: resid logical, intent(out) :: stop Derived Types type, public :: convergence_info Provides information regarding convergence status. Components Type Visibility Attributes Name Initial logical, public :: converge_on_gradient True if convergence on the gradient was achieved; else, false. logical, public :: converge_on_residual_parameter True if convergence on the residual error parameter was achieved; \nelse, false. logical, public :: converge_on_solution_change True if convergence on the change in solution was achieved; else,\nfalse. integer(kind=int32), public :: function_evaluation_count The function evaluation count. real(kind=real64), public :: gradient_value The value of the gradient test parameter. integer(kind=int32), public :: iteration_count The iteration count. logical, public :: reach_function_evaluation_limit True if the solution did not converge in the allowed number of\nfunction evaluations. logical, public :: reach_iteration_limit True if the solution did not converge in the allowed number of \niterations. real(kind=real64), public :: residual_value The value of the residual error parameter. real(kind=real64), public :: solution_change_value The value of the change in solution parameter. logical, public :: user_requested_stop True if the user requested the stop; else, false. type, public :: iteration_controls Provides a collection of iteration control parameters. Components Type Visibility Attributes Name Initial real(kind=real64), public :: change_in_solution_tolerance Defines a tolerance on the change in parameter values. real(kind=real64), public :: gradient_tolerance Defines a tolerance on the gradient of the fitted function. real(kind=real64), public :: iteration_improvement_tolerance Defines a tolerance to ensure adequate improvement on each \niteration. integer(kind=int32), public :: max_function_evaluations Defines the maximum number of function evaluations allowed. integer(kind=int32), public :: max_iteration_between_updates Defines how many iterations can pass before a re-evaluation of \nthe Jacobian matrix is forced. integer(kind=int32), public :: max_iteration_count Defines the maximum number of iterations allowed. real(kind=real64), public :: residual_tolerance Defines a tolerance on the metric associated with the residual \nerror. Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_tolerances type, public :: lm_solver_options Options to control the Levenberg-Marquardt solver. Components Type Visibility Attributes Name Initial real(kind=real64), public :: damping_decrease_factor The factor to use when decreasing the damping parameter. real(kind=real64), public :: damping_increase_factor The factor to use when increasing the damping parameter. real(kind=real64), public :: finite_difference_step_size The step size used for the finite difference calculations of the\nJacobian matrix. integer(kind=int32), public :: method The solver method to utilize.\n- FS_LEVENBERG_MARQUARDT_UPDATE:\n- FS_QUADRATIC_UPDATE:\n- FS_NIELSEN_UDPATE: Type-Bound Procedures procedure, public :: set_to_default => lm_set_default_settings type, public :: regression_statistics A container for regression-related statistical information. Components Type Visibility Attributes Name Initial real(kind=real64), public :: confidence_interval The confidence interval for the parameter at the level \ndetermined by the regression process. Read more… real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. Read more… real(kind=real64), public :: standard_error The standard error for the model coefficient. Read more… real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. Read more… Functions public  function adjusted_r_squared (p, x, xm, err) result(rst) Computes the adjusted R-squared value for a data set. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: p The number of variables. real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result. public  function calculate_regression_statistics (resid, params, c, alpha, err) result(rst) Computes statistics for the quality of fit for a regression \nmodel. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: resid (:) An M-element array containing the model residual errors. real(kind=real64), intent(in) :: params (:) An N-element array containing the model parameters. real(kind=real64), intent(in) :: c (:,:) The N-by-N covariance matrix. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value type( regression_statistics ), allocatable, (:) A regression_statistics object containing the analysis results. public pure function correlation (x, y) result(rst) Computes the sample correlation coefficient (an estimate to the \npopulation Pearson correlation) as follows. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The correlation coefficient. public  function r_squared (x, xm, err) result(rst) Computes the R-squared value for a data set. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the dependent variables from \nthe data set. real(kind=real64), intent(in) :: xm (:) An N-element array containing the corresponding modeled \nvalues. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings\nto the caller.  Possible warning and error codes are as \nfollows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the \n  same size. Return Value real(kind=real64) The result. Subroutines public  subroutine covariance_matrix (x, c, err) Computes the covariance matrix where and is computed\nby design_matrix. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the formatted independent data\n matrix as computed by design_matrix. real(kind=real64), intent(out) :: c (:,:) The N-by-N covariance matrix. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not \n     sized correctly.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine design_matrix (order, intercept, x, c, err) Computes the design matrix for the linear \nleast-squares regression problem of , where is the matrix computed here, is \nthe vector of coefficients to be determined, and is the \nvector of measured dependent variables. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be\nat least one (linear equation), but can be higher as desired. logical, intent(in) :: intercept Set to true if the intercept is being computed\nas part of the regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(out) :: c (:,:) An N-by-K matrix where the results will be written.  K\nmust equal order + 1 in the event intercept is true; \nhowever, if intercept is false, K must equal order. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. public  subroutine jacobian (fun, xdata, params, jac, stop, f0, f1, step, err) Computes the Jacobian matrix for a nonlinear regression problem. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: xdata (:) The M-element array containing x-coordinate data. real(kind=real64), intent(in) :: params (:) The N-element array containing the model parameters. real(kind=real64), intent(out) :: jac (:,:) The M-by-N matrix where the Jacobian will be written. logical, intent(out) :: stop A value that the user can set in fun forcing the\nevaluation process to stop prior to completion. real(kind=real64), intent(in), optional, target :: f0 (:) An optional M-element array containing the model values\n using the current parameters as defined in m.  This input \ncan be used to prevent the routine from performing a \nfunction evaluation at the model parameter state defined in \nparams. real(kind=real64), intent(out), optional, target :: f1 (:) An optional M-element workspace array used for function\nevaluations. real(kind=real64), intent(in), optional :: step The differentiation step size.  The default is the square \nroot of machine precision. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine linear_least_squares (order, intercept, x, y, coeffs, ymod, resid, stats, alpha, err) Computes a linear least-squares regression to fit a set of data. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in) :: x (:) An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in) :: y (:) An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out) :: coeffs (:) An ORDER+1 element array where the coefficients will be written. real(kind=real64), intent(out) :: ymod (:) An N-element array where the modeled data will be written. real(kind=real64), intent(out) :: resid (:) An N-element array where the residual error data will be \nwritten (modeled - actual). type( regression_statistics ), intent(out), optional :: stats (:) An M-element array of regression_statistics items where \nM = ORDER + 1 when intercept is set to true; however, if \nintercept is set to false, M = ORDER. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine nonlinear_least_squares (fun, x, y, params, ymod, resid, weights, maxp, minp, stats, alpha, controls, settings, info, status, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. procedure( iteration_update ), intent(in), optional, pointer :: status An optional pointer to a routine that can be used to extract\niteration information. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed.","tags":"","loc":"module\\fstats_regression.html"},{"title":"fstats – FSTATS","text":"FSTATS is a modern Fortran statistical library containing routines for \ncomputing basic statistical properties, hypothesis testing, regression, \nspecial functions, and experimental design. Uses fstats_smoothing fstats_allan fstats_helper_routines iso_fortran_env fstats_sampling fstats_regression fstats_anova fstats_descriptive_statistics fstats_distributions fstats_bootstrap fstats_special_functions fstats_experimental_design fstats_hypothesis","tags":"","loc":"module\\fstats.html"},{"title":"fstats_types – FSTATS","text":"Uses iso_fortran_env Derived Types type, public :: array_container Provides a container for a real-valued array.  A practical use of\nthis construct is in the construction of jagged arrays. Components Type Visibility Attributes Name Initial real(kind=real64), public, allocatable, dimension(:) :: x The array.","tags":"","loc":"module\\fstats_types.html"},{"title":"fstats_helper_routines – FSTATS","text":"Uses iso_fortran_env Functions public pure function difference (x) result(rst) Computes the difference between elements in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array on which to operate. Return Value real(kind=real64), allocatable, dimension(:) The (N-1)-element array containing the differences between adjacent\nelements. public pure elemental function factorial (x) result(rst) Computes the factorial of X. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value whose factorial is to be computed. Return Value real(kind=real64) The result.","tags":"","loc":"module\\fstats_helper_routines.html"},{"title":"fstats_sampling – FSTATS","text":"Uses fstats_distributions linalg iso_fortran_env Interfaces public        interface box_muller_sample Generates random, normally distributed values via the Box-Muller \ntransform. private  function box_muller_sample_scalar(mu, sigma) result(rst) Generates a pair of independent, standard, normally distributed\nrandom values using the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. Return Value real(kind=real64), (2) The pair of random values. private  function box_muller_array(mu, sigma, n) result(rst) Generates an array of normally distributed random values sampled\nby the Box-Muller transform. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: mu The mean of the distribution. real(kind=real64), intent(in) :: sigma The standard deviation of the distribution. integer(kind=int32), intent(in) :: n The number of Box-Muller pairs to generate. Return Value real(kind=real64), allocatable, dimension(:) A 2N-element array containing the N Box-Muller pairs. Functions public  function rejection_sample (tdist, n, xmin, xmax) result(rst) Uses rejection sampling to randomly sample a target distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: tdist The distribution to sample integer(kind=int32), intent(in) :: n The number of samples to make. real(kind=real64), intent(in) :: xmin The minimum range to explore. real(kind=real64), intent(in) :: xmax The maximum range to explore. Return Value real(kind=real64), allocatable, dimension(:) An N-element array containing the N samples from the \ndistribution.","tags":"","loc":"module\\fstats_sampling.html"},{"title":"fstats_hypothesis – FSTATS","text":"Uses iso_fortran_env fstats_descriptive_statistics fstats_types fstats_distributions fstats_special_functions fstats_errors ieee_arithmetic Interfaces public        interface confidence_interval Computes the confidence interval for the specified distribution. See Also Wikipedia private pure function confidence_interval_scalar(dist, alpha, s, n) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: s The sample standard deviation. integer(kind=int32), intent(in) :: n The number of samples in the data set. Return Value real(kind=real64) The result. private pure function confidence_interval_array(dist, alpha, x) result(rst) Computes the confidence interval for the specified distribution. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: dist The distribution object defining the probability distribution\nto establish the confidence level. real(kind=real64), intent(in) :: alpha The probability value of interest.  For instance, use a value of 0.05\nfor a confidence level of 95%. real(kind=real64), intent(in) :: x (:) An N-element array containing the data to analyze. Return Value real(kind=real64) The result. Subroutines public  subroutine bartletts_test (x, stat, p) Computes Bartlett's test statistic and associated probability. Read more… Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. public  subroutine f_test (x1, x2, stat, p, dof1, dof2) Computes the F-test and returns the probability (two-tailed) that\nthe variances of two data sets are not significantly different. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The F-statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from the two underlying populations that \nhave the same variance. real(kind=real64), intent(out) :: dof1 A measure of the degrees of freedom. real(kind=real64), intent(out) :: dof2 A measure of the degrees of freedom. public  subroutine levenes_test (x, stat, p, err) Computes Levene's test statistic and associated probability. Read more… Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x The arrays of data to analyze. real(kind=real64), intent(out) :: stat The Bartlett's test statistic. real(kind=real64), intent(out) :: p The probability value that the variances of each data set are\nequivalent.  A low p-value, less than some significance level,\nindicates a non-equivalance of variances. class(errors), intent(inout), optional, target :: err public  subroutine t_test_equal_variance (x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed equivalent variances. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. public  subroutine t_test_paired (x1, x2, stat, p, dof, err) Computes the 2-tailed Student's T-Test for two paired data sets. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An N-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from  two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same \n  length. public  subroutine t_test_unequal_variance (x1, x2, stat, p, dof) Computes the 2-tailed Student's T-Test for two data sets of \nassumed non-equivalent variances. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x1 (:) An N-element array containing the first data set. real(kind=real64), intent(in) :: x2 (:) An M-element array containing the second data set. real(kind=real64), intent(out) :: stat The Student-'s T-Test statistic. real(kind=real64), intent(out) :: p The probability value that the two samples are likely to\nhave come from two underlying populations that \nhave the same mean. real(kind=real64), intent(out) :: dof The degrees of freedom.","tags":"","loc":"module\\fstats_hypothesis.html"},{"title":"fstats_descriptive_statistics – FSTATS","text":"Uses linalg ferror iso_fortran_env fstats_types fstats_errors Interfaces public        interface pooled_variance Computes the pooled estimate of variance. private pure function pooled_variance_1(si, ni) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: si An N-element array containing the estimates for each of the N\nvariances. integer(kind=int32), intent(in), dimension(size(si)) :: ni An N-element array containing the number of data points in each\nof the data sets used to compute the variances in si. Return Value real(kind=real64) The pooled variance. private pure function pooled_variance_2(x) result(rst) Computes the pooled estimate of variance. Arguments Type Intent Optional Attributes Name type( array_container ), intent(in), dimension(:) :: x An array of arrays of data. Return Value real(kind=real64) The pooled variance. Functions public pure function covariance (x, y) result(rst) Computes the sample covariance of two data sets. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The first N-element data set. real(kind=real64), intent(in), dimension(size(x)) :: y The second N-element data set. Return Value real(kind=real64) The covariance. public pure function mean (x) result(rst) Computes the mean of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result. public  function median (x) result(rst) Computes the median of the values in an array. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout) :: x (:) The array of values to analyze.  On output, this array is sorted into\nascending order. Return Value real(kind=real64) The result. public pure function quantile (x, q) result(rst) Computes the specified quantile of a data set using the SAS \nMethod 4. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) An N-element array containing the data. real(kind=real64), intent(in) :: q The quantile to compute (e.g. 0.25 computes the 25% quantile). Return Value real(kind=real64) The result. public pure function standard_deviation (x) result(rst) Computes the sample standard deviation of the values in an array. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64) The result. public  function trimmed_mean (x, p) result(rst) Computes the trimmed mean of a data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(inout), dimension(:) :: x An N-element array containing the data.  On output, the\narray is sorted into ascending order. real(kind=real64), intent(in), optional :: p An optional parameter specifying the percentage of values\nfrom either end of the distribution to remove.  The default\nis 0.05 such that the bottom 5% and top 5% are removed. Return Value real(kind=real64) The trimmed mean. public pure function variance (x) result(rst) Computes the sample variance of the values in an array. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:) The array of values to analyze. Return Value real(kind=real64)","tags":"","loc":"module\\fstats_descriptive_statistics.html"},{"title":"fstats_special_functions – FSTATS","text":"Uses ieee_arithmetic iso_fortran_env Functions public pure elemental function beta (a, b) result(rst) Computes the beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. Return Value real(kind=real64) The value of the beta function at and . public pure elemental function digamma (x) result(rst) Computes the digamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function incomplete_beta (a, b, x) result(rst) Computes the incomplete beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the incomplete beta function. public pure elemental function incomplete_gamma_lower (a, x) result(rst) Computes the lower incomplete gamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function incomplete_gamma_upper (a, x) result(rst) Computes the upper incomplete gamma function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The coefficient value. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The function value. public pure elemental function regularized_beta (a, b, x) result(rst) Computes the regularized beta function. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: a The first argument of the function. real(kind=real64), intent(in) :: b The second argument of the function. real(kind=real64), intent(in) :: x The upper limit of the integration. Return Value real(kind=real64) The value of the regularized beta function.","tags":"","loc":"module\\fstats_special_functions.html"},{"title":"fstats_experimental_design – FSTATS","text":"Uses fstats_errors iso_fortran_env Subroutines public  subroutine full_factorial (vars, tbl, err) Computes a table with values scaled from 1 to N describing a \nfull-factorial design. Read more… Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain \nthe number of options for that particular factor to explore. \nThis value must be greater than or equal to 1. integer(kind=int32), intent(out) :: tbl (:,:) A table where the design will be written.  Use \nget_full_factorial_matrix_size to determine the appropriate \ntable size. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.\n- FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized. public  subroutine get_full_factorial_matrix_size (vars, m, n, err) Computes the appropriate size for a full-factorial design table. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: vars (:) An M-element array containing the M factors to study.  Each \nof the M entries to the array is expected to contain the \nnumber of options for that particular factor to explore. This value must be greater than or equal to 1. integer(kind=int32), intent(out) :: m The number of rows for the table. integer(kind=int32), intent(out) :: n The number of columns for the table. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_INVALID_INPUT_ERROR: Occurs if any items in vars are \n     less than 1.","tags":"","loc":"module\\fstats_experimental_design.html"},{"title":"fstats_allan – FSTATS","text":"Uses fstats_errors iso_fortran_env Functions public  function allan_variance (x, dt, err) result(rst) Computes the Allan variance of a data set. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element data set to analyze. real(kind=real64), intent(in), optional :: dt An optional input specifying the time increment between \nsamples in x.  If not specified, this value is set to 1. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. Return Value real(kind=real64), allocatable, dimension(:,:) An M-by-2 array containing the results where M is N / 2 - 1\nif N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The \nfirst column contains the averaging times associated with \nthe M results stored in the second column.","tags":"","loc":"module\\fstats_allan.html"},{"title":"fstats_bootstrap – FSTATS","text":"Uses linalg iso_fortran_env fstats_regression fstats_descriptive_statistics fstats_special_functions fstats_distributions omp_lib fstats_errors Interfaces interface public  subroutine bootstrap_resampling_routine(x, xn) Defines the signature of a subroutine used to compute a \nresampling of data for bootstrapping purposes. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be \nwritten. interface public  function bootstrap_statistic_routine(x) result(rst) Defines the signature of a function for computing the desired\nbootstrap statistic. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The array of data to analyze. Return Value real(kind=real64) The resulting statistic. Derived Types type, public :: bootstrap_regression_statistics A container for regression-related statistical information as \ncomputed in a bootstrap, or equivalent, calculation. Components Type Visibility Attributes Name Initial real(kind=real64), public :: lower_confidence_interval The lower limit of the confidence interval for the parameter. real(kind=real64), public :: probability The probability that the coefficient is not statistically \nimportant.  A statistically important coefficient will have a \nlow probability (p-value), typically 0.05 or lower; however, a \np-value of up to ~0.2 may be acceptable dependent upon the \nproblem.  Typically any p-value larger than ~0.2 indicates the \nparameter is not statistically important for the model. Read more… real(kind=real64), public :: standard_error The standard error for the model coefficient. real(kind=real64), public :: t_statistic The T-statistic for the model coefficient. Read more… real(kind=real64), public :: upper_confidence_interval The upper limit of the confidence interval for the parameter. type, public :: bootstrap_statistics A collection of statistics resulting from the bootstrap process. Components Type Visibility Attributes Name Initial real(kind=real64), public :: bias The bias in the statistic. real(kind=real64), public :: lower_confidence_interval The lower confidence limit on the statistic. real(kind=real64), public, allocatable, dimension(:) :: population An array of the population values generated by the bootstrap\nprocess. real(kind=real64), public :: standard_error The standard error of the statistic. real(kind=real64), public :: statistic_value The value of the statistic of interest. real(kind=real64), public :: upper_confidence_interval The upper confidence limit on the statistic. Functions public  function bootstrap (stat, x, method, nsamples, alpha) result(rst) Performs a bootstrap calculation on the supplied data set for the given\nstatistic.  The default implementation utlizes a random resampling with \nreplacement.  Other resampling methods may be defined by specifying an \nappropriate routine by means of the method input. Arguments Type Intent Optional Attributes Name procedure( bootstrap_statistic_routine ), intent(in), pointer :: stat The routine used to compute the desired statistic. real(kind=real64), intent(in), dimension(:) :: x The N-element data set. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. integer(kind=int32), intent(in), optional :: nsamples An optional input, that if supplied, specifies the number of \nresampling runs to perform.  The default is 10 000. real(kind=real64), intent(in), optional :: alpha An optional input, that if supplied, defines the significance level\nto use for the analysis.  The default is 0.05. Return Value type( bootstrap_statistics ) The resulting bootstrap_statistics type containing the confidence\nintervals, bias, standard error, etc. for the analyzed statistic. Subroutines public  subroutine bootstrap_linear_least_squares (order, intercept, x, y, coeffs, ymod, resid, nsamples, stats, bias, alpha, method, bscoeffs, err) Computes a linear least-squares regression to fit a set of data.\nBootstrapping is utilized to gain insight into the quality of \nthe fit.  Resampling for the bootstrap process is a random resampling \nwith replacement process with the range of values limited by the \nstandard deviation of the original data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: order The order of the equation to fit.  This value must be at \nleast one (linear equation), but can be higher as desired, \nas long as there is sufficient data. logical, intent(in) :: intercept Set to true if the intercept is being computed as part of \nthe regression; else, false. real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable\nmeasurement points. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable\nmeasurement points. real(kind=real64), intent(out), dimension(:) :: coeffs An ORDER+1 element array where the coefficients will\nbe written. real(kind=real64), intent(out), dimension(:) :: ymod An N-element array where the modeled data will be written. real(kind=real64), intent(out), dimension(:) :: resid An N-element array where the residual error data will be \nwritten (modeled - actual). integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. type( bootstrap_regression_statistics ), intent(out), optional, dimension(:) :: stats An M-element array of bootstrap_regression_statistics items \nwhere M = ORDER + 1 when intercept is set to true; however, \nif intercept is set to false, M = ORDER. real(kind=real64), intent(out), optional, dimension(:) :: bias An ORDER+1 element array where an estimate of the bias of\neach coefficient is returned based upon the results of the\nbootstrapping analysis.  The bias is computed as the difference \nbetween the mean of the boostrap population results for the given \nparameter and the original estimate of the given parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_INVALID_INPUT_ERROR: Occurs if order is less than 1.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error. public  subroutine bootstrap_nonlinear_least_squares (fun, x, y, params, ymod, resid, nsamples, weights, maxp, minp, stats, alpha, controls, settings, info, bias, method, bscoeffs, err) Performs a nonlinear regression to fit a model using a version\nof the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain \ninsight into the quality of the fit.  Resampling for the bootstrap \nprocess is a random resampling with replacement process with the \nrange of values limited by the standard deviation of the original \ndata set. Arguments Type Intent Optional Attributes Name procedure( regression_function ), intent(in), pointer :: fun A pointer to the regression_function to evaluate. real(kind=real64), intent(in) :: x (:) The M-element array containing independent data. real(kind=real64), intent(in) :: y (:) The M-element array containing dependent data. real(kind=real64), intent(inout) :: params (:) On input, the N-element array containing the initial estimate\nof the model parameters.  On output, the computed model \nparameters. real(kind=real64), intent(out) :: ymod (:) An M-element array where the modeled dependent data will\nbe written. real(kind=real64), intent(out) :: resid (:) An M-element array where the model residuals will be\nwritten. integer(kind=int32), intent(in), optional :: nsamples The number of bootstrapping samples to utilize. real(kind=real64), intent(in), optional, target :: weights (:) An optional M-element array allowing the weighting of\nindividual points. real(kind=real64), intent(in), optional, target :: maxp (:) An optional N-element array that can be used as upper limits \non the parameter values.  If no upper limit is requested for\na particular parameter, utilize a very large value.  The \ninternal default is to utilize huge() as a value. real(kind=real64), intent(in), optional, target :: minp (:) An optional N-element array that can be used as lower limits \non the parameter values.  If no lower limit is requested for\na particalar parameter, utilize a very large magnitude, but \nnegative, value.  The internal default is to utilize -huge() \nas a value. type( bootstrap_regression_statistics ), intent(out), optional :: stats (:) An optional N-element array that, if supplied, will be used \nto return statistics about the fit for each parameter. real(kind=real64), intent(in), optional :: alpha The significance level at which to evaluate the confidence \nintervals.  The default value is 0.05 such that a 95% \nconfidence interval is calculated. type( iteration_controls ), intent(in), optional :: controls An optional input providing custom iteration controls. type( lm_solver_options ), intent(in), optional :: settings An optional input providing custom settings for the solver. type( convergence_info ), intent(out), optional, target :: info An optional output that can be used to gain information about\nthe iterative solution and the nature of the convergence. real(kind=real64), intent(out), optional, dimension(:) :: bias An optional N-element array that, if supplied, will be used to \nprovide an estimate of the bias of each model parameter based upon\nthe results of the bootstrapping analysis.  The bias is computed as \nthe difference between the mean of the boostrap population results \nfor the given parameter and the original estimate of the given \nparameter. procedure( bootstrap_resampling_routine ), intent(in), optional, pointer :: method An optional pointer to the method to use for resampling of the data.\nIf no method is supplied, a random resampling is utilized. real(kind=real64), intent(out), optional, allocatable, target, dimension(:,:) :: bscoeffs An optional, allocatable matrix, containing the bootstrap \ndistributions for each parameter stored in each row of the matrix\nsuch that the resulting matrix is NCOEFFS -by- NSAMPLES. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     properly sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation \n     error.\n- FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed \n     is underdetetermined (M < N).\n- FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied \n     tolerances are too small to be practical.\n- FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations \n     are allowed. public  subroutine random_resample (x, xn) Random resampling, with replacement, based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written. public  subroutine scaled_random_resample (x, xn) A random resampling, scaled by the standard deviation of the original\ndata, but based upon a normal distribution. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x The N-element array to resample. real(kind=real64), intent(out), dimension(size(x)) :: xn An N-element array where the resampled data set will be written.","tags":"","loc":"module\\fstats_bootstrap.html"},{"title":"fstats_anova – FSTATS","text":"Uses ferror iso_fortran_env fstats_descriptive_statistics fstats_special_functions fstats_errors ieee_arithmetic Interfaces public        interface anova Performs an analysis of variance (ANOVA) on the supplied data \nset. The following example illustrates a single-factor ANOVA on a \ndata set. program example use iso_fortran_env use fstats implicit none ! Local Variables character , parameter :: tab = achar ( 9 ) real ( real64 ) :: x ( 10 , 2 ) type ( single_factor_anova_table ) :: tbl ! Define the data x = reshape ( & [ & 3.086d3 , 3.082d3 , 3.069d3 , 3.072d3 , 3.045d3 , 3.070d3 , 3.079d3 , & 3.050d3 , 3.062d3 , 3.062d3 , 3.075d3 , 3.061d3 , 3.063d3 , 3.038d3 , & 3.070d3 , 3.062d3 , 3.070d3 , 3.049d3 , 3.042d3 , 3.063d3 & ], & [ 10 , 2 ] & ) ! Perform the ANOVA tbl = anova ( x ) ! Print out the table print '(A)' , \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" print '(AF2.0AF5.1AF5.1AF5.3AF5.3)' , \"Main Factor: \" // tab , & tbl % main_factor % dof , tab , & tbl % main_factor % sum_of_squares , tab // tab , & tbl % main_factor % variance , tab // tab , & tbl % main_factor % f_statistic , tab , & tbl % main_factor % probability print '(AF3.0AF6.1AF5.1)' , \"Within: \" // tab , & tbl % within_factor % dof , tab , & tbl % within_factor % sum_of_squares , tab // tab , & tbl % within_factor % variance print '(AF3.0AF6.1AF5.1)' , \"Total: \" // tab // tab , & tbl % total_dof , tab , & tbl % total_sum_of_squares , tab // tab , & tbl % total_variance print '(AF6.1)' , \"Overall Mean: \" , tbl % overall_mean end program The above program produces the following output. Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value\nMain Factor:    1.      352.8           352.8           2.147   0.160\nWithin:         18.     2958.2          164.3\nTotal:          19.     3311.0          174.3\nOverall Mean: 3063.5 See Also Wikipedia SPC Excel Single Factor ANOVA SPC Excel Gage R&R SPC Excel Understanding Regression Statistics NIST - Two Way ANOVA private  function anova_1_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:) An M-by-N matrix containing the M replications of the N test \npoints of interest. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. private  function anova_2_factor(x) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in) :: x (:,:,:) An M-by-N-by-K array containing the M replications of the\nN first factor results, and the K second factor results. Return Value type( two_factor_anova_table ) A two_factor_anova_table instance containing the ANOVA results. private  function anova_model_fit(nmodelparams, ymeas, ymod, err) result(rst) Performs an analysis of variance (ANOVA) on the supplied data set. Arguments Type Intent Optional Attributes Name integer(kind=int32), intent(in) :: nmodelparams The number of model parameters. real(kind=real64), intent(in) :: ymeas (:) An N-element array containing the measured dependent variable data. real(kind=real64), intent(in) :: ymod (:) An N-element array containing the modeled dependent variable data. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the \n  same length.\n- FS_MEMORY_ERROR: Occurs if a memory error is encountered. Return Value type( single_factor_anova_table ) A single_factor_anova_table instance containing the ANOVA results. Derived Types type, public :: anova_factor Defines an ANOVA factor result. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedome. real(kind=real64), public :: f_statistic The F-statistic. real(kind=real64), public :: probability The variance probability term. real(kind=real64), public :: sum_of_squares The sum of the squares. real(kind=real64), public :: variance The estimate of variance. type, public :: single_factor_anova_table Defines a single-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: main_factor The main, or main factor, results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within-treatement (error) results. type, public :: two_factor_anova_table Defines a two-factor ANOVA results table. Components Type Visibility Attributes Name Initial type( anova_factor ), public :: interaction The interaction effects. type( anova_factor ), public :: main_factor_1 The first main-factor results. type( anova_factor ), public :: main_factor_2 The second main-factor results. real(kind=real64), public :: overall_mean The overall mean value. real(kind=real64), public :: total_dof The total number of degrees of freedom. real(kind=real64), public :: total_sum_of_squares The total sum of squares. real(kind=real64), public :: total_variance The total variance estimate. type( anova_factor ), public :: within_factor The within (error) factor results.","tags":"","loc":"module\\fstats_anova.html"},{"title":"fstats_errors – FSTATS","text":"Uses ferror iso_fortran_env Variables Type Visibility Attributes Name Initial integer(kind=int32), public, parameter :: FS_ARRAY_SIZE_ERROR = 10000 integer(kind=int32), public, parameter :: FS_INVALID_INPUT_ERROR = 10002 integer(kind=int32), public, parameter :: FS_MATRIX_SIZE_ERROR = 10001 integer(kind=int32), public, parameter :: FS_MEMORY_ERROR = 10003 integer(kind=int32), public, parameter :: FS_NO_ERROR = 0 integer(kind=int32), public, parameter :: FS_TOLERANCE_TOO_SMALL_ERROR = 10005 integer(kind=int32), public, parameter :: FS_TOO_FEW_ITERATION_ERROR = 10006 integer(kind=int32), public, parameter :: FS_UNDERDEFINED_PROBLEM_ERROR = 10004 Subroutines public  subroutine report_array_size_error (err, fname, name, expect, actual) Reports an array size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the array. integer(kind=int32), intent(in) :: expect The expected size of the array. integer(kind=int32), intent(in) :: actual The actual size of the array. public  subroutine report_arrays_not_same_size_error (err, fname, name1, name2, size1, size2) Reports an error relating to two arrays not being the same size\nwhen they should be the same size. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name1 The name of the first array. character(len=*), intent(in) :: name2 The name of the second array. integer(kind=int32), intent(in) :: size1 The size of the first array. integer(kind=int32), intent(in) :: size2 The size of the second array. public  subroutine report_iteration_count_error (err, fname, msg, mincount) Reports an iteration count error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*) :: fname The name of the routine in which the error occurred. character(len=*) :: msg The error message. integer(kind=int32), intent(in) :: mincount The minimum iteration count expected. public  subroutine report_matrix_size_error (err, fname, name, expect_rows, expect_cols, actual_rows, actual_cols) Reports a matrix size error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. character(len=*), intent(in) :: name The name of the matrix. integer(kind=int32), intent(in) :: expect_rows The expected number of rows. integer(kind=int32), intent(in) :: expect_cols The expected number of columns. integer(kind=int32), intent(in) :: actual_rows The actual number of rows. integer(kind=int32), intent(in) :: actual_cols The actual number of columns. public  subroutine report_memory_error (err, fname, code) Reports a memory allocation related error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: code The error code returned by the allocation routine. public  subroutine report_underdefined_error (err, fname, expect, actual) Reports an underdefined problem error. Arguments Type Intent Optional Attributes Name class(errors), intent(inout) :: err The error handling object. character(len=*), intent(in) :: fname The name of the routine in which the error occurred. integer(kind=int32), intent(in) :: expect The expected minimum number of equations. integer(kind=int32), intent(in) :: actual The actual number of equations.","tags":"","loc":"module\\fstats_errors.html"},{"title":"fstats_distributions – FSTATS","text":"Uses fstats_special_functions fstats_helper_routines ieee_arithmetic iso_fortran_env Interfaces interface public pure elemental function distribution_function(this, x) result(rst) Defines the interface for a probability distribution function. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. real(kind=real64), intent(in) :: x The value at which to evaluate the function. Return Value real(kind=real64) The value of the function. interface public pure function distribution_property(this) result(rst) Computes the value of a distribution property. Arguments Type Intent Optional Attributes Name class( distribution ), intent(in) :: this The distribution object. Return Value real(kind=real64) The property value. Derived Types type, public, extends( distribution ) :: binomial_distribution Defines a binomial distribution.  The binomial distribution describes\nthe probability p of getting k successes in n independent trials. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: n The number of independent trials. real(kind=real64), public :: p The success probability for each trial.  This parameter must\nexist on the set [0, 1]. Type-Bound Procedures procedure, public :: cdf => bd_cdf procedure, public :: mean => bd_mean procedure, public :: median => bd_median procedure, public :: mode => bd_mode procedure, public :: pdf => bd_pdf procedure, public :: variance => bd_variance type, public, extends( distribution ) :: chi_squared_distribution Defines a Chi-squared distribution. Components Type Visibility Attributes Name Initial integer(kind=int32), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => cs_cdf procedure, public :: mean => cs_mean procedure, public :: median => cs_median procedure, public :: mode => cs_mode procedure, public :: pdf => cs_pdf procedure, public :: variance => cs_variance type, public :: distribution Defines a probability distribution. Type-Bound Procedures procedure( distribution_function ), public, deferred, pass :: cdf ..\\..\\ Computes the cumulative distribution function.<\\p> procedure( distribution_property ), public, deferred, pass :: mean ..\\..\\ Computes the mean of the distribution.<\\p> procedure( distribution_property ), public, deferred, pass :: median ..\\..\\ Computes the median of the distribution.<\\p> procedure( distribution_property ), public, deferred, pass :: mode ..\\..\\ Computes the mode of the distribution.<\\p> procedure( distribution_function ), public, deferred, pass :: pdf ..\\..\\ Computes the probability density function.<\\p> procedure( distribution_property ), public, deferred, pass :: variance ..\\..\\ Computes the variance of the distribution.<\\p> type, public, extends( distribution ) :: f_distribution Defines an F-distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: d1 The measure of degrees of freedom for the first data set. real(kind=real64), public :: d2 The measure of degrees of freedom for the second data set. Type-Bound Procedures procedure, public :: cdf => fd_cdf procedure, public :: mean => fd_mean procedure, public :: median => fd_median procedure, public :: mode => fd_mode procedure, public :: pdf => fd_pdf procedure, public :: variance => fd_variance type, public, extends( distribution ) :: normal_distribution Defines a normal distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: mean_value The mean value of the distribution. real(kind=real64), public :: standard_deviation The standard deviation of the distribution. Type-Bound Procedures procedure, public :: cdf => nd_cdf procedure, public :: mean => nd_mean procedure, public :: median => nd_median procedure, public :: mode => nd_mode procedure, public :: pdf => nd_pdf procedure, public :: standardize => nd_standardize procedure, public :: variance => nd_variance type, public, extends( distribution ) :: t_distribution Defines Student's T-Distribution. Components Type Visibility Attributes Name Initial real(kind=real64), public :: dof The number of degrees of freedom. Type-Bound Procedures procedure, public :: cdf => td_cdf procedure, public :: mean => td_mean procedure, public :: median => td_median procedure, public :: mode => td_mode procedure, public :: pdf => td_pdf procedure, public :: variance => td_variance","tags":"","loc":"module\\fstats_distributions.html"},{"title":"fstats_smoothing – FSTATS","text":"Uses linalg ferror fstats_errors iso_fortran_env Subroutines public  subroutine lowess (x, y, ys, fsmooth, nstps, del, rweights, resid, err) Computes the smoothing of a data set using a robust locally weighted\nscatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at\neach of the supplied x values. Read more… Arguments Type Intent Optional Attributes Name real(kind=real64), intent(in), dimension(:) :: x An N-element array containing the independent variable data.  This\narray must be monotonically increasing. real(kind=real64), intent(in), dimension(:) :: y An N-element array containing the dependent variable data. real(kind=real64), intent(out), dimension(:) :: ys An N-element array where the smoothed results will be written. real(kind=real64), intent(in), optional :: fsmooth An optional input that specifies the amount of smoothing. Specifically, this value is the fraction of points used to compute\neach value.  As this value increases, the output becomes smoother.\nChoosing a value in the range of 0.2 to 0.8 typically results in a\ngood fit.  The default value is 0.2. integer(kind=int32), intent(in), optional :: nstps An optional input that specifies the numb of iterations.  If set to\nzero, a non-robust fit is returned.  The default value is set to 2. real(kind=real64), intent(in), optional :: del real(kind=real64), intent(out), optional, dimension(:), target :: rweights An optional N-element array, that if supplied, will be used to\nreturn the weights given to each data point. real(kind=real64), intent(out), optional, dimension(:), target :: resid An optional N-element array, that if supplied, will be used to \nreturn the residual. class(errors), intent(inout), optional, target :: err A mechanism for communicating errors and warnings to the \ncaller.  Possible warning and error codes are as follows.\n- FS_NO_ERROR: No errors encountered.\n- FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not \n     approriately sized.\n- FS_MEMORY_ERROR: Occurs if there is a memory allocation error.","tags":"","loc":"module\\fstats_smoothing.html"},{"title":"fstats_regression.f90 – FSTATS","text":"Source Code module fstats_regression use iso_fortran_env use linalg use fstats_errors use blas use ferror use fstats_descriptive_statistics use fstats_distributions use fstats_special_functions use fstats_hypothesis implicit none private public :: iteration_controls public :: convergence_info public :: lm_solver_options public :: regression_function public :: iteration_update public :: regression_statistics public :: r_squared public :: adjusted_r_squared public :: correlation public :: design_matrix public :: covariance_matrix public :: linear_least_squares public :: calculate_regression_statistics public :: jacobian public :: nonlinear_least_squares public :: FS_LEVENBERG_MARQUARDT_UPDATE public :: FS_QUADRATIC_UPDATE public :: FS_NIELSEN_UPDATE ! ****************************************************************************** ! CONSTANTS ! ------------------------------------------------------------------------------ integer ( int32 ), parameter :: FS_LEVENBERG_MARQUARDT_UPDATE = 1 integer ( int32 ), parameter :: FS_QUADRATIC_UPDATE = 2 integer ( int32 ), parameter :: FS_NIELSEN_UPDATE = 3 ! ****************************************************************************** ! TYPES ! ------------------------------------------------------------------------------ type regression_statistics !! A container for regression-related statistical information. real ( real64 ) :: standard_error !! The standard error for the model coefficient. !! !!  E_{s}(\\beta_{i}) = \\sqrt{\\sigma&#94;{2} C_{ii}}  real ( real64 ) :: t_statistic !! The T-statistic for the model coefficient. !! !!  t_o = \\frac{ \\beta_{i} }{E_{s}(\\beta_{i})}  real ( real64 ) :: probability !! The probability that the coefficient is not statistically !! important.  A statistically important coefficient will have a !! low probability (p-value), typically 0.05 or lower; however, a !! p-value of up to ~0.2 may be acceptable dependent upon the !! problem.  Typically any p-value larger than ~0.2 indicates the !! parameter is not statistically important for the model. !! !!  p = t_{|t_o|, df_{residual}}  real ( real64 ) :: confidence_interval !! The confidence interval for the parameter at the level !! determined by the regression process. !! !!  c = t_{\\alpha, df} E_{s}(\\beta_{i})  end type type iteration_controls !! Provides a collection of iteration control parameters. integer ( int32 ) :: max_iteration_count !! Defines the maximum number of iterations allowed. integer ( int32 ) :: max_function_evaluations !! Defines the maximum number of function evaluations allowed. real ( real64 ) :: gradient_tolerance !! Defines a tolerance on the gradient of the fitted function. real ( real64 ) :: change_in_solution_tolerance !! Defines a tolerance on the change in parameter values. real ( real64 ) :: residual_tolerance !! Defines a tolerance on the metric associated with the residual !! error. real ( real64 ) :: iteration_improvement_tolerance !! Defines a tolerance to ensure adequate improvement on each !! iteration. integer ( int32 ) :: max_iteration_between_updates !! Defines how many iterations can pass before a re-evaluation of !! the Jacobian matrix is forced. contains procedure , public :: set_to_default => lm_set_default_tolerances end type type convergence_info !! Provides information regarding convergence status. logical :: converge_on_gradient !! True if convergence on the gradient was achieved; else, false. real ( real64 ) :: gradient_value !! The value of the gradient test parameter. logical :: converge_on_solution_change !! True if convergence on the change in solution was achieved; else, !! false. real ( real64 ) :: solution_change_value !! The value of the change in solution parameter. logical :: converge_on_residual_parameter !! True if convergence on the residual error parameter was achieved; !! else, false. real ( real64 ) :: residual_value !! The value of the residual error parameter. logical :: reach_iteration_limit !! True if the solution did not converge in the allowed number of !! iterations. integer ( int32 ) :: iteration_count !! The iteration count. logical :: reach_function_evaluation_limit !! True if the solution did not converge in the allowed number of !! function evaluations. integer ( int32 ) :: function_evaluation_count !! The function evaluation count. logical :: user_requested_stop !! True if the user requested the stop; else, false. end type type lm_solver_options !! Options to control the Levenberg-Marquardt solver. integer ( int32 ) :: method !! The solver method to utilize. !! - FS_LEVENBERG_MARQUARDT_UPDATE: !! - FS_QUADRATIC_UPDATE: !! - FS_NIELSEN_UDPATE: real ( real64 ) :: finite_difference_step_size !! The step size used for the finite difference calculations of the !! Jacobian matrix. real ( real64 ) :: damping_increase_factor !! The factor to use when increasing the damping parameter. real ( real64 ) :: damping_decrease_factor !! The factor to use when decreasing the damping parameter. contains procedure , public :: set_to_default => lm_set_default_settings end type interface subroutine regression_function ( xdata , params , resid , stop ) use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: xdata , params real ( real64 ), intent ( out ), dimension (:) :: resid logical , intent ( out ) :: stop end subroutine subroutine iteration_update ( iter , funvals , resid , params , step ) use iso_fortran_env , only : int32 , real64 integer ( int32 ), intent ( in ) :: iter real ( real64 ), intent ( in ) :: funvals (:), resid (:), params (:), step (:) end subroutine end interface contains ! ------------------------------------------------------------------------------ function r_squared ( x , xm , err ) result ( rst ) !! Computes the R-squared value for a data set. !! !! The R-squared value is computed by determining the sum of the squares !! of the residuals: !!  SS_{res} = \\Sigma \\left( y_i - f_i \\right)&#94;2  !! The total sum of the squares: !!  SS_{tot} = \\Sigma \\left( y_i - \\bar{y} \\right)&#94;2 . !! The R-squared value is then: !!  R&#94;2 = 1 - \\frac{SS_{res}}{SS_{tot}} . !! !! See Also: !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Coefficient_of_determination) real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the dependent variables from !! the data set. real ( real64 ), intent ( in ) :: xm (:) !! An N-element array containing the corresponding modeled !! values. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings !! to the caller.  Possible warning and error codes are as !! follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the !!   same size. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: esum , vt class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Input Check n = size ( x ) if ( size ( xm ) /= n ) then call report_array_size_error ( errmgr , \"r_squared_real64\" , \"XM\" , n , & size ( xm )) return end if ! Process esum = zero do i = 1 , n esum = esum + ( x ( i ) - xm ( i )) ** 2 end do vt = variance ( x ) * ( n - one ) rst = one - esum / vt end function ! ------------------------------------------------------------------------------ function adjusted_r_squared ( p , x , xm , err ) result ( rst ) !! Computes the adjusted R-squared value for a data set. !! !! The adjusted R-squared provides a mechanism for tempering the effects !! of extra explanatory variables on the traditional R-squared !! calculation.  It is computed by noting the sample size  n  and !! the number of variables  p . !!  \\bar{R}&#94;2 = 1 - \\left( 1 - R&#94;2 \\right) \\frac{n - 1}{n - p} . !! !! See Also: !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Coefficient_of_determination#Adjusted_R2) integer ( int32 ), intent ( in ) :: p !! The number of variables. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the dependent variables from !! the data set. real ( real64 ), intent ( in ) :: xm (:) !! An N-element array containing the corresponding modeled !! values. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings !! to the caller.  Possible warning and error codes are as !! follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x and xm are not the !!   same size. real ( real64 ) :: rst !! The result. ! Local Variables integer ( int32 ) :: n real ( real64 ) :: r2 class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n = size ( x ) ! Process r2 = r_squared ( x , xm , errmgr ) if ( errmgr % has_error_occurred ()) return rst = one - ( one - r2 ) * ( n - one ) / ( n - p - one ) end function ! ------------------------------------------------------------------------------ pure function correlation ( x , y ) result ( rst ) !! Computes the sample correlation coefficient (an estimate to the !! population Pearson correlation) as follows. !! !!  r_{xy} = \\frac{cov(x, y)}{s_{x} s_{y}} . !! !! Where,  s_{x}  &  s_{y}  are the sample standard deviations of !! x and y respectively. real ( real64 ), intent ( in ), dimension (:) :: x !! The first N-element data set. real ( real64 ), intent ( in ), dimension ( size ( x )) :: y !! The second N-element data set. real ( real64 ) :: rst !! The correlation coefficient. ! Process rst = covariance ( x , y ) / ( standard_deviation ( x ) * standard_deviation ( y )) end function ! ------------------------------------------------------------------------------ subroutine design_matrix ( order , intercept , x , c , err ) !! Computes the design matrix  X  for the linear !! least-squares regression problem of  X \\beta = y , where !!  X  is the matrix computed here,  \\beta  is !! the vector of coefficients to be determined, and  y  is the !! vector of measured dependent variables. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Linear_regression) !! - [Wikipedia](https://en.wikipedia.org/wiki/Vandermonde_matrix) !! - [Wikipedia](https://en.wikipedia.org/wiki/Design_matrix) integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be !! at least one (linear equation), but can be higher as desired. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed !! as part of the regression; else, false. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( out ) :: c (:,:) !! An N-by-K matrix where the results will be written.  K !! must equal order + 1 in the event intercept is true; !! however, if intercept is false, K must equal order. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if c is not properly sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , start , npts , ncols class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x ) ncols = order if ( intercept ) ncols = ncols + 1 ! Input Check if ( order < 1 ) then call errmgr % report_error ( \"design_matrix\" , & \"The model order must be at least one.\" , FS_INVALID_INPUT_ERROR ) return end if if ( size ( c , 1 ) /= npts . or . size ( c , 2 ) /= ncols ) then call report_matrix_size_error ( errmgr , \"design_matrix\" , & \"c\" , npts , ncols , size ( c , 1 ), size ( c , 2 )) return end if ! Process if ( intercept ) then c (:, 1 ) = one c (:, 2 ) = x start = 3 else c (:, 1 ) = x start = 2 end if if ( start >= ncols ) return do i = start , ncols c (:, i ) = c (:, i - 1 ) * x end do end subroutine ! ------------------------------------------------------------------------------ subroutine covariance_matrix ( x , c , err ) !! Computes the covariance matrix  C  where !!  C = \\left( X&#94;{T} X \\right)&#94;{-1}  and  X  is computed !! by design_matrix. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Covariance_matrix) !! - [Wikipedia - Regression](https://en.wikipedia.org/wiki/Linear_regression) real ( real64 ), intent ( in ) :: x (:,:) !! An M-by-N matrix containing the formatted independent data !!  matrix  X  as computed by design_matrix. real ( real64 ), intent ( out ) :: c (:,:) !! The N-by-N covariance matrix. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the matrices are not !!      sized correctly. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr integer ( int32 ) :: npts , ncoeffs , flag real ( real64 ), allocatable :: xtx (:,:) ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x , 1 ) ncoeffs = size ( x , 2 ) ! Input Checking if ( size ( c , 1 ) /= ncoeffs . or . size ( c , 2 ) /= ncoeffs ) then call report_matrix_size_error ( errmgr , \"covariance_matrix\" , & \"c\" , ncoeffs , ncoeffs , size ( c , 1 ), size ( c , 2 )) return end if ! Local Memory Allocation allocate ( xtx ( ncoeffs , ncoeffs ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"covariance_matrix\" , flag ) return end if ! Compute X**T * X call DGEMM ( \"T\" , \"N\" , ncoeffs , ncoeffs , npts , one , x , npts , x , npts , & zero , xtx , ncoeffs ) ! Compute the inverse of X**T * X to obtain the covariance matrix call mtx_pinverse ( xtx , c , err = errmgr ) if ( errmgr % has_error_occurred ()) return end subroutine ! ------------------------------------------------------------------------------ subroutine linear_least_squares ( order , intercept , x , y , coeffs , & ymod , resid , stats , alpha , err ) !! Computes a linear least-squares regression to fit a set of data. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Linear_regression) !! - [SPC Excel Understanding Regression Statistics](https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1) integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be at !! least one (linear equation), but can be higher as desired, !! as long as there is sufficient data. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed as part of !! the regression; else, false. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( in ) :: y (:) !! An N-element array containing the dependent variable !! measurement points. real ( real64 ), intent ( out ) :: coeffs (:) !! An ORDER+1 element array where the coefficients will be written. real ( real64 ), intent ( out ) :: ymod (:) !! An N-element array where the modeled data will be written. real ( real64 ), intent ( out ) :: resid (:) !! An N-element array where the residual error data will be !! written (modeled - actual). type ( regression_statistics ), intent ( out ), optional :: stats (:) !! An M-element array of regression_statistics items where !! M = ORDER + 1 when intercept is set to true; however, if !! intercept is set to false, M = ORDER. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , npts , ncols , ncoeffs , flag real ( real64 ) :: alph , var , df , ssr , talpha real ( real64 ), allocatable :: a (:,:), c (:,:), cxt (:,:) type ( t_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if npts = size ( x ) ncoeffs = order + 1 ncols = order if ( intercept ) ncols = ncols + 1 alph = 0.05d0 if ( present ( alpha )) alph = alpha ! Input Check if ( order < 1 ) then call errmgr % report_error ( \"linear_least_squares\" , & \"The model order must be at least one.\" , FS_INVALID_INPUT_ERROR ) return end if if ( size ( y ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"y\" , npts , size ( y )) return end if if ( size ( coeffs ) /= ncoeffs ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"coeffs\" , ncoeffs , size ( coeffs )) return end if if ( size ( ymod ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"ymod\" , npts , size ( ymod )) return end if if ( size ( resid ) /= npts ) then call report_array_size_error ( errmgr , \"linear_least_squares\" , & \"resid\" , npts , size ( resid )) return end if if ( present ( stats )) then if ( size ( stats ) /= ncols ) then call report_array_size_error ( errmgr , & \"linear_least_squares\" , \"stats\" , ncols , size ( stats )) return end if end if ! Memory Allocation allocate ( a ( npts , ncols ), stat = flag ) if ( flag == 0 ) allocate ( c ( ncols , ncols ), stat = flag ) if ( flag == 0 ) allocate ( cxt ( ncols , npts ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"linear_least_squares\" , flag ) return end if ! Compute the coefficient matrix call design_matrix ( order , intercept , x , a , errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the covariance matrix call covariance_matrix ( a , c , errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the coefficients (NCOLS-by-1) call DGEMM ( \"N\" , \"T\" , ncols , npts , ncols , one , c , ncols , a , npts , zero , & cxt , ncols ) ! C * X**T i = 2 coeffs ( 1 ) = zero if ( intercept ) i = 1 call DGEMM ( \"N\" , \"N\" , ncols , 1 , npts , one , cxt , ncols , y , npts , zero , & coeffs ( i :), ncols ) ! (C * X**T) * Y ! Evaluate the model and compute the residuals call DGEMM ( \"N\" , \"N\" , npts , 1 , ncols , one , a , npts , coeffs ( i :), & ncols , zero , ymod , npts ) resid = ymod - y ! If the user doesn't want the statistics calculations we can stop now if (. not . present ( stats )) return ! Start the process of computing statistics stats = calculate_regression_statistics ( resid , coeffs ( i :), c , alph , & errmgr ) end subroutine ! ------------------------------------------------------------------------------ function calculate_regression_statistics ( resid , params , c , alpha , err ) & result ( rst ) !! Computes statistics for the quality of fit for a regression !! model. real ( real64 ), intent ( in ) :: resid (:) !! An M-element array containing the model residual errors. real ( real64 ), intent ( in ) :: params (:) !! An N-element array containing the model parameters. real ( real64 ), intent ( in ) :: c (:,:) !! The N-by-N covariance matrix. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if c is not sized correctly. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. type ( regression_statistics ), allocatable :: rst (:) !! A regression_statistics object containing the analysis results. ! Parameters real ( real64 ), parameter :: p05 = 0.05d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , m , n , dof , flag real ( real64 ) :: a , ssr , var , talpha type ( t_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Initialization m = size ( resid ) n = size ( params ) dof = m - n if ( present ( alpha )) then a = alpha else a = p05 end if allocate ( rst ( n ), stat = flag ) if ( flag /= 0 ) then end if ! Input Checking if ( size ( c , 1 ) /= n . or . size ( c , 2 ) /= n ) then end if ! Process ssr = norm2 ( resid ) ** 2 ! sum of the squares of the residual var = ssr / dof dist % dof = real ( dof , real64 ) talpha = confidence_interval ( dist , a , one , 1 ) do i = 1 , n rst ( i )% standard_error = sqrt ( var * c ( i , i )) rst ( i )% t_statistic = params ( i ) / rst ( i )% standard_error rst ( i )% probability = regularized_beta ( & half * dof , & half , & real ( dof , real64 ) / ( dof + ( rst ( i )% t_statistic ) ** 2 ) & ) rst ( i )% confidence_interval = talpha * rst ( i )% standard_error end do end function ! ------------------------------------------------------------------------------ subroutine jacobian ( fun , xdata , params , & jac , stop , f0 , f1 , step , err ) !! Computes the Jacobian matrix for a nonlinear regression problem. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: xdata (:) !! The M-element array containing x-coordinate data. real ( real64 ), intent ( in ) :: params (:) !! The N-element array containing the model parameters. real ( real64 ), intent ( out ) :: jac (:,:) !! The M-by-N matrix where the Jacobian will be written. logical , intent ( out ) :: stop !! A value that the user can set in fun forcing the !! evaluation process to stop prior to completion. real ( real64 ), intent ( in ), optional , target :: f0 (:) !! An optional M-element array containing the model values !!  using the current parameters as defined in m.  This input !! can be used to prevent the routine from performing a !! function evaluation at the model parameter state defined in !! params. real ( real64 ), intent ( out ), optional , target :: f1 (:) !! An optional M-element workspace array used for function !! evaluations. real ( real64 ), intent ( in ), optional :: step !! The differentiation step size.  The default is the square !! root of machine precision. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Local Variables real ( real64 ) :: h integer ( int32 ) :: m , n , flag , expected , actual real ( real64 ), pointer :: f1p (:), f0p (:) real ( real64 ), allocatable , target :: f1a (:), f0a (:), work (:) class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( step )) then h = step else h = sqrt ( epsilon ( h )) end if m = size ( xdata ) n = size ( params ) ! Input Size Checking if ( size ( jac , 1 ) /= m . or . size ( jac , 2 ) /= n ) then call report_matrix_size_error ( errmgr , \"jacobian\" , & \"JAC\" , m , n , size ( jac , 1 ), size ( jac , 2 )) return end if if ( present ( f0 )) then ! Check Size if ( size ( f0 ) /= m ) then call report_array_size_error ( errmgr , \"jacobian\" , & \"F0\" , m , size ( f0 )) return end if f0p ( 1 : m ) => f0 else ! Allocate space, and fill the array with the current function ! results allocate ( f0a ( m ), stat = flag ) if ( flag /= 0 ) go to 20 f0p ( 1 : m ) => f0a call fun ( xdata , params , f0p , stop ) if ( stop ) return end if if ( present ( f1 )) then ! Check Size if ( size ( f1 ) /= m ) then call report_array_size_error ( errmgr , \"jacobian\" , & \"F1\" , m , size ( f1 )) return end if f1p ( 1 : m ) => f1 else ! Allocate space allocate ( f1a ( m ), stat = flag ) if ( flag /= 0 ) go to 20 f1p ( 1 : m ) => f1a end if ! Allocate a workspace array the same size as params allocate ( work ( n ), stat = flag ) if ( flag /= 0 ) go to 20 ! Compute the Jacobian call jacobian_finite_diff ( fun , xdata , params , f0p , jac , f1p , & stop , h , work ) ! End return ! Memroy Allocation Error Handling 20 continue call report_memory_error ( errmgr , \"jacobian\" , flag ) return end subroutine ! ------------------------------------------------------------------------------ subroutine nonlinear_least_squares ( fun , x , y , params , ymod , & resid , weights , maxp , minp , stats , alpha , controls , settings , info , & status , err ) !! Performs a nonlinear regression to fit a model using a version !! of the Levenberg-Marquardt algorithm. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: x (:) !! The M-element array containing independent data. real ( real64 ), intent ( in ) :: y (:) !! The M-element array containing dependent data. real ( real64 ), intent ( inout ) :: params (:) !! On input, the N-element array containing the initial estimate !! of the model parameters.  On output, the computed model !! parameters. real ( real64 ), intent ( out ) :: ymod (:) !! An M-element array where the modeled dependent data will !! be written. real ( real64 ), intent ( out ) :: resid (:) !! An M-element array where the model residuals will be !! written. real ( real64 ), intent ( in ), optional , target :: weights (:) !! An optional M-element array allowing the weighting of !! individual points. real ( real64 ), intent ( in ), optional , target :: maxp (:) !! An optional N-element array that can be used as upper limits !! on the parameter values.  If no upper limit is requested for !! a particular parameter, utilize a very large value.  The !! internal default is to utilize huge() as a value. real ( real64 ), intent ( in ), optional , target :: minp (:) !! An optional N-element array that can be used as lower limits !! on the parameter values.  If no lower limit is requested for !! a particalar parameter, utilize a very large magnitude, but !! negative, value.  The internal default is to utilize -huge() !! as a value. type ( regression_statistics ), intent ( out ), optional :: stats (:) !! An optional N-element array that, if supplied, will be used !! to return statistics about the fit for each parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. type ( iteration_controls ), intent ( in ), optional :: controls !! An optional input providing custom iteration controls. type ( lm_solver_options ), intent ( in ), optional :: settings !! An optional input providing custom settings for the solver. type ( convergence_info ), intent ( out ), optional , target :: info !! An optional output that can be used to gain information about !! the iterative solution and the nature of the convergence. procedure ( iteration_update ), intent ( in ), pointer , optional :: status !! An optional pointer to a routine that can be used to extract !! iteration information. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. !! - FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed !!      is underdetetermined (M < N). !! - FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied !!      tolerances are too small to be practical. !! - FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations !!      are allowed. ! Parameters real ( real64 ), parameter :: too_small = 1.0d-14 integer ( int32 ), parameter :: min_iter_count = 2 integer ( int32 ), parameter :: min_fun_count = 10 integer ( int32 ), parameter :: min_update_count = 1 ! Local Variables logical :: stop integer ( int32 ) :: m , n , actual , expected , flag real ( real64 ), pointer :: w (:), pmax (:), pmin (:) real ( real64 ), allocatable , target :: defaultWeights (:), maxparam (:), & minparam (:), JtWJ (:,:) type ( iteration_controls ) :: tol type ( lm_solver_options ) :: opt type ( convergence_info ) :: cInfo class ( errors ), pointer :: errmgr type ( errors ), target :: deferr type ( convergence_info ), target :: defaultinfo type ( convergence_info ), pointer :: inf ! Initialization stop = . false . m = size ( x ) n = size ( params ) if ( present ( info )) then inf => info else inf => defaultinfo end if if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( controls )) then tol = controls else call tol % set_to_default () end if if ( present ( settings )) then opt = settings else call opt % set_to_default () end if ! Input Checking if ( size ( y ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"y\" , m , size ( y )) return end if if ( size ( ymod ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"ymod\" , m , size ( ymod )) return end if if ( size ( resid ) /= m ) then call report_array_size_error ( errmgr , \"nonlinear_least_squares\" , & \"resid\" , m , size ( resid )) return end if if ( m < n ) then call report_underdefined_error ( errmgr , & \"nonlinear_least_squares\" , n , m ) return end if ! Tolerance Checking if ( tol % gradient_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The gradient tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % change_in_solution_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The change in solution tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % residual_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The residual error tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if if ( tol % iteration_improvement_tolerance < too_small ) then call errmgr % report_error ( \"nonlinear_least_squares\" , & \"The iteration improvement tolerance was found to be too small.\" , & FS_TOLERANCE_TOO_SMALL_ERROR ) return end if ! Iteration Count Checking if ( tol % max_iteration_count < min_iter_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few iterations were specified.\" , & min_iter_count ) return end if if ( tol % max_function_evaluations < min_fun_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few function evaluations were specified.\" , & min_fun_count ) return end if if ( tol % max_iteration_between_updates < min_update_count ) then call report_iteration_count_error ( errmgr , & \"nonlinear_least_squares\" , & \"Too few iterations between updates were specified.\" , & min_update_count ) return end if ! Optional Array Arguments (weights, parameter limits, etc.) if ( present ( weights )) then if ( size ( weights ) < m ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"weights\" , m , size ( weights )) return end if w ( 1 : m ) => weights ( 1 : m ) else allocate ( defaultWeights ( m ), source = 1.0d0 , stat = flag ) if ( flag /= 0 ) go to 50 w ( 1 : m ) => defaultWeights ( 1 : m ) end if if ( present ( maxp )) then if ( size ( maxp ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"maxp\" , n , size ( maxp )) return end if pmax ( 1 : n ) => maxp ( 1 : n ) else allocate ( maxparam ( n ), source = huge ( 1.0d0 ), stat = flag ) if ( flag /= 0 ) go to 50 pmax ( 1 : n ) => maxparam ( 1 : n ) end if if ( present ( minp )) then if ( size ( minp ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"minp\" , n , size ( minp )) return end if pmin ( 1 : n ) => minp ( 1 : n ) else allocate ( minparam ( n ), source = - huge ( 1.0d0 ), stat = flag ) if ( flag /= 0 ) go to 50 pmin ( 1 : n ) => minparam ( 1 : n ) end if ! Local Memory Allocations allocate ( JtWJ ( n , n ), stat = flag ) if ( flag /= 0 ) go to 50 ! Process call lm_solve ( fun , x , y , params , w , pmax , pmin , tol , opt , ymod , & resid , JtWJ , inf , stop , errmgr , status ) ! Statistical Parameters if ( present ( stats )) then if ( size ( stats ) /= n ) then call report_array_size_error ( errmgr , & \"nonlinear_least_squares\" , \"stats\" , n , size ( stats )) return end if ! Compute the covariance matrix call mtx_inverse ( JtWJ , err = errmgr ) if ( errmgr % has_error_occurred ()) return ! Compute the statistics stats = calculate_regression_statistics ( resid , params , JtWJ , & alpha , errmgr ) end if ! End return ! Memory Error Handler 50 continue call report_memory_error ( errmgr , \"nonlinear_least_squares\" , flag ) return end subroutine ! ****************************************************************************** ! SETTINGS DEFAULTS ! ------------------------------------------------------------------------------ ! Sets up default tolerances. subroutine lm_set_default_tolerances ( x ) ! Arguments class ( iteration_controls ), intent ( inout ) :: x ! Set defaults x % max_iteration_count = 500 x % max_function_evaluations = 5000 x % max_iteration_between_updates = 10 x % gradient_tolerance = 1.0d-8 x % residual_tolerance = 0.5d-2 x % change_in_solution_tolerance = 1.0d-6 x % iteration_improvement_tolerance = 1.0d-1 end subroutine ! ------------------------------------------------------------------------------ ! Sets up default solver settings. subroutine lm_set_default_settings ( x ) ! Arguments class ( lm_solver_options ), intent ( inout ) :: x ! Set defaults x % method = FS_LEVENBERG_MARQUARDT_UPDATE x % finite_difference_step_size = sqrt ( epsilon ( 1.0d0 )) x % damping_increase_factor = 1 1.0d0 x % damping_decrease_factor = 9.0d0 end subroutine ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ ! Computes the Jacobian matrix via a forward difference. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - params: The model parameters (N-by-1) ! - f0: The current model estimate (M-by-1) ! - step: The differentiation step size ! ! Outputs: ! - jac: The Jacobian matrix (M-by-N) ! - f1: A workspace array for the model output (M-by-1) ! - stop: A flag allowing the user to terminate model execution ! - work: A workspace array for the model parameters (N-by-1) subroutine jacobian_finite_diff ( fun , xdata , params , f0 , jac , f1 , & stop , step , work ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), params (:) real ( real64 ), intent ( in ) :: f0 (:) real ( real64 ), intent ( out ) :: jac (:,:) real ( real64 ), intent ( out ) :: f1 (:), work (:) logical , intent ( out ) :: stop real ( real64 ), intent ( in ) :: step ! Local Variables integer ( int32 ) :: i , n ! Initialization n = size ( params ) ! Cycle over each column of the Jacobian and calculate the derivative ! via a forward difference scheme ! ! J(i,j) = df(i) / dx(j) work = params do i = 1 , n work ( i ) = work ( i ) + step call fun ( xdata , work , f1 , stop ) if ( stop ) return jac (:, i ) = ( f1 - f0 ) / step work ( i ) = params ( i ) end do end subroutine ! ------------------------------------------------------------------------------ ! Computes a rank-1 update to the Jacobian matrix ! ! Inputs: ! - pOld: previous set of parameters (N-by-1) ! - yOld: model evaluation at previous set of parameters (M-by-1) ! - jac: current Jacobian estimate (M-by-N) ! - p: current set of parameters (N-by-1) ! - y: model evaluation at current set of parameters (M-by-1) ! ! Outputs: ! - jac: updated Jacobian matrix (M-by-N) (dy * dp**T + J) ! - dp: p - pOld (N-by-1) ! - dy: (y - yOld - J * dp) / (dp' * dp) (M-by-1) subroutine broyden_update ( pOld , yOld , jac , p , y , dp , dy ) ! Arguments real ( real64 ), intent ( in ) :: pOld (:), yOld (:), p (:), y (:) real ( real64 ), intent ( inout ) :: jac (:,:) real ( real64 ), intent ( out ) :: dp (:), dy (:) ! Local Variables real ( real64 ) :: h2 ! Process dp = p - pOld h2 = dot_product ( dp , dp ) dy = y - yOld - matmul ( jac , dp ) dy = dy / h2 call rank1_update ( 1.0d0 , dy , dp , jac ) end subroutine ! ------------------------------------------------------------------------------ ! Updates the Levenberg-Marquardt matrix by either computing a new Jacobian ! matrix or performing a rank-1 update to the existing Jacobian matrix. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - pOld: previous set of parameters (N-by-1) ! - yOld: model evaluation at previous set of parameters (M-by-1) ! - dX2: The previous change in the Chi-squared criteria ! - jac: current Jacobian estimate (M-by-N) ! - p: current set of parameters (N-by-1) ! - weights: A weighting vector (M-by-1) ! - neval: Current number of function evaluations ! - update: Set to true to force an update of the Jacobian; else, set to !       false to let the program choose based upon the change in the !       Chi-squared parameter. ! - step: The differentiation step size ! ! Outputs: ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - JtWdy: linearized fitting vector (N-by-1) ! - X2: Updated Chi-squared criteria ! - yNew: model evaluated with parameters of p (M-by-1) ! - jac: updated Jacobian matrix (M-by-N) ! - neval: updated count of function evaluations ! - stop: A flag allowing the user to terminate model execution ! - work: A workspace array (N+M-by-1) ! - mwork: A workspace matrix (N-by-M) ! - update: Reset to false if a Jacobian evaluation was performed. subroutine lm_matrix ( fun , xdata , ydata , pOld , yOld , dX2 , jac , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , yNew , stop , work , mwork ) ! Arguments procedure ( regression_function ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), pOld (:), yOld (:), & p (:), weights (:) real ( real64 ), intent ( in ) :: dX2 , step real ( real64 ), intent ( inout ) :: jac (:,:) integer ( int32 ), intent ( inout ) :: neval logical , intent ( inout ) :: update real ( real64 ), intent ( out ) :: JtWJ (:,:), JtWdy (:) real ( real64 ), intent ( out ) :: X2 , mwork (:,:), yNew (:) logical , intent ( out ) :: stop real ( real64 ), intent ( out ), target :: work (:) ! Local Variables integer ( int32 ) :: m , n real ( real64 ), pointer :: w1 (:), w2 (:) ! Initialization m = size ( xdata ) n = size ( p ) w1 ( 1 : m ) => work ( 1 : m ) w2 ( 1 : n ) => work ( m + 1 : n + m ) ! Perform the next function evaluation call fun ( xdata , p , yNew , stop ) neval = neval + 1 if ( stop ) return ! Update or recompute the Jacobian matrix if ( dX2 > 0 . or . update ) then ! Recompute the Jacobian call jacobian_finite_diff ( fun , xdata , p , yNew , jac , w1 , & stop , step , w2 ) neval = neval + n if ( stop ) return update = . false . else ! Simply perform a rank-1 update to the Jacobian call broyden_update ( pOld , yOld , jac , p , yNew , w2 , w1 ) end if ! Update the Chi-squared estimate w1 = ydata - yNew X2 = dot_product ( w1 , w1 * weights ) ! Compute J**T * (W .* dY) w1 = w1 * weights call mtx_mult (. true ., 1.0d0 , jac , w1 , 0.0d0 , JtWdy ) ! Update the Hessian ! First: J**T * W = MWORK ! Second: (J**T * W) * J call diag_mtx_mult (. false ., . true ., 1.0d0 , weights , jac , 0.0d0 , mwork ) call mtx_mult (. false ., . false ., 1.0d0 , mwork , jac , 0.0d0 , JtWJ ) end subroutine ! ------------------------------------------------------------------------------ ! Performs a single iteration of the Levenberg-Marquardt algorithm. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - p: current set of parameters (N-by-1) ! - neval: current number of function evaluations ! - niter: current iteration number ! - update: set to 1 to use Marquardt's modification; else, ! - step: the differentiation step size ! - lambda: LM damping parameter ! - maxP: maximum limits on the parameters.  Use huge() or larger for no constraints (N-by-1) ! - minP: minimum limits on the parameters.  Use -huge() or smaller for no constraints (N-by-1) ! - weights: a weighting vector (M-by-1) ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - JtWdy: linearized fitting vector (N-by-1) ! ! Outputs: ! - JtWJ: overwritten LU factorization of the original matrix (N-by-N) ! - h: The new estimate of the change in parameter (N-by-1) ! - pNew: The new parameter estimates (N-by-1) ! - deltaY: The new difference between data and model (M-by-1) ! - yNew: model evaluated with parameters of pNew (M-by-1) ! - neval: updated count of function evaluations ! - niter: updated current iteration number ! - X2: updated Chi-squared criteria ! - stop: A flag allowing the user to terminate model execution ! - iwork: A workspace array (N-by-1) ! - err: An error handling mechanism subroutine lm_iter ( fun , xdata , ydata , p , neval , niter , update , lambda , & maxP , minP , weights , JtWJ , JtWdy , h , pNew , deltaY , yNew , X2 , X2Old , & alpha , stop , iwork , err , status ) ! Arguments procedure ( regression_function ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), p (:), maxP (:), & minP (:), weights (:), JtWdy (:) real ( real64 ), intent ( in ) :: lambda , X2Old integer ( int32 ), intent ( inout ) :: neval , niter integer ( int32 ), intent ( in ) :: update real ( real64 ), intent ( inout ) :: JtWJ (:,:) real ( real64 ), intent ( out ) :: h (:), pNew (:), deltaY (:), yNew (:) real ( real64 ), intent ( out ) :: X2 , alpha logical , intent ( out ) :: stop integer ( int32 ), intent ( out ) :: iwork (:) class ( errors ), intent ( inout ) :: err procedure ( iteration_update ), intent ( in ), pointer , optional :: status ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: dpJh ! Initialization n = size ( p ) ! Increment the iteration counter niter = niter + 1 ! Solve the linear system to determine the change in parameters ! A is N-by-N and is stored in JtWJ ! b is N-by-1 if ( update == FS_LEVENBERG_MARQUARDT_UPDATE ) then ! Compute: h = A \\ b ! A = J**T * W * J + lambda * diag(J**T * W * J) ! b = J**T * W * dy do i = 1 , n JtWJ ( i , i ) = JtWJ ( i , i ) * ( 1.0d0 + lambda ) h ( i ) = JtWdy ( i ) end do else ! Compute: h = A \\ b ! A = J**T * W * J + lambda * I ! b = J**T * W * dy do i = 1 , n JtWJ ( i , i ) = JtWJ ( i , i ) + lambda h ( i ) = JtWdy ( i ) end do end if call lu_factor ( JtWJ , iwork , err ) ! overwrites JtWJ with [L\\U] if ( err % has_error_occurred ()) return ! if JtWJ is singular call solve_lu ( JtWJ , iwork , h ) ! solution stored in h ! Compute the new attempted solution, and apply any constraints do i = 1 , n pNew ( i ) = min ( max ( minP ( i ), h ( i ) + p ( i )), maxP ( i )) end do ! Update the residual error call fun ( xdata , pNew , yNew , stop ) neval = neval + 1 deltaY = ydata - yNew if ( stop ) return ! Update the Chi-squared estimate X2 = dot_product ( deltaY , deltaY * weights ) ! Perform a quadratic line update in the H direction, if necessary if ( update == FS_QUADRATIC_UPDATE ) then dpJh = dot_product ( JtWdy , h ) alpha = abs ( dpJh / ( 0.5d0 * ( X2 - X2Old ) + 2.0d0 * dpJh )) h = alpha * h do i = 1 , n pNew ( i ) = min ( max ( minP ( i ), p ( i ) + h ( i )), maxP ( i )) end do call fun ( xdata , pNew , yNew , stop ) if ( stop ) return neval = neval + 1 deltaY = ydata - yNew X2 = dot_product ( deltaY , deltaY * weights ) end if ! Update the status of the iteration, if needed if ( present ( status )) then call status ( niter , yNew , deltaY , pNew , h ) end if end subroutine ! ------------------------------------------------------------------------------ ! A Levenberg-Marquardt solver. ! ! Inputs: ! - fun: The function to evaluate ! - xdata: The independent coordinate data to fit (M-by-1) ! - ydata: The dependent coordinate data to fit (M-by-1) ! - p: current set of parameters (N-by-1) ! - weights: a weighting vector (M-by-1) ! - maxP: maximum limits on the parameters.  Use huge() or larger for no constraints (N-by-1) ! - minP: minimum limits on the parameters.  Use -huge() or smaller for no constraints (N-by-1) ! - controls: an iteration_controls instance containing solution tolerances ! ! Outputs: ! - p: solution (N-by-1) ! - y: model results at p (M-by-1) ! - resid: residual (ydata - y) (M-by-1) ! - JtWJ: linearized Hessian matrix (inverse of the covariance matrix) (N-by-N) ! - opt: a convergence_info object containing information regarding !       convergence of the iteration ! - stop: A flag allowing the user to terminate model execution ! - err: An error handling object subroutine lm_solve ( fun , xdata , ydata , p , weights , maxP , minP , controls , & opt , y , resid , JtWJ , info , stop , err , status ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), weights (:), maxP (:), & minP (:) real ( real64 ), intent ( inout ) :: p (:) class ( iteration_controls ), intent ( in ) :: controls class ( lm_solver_options ), intent ( in ) :: opt real ( real64 ), intent ( out ) :: y (:), resid (:), JtWJ (:,:) class ( convergence_info ), intent ( out ) :: info logical , intent ( out ) :: stop class ( errors ), intent ( inout ) :: err procedure ( iteration_update ), intent ( in ), pointer , optional :: status ! Local Variables logical :: update integer ( int32 ) :: i , m , n , dof , flag , neval , niter , nupdate real ( real64 ) :: dX2 , X2 , X2Old , X2Try , lambda , alpha , nu , step real ( real64 ), allocatable :: pOld (:), yOld (:), J (:,:), JtWdy (:), & work (:), mwork (:,:), pTry (:), yTemp (:), JtWJc (:,:), h (:) integer ( int32 ), allocatable :: iwork (:) character ( len = :), allocatable :: errmsg ! Initialization update = . true . m = size ( xdata ) n = size ( p ) dof = m - n niter = 0 step = opt % finite_difference_step_size stop = . false . info % user_requested_stop = . false . nupdate = 0 ! Local Memory Allocation allocate ( pOld ( n ), source = 0.0d0 , stat = flag ) if ( flag == 0 ) allocate ( yOld ( m ), source = 0.0d0 , stat = flag ) if ( flag == 0 ) allocate ( J ( m , n ), stat = flag ) if ( flag == 0 ) allocate ( JtWdy ( n ), stat = flag ) if ( flag == 0 ) allocate ( work ( m + n ), stat = flag ) if ( flag == 0 ) allocate ( mwork ( n , m ), stat = flag ) if ( flag == 0 ) allocate ( pTry ( n ), stat = flag ) if ( flag == 0 ) allocate ( h ( n ), stat = flag ) if ( flag == 0 ) allocate ( yTemp ( m ), stat = flag ) if ( flag == 0 ) allocate ( JtWJc ( n , n ), stat = flag ) if ( flag == 0 ) allocate ( iwork ( n ), stat = flag ) if ( flag /= 0 ) go to 10 ! Perform an initial function evaluation call fun ( xdata , p , y , stop ) neval = 1 ! Evaluate the problem matrices call lm_matrix ( fun , xdata , ydata , pOld , yOld , 1.0d0 , J , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , y , stop , work , mwork ) if ( stop ) go to 5 X2Old = X2 JtWJc = JtWJ ! Determine an initial value for lambda if ( opt % method == FS_LEVENBERG_MARQUARDT_UPDATE ) then lambda = 1.0d-2 else call extract_diagonal ( JtWJ , work ( 1 : n )) lambda = 1.0d-2 * maxval ( work ( 1 : n )) nu = 2.0d0 end if ! Main Loop main : do while ( niter < controls % max_iteration_count ) ! Compute the linear solution at the current solution estimate and ! update the new parameter estimates call lm_iter ( fun , xdata , ydata , p , neval , niter , opt % method , & lambda , maxP , minP , weights , JtWJc , JtWdy , h , pTry , resid , & yTemp , X2Try , X2Old , alpha , stop , iwork , err , status ) if ( stop ) go to 5 if ( err % has_error_occurred ()) return ! Update the Chi-squared estimate, update the damping parameter ! lambda, and, if necessary, update the matrices call lm_update ( fun , xdata , ydata , pOld , p , pTry , yOld , y , h , dX2 , & X2Old , X2 , X2Try , lambda , alpha , nu , JtWdy , JtWJ , J , weights , & niter , neval , update , step , work , mwork , controls , opt , stop ) if ( stop ) go to 5 JtWJc = JtWJ ! Determine the matrix update scheme nupdate = nupdate + 1 if ( opt % method == FS_QUADRATIC_UPDATE ) then update = mod ( niter , 2 * n ) > 0 else if ( nupdate >= controls % max_iteration_between_updates ) then update = . true . nupdate = 0 end if ! Test for convergence if ( lm_check_convergence ( controls , dof , resid , niter , neval , & JtWdy , h , p , X2 , info )) & then exit main end if end do main ! End return ! User Requested End 5 continue info % user_requested_stop = . true . return ! Memory Error Handling 10 continue allocate ( character ( len = 512 ) :: errmsg ) write ( errmsg , 100 ) \"Memory allocation error code \" , flag , \".\" call err % report_error ( \"lm_solve\" , & trim ( errmsg ), FS_MEMORY_ERROR ) return ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ ! subroutine lm_update ( fun , xdata , ydata , pOld , p , pTry , yOld , y , h , dX2 , & X2old , X2 , X2try , lambda , alpha , nu , JtWdy , JtWJ , J , weights , niter , & neval , update , step , work , mwork , controls , opt , stop ) ! Arguments procedure ( regression_function ), intent ( in ), pointer :: fun real ( real64 ), intent ( in ) :: xdata (:), ydata (:), X2try , h (:), step , & pTry (:), weights (:), alpha real ( real64 ), intent ( inout ) :: pOld (:), p (:), yOld (:), y (:), lambda , & JtWdy (:), dX2 , X2 , X2old , JtWJ (:,:), J (:,:), nu real ( real64 ), intent ( out ) :: work (:), mwork (:,:) integer ( int32 ), intent ( in ) :: niter integer ( int32 ), intent ( inout ) :: neval logical , intent ( inout ) :: update class ( iteration_controls ), intent ( in ) :: controls class ( lm_solver_options ), intent ( in ) :: opt logical , intent ( out ) :: stop ! Local Variables integer ( int32 ) :: n real ( real64 ) :: rho ! Initialization n = size ( p ) ! Process if ( opt % method == FS_LEVENBERG_MARQUARDT_UPDATE ) then call extract_diagonal ( JtWJ , work ( 1 : n )) work ( 1 : n ) = lambda * work ( 1 : n ) * h + JtWdy else work ( 1 : n ) = lambda * h + JtWdy end if rho = ( X2 - X2try ) / abs ( dot_product ( h , work ( 1 : n ))) if ( rho > controls % iteration_improvement_tolerance ) then ! Things are getting better at an acceptable rate dX2 = X2 - X2old X2old = X2 pOld = p yOld = y p = pTry ! Recompute the matrices call lm_matrix ( fun , xdata , ydata , pOld , yOld , dX2 , J , p , weights , & neval , update , step , JtWJ , JtWdy , X2 , y , stop , work , mwork ) if ( stop ) return ! Decrease lambda select case ( opt % method ) case ( FS_LEVENBERG_MARQUARDT_UPDATE ) lambda = max ( lambda / opt % damping_decrease_factor , 1.0d-7 ) case ( FS_QUADRATIC_UPDATE ) lambda = max ( lambda / ( 1.0d0 + alpha ), 1.0d-7 ) case ( FS_NIELSEN_UPDATE ) lambda = lambda * max ( 1.0d0 / 3.0d0 , & 1.0d0 - ( 2.0d0 * rho - 1.0d0 ** 3 )) nu = 2.0d0 end select else ! The iteration is not improving in a satisfactory manner X2 = X2old if ( mod ( niter , 2 * n ) /= 0 ) then call lm_matrix ( fun , xdata , ydata , pOld , yOld , - 1.0d0 , J , p , & weights , neval , update , step , JtWJ , JtWdy , dX2 , y , stop , & work , mwork ) if ( stop ) return end if ! Increase lambda select case ( opt % method ) case ( FS_LEVENBERG_MARQUARDT_UPDATE ) lambda = min ( lambda * opt % damping_increase_factor , 1.0d7 ) case ( FS_QUADRATIC_UPDATE ) lambda = lambda + abs (( X2try - X2 ) / 2.0d0 / alpha ) case ( FS_NIELSEN_UPDATE ) lambda = lambda * nu nu = 2.0d0 * nu end select end if end subroutine ! ------------------------------------------------------------------------------ ! Checks the Levenberg-Marquardt solution against the convergence criteria. ! ! Inputs: ! - controls: the solution controls and convergence criteria ! - dof: the statistical degrees of freedom of the system (M - N) ! - resid: the residual error (M-by-1) ! - niter: the number of iterations ! - neval: the number of function evaluations ! - JtWdy: linearized fitting vector (N-by-1) ! - h: the change in parameter (solution) values (N-by-1) ! - p: the parameter (solution) values (N-by-1) ! - X2: the Chi-squared estimate ! ! Outputs: ! - info: The convergence information. ! - rst: True if convergence was achieved; else, false. function lm_check_convergence ( controls , dof , resid , niter , neval , & JtWdy , h , p , X2 , info ) result ( rst ) ! Arguments class ( iteration_controls ), intent ( in ) :: controls real ( real64 ), intent ( in ) :: resid (:), JtWdy (:), h (:), p (:), X2 integer ( int32 ), intent ( in ) :: dof , niter , neval class ( convergence_info ), intent ( out ) :: info logical :: rst ! Initialization rst = . false . ! Iteration Checks info % iteration_count = niter if ( niter >= controls % max_iteration_count ) then info % reach_iteration_limit = . true . rst = . true . else info % reach_iteration_limit = . false . end if info % function_evaluation_count = neval if ( neval >= controls % max_function_evaluations ) then info % reach_function_evaluation_limit = . true . rst = . true . else info % reach_function_evaluation_limit = . false . end if info % gradient_value = maxval ( abs ( JtWdy )) if ( info % gradient_value < controls % gradient_tolerance . and . niter > 2 ) & then info % converge_on_gradient = . true . rst = . true . else info % converge_on_gradient = . false . end if info % solution_change_value = maxval ( abs ( h ) / ( abs ( p ) + 1.0d-12 )) if ( info % solution_change_value < & controls % change_in_solution_tolerance . and . niter > 2 ) & then info % converge_on_solution_change = . true . rst = . true . else info % converge_on_solution_change = . false . end if info % residual_value = X2 / dof if ( info % residual_value < controls % residual_tolerance . and . niter > 2 ) & then info % converge_on_residual_parameter = . true . rst = . true . else info % converge_on_residual_parameter = . false . end if end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_regression.f90.html"},{"title":"fstats.f90 – FSTATS","text":"Source Code module fstats !! FSTATS is a modern Fortran statistical library containing routines for !! computing basic statistical properties, hypothesis testing, regression, !! special functions, and experimental design. use iso_fortran_env use fstats_special_functions use fstats_descriptive_statistics use fstats_hypothesis use fstats_distributions use fstats_anova use fstats_helper_routines use fstats_regression use fstats_experimental_design use fstats_allan use fstats_bootstrap use fstats_sampling use fstats_smoothing implicit none private public :: distribution public :: distribution_function public :: distribution_property public :: t_distribution public :: normal_distribution public :: f_distribution public :: chi_squared_distribution public :: binomial_distribution public :: mean public :: variance public :: standard_deviation public :: median public :: covariance public :: r_squared public :: adjusted_r_squared public :: correlation public :: quantile public :: t_test_equal_variance public :: t_test_unequal_variance public :: t_test_paired public :: f_test public :: anova public :: anova_factor public :: single_factor_anova_table public :: two_factor_anova_table public :: confidence_interval public :: beta public :: regularized_beta public :: incomplete_beta public :: digamma public :: incomplete_gamma_upper public :: incomplete_gamma_lower public :: design_matrix public :: covariance_matrix public :: linear_least_squares public :: regression_statistics public :: get_full_factorial_matrix_size public :: full_factorial public :: iteration_controls public :: lm_solver_options public :: convergence_info public :: regression_function public :: iteration_update public :: jacobian public :: nonlinear_least_squares public :: allan_variance public :: trimmed_mean public :: difference public :: factorial public :: bootstrap_resampling_routine public :: bootstrap_statistic_routine public :: random_resample public :: scaled_random_resample public :: bootstrap_statistics public :: bootstrap public :: bootstrap_regression_statistics public :: bootstrap_linear_least_squares public :: bootstrap_nonlinear_least_squares public :: box_muller_sample public :: rejection_sample public :: lowess public :: pooled_variance public :: bartletts_test public :: levenes_test public :: FS_LEVENBERG_MARQUARDT_UPDATE public :: FS_QUADRATIC_UPDATE public :: FS_NIELSEN_UPDATE end module","tags":"","loc":"sourcefile\\fstats.f90.html"},{"title":"fstats_types.f90 – FSTATS","text":"Source Code module fstats_types use iso_fortran_env implicit none type array_container !! Provides a container for a real-valued array.  A practical use of !! this construct is in the construction of jagged arrays. real ( real64 ), allocatable , dimension (:) :: x !! The array. end type end module","tags":"","loc":"sourcefile\\fstats_types.f90.html"},{"title":"fstats_helper_routines.f90 – FSTATS","text":"Source Code module fstats_helper_routines use iso_fortran_env implicit none private public :: difference public :: factorial contains ! ------------------------------------------------------------------------------ pure function difference ( x ) result ( rst ) !! Computes the difference between elements in an array. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array on which to operate. real ( real64 ), allocatable , dimension (:) :: rst !! The (N-1)-element array containing the differences between adjacent !! elements. ! Local Variables integer ( int32 ) :: i , n ! Process n = size ( x ) allocate ( rst ( n - 1 )) do i = 1 , n - 1 rst ( i ) = x ( i + 1 ) - x ( i ) end do end function ! ------------------------------------------------------------------------------ pure elemental function factorial ( x ) result ( rst ) !! Computes the factorial of X. real ( real64 ), intent ( in ) :: x !! The value whose factorial is to be computed. real ( real64 ) :: rst !! The result. rst = gamma ( x + 1.0d0 ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_helper_routines.f90.html"},{"title":"fstats_sampling.f90 – FSTATS","text":"Source Code module fstats_sampling use iso_fortran_env use linalg , only : sort use fstats_distributions implicit none private public :: box_muller_sample public :: rejection_sample real ( real64 ), parameter :: pi = 2.0d0 * acos ( 0.0d0 ) real ( real64 ), parameter :: twopi = 2.0d0 * pi real ( real64 ), parameter :: pi_f = 2.0 * acos ( 0.0 ) real ( real64 ), parameter :: twopi_f = 2.0 * pi_f interface box_muller_sample !! Generates random, normally distributed values via the Box-Muller !! transform. module procedure :: box_muller_sample_scalar module procedure :: box_muller_array end interface contains ! ------------------------------------------------------------------------------ function box_muller_sample_scalar ( mu , sigma ) result ( rst ) !! Generates a pair of independent, standard, normally distributed !! random values using the Box-Muller transform. real ( real64 ), intent ( in ) :: mu !! The mean of the distribution. real ( real64 ), intent ( in ) :: sigma !! The standard deviation of the distribution. real ( real64 ) :: rst ( 2 ) !! The pair of random values. ! Parameters complex ( real64 ), parameter :: j = ( 0.0d0 , 1.0d0 ) ! Local Variables real ( real64 ) :: u1 , u2 complex ( real64 ) :: z ! Process call random_number ( u1 ) call random_number ( u2 ) z = sqrt ( - log ( u1 )) * exp ( j * twopi * u2 ) rst = [ real ( z , real64 ), aimag ( z )] end function ! ------------------------------------------------------------------------------ function box_muller_array ( mu , sigma , n ) result ( rst ) !! Generates an array of normally distributed random values sampled !! by the Box-Muller transform. real ( real64 ), intent ( in ) :: mu !! The mean of the distribution. real ( real64 ), intent ( in ) :: sigma !! The standard deviation of the distribution. integer ( int32 ), intent ( in ) :: n !! The number of Box-Muller pairs to generate. real ( real64 ), allocatable , dimension (:) :: rst !! A 2N-element array containing the N Box-Muller pairs. ! Local Variables integer ( int32 ) :: i ! Process if ( n < 1 ) then allocate ( rst ( 0 )) return end if allocate ( rst ( 2 * n )) do i = 1 , n rst ( 2 * i - 1 : 2 * i ) = box_muller_sample ( mu , sigma ) end do end function ! ****************************************************************************** ! REJECTION SAMPLING ! ------------------------------------------------------------------------------ function rejection_sample ( tdist , n , xmin , xmax ) result ( rst ) !! Uses rejection sampling to randomly sample a target distribution. class ( distribution ), intent ( in ) :: tdist !! The distribution to sample integer ( int32 ), intent ( in ) :: n !! The number of samples to make. real ( real64 ), intent ( in ) :: xmin !! The minimum range to explore. real ( real64 ), intent ( in ) :: xmax !! The maximum range to explore. real ( real64 ), allocatable , dimension (:) :: rst !! An N-element array containing the N samples from the !! distribution. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: c_start = 1.01d0 ! Local Variables integer ( int32 ) :: i , j , jmax real ( real64 ) :: u , c , g , f , rng ! Quick Return if ( n < 1 ) then allocate ( rst ( 0 ), source = zero ) end if ! Process i = 0 j = 0 jmax = min ( 1000 * n , huge ( j )) ! Guard against insanity rng = xmax - xmin c = c_start allocate ( rst ( n ), source = zero ) do while ( i <= n ) ! Update the acceptance threshold call random_number ( u ) ! Sample from the proposal distribution call random_number ( g ) g = g * rng + xmin ! Sample the target distribution f = tdist % pdf ( g ) ! Test if ( u <= f / ( c * g )) then i = i + 1 rst ( i ) = g end if ! Update C c = max ( c , f / g ) ! Update the infinite loop guard variable j = j + 1 if ( j == jmax ) exit end do end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_sampling.f90.html"},{"title":"fstats_hypothesis.f90 – FSTATS","text":"Source Code module fstats_hypothesis use iso_fortran_env use ieee_arithmetic use fstats_errors use fstats_special_functions use fstats_distributions use fstats_descriptive_statistics use fstats_types private public :: confidence_interval public :: t_test_equal_variance public :: t_test_unequal_variance public :: t_test_paired public :: f_test public :: bartletts_test public :: levenes_test interface confidence_interval !! Computes the confidence interval for the specified distribution. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Confidence_interval) module procedure :: confidence_interval_scalar module procedure :: confidence_interval_array end interface contains ! ------------------------------------------------------------------------------ pure function confidence_interval_scalar ( dist , alpha , s , n ) result ( rst ) !! Computes the confidence interval for the specified distribution. class ( distribution ), intent ( in ) :: dist !! The distribution object defining the probability distribution !! to establish the confidence level. real ( real64 ), intent ( in ) :: alpha !! The probability value of interest.  For instance, use a value of 0.05 !! for a confidence level of 95%. real ( real64 ), intent ( in ) :: s !! The sample standard deviation. integer ( int32 ), intent ( in ) :: n !! The number of samples in the data set. real ( real64 ) :: rst !! The result. ! Local Variables integer ( int32 ), parameter :: maxiter = 100 real ( real64 ), parameter :: tol = 1.0d-6 integer ( int32 ) :: i real ( real64 ) :: x , f , df , h , twoh , dy ! Process ! ! We use a simplified Newton's method to solve for the independent variable ! of the CDF function where it equals 1 - alpha / 2. h = 1.0d-6 twoh = 2.0d0 * h x = 1.0d0 - alpha / 2.0d0 rst = 0.5d0 do i = 1 , maxiter ! Compute the CDF and its derivative at y f = dist % cdf ( rst ) - x df = ( dist % cdf ( rst + h ) - dist % cdf ( rst - h )) / twoh dy = f / df rst = rst - dy if ( abs ( dy ) < tol ) exit end do ! Determine the actual interval rst = rst * s / sqrt ( real ( n , real64 )) end function ! ------------------------------------------------------------------------------ pure function confidence_interval_array ( dist , alpha , x ) result ( rst ) !! Computes the confidence interval for the specified distribution. class ( distribution ), intent ( in ) :: dist !! The distribution object defining the probability distribution !! to establish the confidence level. real ( real64 ), intent ( in ) :: alpha !! The probability value of interest.  For instance, use a value of 0.05 !! for a confidence level of 95%. real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the data to analyze. real ( real64 ) :: rst !! The result. ! Process rst = confidence_interval ( dist , alpha , standard_deviation ( x ), size ( x )) end function ! ------------------------------------------------------------------------------ subroutine t_test_equal_variance ( x1 , x2 , stat , p , dof ) !! Computes the 2-tailed Student's T-Test for two data sets of !! assumed equivalent variances. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables real ( real64 ) :: v1 , v2 , m1 , m2 , sv , a , b , x integer ( int32 ) :: n1 , n2 ! Compute the T-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = n1 + n2 - two sv = (( n1 - one ) * v1 + ( n2 - one ) * v2 ) / dof stat = abs ( m1 - m2 ) / sqrt ( sv * ( one / real ( n1 ) + one / real ( n2 ))) ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine t_test_unequal_variance ( x1 , x2 , stat , p , dof ) !! Computes the 2-tailed Student's T-Test for two data sets of !! assumed non-equivalent variances. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables real ( real64 ) :: v1 , v2 , m1 , m2 , sv , a , b , x integer ( int32 ) :: n1 , n2 ! Compute the T-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = ( v1 / real ( n1 ) + v2 / real ( n2 )) ** 2 / (( v1 / n1 ) ** 2 / ( n1 - one ) + & ( v2 / n2 ) ** 2 / ( n2 - one )) sv = sqrt ( v1 / n1 + v2 / n2 ) stat = ( m1 - m2 ) / sv ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine t_test_paired ( x1 , x2 , stat , p , dof , err ) !! Computes the 2-tailed Student's T-Test for two paired data sets. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Student%27s_t-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An N-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The Student-'s T-Test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from  two underlying populations that !! have the same mean. real ( real64 ), intent ( out ) :: dof !! The degrees of freedom. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if x1 and x2 are not the same !!   length. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr real ( real64 ) :: v1 , v2 , m1 , m2 , sd , cov , a , b , x integer ( int32 ) :: i , n1 , n2 , n ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n1 = size ( x1 ) n2 = size ( x2 ) n = min ( n1 , n2 ) ! Input Checking if ( n1 /= n2 ) then call report_arrays_not_same_size_error ( errmgr , \"t_test_paired_real64\" , & \"X1\" , \"X2\" , n1 , n2 ) return end if ! Compute the T-statistic m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) dof = real ( n1 ) - one cov = zero do i = 1 , n cov = cov + ( x1 ( i ) - m1 ) * ( x2 ( i ) - m2 ) end do cov = cov / dof sd = sqrt (( v1 + v2 - two * cov ) / n ) stat = ( m1 - m2 ) / sd ! Compute the probability a = half * dof b = half x = dof / ( dof + stat ** 2 ) p = regularized_beta ( a , b , x ) end subroutine ! ------------------------------------------------------------------------------ subroutine f_test ( x1 , x2 , stat , p , dof1 , dof2 ) !! Computes the F-test and returns the probability (two-tailed) that !! the variances of two data sets are not significantly different. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/F-test) real ( real64 ), intent ( in ) :: x1 (:) !! An N-element array containing the first data set. real ( real64 ), intent ( in ) :: x2 (:) !! An M-element array containing the second data set. real ( real64 ), intent ( out ) :: stat !! The F-statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the two samples are likely to !! have come from the two underlying populations that !! have the same variance. real ( real64 ), intent ( out ) :: dof1 !! A measure of the degrees of freedom. real ( real64 ), intent ( out ) :: dof2 !! A measure of the degrees of freedom. ! Parameters real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 ! Local Variables integer ( int32 ) :: n1 , n2 real ( real64 ) :: v1 , v2 , m1 , m2 type ( f_distribution ) :: dist ! Compute the F-statistic n1 = size ( x1 ) n2 = size ( x2 ) m1 = mean ( x1 ) m2 = mean ( x2 ) v1 = variance ( x1 ) v2 = variance ( x2 ) if ( v1 > v2 ) then stat = v1 / v2 dof1 = n1 - one dof2 = n2 - one else stat = v2 / v1 dof1 = n2 - one dof2 = n1 - one end if dist % d1 = dof1 dist % d2 = dof2 p = two * ( one - dist % cdf ( stat )) end subroutine ! ------------------------------------------------------------------------------ subroutine bartletts_test ( x , stat , p ) !! Computes Bartlett's test statistic and associated probability. !! !! The statistic is calculated as follows. !! !!  \\chi&#94;{2} = \\frac{(N - k) \\ln(S_{p}&#94;{2}) \\sum_{i = 1}&#94;{k} !! \\left(n_{i} - 1 \\right) \\ln(S_{i}&#94;{2})}{1 + !! \\frac{1}{3 \\left( k - 1 \\right)} \\left( \\sum_{i = 1}&#94;{k} !! \\left( \\frac{1}{n_{i} - 1} \\right) - \\frac{1}{N - k} \\right)}  !! !! Where  N = \\sum_{i = 1}&#94;{k} n_{i}  and  S_{p}&#94;{2}  is the pooled !! variance. !! !! The probability is calculated as the right-tail probability of the !! chi-squared distribution. !! !! Bartlett's test is most relevant for distributions showing strong !! normality.  For distributions lacking strong normality, consider !! Levene's test instead. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Bartlett%27s_test) type ( array_container ), intent ( in ), dimension (:) :: x !! The arrays of data to analyze. real ( real64 ), intent ( out ) :: stat !! The Bartlett's test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the variances of each data set are !! equivalent.  A low p-value, less than some significance level, !! indicates a non-equivalance of variances. ! Local Variables integer ( int32 ) :: i , n , k , ni real ( real64 ) :: si , sp , numer , denom type ( chi_squared_distribution ) :: dist ! Initialization k = size ( x ) n = 0 do i = 1 , k n = n + size ( x ( i )% x ) end do ! Compute the statistic n = 0 sp = 0.0d0 numer = 0.0d0 denom = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni si = variance ( x ( i )% x ) sp = sp + ( ni - 1.0d0 ) * si numer = numer + ( ni - 1.0d0 ) * log ( variance ( x ( i )% x )) denom = denom + 1.0d0 / ( ni - 1.0d0 ) end do sp = sp / real ( n - k , real64 ) stat = (( n - k ) * log ( sp ) - numer ) / & ( 1.0d0 + ( 1.0d0 / ( 3.0d0 * k - 3.0d0 )) * & ( denom - 1.0d0 / real ( n - k , real64 ))) ! Compute the p-value dist % dof = k - 1 p = 1.0d0 - dist % cdf ( stat ) end subroutine ! ------------------------------------------------------------------------------ subroutine levenes_test ( x , stat , p , err ) !! Computes Levene's test statistic and associated probability. !! !! The statistic is calculated as follows. !!  W = \\frac{N - k}{k - 1} \\frac{ \\sum_{i = 1}&#94;{k} N_{i} \\left( Z_{i.} - !! Z{..} \\right)&#94;{2}}{ \\sum_{i = 1}&#94;{k} \\sum_{j = 1}&#94;{n_{i}} \\left( Z_{ij} - !! Z_{i.} \\right)&#94;{2} }  !! !! Where: !!  Z_{ij} = |X_{ij} - \\overline{X_{i.}}|  !!  Z_{i.} = \\frac{1}{n_{i}} \\sum_{j = 1}&#94;{n_{i}} Z_{ij}  !!  Z_{..} = \\frac{1}{N} \\sum_{i = 1}&#94;{k} \\sum_{j = 1}&#94;{n_{i}} Z_{ij}  !! !! As the test statistic is approximately F-distributed, the F-distribution !! is used to calculate the probability term. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Levene%27s_test) type ( array_container ), intent ( in ), dimension (:) :: x !! The arrays of data to analyze. real ( real64 ), intent ( out ) :: stat !! The Bartlett's test statistic. real ( real64 ), intent ( out ) :: p !! The probability value that the variances of each data set are !! equivalent.  A low p-value, less than some significance level, !! indicates a non-equivalance of variances. class ( errors ), intent ( inout ), optional , target :: err ! Local Variables integer ( int32 ) :: i , j , k , n , ni , flag real ( real64 ) :: numer , denom , inner , yi , z , zij real ( real64 ), allocatable , dimension (:) :: y , zt , zi type ( f_distribution ) :: dist class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if k = size ( x ) ! Local Memory Allocations allocate ( y ( k ), zi ( k ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"levenes_test\" , flag ) return end if ! Compute the total mean z = 0.0d0 n = 0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni y ( i ) = mean ( x ( i )% x ) zt = abs ( x ( i )% x - y ( i )) zi ( i ) = mean ( zt ) z = z + zi ( i ) * ni end do z = z / n ! Process numer = 0.0d0 denom = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) yi = y ( i ) numer = numer + ni * ( zi ( i ) - z ) ** 2 inner = 0.0d0 do j = 1 , ni zij = abs ( x ( i )% x ( j ) - yi ) inner = inner + ( zij - zi ( i )) ** 2 end do denom = denom + inner end do stat = real (( N - k ) / ( k - 1 ), real64 ) * ( numer / denom ) dist % d1 = k - 1.0d0 dist % d2 = real ( n - k , real64 ) p = 1.0d0 - dist % cdf ( stat ) end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_hypothesis.f90.html"},{"title":"fstats_descriptive_statistics.f90 – FSTATS","text":"Source Code module fstats_descriptive_statistics use iso_fortran_env use linalg , only : sort use ferror use fstats_errors use fstats_types implicit none private public :: mean public :: variance public :: standard_deviation public :: median public :: quantile public :: trimmed_mean public :: covariance public :: pooled_variance interface pooled_variance !! Computes the pooled estimate of variance. module procedure :: pooled_variance_1 module procedure :: pooled_variance_2 end interface contains ! ------------------------------------------------------------------------------ pure function mean ( x ) result ( rst ) !! Computes the mean of the values in an array. real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 ! Local Variables integer ( int32 ) :: i , n ! Process n = size ( x ) if ( n == 0 ) then rst = zero else rst = x ( 1 ) do i = 2 , n rst = rst + ( x ( i ) - rst ) / i end do end if end function ! ------------------------------------------------------------------------------ pure function variance ( x ) result ( rst ) !! Computes the sample variance of the values in an array. !! !! The variance computed is the sample variance such that !!  s&#94;2 = \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right)&#94;2}{n - 1} . real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: oldMean , newMean ! Process n = size ( x ) if ( n <= 1 ) then rst = zero else oldMean = x ( 1 ) rst = zero do i = 2 , n newMean = oldMean + ( x ( i ) - oldMean ) / i rst = rst + ( x ( i ) - oldMean ) * ( x ( i ) - newMean ) oldMean = newMean end do rst = rst / ( n - one ) end if end function ! ------------------------------------------------------------------------------ pure function standard_deviation ( x ) result ( rst ) !! Computes the sample standard deviation of the values in an array. !! !! The value computed is the sample standard deviation. !!  s = \\sqrt{ \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right)&#94;2}{n - 1} }  real ( real64 ), intent ( in ) :: x (:) !! The array of values to analyze. real ( real64 ) :: rst !! The result. ! Process rst = sqrt ( variance ( x )) end function ! ------------------------------------------------------------------------------ function median ( x ) result ( rst ) !! Computes the median of the values in an array. real ( real64 ), intent ( inout ) :: x (:) !! The array of values to analyze.  On output, this array is sorted into !! ascending order. real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: n , nmid , nmidp1 , flag , iflag ! Initialization n = size ( x ) nmid = n / 2 nmidp1 = nmid + 1 iflag = n - 2 * nmid ! Sort the array in ascending order call sort ( x , . true .) ! Find the median if ( iflag == 0 ) then rst = half * ( x ( nmid ) + x ( nmidp1 )) else rst = x ( nmidp1 ) end if end function ! ------------------------------------------------------------------------------ ! REF: https://fortranwiki.org/fortran/show/Quartiles ! ! This is the method used by Minitab pure function quantile ( x , q ) result ( rst ) !! Computes the specified quantile of a data set using the SAS !! Method 4. !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Quantile) real ( real64 ), intent ( in ) :: x (:) !! An N-element array containing the data. real ( real64 ), intent ( in ) :: q !! The quantile to compute (e.g. 0.25 computes the 25% quantile). real ( real64 ) :: rst !! The result. ! Parameters real ( real64 ), parameter :: one = 1.0d0 ! Local Variables real ( real64 ) :: a , b , c , tol integer ( int32 ) :: n , ib ! Initialization tol = sqrt ( epsilon ( tol )) n = size ( x ) ! Process a = ( n + one ) * q b = mod ( a , one ) c = a - b ib = int ( c , int32 ) if (( ib + 1 ) > n ) then rst = ( one - b ) * x ( ib ) + b * x ( n ) else rst = ( one - b ) * x ( ib ) + b * x ( ib + 1 ) end if end function ! ------------------------------------------------------------------------------ function trimmed_mean ( x , p ) result ( rst ) !! Computes the trimmed mean of a data set. real ( real64 ), intent ( inout ), dimension (:) :: x !! An N-element array containing the data.  On output, the !! array is sorted into ascending order. real ( real64 ), intent ( in ), optional :: p !! An optional parameter specifying the percentage of values !! from either end of the distribution to remove.  The default !! is 0.05 such that the bottom 5% and top 5% are removed. real ( real64 ) :: rst !! The trimmed mean. ! Local Variables integer ( int32 ) :: i1 , i2 , n real ( real64 ) :: pv ! Initialization if ( present ( p )) then pv = abs ( p ) else pv = 0.05d0 end if ! Sort the array into ascending order call sort ( x , . true .) ! Find the limiting indices n = size ( x ) i1 = max ( floor ( n * pv , int32 ), 1 ) i2 = min ( n , n - i1 + 1 ) rst = mean ( x ( i1 : i2 )) end function ! ------------------------------------------------------------------------------ pure function covariance ( x , y ) result ( rst ) !! Computes the sample covariance of two data sets. !! !! The covariance computed is the sample covariance such that !!  q_{jk} = \\frac{\\Sigma \\left( x_{i} - \\bar{x} \\right) !! \\left( y_{i} - \\bar{y} \\right)}{n - 1} . real ( real64 ), intent ( in ), dimension (:) :: x !! The first N-element data set. real ( real64 ), intent ( in ), dimension ( size ( x )) :: y !! The second N-element data set. real ( real64 ) :: rst !! The covariance. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: meanX , meanY ! Process n = size ( x ) if ( n <= 1 ) then rst = zero else ! Compute the means meanX = x ( 1 ) meanY = y ( 1 ) do i = 2 , n meanX = meanX + ( x ( i ) - meanX ) / i meanY = meanY + ( y ( i ) - meanY ) / i end do ! Compute the covariance rst = sum (( x - meanX ) * ( y - meanY )) / ( n - one ) end if end function ! ------------------------------------------------------------------------------ pure function pooled_variance_1 ( si , ni ) result ( rst ) !! Computes the pooled estimate of variance. real ( real64 ), intent ( in ), dimension (:) :: si !! An N-element array containing the estimates for each of the N !! variances. integer ( int32 ), intent ( in ), dimension ( size ( si )) :: ni !! An N-element array containing the number of data points in each !! of the data sets used to compute the variances in si. real ( real64 ) :: rst !! The pooled variance. ! Local Variables integer ( int32 ) :: i , k , n ! Process k = size ( si ) rst = 0.0d0 n = 0 do i = 1 , k n = n + ni ( i ) rst = rst + ( ni ( i ) - 1.0d0 ) * si ( i ) end do rst = rst / real ( n - k , real64 ) end function pure function pooled_variance_2 ( x ) result ( rst ) !! Computes the pooled estimate of variance. type ( array_container ), intent ( in ), dimension (:) :: x !! An array of arrays of data. real ( real64 ) :: rst !! The pooled variance. ! Local Variables integer ( int32 ) :: i , k , n , ni ! Process k = size ( x ) n = 0 rst = 0.0d0 do i = 1 , k ni = size ( x ( i )% x ) n = n + ni rst = rst + variance ( x ( i )% x ) * ( ni - 1.0 ) end do rst = rst / real ( n - k , real64 ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_descriptive_statistics.f90.html"},{"title":"fstats_special_functions.f90 – FSTATS","text":"Source Code module fstats_special_functions use iso_fortran_env use ieee_arithmetic implicit none private public :: beta public :: regularized_beta public :: incomplete_beta public :: incomplete_gamma_lower public :: incomplete_gamma_upper public :: digamma contains ! ------------------------------------------------------------------------------ pure elemental function beta ( a , b ) result ( rst ) !! Computes the beta function. !! !! The beta function is related to the gamma function !! by the following relationship. !!  \\beta(a,b) = \\frac{\\Gamma(a) \\Gamma(b)}{\\Gamma(a + b)} . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ) :: rst !! The value of the beta function at  a  and  b . ! Process ! REF: https://en.wikipedia.org/wiki/Beta_function rst = exp ( log_gamma ( a ) + log_gamma ( b ) - log_gamma ( a + b )) end function ! ------------------------------------------------------------------------------ ! source: https://people.math.sc.edu/Burkardt/f_src/special_functions/special_functions.f90 pure elemental function regularized_beta ( a , b , x ) result ( rst ) !! Computes the regularized beta function. !! !! The regularized beta function is defined as the ratio between !! the incomplete beta function and the beta function. !!  I_{x}(a,b) = \\frac{\\beta(x;a,b)}{\\beta(a,b)} . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ), intent ( in ) :: x !! The upper limit of the integration. real ( real64 ) :: rst !! The value of the regularized beta function. ! Local Variables real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: two = 2.0d0 real ( real64 ) :: bt , dk ( 51 ), fk ( 51 ), s0 , t1 , t2 , ta , tb integer ( int32 ) :: k ! Process s0 = ( a + one ) / ( a + b + two ) bt = beta ( a , b ) if ( x <= s0 ) then do k = 1 , 20 dk ( 2 * k ) = k * ( b - k ) * x / ( a + two * k - one ) / ( a + two * k ) end do do k = 0 , 20 dk ( 2 * k + 1 ) = - ( a + k ) * ( a + b + k ) * x / ( a + two * k ) / & ( a + two * k + one ) end do t1 = zero do k = 20 , 1 , - 1 t1 = dk ( k ) / ( one + t1 ) end do ta = one / ( one + t1 ) rst = x ** a * ( one - x ) ** b / ( a * bt ) * ta else do k = 1 , 20 fk ( 2 * k ) = k * ( a - k ) * ( one - x ) / ( b + two * k - one ) / & ( b + two * k ) end do do k = 0 , 20 fk ( 2 * k + 1 ) = - ( b + k ) * ( a + b + k ) * ( one - x ) / ( b + two * k ) / & ( b + two * k + one ) end do t2 = zero do k = 20 , 1 , - 1 t2 = fk ( k ) / ( one + t2 ) end do tb = one / ( one + t2 ) rst = one - x ** a * ( one - x ) ** b / ( b * bt ) * tb end if end function ! ------------------------------------------------------------------------------ pure elemental function incomplete_beta ( a , b , x ) result ( rst ) !! Computes the incomplete beta function. !! !! The incomplete beta function is defind as: !!  \\beta(x;a,b) = \\int_{0}&#94;{x} t&#94;{a-1} (1 - t)&#94;{b-1} dt . !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) real ( real64 ), intent ( in ) :: a !! The first argument of the function. real ( real64 ), intent ( in ) :: b !! The second argument of the function. real ( real64 ), intent ( in ) :: x !! The upper limit of the integration. real ( real64 ) :: rst !! The value of the incomplete beta function. ! Process rst = beta ( a , b ) * regularized_beta ( a , b , x ) end function ! ------------------------------------------------------------------------------ ! REF: https://people.math.sc.edu/Burkardt/f_src/special_functions/special_functions.f90 pure elemental function incomplete_gamma_upper ( a , x ) result ( rst ) !! Computes the upper incomplete gamma function. !! !! The upper incomplete gamma function is defined as: !!  \\Gamma(a, x) = \\int_{x}&#94;{\\infty} t&#94;{a-1} e&#94;{-t} \\,dt  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function) real ( real64 ), intent ( in ) :: a !! The coefficient value. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: ten = 1.0d1 ! Local Variables real ( real64 ) :: ga , gin , gip , r , s , t0 , xam , small integer ( int32 ) :: k ! Process small = ten * epsilon ( small ) xam = - x + a * log ( x ) if ( xam > 7.0d2 . or . a > 1.7d2 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) return end if if ( x == zero ) then rst = gamma ( a ) else if ( x <= one + a ) then s = one / a r = s do k = 1 , 60 r = r * x / ( a + k ) s = s + r if ( abs ( r / s ) < small ) then exit end if end do gin = exp ( xam ) * s ga = gamma ( a ) gip = gin / ga rst = ga - gin else if ( one + a < x ) then t0 = zero do k = 60 , 1 , - 1 t0 = ( k - a ) / ( one + k / ( x + t0 )) end do rst = exp ( xam ) / ( x + t0 ) end if end function ! ------------------------------------------------------------------------------ pure elemental function incomplete_gamma_lower ( a , x ) result ( rst ) !! Computes the lower incomplete gamma function. !! !! The lower incomplete gamma function is defined as: !!  \\gamma(a, x) = \\int_{0}&#94;{x} t&#94;{a-1} e&#94;{-t} \\,dt  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function) real ( real64 ), intent ( in ) :: a !! The coefficient value. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: ten = 1.0d1 ! Local Variables real ( real64 ) :: ga , gim , r , s , t0 , xam , small integer ( int32 ) :: k ! Process small = ten * epsilon ( small ) xam = - x + a * log ( x ) if ( xam > 7.0d2 . or . a > 1.7d2 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) return end if if ( x == zero ) then rst = 0.0d0 else if ( x <= one + a ) then s = one / a r = s do k = 1 , 60 r = r * x / ( a + k ) s = s + r if ( abs ( r / s ) < small ) then exit end if end do rst = exp ( xam ) * s else if ( one + a < x ) then t0 = zero do k = 60 , 1 , - 1 t0 = ( k - a ) / ( one + k / ( x + t0 )) end do gim = exp ( xam ) / ( x + t0 ) ga = gamma ( a ) rst = ga - gim end if end function ! ------------------------------------------------------------------------------ pure elemental function digamma ( x ) result ( rst ) !! Computes the digamma function. !! !! The digamma function is defined as: !!  \\psi(x) = !! \\frac{d}{dx}\\left( \\ln \\left( \\Gamma \\left( x \\right) \\right) !! \\right)  !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Digamma_function) real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The function value. ! Parameters real ( real64 ), parameter :: c = 8.5d0 real ( real64 ), parameter :: euler_mascheroni = 0.57721566490153286060d0 ! Local Variables real ( real64 ) :: r , x2 , nan ! REF: ! - https://people.sc.fsu.edu/~jburkardt/f_src/asa103/asa103.f90 ! If x <= 0.0 if ( x <= 0.0 ) then nan = ieee_value ( nan , IEEE_QUIET_NAN ) rst = nan return end if ! Approximation for a small argument if ( x <= 1.0d-6 ) then rst = - euler_mascheroni - 1.0d0 / x + 1.6449340668482264365d0 * x return end if ! Process rst = 0.0d0 x2 = x do while ( x2 < c ) rst = rst - 1.0d0 / x2 x2 = x2 + 1.0d0 end do r = 1.0d0 / x2 rst = rst + log ( x2 ) - 0.5d0 * r r = r * r rst = rst & - r * ( 1.0d0 / 1 2.0d0 & - r * ( 1.0d0 / 12 0.0d0 & - r * ( 1.0d0 / 25 2.0d0 & - r * ( 1.0d0 / 24 0.0d0 & - r * ( 1.0d0 / 13 2.0d0 ) & )))) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_special_functions.f90.html"},{"title":"fstats_experimental_design.f90 – FSTATS","text":"Source Code module fstats_experimental_design use iso_fortran_env use fstats_errors implicit none private public :: get_full_factorial_matrix_size public :: full_factorial contains ! ------------------------------------------------------------------------------ subroutine get_full_factorial_matrix_size ( vars , m , n , err ) !! Computes the appropriate size for a full-factorial design table. integer ( int32 ), intent ( in ) :: vars (:) !! An M-element array containing the M factors to study.  Each !! of the M entries to the array is expected to contain the !! number of options for that particular factor to explore. !! This value must be greater than or equal to 1. integer ( int32 ), intent ( out ) :: m !! The number of rows for the table. integer ( int32 ), intent ( out ) :: n !! The number of columns for the table. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_INVALID_INPUT_ERROR: Occurs if any items in vars are !!      less than 1. ! Local Variables integer ( int32 ) :: i class ( errors ), pointer :: errmgr type ( errors ), target :: deferr character ( len = 256 ) :: errmsg ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if m = 0 n = 0 ! Ensure every value is greater than 1 do i = 1 , size ( vars ) if ( vars ( i ) < 1 ) then write ( errmsg , 100 ) \"A value less than 1 was found at index \" , & i , \" of the input array.  All values must be greater \" // & \"than or equal to 1.\" call errmgr % report_error ( \"get_full_factorial_matrix_size\" , & trim ( errmsg ), FS_INVALID_INPUT_ERROR ) return end if end do ! Process m = product ( vars ) n = size ( vars ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine full_factorial ( vars , tbl , err ) !! Computes a table with values scaled from 1 to N describing a !! full-factorial design. !! !! ```fortran !! program example !!     use iso_fortran_env !!     use fstats !!     implicit none !! !!     ! Local Variables !!     integer(int32) :: i, vars(3), tbl(24, 3) !! !!     ! Define the number of design points for each of the 3 factors to study !!     vars = [2, 4, 3] !! !!     ! Determine the design table !!     call full_factorial(vars, tbl) !! !!     ! Display the table !!     do i = 1, size(tbl, 1) !!         print *, tbl(i,:) !!     end do !! end program !! ``` !! The above program produces the following output. !! ```text !! 1           1           1 !! 1           1           2 !! 1           1           3 !! 1           2           1 !! 1           2           2 !! 1           2           3 !! 1           3           1 !! 1           3           2 !! 1           3           3 !! 1           4           1 !! 1           4           2 !! 1           4           3 !! 2           1           1 !! 2           1           2 !! 2           1           3 !! 2           2           1 !! 2           2           2 !! 2           2           3 !! 2           3           1 !! 2           3           2 !! 2           3           3 !! 2           4           1 !! 2           4           2 !! 2           4           3 !! ``` integer ( int32 ), intent ( in ) :: vars (:) !! An M-element array containing the M factors to study. !! Each of the M entries to the array is expected to contain !! the number of options for that particular factor to explore. !! This value must be greater than or equal to 1. integer ( int32 ), intent ( out ) :: tbl (:,:) !! A table where the design will be written.  Use !! get_full_factorial_matrix_size to determine the appropriate !! table size. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_INVALID_INPUT_ERROR: Occurs if any items in vars are !!      less than 1. !! - FS_ARRAY_SIZE_ERROR: Occurs if tbl is not properly sized. ! Local Variables integer ( int32 ) :: i , col , stride , last , val , m , n class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if ! Verify the size of the input table call get_full_factorial_matrix_size ( vars , m , n , errmgr ) if ( errmgr % has_error_occurred ()) return if ( size ( tbl , 1 ) /= m . or . size ( tbl , 2 ) /= n ) then call report_matrix_size_error ( errmgr , \"full_factorial\" , & \"tbl\" , m , n , size ( tbl , 1 ), size ( tbl , 2 )) return end if ! Process do col = 1 , n stride = 1 if ( col /= n ) stride = product ( vars ( col + 1 : n )) val = 1 do i = 1 , m , stride last = i + stride - 1 tbl ( i : last , col ) = val val = val + 1 if ( val > vars ( col )) val = 1 end do end do end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_experimental_design.f90.html"},{"title":"fstats_allan.f90 – FSTATS","text":"Source Code module fstats_allan use iso_fortran_env use fstats_errors implicit none private public :: allan_variance contains ! ------------------------------------------------------------------------------ ! REF: Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, ! Viraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm for ! Fully Overlapped Allan Variance and Total Variance for Analysis and Modeling ! of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. ! 10.1109/LSENS.2018.2829799. ! ! https://www.researchgate.net/publication/324738301_A_Fast_Parallel_Algorithm_for_Fully_Overlapped_Allan_Variance_and_Total_Variance_for_Analysis_and_Modeling_of_Noise_in_Inertial_Sensors ! https://github.com/shrikanth95/Fast-Parallel-Fully-Overlapped-Allan-Variance-and-Total-Variance/blob/master/fast_FOAV.m function allan_variance ( x , dt , err ) result ( rst ) !! Computes the Allan variance of a data set. !! !! Remarks !! !! This implementation computes the fully overlapped Allan variance !! using the method presented by Yadav et. al. !! !! Yadav, Shrikanth & Shastri, Saurav & Chakravarthi, Ghanashyam & Kumar, !! Viraj & Rao, Divya & Agrawal, Vinod. (2018). A Fast, Parallel Algorithm !! for Fully Overlapped Allan Variance and Total Variance for Analysis and !! Modeling of Noise in Inertial Sensors. IEEE Sensors Letters. PP. 1-1. !! 10.1109/LSENS.2018.2829799. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element data set to analyze. real ( real64 ), intent ( in ), optional :: dt !! An optional input specifying the time increment between !! samples in x.  If not specified, this value is set to 1. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. real ( real64 ), allocatable , dimension (:,:) :: rst !! An M-by-2 array containing the results where M is N / 2 - 1 !! if N is even; else, M is (N - 1) / 2 - 1 if N is odd.  The !! first column contains the averaging times associated with !! the M results stored in the second column. ! Local Variables class ( errors ), pointer :: errmgr type ( errors ), target :: deferr integer ( int32 ) :: flag , j , m , n , limit , nr real ( real64 ), allocatable , dimension (:) :: tall1 , tall2 real ( real64 ) :: temp , deltaT ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( dt )) then deltaT = dt else deltaT = 1.0d0 end if ! Initialization n = size ( x ) limit = n nr = floor ( 0.5 * n ) - 1 allocate ( tall1 ( n - 1 ), source = x (: n - 1 ), stat = flag ) if ( flag == 0 ) allocate ( tall2 ( n - 1 ), source = x ( 2 : n )) if ( flag == 0 ) allocate ( rst ( nr , 2 ), source = 0.0d0 ) if ( flag /= 0 ) go to 10 ! Process do m = 1 , nr temp = 0.0d0 do j = 1 , limit - 1 temp = temp + ( tall2 ( j ) - tall1 ( j )) ** 2 tall1 ( j ) = tall1 ( j ) + x ( min ( n , m + j )) tall2 ( j ) = tall2 ( min ( n - 1 , j + 1 )) + x ( min ( n , 2 * m + j + 1 )) end do limit = limit - 2 rst ( m , 1 ) = dt * m rst ( m , 2 ) = temp / ( 2.0d0 * ( n - 2 * m + 1 ) * m ** 2 ) end do ! End return ! Memory Error Handling 10 continue call report_memory_error ( errmgr , \"allan_variance\" , flag ) return end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_allan.f90.html"},{"title":"fstats_bootstrap.f90 – FSTATS","text":"Source Code module fstats_bootstrap use iso_fortran_env use fstats_errors use omp_lib use fstats_distributions use fstats_descriptive_statistics use fstats_special_functions use fstats_regression use linalg , only : sort implicit none private public :: bootstrap_resampling_routine public :: bootstrap_statistic_routine public :: random_resample public :: scaled_random_resample public :: bootstrap_statistics public :: bootstrap public :: bootstrap_regression_statistics public :: bootstrap_linear_least_squares public :: bootstrap_nonlinear_least_squares ! REFERENCES: ! - https://medium.com/@m21413108/bootstrapping-maximum-entropy-non-parametric-boot-python-3b1e23ea589d ! - https://cran.r-project.org/web/packages/meboot/vignettes/meboot.pdf ! - https://gist.github.com/christianjauregui/314456688a3c2fead43a48be3a47dad6 type bootstrap_regression_statistics !! A container for regression-related statistical information as !! computed in a bootstrap, or equivalent, calculation. real ( real64 ) :: standard_error !! The standard error for the model coefficient. real ( real64 ) :: t_statistic !! The T-statistic for the model coefficient. !! !!  t_o = \\frac{ \\beta_{i} }{E_{s}(\\beta_{i})}  real ( real64 ) :: probability !! The probability that the coefficient is not statistically !! important.  A statistically important coefficient will have a !! low probability (p-value), typically 0.05 or lower; however, a !! p-value of up to ~0.2 may be acceptable dependent upon the !! problem.  Typically any p-value larger than ~0.2 indicates the !! parameter is not statistically important for the model. !! !!  p = t_{|t_o|, df_{residual}}  real ( real64 ) :: upper_confidence_interval !! The upper limit of the confidence interval for the parameter. real ( real64 ) :: lower_confidence_interval !! The lower limit of the confidence interval for the parameter. end type type bootstrap_statistics !! A collection of statistics resulting from the bootstrap process. real ( real64 ) :: statistic_value !! The value of the statistic of interest. real ( real64 ) :: upper_confidence_interval !! The upper confidence limit on the statistic. real ( real64 ) :: lower_confidence_interval !! The lower confidence limit on the statistic. real ( real64 ) :: bias !! The bias in the statistic. real ( real64 ) :: standard_error !! The standard error of the statistic. real ( real64 ), allocatable , dimension (:) :: population !! An array of the population values generated by the bootstrap !! process. end type interface subroutine bootstrap_resampling_routine ( x , xn ) !! Defines the signature of a subroutine used to compute a !! resampling of data for bootstrapping purposes. use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be !! written. end subroutine function bootstrap_statistic_routine ( x ) result ( rst ) !! Defines the signature of a function for computing the desired !! bootstrap statistic. use iso_fortran_env , only : real64 real ( real64 ), intent ( in ), dimension (:) :: x !! The array of data to analyze. real ( real64 ) :: rst !! The resulting statistic. end function end interface contains ! ****************************************************************************** ! RESAMPLING ! ------------------------------------------------------------------------------ subroutine random_resample ( x , xn ) !! Random resampling, with replacement, based upon a normal distribution. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be written. ! Parameters real ( real64 ), parameter :: scale = 1.25d0 ! Local Variables integer ( int32 ) :: i , n real ( real64 ) :: xmin , xmax , rng ! Process n = size ( x ) xmin = x ( 1 ) xmax = x ( 1 ) do i = 2 , n xmin = min ( xmin , x ( i )) xmax = max ( xmax , x ( i )) end do rng = ( xmax - xmin ) call random_number ( xn ) xn = xn * rng + xmin end subroutine ! ------------------------------------------------------------------------------ subroutine scaled_random_resample ( x , xn ) !! A random resampling, scaled by the standard deviation of the original !! data, but based upon a normal distribution. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element array to resample. real ( real64 ), intent ( out ), dimension ( size ( x )) :: xn !! An N-element array where the resampled data set will be written. ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: n real ( real64 ) :: eps ! Process n = size ( x ) eps = standard_deviation ( x ) / sqrt ( real ( n , real64 )) call random_number ( xn ) xn = eps * ( xn - half ) + x end subroutine ! ****************************************************************************** ! BOOTSTRAPPING ! ------------------------------------------------------------------------------ function bootstrap ( stat , x , method , nsamples , alpha ) result ( rst ) !! Performs a bootstrap calculation on the supplied data set for the given !! statistic.  The default implementation utlizes a random resampling with !! replacement.  Other resampling methods may be defined by specifying an !! appropriate routine by means of the method input. procedure ( bootstrap_statistic_routine ), pointer , intent ( in ) :: stat !! The routine used to compute the desired statistic. real ( real64 ), intent ( in ), dimension (:) :: x !! The N-element data set. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. integer ( int32 ), intent ( in ), optional :: nsamples !! An optional input, that if supplied, specifies the number of !! resampling runs to perform.  The default is 10 000. real ( real64 ), intent ( in ), optional :: alpha !! An optional input, that if supplied, defines the significance level !! to use for the analysis.  The default is 0.05. type ( bootstrap_statistics ) :: rst !! The resulting bootstrap_statistics type containing the confidence !! intervals, bias, standard error, etc. for the analyzed statistic. ! Parameters real ( real64 ), parameter :: half = 0.5d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , i1 , i2 , n , ns real ( real64 ) :: a real ( real64 ), allocatable , dimension (:) :: xn procedure ( bootstrap_resampling_routine ), pointer :: resample ! Initialization n = size ( x ) if ( present ( method )) then resample => method else resample => random_resample end if if ( present ( nsamples )) then ns = nsamples else ns = 10000 end if if ( present ( alpha )) then a = alpha else a = p05 end if allocate ( rst % population ( ns )) i1 = floor ( half * a * ns , int32 ) i2 = ns - i1 + 1 ! Analyze the basic data set rst % statistic_value = stat ( x ) rst % population ( 1 ) = rst % statistic_value ! Resampling Process #ifdef USEOPENMP ! Use OpenMP to run operations in parallel !$OMP PARALLEL DO PRIVATE(xn) SHARED(rst) do i = 2 , ns ! Per-thread memory allocation if (. not . allocated ( xn )) allocate ( xn ( n )) ! Resample the data call resample ( x , xn ) ! Compute the statistic rst % population ( i ) = stat ( xn ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( xn ( n )) do i = 2 , ns ! Resample the data call resample ( x , xn ) ! Compute the statistic for the resampled data rst % population ( i ) = stat ( xn ) end do #endif ! Compute the relevant quantities on the resampled statistic call sort ( rst % population , . true .) rst % upper_confidence_interval = rst % population ( i2 ) rst % lower_confidence_interval = rst % population ( i1 ) rst % bias = mean ( rst % population ) - rst % statistic_value rst % standard_error = standard_deviation ( rst % population ) end function ! ****************************************************************************** ! LINEAR REGRESSION ! ------------------------------------------------------------------------------ subroutine bootstrap_linear_least_squares ( order , intercept , x , y , & coeffs , ymod , resid , nsamples , stats , bias , alpha , method , bscoeffs , err ) !! Computes a linear least-squares regression to fit a set of data. !! Bootstrapping is utilized to gain insight into the quality of !! the fit.  Resampling for the bootstrap process is a random resampling !! with replacement process with the range of values limited by the !! standard deviation of the original data set. integer ( int32 ), intent ( in ) :: order !! The order of the equation to fit.  This value must be at !! least one (linear equation), but can be higher as desired, !! as long as there is sufficient data. logical , intent ( in ) :: intercept !! Set to true if the intercept is being computed as part of !! the regression; else, false. real ( real64 ), intent ( in ), dimension (:) :: x !! An N-element array containing the independent variable !! measurement points. real ( real64 ), intent ( in ), dimension (:) :: y !! An N-element array containing the dependent variable !! measurement points. real ( real64 ), intent ( out ), dimension (:) :: coeffs !! An ORDER+1 element array where the coefficients will !! be written. real ( real64 ), intent ( out ), dimension (:) :: ymod !! An N-element array where the modeled data will be written. real ( real64 ), intent ( out ), dimension (:) :: resid !! An N-element array where the residual error data will be !! written (modeled - actual). integer ( int32 ), intent ( in ), optional :: nsamples !! The number of bootstrapping samples to utilize. type ( bootstrap_regression_statistics ), intent ( out ), optional , & dimension (:) :: stats !! An M-element array of bootstrap_regression_statistics items !! where M = ORDER + 1 when intercept is set to true; however, !! if intercept is set to false, M = ORDER. real ( real64 ), intent ( out ), optional , dimension (:) :: bias !! An ORDER+1 element array where an estimate of the bias of !! each coefficient is returned based upon the results of the !! bootstrapping analysis.  The bias is computed as the difference !! between the mean of the boostrap population results for the given !! parameter and the original estimate of the given parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. real ( real64 ), intent ( out ), optional , allocatable , target , dimension (:,:) :: bscoeffs !! An optional, allocatable matrix, containing the bootstrap !! distributions for each parameter stored in each row of the matrix !! such that the resulting matrix is NCOEFFS -by- NSAMPLES. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_INVALID_INPUT_ERROR: Occurs if order is less than 1. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , j , n , ns , nc , ncoeffs , flag , nthreads , thread real ( real64 ) :: alph real ( real64 ), allocatable , dimension (:) :: fLocal , yLocal , rLocal real ( real64 ), allocatable , target , dimension (:,:) :: coeffstorage real ( real64 ), pointer , dimension (:,:) :: allcoeffs class ( errors ), pointer :: errmgr type ( errors ), target :: deferr procedure ( bootstrap_resampling_routine ), pointer :: resample ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( nsamples )) then ns = nsamples else ns = 1000 end if if ( present ( alpha )) then alph = alpha else alph = p05 end if if ( present ( method )) then resample => method else resample => scaled_random_resample end if n = size ( x ) ncoeffs = order + 1 nc = order if ( intercept ) nc = nc + 1 nthreads = omp_get_num_threads () ! Compute the fit call linear_least_squares ( order , intercept , x , y , coeffs , & ymod , resid , alpha = alph , err = errmgr ) if ( errmgr % has_error_occurred ()) return ! Memory Allocations if ( present ( bscoeffs )) then allocate ( bscoeffs ( ncoeffs , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"bootstrap_linear_least_squares\" , & flag ) return end if allcoeffs => bscoeffs else allocate ( coeffstorage ( ncoeffs , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"bootstrap_linear_least_squares\" , & flag ) return end if allcoeffs => coeffstorage end if allcoeffs (:, 1 ) = coeffs ! Cycle over each data set and perform the fit #ifdef USEOPENMP !$OMP PARALLEL DO PRIVATE(fLocal, yLocal, rLocal) SHARED(allcoeffs) do i = 2 , ns ! Get the current thread number ! The +1 is because OpenMP is zero-based for thread numbering thread = omp_get_thread_num () + 1 ! Allocate local arrays on a per-thread basis if (. not . allocated ( fLocal )) allocate ( fLocal ( n )) if (. not . allocated ( yLocal )) allocate ( yLocal ( n )) if (. not . allocated ( rLocal )) allocate ( rLocal ( n )) ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call linear_least_squares ( order , intercept , x , yLocal , & allcoeffs (:, i ), fLocal , rLocal , alpha = alph ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( fLocal ( n ), yLocal ( n ), rLocal ( n )) do i = 2 , ns ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call linear_least_squares ( order , intercept , x , yLocal , & allcoeffs (:, i ), fLocal , rLocal , alpha = alph ) end do #endif ! Perform statistics calculations, if needed if ( present ( stats )) then call compute_stats ( coeffs , allcoeffs , alph , intercept , stats ) end if ! Compute the bias for each parameter, if needed if ( present ( bias )) then ! Verify the size of the array if ( size ( bias ) /= ncoeffs ) then call report_array_size_error ( errmgr , & \"bootstrap_linear_least_squares\" , \"bias\" , ncoeffs , size ( bias )) return end if ! Perform the calculations do i = 1 , ncoeffs bias ( i ) = mean ( allcoeffs ( i ,:)) - coeffs ( i ) end do end if end subroutine ! ****************************************************************************** ! NONLINEAR REGRESSION ! ------------------------------------------------------------------------------ subroutine bootstrap_nonlinear_least_squares ( fun , x , y , params , ymod , resid , & nsamples , weights , maxp , minp , stats , alpha , controls , settings , info , & bias , method , bscoeffs , err ) !! Performs a nonlinear regression to fit a model using a version !! of the Levenberg-Marquardt algorithm.  Bootstrapping is utilized to gain !! insight into the quality of the fit.  Resampling for the bootstrap !! process is a random resampling with replacement process with the !! range of values limited by the standard deviation of the original !! data set. procedure ( regression_function ), intent ( in ), pointer :: fun !! A pointer to the regression_function to evaluate. real ( real64 ), intent ( in ) :: x (:) !! The M-element array containing independent data. real ( real64 ), intent ( in ) :: y (:) !! The M-element array containing dependent data. real ( real64 ), intent ( inout ) :: params (:) !! On input, the N-element array containing the initial estimate !! of the model parameters.  On output, the computed model !! parameters. real ( real64 ), intent ( out ) :: ymod (:) !! An M-element array where the modeled dependent data will !! be written. real ( real64 ), intent ( out ) :: resid (:) !! An M-element array where the model residuals will be !! written. integer ( int32 ), intent ( in ), optional :: nsamples !! The number of bootstrapping samples to utilize. real ( real64 ), intent ( in ), optional , target :: weights (:) !! An optional M-element array allowing the weighting of !! individual points. real ( real64 ), intent ( in ), optional , target :: maxp (:) !! An optional N-element array that can be used as upper limits !! on the parameter values.  If no upper limit is requested for !! a particular parameter, utilize a very large value.  The !! internal default is to utilize huge() as a value. real ( real64 ), intent ( in ), optional , target :: minp (:) !! An optional N-element array that can be used as lower limits !! on the parameter values.  If no lower limit is requested for !! a particalar parameter, utilize a very large magnitude, but !! negative, value.  The internal default is to utilize -huge() !! as a value. type ( bootstrap_regression_statistics ), intent ( out ), optional :: stats (:) !! An optional N-element array that, if supplied, will be used !! to return statistics about the fit for each parameter. real ( real64 ), intent ( in ), optional :: alpha !! The significance level at which to evaluate the confidence !! intervals.  The default value is 0.05 such that a 95% !! confidence interval is calculated. type ( iteration_controls ), intent ( in ), optional :: controls !! An optional input providing custom iteration controls. type ( lm_solver_options ), intent ( in ), optional :: settings !! An optional input providing custom settings for the solver. type ( convergence_info ), intent ( out ), optional , target :: info !! An optional output that can be used to gain information about !! the iterative solution and the nature of the convergence. real ( real64 ), intent ( out ), optional , dimension (:) :: bias !! An optional N-element array that, if supplied, will be used to !! provide an estimate of the bias of each model parameter based upon !! the results of the bootstrapping analysis.  The bias is computed as !! the difference between the mean of the boostrap population results !! for the given parameter and the original estimate of the given !! parameter. procedure ( bootstrap_resampling_routine ), pointer , intent ( in ), optional :: method !! An optional pointer to the method to use for resampling of the data. !! If no method is supplied, a random resampling is utilized. real ( real64 ), intent ( out ), optional , allocatable , target , dimension (:,:) :: bscoeffs !! An optional, allocatable matrix, containing the bootstrap !! distributions for each parameter stored in each row of the matrix !! such that the resulting matrix is NCOEFFS -by- NSAMPLES. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      properly sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation !!      error. !! - FS_UNDERDEFINED_PROBLEM_ERROR: Occurs if the problem posed !!      is underdetetermined (M < N). !! - FS_TOLERANCE_TOO_SMALL_ERROR: Occurs if any supplied !!      tolerances are too small to be practical. !! - FS_TOO_FEW_ITERATION_ERROR: Occurs if too few iterations !!      are allowed. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p05 = 5.0d-2 ! Local Variables integer ( int32 ) :: i , n , ns , nparams , flag real ( real64 ) :: alph real ( real64 ), allocatable , dimension (:) :: fLocal , yLocal , rLocal real ( real64 ), allocatable , target , dimension (:,:) :: coeffstorage real ( real64 ), pointer , dimension (:,:) :: allcoeffs procedure ( bootstrap_resampling_routine ), pointer :: resample class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if if ( present ( nsamples )) then ns = nsamples else ns = 1000 end if if ( present ( alpha )) then alph = alpha else alph = p05 end if if ( present ( method )) then resample => method else resample => scaled_random_resample end if n = size ( x ) nparams = size ( params ) ! Compute the fit call nonlinear_least_squares ( fun , x , y , params , ymod , resid , & weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info , err = err ) ! Memory Allocations if ( present ( bscoeffs )) then allocate ( bscoeffs ( nparams , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , flag ) return end if allcoeffs => bscoeffs else allocate ( coeffstorage ( nparams , ns ), source = zero , stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , flag ) return end if allcoeffs => coeffstorage end if allcoeffs (:, 1 ) = params ! Define initial guesses for each step.  Base upon the results of the ! initial analysis as this should provide a strong starting point for ! subsequent analysis do i = 1 , nparams allcoeffs ( i ,:) = params ( i ) end do ! Cycle over each data set and perform the fit #ifdef USEOPENMP !$OMP PARALLEL DO PRIVATE(fLocal, yLocal, rLocal) do i = 2 , ns ! Allocate local arrays on a per-thread basis if (. not . allocated ( fLocal )) allocate ( fLocal ( n )) if (. not . allocated ( yLocal )) allocate ( yLocal ( n )) if (. not . allocated ( rLocal )) allocate ( rLocal ( n )) ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call nonlinear_least_squares ( fun , x , yLocal , allcoeffs (:, i ), fLocal , & rLocal , weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info ) end do !$OMP END PARALLEL DO #else ! OpenMP is not available - run in a serial manner allocate ( fLocal ( n ), yLocal ( n ), rLocal ( n )) do i = 2 , ns ! Compute a random data set call resample ( y , yLocal ) ! Compute the fit of the perturbed data set call nonlinear_least_squares ( fun , x , yLocal , allcoeffs (:, i ), fLocal , & rLocal , weights = weights , maxp = maxp , minp = minp , alpha = alph , & controls = controls , settings = settings , info = info ) end do #endif ! Perform the statistics calculations, if needed if ( present ( stats )) then ! Verify the size of stats if ( size ( stats ) /= nparams ) then call report_array_size_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , \"stats\" , & nparams , size ( stats )) return end if ! Perform the calculations call compute_stats ( params , allcoeffs , alph , . true ., stats ) end if ! Compute the bias for each parameter, if needed if ( present ( bias )) then ! Verify the size of the array if ( size ( bias ) /= nparams ) then call report_array_size_error ( errmgr , & \"bootstrap_nonlinear_least_squares\" , \"bias\" , & nparams , size ( bias )) return end if ! Perform the calculations do i = 1 , nparams bias ( i ) = mean ( allcoeffs ( i ,:)) - params ( i ) end do end if end subroutine ! ****************************************************************************** ! PRIVATE HELPER ROUTINES ! ------------------------------------------------------------------------------ subroutine compute_stats ( mdl , coeffs , alpha , intercept , stats ) ! Arguments real ( real64 ), intent ( in ), dimension (:) :: mdl real ( real64 ), intent ( inout ), dimension (:,:) :: coeffs real ( real64 ), intent ( in ) :: alpha logical , intent ( in ) :: intercept type ( bootstrap_regression_statistics ), intent ( out ), dimension (:) :: stats ! Parameters real ( real64 ), parameter :: half = 0.5d0 ! Local Variables integer ( int32 ) :: i , j , i1 , i2 , ncoeffs , nc , nsamples real ( real64 ) :: ms type ( t_distribution ) :: dist ! Initialization ncoeffs = size ( coeffs , 1 ) nsamples = size ( coeffs , 2 ) nc = ncoeffs if (. not . intercept ) nc = ncoeffs - 1 i1 = floor ( half * alpha * nsamples , int32 ) i2 = nsamples - i1 + 1 dist % dof = real ( nsamples - nc ) ! Process j = 1 if ( intercept ) j = 0 do i = 1 , nc j = j + 1 ms = trimmed_mean ( coeffs ( j ,:), p = half * alpha ) ! As we have a distribution of mean values, the standard deviation ! of this population yields the standard error estimate for the ! overall problem stats ( i )% standard_error = standard_deviation ( coeffs ( j ,:)) ! As before, this is a distribution of mean values.  The CI can ! be directly estimated by considering the values of the bottom ! alpha/2 and top alpha/2 terms. stats ( i )% upper_confidence_interval = coeffs ( j , i2 ) stats ( i )% lower_confidence_interval = coeffs ( j , i1 ) ! Compute the remaining parameters stats ( i )% t_statistic = mdl ( j ) / stats ( i )% standard_error stats ( i )% probability = regularized_beta ( half * dist % dof , half , & dist % dof / ( dist % dof + ( stats ( i )% t_statistic ) ** 2 )) end do end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_bootstrap.f90.html"},{"title":"fstats_anova.f90 – FSTATS","text":"Source Code module fstats_anova use iso_fortran_env use ieee_arithmetic use fstats_special_functions use fstats_descriptive_statistics use ferror use fstats_errors implicit none private public :: anova_factor public :: single_factor_anova_table public :: two_factor_anova_table public :: anova type anova_factor !! Defines an ANOVA factor result. real ( real64 ) :: dof !! The number of degrees of freedome. real ( real64 ) :: variance !! The estimate of variance. real ( real64 ) :: sum_of_squares !! The sum of the squares. real ( real64 ) :: f_statistic !! The F-statistic. real ( real64 ) :: probability !! The variance probability term. end type type single_factor_anova_table !! Defines a single-factor ANOVA results table. type ( anova_factor ) :: main_factor !! The main, or main factor, results. type ( anova_factor ) :: within_factor !! The within-treatement (error) results. real ( real64 ) :: total_dof !! The total number of degrees of freedom. real ( real64 ) :: total_sum_of_squares !! The total sum of squares. real ( real64 ) :: total_variance !! The total variance estimate. real ( real64 ) :: overall_mean !! The overall mean value. end type type two_factor_anova_table !! Defines a two-factor ANOVA results table. type ( anova_factor ) :: main_factor_1 !! The first main-factor results. type ( anova_factor ) :: main_factor_2 !! The second main-factor results. type ( anova_factor ) :: interaction !! The interaction effects. type ( anova_factor ) :: within_factor !! The within (error) factor results. real ( real64 ) :: total_dof !! The total number of degrees of freedom. real ( real64 ) :: total_sum_of_squares !! The total sum of squares. real ( real64 ) :: total_variance !! The total variance estimate. real ( real64 ) :: overall_mean !! The overall mean value. end type interface anova !! Performs an analysis of variance (ANOVA) on the supplied data !! set. !! !! The following example illustrates a single-factor ANOVA on a !! data set. !! ```fortran !! program example !!     use iso_fortran_env !!     use fstats !!     implicit none !! !!     ! Local Variables !!     character, parameter :: tab = achar(9) !!     real(real64) :: x(10, 2) !!     type(single_factor_anova_table) :: tbl !! !!     ! Define the data !!     x = reshape( & !!         [ & !!             3.086d3, 3.082d3, 3.069d3, 3.072d3, 3.045d3, 3.070d3, 3.079d3, & !!             3.050d3, 3.062d3, 3.062d3, 3.075d3, 3.061d3, 3.063d3, 3.038d3, & !!             3.070d3, 3.062d3, 3.070d3, 3.049d3, 3.042d3, 3.063d3 & !!         ], & !!         [10, 2] & !!     ) !! !!     ! Perform the ANOVA !!     tbl = anova(x) !! !!     ! Print out the table !!     print '(A)', \"Description\" // tab // \"DOF\" // tab // \"Sum of Sq.\" // & !!         tab // \"Variance\" // tab // \"F-Stat\" // tab // \"P-Value\" !!     print '(AF2.0AF5.1AF5.1AF5.3AF5.3)', \"Main Factor: \" // tab, & !!         tbl%main_factor%dof, tab, & !!         tbl%main_factor%sum_of_squares, tab // tab, & !!         tbl%main_factor%variance, tab // tab, & !!         tbl%main_factor%f_statistic, tab, & !!         tbl%main_factor%probability !! !!     print '(AF3.0AF6.1AF5.1)', \"Within: \" // tab, & !!         tbl%within_factor%dof, tab, & !!         tbl%within_factor%sum_of_squares, tab // tab, & !!         tbl%within_factor%variance !! !!     print '(AF3.0AF6.1AF5.1)', \"Total: \" // tab // tab, & !!         tbl%total_dof, tab, & !!         tbl%total_sum_of_squares, tab // tab, & !!         tbl%total_variance !! !!     print '(AF6.1)', \"Overall Mean: \", tbl%overall_mean !! end program !! ``` !! The above program produces the following output. !! ```text !! Description     DOF     Sum of Sq.      Variance        F-Stat  P-Value !! Main Factor:    1.      352.8           352.8           2.147   0.160 !! Within:         18.     2958.2          164.3 !! Total:          19.     3311.0          174.3 !! Overall Mean: 3063.5 !! ``` !! !! See Also !! !! - [Wikipedia](https://en.wikipedia.org/wiki/Analysis_of_variance) !! - [SPC Excel Single Factor ANOVA](https://www.spcforexcel.com/knowledge/root-cause-analysis/single-factor-anova) !! - [SPC Excel Gage R&R](https://www.spcforexcel.com/knowledge/measurement-systems-analysis/anova-gage-rr-part-1) !! - [SPC Excel Understanding Regression Statistics](https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1) !! - [NIST - Two Way ANOVA](https://www.itl.nist.gov/div898/handbook/prc/section4/prc427.htm) module procedure :: anova_1_factor module procedure :: anova_2_factor module procedure :: anova_model_fit end interface contains ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/root-cause-analysis/single-factor-anova function anova_1_factor ( x ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. real ( real64 ), intent ( in ) :: x (:,:) !! An M-by-N matrix containing the M replications of the N test !! points of interest. type ( single_factor_anova_table ) :: rst !! A single_factor_anova_table instance containing the ANOVA results. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 ! Local Variables integer ( int32 ) :: j , a , n , nt real ( real64 ) :: sum_all , tssq , essq , bssq ! Initialization a = size ( x , 2 ) nt = size ( x , 1 ) n = nt * a rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Determine the degrees of freedom rst % main_factor % dof = a - 1 rst % within_factor % dof = n - a rst % total_dof = n - 1 ! Quick Return if ( a == 1 . or . nt == 1 ) then rst % main_factor % sum_of_squares = zero rst % main_factor % variance = zero rst % main_factor % f_statistic = zero rst % main_factor % probability = zero rst % within_factor % sum_of_squares = zero rst % within_factor % variance = zero rst % total_variance = variance ( pack ( x , . true .)) rst % total_sum_of_squares = rst % total_variance * rst % total_dof rst % overall_mean = mean ( pack ( x , . true .)) return end if ! Compute the sum of squares for all factors sum_all = sum ( x ) tssq = sum ( x ** 2 ) - ( sum_all ** 2 / n ) bssq = zero do j = 1 , a bssq = bssq + sum ( x (:, j )) ** 2 end do bssq = ( bssq / nt ) - ( sum_all ** 2 / n ) essq = tssq - bssq rst % main_factor % sum_of_squares = bssq rst % within_factor % sum_of_squares = essq rst % total_sum_of_squares = tssq ! Compute the variance terms rst % main_factor % variance = bssq / rst % main_factor % dof rst % within_factor % variance = essq / rst % within_factor % dof rst % total_variance = tssq / rst % total_dof ! Compute the overall mean rst % overall_mean = mean ( pack ( x , . true .)) ! Compute the F-statistic and probability term call anova_probability ( & rst % main_factor % variance , & rst % within_factor % variance , & rst % main_factor % dof , & rst % within_factor % dof , & rst % main_factor % f_statistic , & rst % main_factor % probability & ) end function ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/measurement-systems-analysis/anova-gage-rr-part-1 ! REF: https://www.itl.nist.gov/div898/handbook/prc/section4/prc427.htm ! Data set is expected as a 3D array with each of the K pages containing the R !   treatments of N tests such that the array size is N-by-R-by-K function anova_2_factor ( x ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. real ( real64 ), intent ( in ) :: x (:,:,:) !! An M-by-N-by-K array containing the M replications of the !! N first factor results, and the K second factor results. type ( two_factor_anova_table ) :: rst !! A two_factor_anova_table instance containing the ANOVA results. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 ! Local Variables integer ( int32 ) :: i , j , jj , k , r , n real ( real64 ) :: factorMean , sum_all real ( real64 ), allocatable :: xpack (:) ! Initialization n = size ( x , 3 ) k = size ( x , 2 ) r = size ( x , 1 ) rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Quick Return if ( k == 1 ) then ! This is a one-factor anova end if ! Determine the number of DOF rst % main_factor_1 % dof = k - one rst % main_factor_2 % dof = n - 1 rst % interaction % dof = ( k - 1 ) * ( n - 1 ) rst % within_factor % dof = n * k * ( r - 1 ) rst % total_dof = n * k * r - 1 ! Compute the overall mean, sum of squares, and variance xpack = pack ( x , . true .) rst % overall_mean = mean ( xpack ) rst % total_sum_of_squares = sum (( xpack - rst % overall_mean ) ** 2 ) rst % total_variance = rst % total_sum_of_squares / rst % total_dof ! Compute factor 1 results rst % main_factor_1 % sum_of_squares = zero do i = 1 , k factorMean = mean ( pack ( x (:, i ,:), . true .)) rst % main_factor_1 % sum_of_squares = rst % main_factor_1 % sum_of_squares + & ( factorMean - rst % overall_mean ) ** 2 end do rst % main_factor_1 % sum_of_squares = n * r * rst % main_factor_1 % sum_of_squares rst % main_factor_1 % variance = rst % main_factor_1 % sum_of_squares / & rst % main_factor_1 % dof ! Compute factor 2 results rst % main_factor_2 % sum_of_squares = zero do i = 1 , n factorMean = mean ( pack ( x (:,:, i ), . true .)) rst % main_factor_2 % sum_of_squares = rst % main_factor_2 % sum_of_squares + & ( factorMean - rst % overall_mean ) ** 2 end do rst % main_factor_2 % sum_of_squares = k * r * rst % main_factor_2 % sum_of_squares rst % main_factor_2 % variance = rst % main_factor_2 % sum_of_squares / & rst % main_factor_2 % dof ! Compute the within (error) term rst % within_factor % sum_of_squares = zero do j = 1 , k do i = 1 , n factorMean = mean ( x (:, j , i )) do jj = 1 , r rst % within_factor % sum_of_squares = & rst % within_factor % sum_of_squares + & ( x ( jj , j , i ) - factorMean ) ** 2 end do end do end do rst % within_factor % variance = rst % within_factor % sum_of_squares / & rst % within_factor % dof ! Compute the interaction term rst % interaction % sum_of_squares = rst % total_sum_of_squares - ( & rst % main_factor_1 % sum_of_squares + & rst % main_factor_2 % sum_of_squares + & rst % within_factor % sum_of_squares & ) rst % interaction % variance = rst % interaction % sum_of_squares / & rst % interaction % dof ! Compute the F-statistics call anova_probability ( & rst % main_factor_1 % variance , & rst % within_factor % variance , & rst % main_factor_1 % dof , & rst % within_factor % dof , & rst % main_factor_1 % f_statistic , & rst % main_factor_1 % probability & ) call anova_probability ( & rst % main_factor_2 % variance , & rst % within_factor % variance , & rst % main_factor_2 % dof , & rst % within_factor % dof , & rst % main_factor_2 % f_statistic , & rst % main_factor_2 % probability & ) call anova_probability ( & rst % interaction % variance , & rst % within_factor % variance , & rst % interaction % dof , & rst % within_factor % dof , & rst % interaction % f_statistic , & rst % interaction % probability & ) end function ! ------------------------------------------------------------------------------ ! REF: https://www.spcforexcel.com/knowledge/root-cause-analysis/understanding-regression-statistics-part-1 function anova_model_fit ( nmodelparams , ymeas , ymod , err ) result ( rst ) !! Performs an analysis of variance (ANOVA) on the supplied data set. integer ( int32 ), intent ( in ) :: nmodelparams !! The number of model parameters. real ( real64 ), intent ( in ) :: ymeas (:) !! An N-element array containing the measured dependent variable data. real ( real64 ), intent ( in ) :: ymod (:) !! An N-element array containing the modeled dependent variable data. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if ymeas and ymod are not the !!   same length. !! - FS_MEMORY_ERROR: Occurs if a memory error is encountered. type ( single_factor_anova_table ) :: rst !! A single_factor_anova_table instance containing the ANOVA results. ! Local Variables integer ( int32 ) :: n , flag real ( real64 ), allocatable :: ypack (:) real ( real64 ) :: sum_all class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization n = size ( ymeas ) if ( present ( err )) then errmgr => err else errmgr => deferr end if rst % within_factor % f_statistic = ieee_value ( sum_all , IEEE_QUIET_NAN ) rst % within_factor % probability = ieee_value ( sum_all , IEEE_QUIET_NAN ) ! Input Checking if ( size ( ymod ) /= n ) then call report_arrays_not_same_size_error ( errmgr , \"anova_model_fit\" , & \"YMEAS\" , \"YMOD\" , n , size ( ymod )) return end if ! Memory Allocation allocate ( ypack ( 2 * n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"anova_model_fit\" , flag ) return end if ! Determine the number of DOF rst % main_factor % dof = nmodelparams - 1 rst % within_factor % dof = n - rst % main_factor % dof - 1 rst % total_dof = n - 1 ! Process ypack ( 1 : n ) = ymeas ypack ( n + 1 : 2 * n ) = ymod rst % overall_mean = mean ( ypack ) rst % total_sum_of_squares = sum (( ymeas - rst % overall_mean ) ** 2 ) rst % main_factor % sum_of_squares = sum (( ymod - rst % overall_mean ) ** 2 ) rst % within_factor % sum_of_squares = sum (( ymeas - ymod ) ** 2 ) rst % total_variance = rst % total_sum_of_squares / rst % total_dof rst % main_factor % variance = rst % main_factor % sum_of_squares / & rst % main_factor % dof rst % within_factor % variance = rst % within_factor % sum_of_squares / & rst % within_factor % dof ! Compute the F-statistic and probability term call anova_probability ( & rst % main_factor % variance , & rst % within_factor % variance , & rst % main_factor % dof , & rst % within_factor % dof , & rst % main_factor % f_statistic , & rst % main_factor % probability & ) ! Formatting 100 format ( A , I0 , A , I0 , A ) 101 format ( A , I0 , A ) end function ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ subroutine anova_probability ( v1 , v2 , dof1 , dof2 , f , p ) ! Arguments real ( real64 ), intent ( in ) :: v1 , v2 , dof1 , dof2 real ( real64 ), intent ( out ) :: f , p ! Local Variables real ( real64 ) :: d1 , d2 , a , b , x ! Process f = v1 / v2 d1 = dof1 d2 = dof2 a = 0.5d0 * d2 b = 0.5d0 * d1 x = d2 / ( d2 + d1 * f ) p = regularized_beta ( a , b , x ) if ( p > 1.0d0 ) then p = 2.0d0 - p end if end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_anova.f90.html"},{"title":"fstats_errors.f90 – FSTATS","text":"Source Code ! A module providing a set of routines to handle errors for the FSTATS library. module fstats_errors use ferror use iso_fortran_env , only : int32 implicit none ! ****************************************************************************** ! ERROR CODES ! ------------------------------------------------------------------------------ integer ( int32 ), parameter :: FS_NO_ERROR = 0 integer ( int32 ), parameter :: FS_ARRAY_SIZE_ERROR = 10000 integer ( int32 ), parameter :: FS_MATRIX_SIZE_ERROR = 10001 integer ( int32 ), parameter :: FS_INVALID_INPUT_ERROR = 10002 integer ( int32 ), parameter :: FS_MEMORY_ERROR = 10003 integer ( int32 ), parameter :: FS_UNDERDEFINED_PROBLEM_ERROR = 10004 integer ( int32 ), parameter :: FS_TOLERANCE_TOO_SMALL_ERROR = 10005 integer ( int32 ), parameter :: FS_TOO_FEW_ITERATION_ERROR = 10006 ! ------------------------------------------------------------------------------ integer ( int32 ), private , parameter :: MESSAGE_SIZE = 1024 contains ! ------------------------------------------------------------------------------ subroutine report_memory_error ( err , fname , code ) !! Reports a memory allocation related error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. integer ( int32 ), intent ( in ) :: code !! The error code returned by the allocation routine. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) & \"A memory allocation error occurred with code \" , code , \".\" call err % report_error ( fname , trim ( msg ), FS_MEMORY_ERROR ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_array_size_error ( err , fname , name , expect , actual ) !! Reports an array size error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name !! The name of the array. integer ( int32 ), intent ( in ) :: expect !! The expected size of the array. integer ( int32 ), intent ( in ) :: actual !! The actual size of the array. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Expected array \" // name // \" to be of length \" , & expect , \", but found it to be of length \" , actual , \".\" call err % report_error ( fname , trim ( msg ), FS_ARRAY_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_matrix_size_error ( err , fname , name , expect_rows , & expect_cols , actual_rows , actual_cols ) !! Reports a matrix size error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name !! The name of the matrix. integer ( int32 ), intent ( in ) :: expect_rows !! The expected number of rows. integer ( int32 ), intent ( in ) :: expect_cols !! The expected number of columns. integer ( int32 ), intent ( in ) :: actual_rows !! The actual number of rows. integer ( int32 ), intent ( in ) :: actual_cols !! The actual number of columns. ! Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Expected matrix \" // name // \" to be of size (\" , & expect_rows , \", \" , expect_cols , \"), but found it to be of size (\" , & actual_rows , \", \" , actual_cols , \").\" call err % report_error ( fname , trim ( msg ), FS_MATRIX_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_arrays_not_same_size_error ( err , fname , name1 , name2 , & size1 , size2 ) !! Reports an error relating to two arrays not being the same size !! when they should be the same size. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. character ( len = * ), intent ( in ) :: name1 !! The name of the first array. character ( len = * ), intent ( in ) :: name2 !! The name of the second array. integer ( int32 ), intent ( in ) :: size1 !! The size of the first array. integer ( int32 ), intent ( in ) :: size2 !! The size of the second array. ! Local Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"Array \" // name1 // \" and array \" // name2 // & \"were expected to be the same size, but instead were found \" // & \"to be sized \" , size1 , \" and \" , size2 , \" respectively.\" call err % report_error ( fname , trim ( msg ), FS_ARRAY_SIZE_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_underdefined_error ( err , fname , expect , actual ) !! Reports an underdefined problem error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ), intent ( in ) :: fname !! The name of the routine in which the error occurred. integer ( int32 ), intent ( in ) :: expect !! The expected minimum number of equations. integer ( int32 ), intent ( in ) :: actual !! The actual number of equations. ! Local Variables character ( len = MESSAGE_SIZE ) :: msg ! Process write ( msg , 100 ) \"The problem is underdefined.  The number of \" // & \"equations was found to be \" , actual , & \", but must be at least equal to the number of unknowns \" , & expect , \".\" call err % report_error ( fname , trim ( msg ), FS_UNDERDEFINED_PROBLEM_ERROR ) ! Formatting 100 format ( A , I0 , A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ subroutine report_iteration_count_error ( err , fname , msg , mincount ) !! Reports an iteration count error. class ( errors ), intent ( inout ) :: err !! The error handling object. character ( len = * ) :: fname !! The name of the routine in which the error occurred. character ( len = * ) :: msg !! The error message. integer ( int32 ), intent ( in ) :: mincount !! The minimum iteration count expected. ! Local Variables character ( len = MESSAGE_SIZE ) :: emsg ! Process write ( emsg , 100 ) msg // \"  A minimum of \" , mincount , \" is expected.\" call err % report_error ( fname , trim ( emsg ), FS_TOO_FEW_ITERATION_ERROR ) ! Formatting 100 format ( A , I0 , A ) end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_errors.f90.html"},{"title":"fstats_distributions.f90 – FSTATS","text":"Source Code module fstats_distributions use iso_fortran_env use ieee_arithmetic use fstats_special_functions use fstats_helper_routines implicit none private public :: distribution public :: distribution_function public :: distribution_property public :: t_distribution public :: normal_distribution public :: f_distribution public :: chi_squared_distribution public :: binomial_distribution real ( real64 ), parameter :: pi = 2.0d0 * acos ( 0.0d0 ) type , abstract :: distribution !! Defines a probability distribution. contains procedure ( distribution_function ), deferred , pass :: pdf !! Computes the probability density function. procedure ( distribution_function ), deferred , pass :: cdf !! Computes the cumulative distribution function. procedure ( distribution_property ), deferred , pass :: mean !! Computes the mean of the distribution. procedure ( distribution_property ), deferred , pass :: median !! Computes the median of the distribution. procedure ( distribution_property ), deferred , pass :: mode !! Computes the mode of the distribution. procedure ( distribution_property ), deferred , pass :: variance !! Computes the variance of the distribution. end type interface pure elemental function distribution_function ( this , x ) result ( rst ) !! Defines the interface for a probability distribution function. use iso_fortran_env , only : real64 import distribution class ( distribution ), intent ( in ) :: this !! The distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. end function pure function distribution_property ( this ) result ( rst ) !! Computes the value of a distribution property. use iso_fortran_env , only : real64 import distribution class ( distribution ), intent ( in ) :: this !! The distribution object. real ( real64 ) :: rst !! The property value. end function end interface ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: t_distribution !! Defines Student's T-Distribution. real ( real64 ) :: dof !! The number of degrees of freedom. contains procedure , public :: pdf => td_pdf procedure , public :: cdf => td_cdf procedure , public :: mean => td_mean procedure , public :: median => td_median procedure , public :: mode => td_mode procedure , public :: variance => td_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: normal_distribution !! Defines a normal distribution. real ( real64 ) :: standard_deviation !! The standard deviation of the distribution. real ( real64 ) :: mean_value !! The mean value of the distribution. contains procedure , public :: pdf => nd_pdf procedure , public :: cdf => nd_cdf procedure , public :: mean => nd_mean procedure , public :: median => nd_median procedure , public :: mode => nd_mode procedure , public :: variance => nd_variance procedure , public :: standardize => nd_standardize end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: f_distribution !! Defines an F-distribution. real ( real64 ) :: d1 !! The measure of degrees of freedom for the first data set. real ( real64 ) :: d2 !! The measure of degrees of freedom for the second data set. contains procedure , public :: pdf => fd_pdf procedure , public :: cdf => fd_cdf procedure , public :: mean => fd_mean procedure , public :: median => fd_median procedure , public :: mode => fd_mode procedure , public :: variance => fd_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: chi_squared_distribution !! Defines a Chi-squared distribution. integer ( int32 ) :: dof !! The number of degrees of freedom. contains procedure , public :: pdf => cs_pdf procedure , public :: cdf => cs_cdf procedure , public :: mean => cs_mean procedure , public :: median => cs_median procedure , public :: mode => cs_mode procedure , public :: variance => cs_variance end type ! ------------------------------------------------------------------------------ type , extends ( distribution ) :: binomial_distribution !! Defines a binomial distribution.  The binomial distribution describes !! the probability p of getting k successes in n independent trials. integer ( int32 ) :: n !! The number of independent trials. real ( real64 ) :: p !! The success probability for each trial.  This parameter must !! exist on the set [0, 1]. contains procedure , public :: pdf => bd_pdf procedure , public :: cdf => bd_cdf procedure , public :: mean => bd_mean procedure , public :: median => bd_median procedure , public :: mode => bd_mode procedure , public :: variance => bd_variance end type ! ------------------------------------------------------------------------------ contains ! ****************************************************************************** ! STUDENT'S T-DISTRIBUTION ! ------------------------------------------------------------------------------ ! REF: https://en.wikipedia.org/wiki/Student%27s_t-distribution pure elemental function td_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for Student's T-Distribution is given as !!  f(t) = \\frac{ \\Gamma \\left( \\frac{\\nu + 1}{2} \\right) } !! { \\sqrt{\\nu \\pi} \\Gamma \\left( \\frac{\\nu}{2} \\right) } !! \\left( 1 + \\frac{t&#94;2}{\\nu} \\right)&#94;{-(\\nu + 1) / 2} . class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process rst = gamma (( this % dof + 1.0d0 ) / 2.0d0 ) / & ( sqrt ( this % dof * pi ) * gamma ( this % dof / 2.0d0 )) * & ( 1.0d0 + x ** 2 / this % dof ) ** ( - 0.5d0 * ( 1.0d0 + this % dof )) end function ! ------------------------------------------------------------------------------ pure elemental function td_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for Student's T-Distribution is given as !!  F(t) = \\int_{-\\infty}&#94;{t} f(u) \\,du = 1 - \\frac{1}{2} I_{x(t)} !! \\left( \\frac{\\nu}{2}, \\frac{1}{2} \\right)  !! where  x(t) = \\frac{\\nu}{\\nu + t&#94;2} . class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: t t = this % dof / ( this % dof + x ** 2 ) rst = 1.0d0 - 0.5d0 * regularized_beta ( 0.5d0 * this % dof , 0.5d0 , t ) if ( x < 0 ) rst = 1.0d0 - rst end function ! ------------------------------------------------------------------------------ pure function td_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The mean. ! Process if ( this % dof < 1.0d0 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) else rst = 0.0d0 end if end function ! ------------------------------------------------------------------------------ pure function td_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst ! Process rst = 0.0d0 end function ! ------------------------------------------------------------------------------ pure function td_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The mode. ! Process rst = 0.0d0 end function ! ------------------------------------------------------------------------------ pure function td_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( t_distribution ), intent ( in ) :: this !! The t_distribution object. real ( real64 ) :: rst !! The variance. ! Process if ( this % dof <= 1.0d0 ) then rst = ieee_value ( rst , IEEE_QUIET_NAN ) else if ( this % dof > 1.0d0 . and . this % dof <= 2.0d0 ) then rst = ieee_value ( rst , IEEE_POSITIVE_INF ) else rst = this % dof / ( this % dof - 2.0d0 ) end if end function ! ****************************************************************************** ! NORMAL DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function nd_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a normal distribution is given as !!  f(x) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\exp \\left(-\\frac{1}{2} !! \\left( \\frac{x - \\mu}{\\sigma} \\right)&#94;2 \\right) . class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. rst = exp ( - 0.5d0 * (( x - this % mean_value ) / this % standard_deviation ) ** 2 ) / & ( this % standard_deviation * sqrt ( 2.0d0 * pi )) end function ! ------------------------------------------------------------------------------ pure elemental function nd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a normal distribution is given as !!  F(x) = \\frac{1}{2} \\left( 1 + erf \\left( \\frac{x - \\mu} !! {\\sigma \\sqrt{2}} \\right) \\right) . class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. rst = 0.5d0 * ( 1.0d0 + erf (( x - this % mean_value ) / & ( this % standard_deviation * sqrt ( 2.0d0 )))) end function ! ------------------------------------------------------------------------------ pure function nd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The mean rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The median. rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The mode. rst = this % mean_value end function ! ------------------------------------------------------------------------------ pure function nd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( normal_distribution ), intent ( in ) :: this !! The normal_distribution object. real ( real64 ) :: rst !! The variance. rst = this % standard_deviation ** 2 end function ! ------------------------------------------------------------------------------ subroutine nd_standardize ( this ) !! Standardizes the normal distribution to a mean of 0 and a !! standard deviation of 1. class ( normal_distribution ), intent ( inout ) :: this !! The normal_distribution object. this % mean_value = 0.0d0 this % standard_deviation = 1.0d0 end subroutine ! ****************************************************************************** ! F DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function fd_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a F distribution is given as !!  f(x) = !! \\sqrt{ \\frac{ (d_1 x)&#94;{d_1} d_{2}&#94;{d_2} }{ (d_1 x + d_2)&#94;{d_1 + d_2} } } !! \\frac{1}{x \\beta \\left( \\frac{d_1}{2}, \\frac{d_2}{2} \\right) } . class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 rst = ( 1.0d0 / beta ( 0.5d0 * d1 , 0.5d0 * d2 )) * ( d1 / d2 ) ** ( 0.5d0 * d1 ) * & x ** ( 0.5d0 * d1 - 1.0d0 ) * ( 1.0d0 + d1 * x / d2 ) ** ( - 0.5d0 * ( d1 + d2 )) end function ! ------------------------------------------------------------------------------ pure elemental function fd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a F distribution is given as !!  F(x) = I_{d_1 x/(d_1 x + d_2)} \\left( \\frac{d_1}{2}, !! \\frac{d_2}{2} \\right) . class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 rst = regularized_beta ( 0.5d0 * d1 , 0.5d0 * d2 , d1 * x / ( d1 * x + d2 )) end function ! ------------------------------------------------------------------------------ pure function fd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The mean. ! Process if ( this % d2 > 2.0d0 ) then rst = this % d2 / ( this % d2 - 2.0d0 ) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ------------------------------------------------------------------------------ pure function fd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The median. rst = ieee_value ( rst , IEEE_QUIET_NAN ) end function ! ------------------------------------------------------------------------------ pure function fd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The mode. ! Process if ( this % d1 > 2.0d0 ) then rst = (( this % d1 - 2.0d0 ) / this % d1 ) * ( this % d2 / ( this % d2 + 2.0d0 )) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ------------------------------------------------------------------------------ pure function fd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( f_distribution ), intent ( in ) :: this !! The f_distribution object. real ( real64 ) :: rst !! The variance. ! Process real ( real64 ) :: d1 , d2 d1 = this % d1 d2 = this % d2 if ( d2 > 4.0d0 ) then rst = ( 2.0d0 * d2 ** 2 * ( d1 + d2 - 2.0d0 )) / & ( d1 * ( d2 - 2.0d0 ) ** 2 * ( d2 - 4.0d0 )) else rst = ieee_value ( rst , IEEE_QUIET_NAN ) end if end function ! ****************************************************************************** ! CHI-SQUARED DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function cs_pdf ( this , x ) result ( rst ) !! Computes the probability density function. !! !! The PDF for a Chi-squared distribution is given as !!  f(x) = \\frac{x&#94;{k/2 - 1} \\exp{-x / 2}} {2&#94;{k / 2} !! \\Gamma \\left( \\frac{k}{2} \\right)} . class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: arg ! Process arg = 0.5d0 * this % dof rst = 1.0d0 / ( 2.0d0 ** arg * gamma ( arg )) * x ** ( arg - 1.0d0 ) * exp ( - 0.5d0 * x ) end function ! ------------------------------------------------------------------------------ pure elemental function cs_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution function. !! !! The CDF for a Chi-squared distribution is given as !!  F(x) = \\frac{ \\gamma \\left( \\frac{k}{2}, \\frac{x}{2} \\right) } !! { \\Gamma \\left( \\frac{k}{2} \\right)} . class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: arg ! Process arg = 0.5d0 * this % dof rst = incomplete_gamma_lower ( arg , 0.5d0 * x ) / gamma ( arg ) end function ! ------------------------------------------------------------------------------ pure function cs_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The mean. ! Process rst = real ( this % dof , real64 ) end function ! ------------------------------------------------------------------------------ pure function cs_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The median. ! Process rst = this % dof * ( 1.0d0 - 2.0d0 / ( 9.0d0 * this % dof )) ** 3 end function ! ------------------------------------------------------------------------------ pure function cs_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The mode. ! Process rst = max ( this % dof - 2.0d0 , 0.0d0 ) end function ! ------------------------------------------------------------------------------ pure function cs_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( chi_squared_distribution ), intent ( in ) :: this !! The chi_squared_distribution object. real ( real64 ) :: rst !! The variance. ! Process rst = 2.0d0 * this % dof end function ! ****************************************************************************** ! BINOMIAL DISTRIBUTION ! ------------------------------------------------------------------------------ pure elemental function bd_pdf ( this , x ) result ( rst ) !! Computes the probability mass function. !! !! The PMF for a binomial distribution is given as !!  f(k,n,p) = \\frac{n!}{k! \\left( n - k! \\right)} p&#94;k !! \\left( 1 - p \\right)&#94;{n-k} . class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function.  This parameter !! is the number k successes in the n independent trials.  As !! such, this parameter must exist on the set [0, n]. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: dn ! Process dn = real ( this % n , real64 ) rst = ( factorial ( dn ) / ( factorial ( x ) * factorial ( dn - x ))) * ( this % p ** x ) * ( 1.0d0 - this % p ) ** ( dn - x ) end function ! ------------------------------------------------------------------------------ pure elemental function bd_cdf ( this , x ) result ( rst ) !! Computes the cumulative distribution funtion. !! !! The CDF for a binomial distribution is given as !!  F(k,n,p) = I_{1-p} \\left( n - k, 1 + k \\right) , which is simply !! the regularized incomplete beta function. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ), intent ( in ) :: x !! The value at which to evaluate the function.  This parameter !! is the number k successes in the n independent trials.  As !! such, this parameter must exist on the set [0, n]. real ( real64 ) :: rst !! The value of the function. ! Local Variables real ( real64 ) :: dn ! Process dn = real ( this % n , real64 ) rst = regularized_beta ( dn - x , x + 1.0d0 , 1.0d0 - this % p ) end function ! ------------------------------------------------------------------------------ pure function bd_mean ( this ) result ( rst ) !! Computes the mean of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The mean. rst = real ( this % n * this % p , real64 ) end function ! ------------------------------------------------------------------------------ pure function bd_median ( this ) result ( rst ) !! Computes the median of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The median. rst = real ( this % n * this % p , real64 ) end function ! ------------------------------------------------------------------------------ pure function bd_mode ( this ) result ( rst ) !! Computes the mode of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The mode. rst = ( this % n + 1.0d0 ) * this % p end function ! ------------------------------------------------------------------------------ pure function bd_variance ( this ) result ( rst ) !! Computes the variance of the distribution. class ( binomial_distribution ), intent ( in ) :: this !! The binomial_distribution object. real ( real64 ) :: rst !! The variance. rst = this % n * this % p * ( 1.0d0 - this % p ) end function ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_distributions.f90.html"},{"title":"fstats_smoothing.f90 – FSTATS","text":"Source Code module fstats_smoothing use iso_fortran_env use ferror use fstats_errors use linalg , only : sort implicit none private public :: lowess contains ! ------------------------------------------------------------------------------ subroutine lowess ( x , y , ys , fsmooth , nstps , del , rweights , resid , err ) !! Computes the smoothing of a data set using a robust locally weighted !! scatterplot smoothing (LOWESS) algorithm.  Fitted values are computed at !! each of the supplied x values. !! !! Remarks !! !! The code is a reimplementation of the LOWESS library.  For a detailed !! understanding, see [this] !! (http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf) !! paper by William Cleveland. real ( real64 ), intent ( in ), dimension (:) :: x !! An N-element array containing the independent variable data.  This !! array must be monotonically increasing. real ( real64 ), intent ( in ), dimension (:) :: y !! An N-element array containing the dependent variable data. real ( real64 ), intent ( out ), dimension (:) :: ys !! An N-element array where the smoothed results will be written. real ( real64 ), intent ( in ), optional :: fsmooth !! An optional input that specifies the amount of smoothing. !! Specifically, this value is the fraction of points used to compute !! each value.  As this value increases, the output becomes smoother. !! Choosing a value in the range of 0.2 to 0.8 typically results in a !! good fit.  The default value is 0.2. integer ( int32 ), intent ( in ), optional :: nstps !! An optional input that specifies the numb of iterations.  If set to !! zero, a non-robust fit is returned.  The default value is set to 2. real ( real64 ), intent ( in ), optional :: del !! real ( real64 ), intent ( out ), optional , dimension (:), target :: rweights !! An optional N-element array, that if supplied, will be used to !! return the weights given to each data point. real ( real64 ), intent ( out ), optional , dimension (:), target :: resid !! An optional N-element array, that if supplied, will be used to !! return the residual. class ( errors ), intent ( inout ), optional , target :: err !! A mechanism for communicating errors and warnings to the !! caller.  Possible warning and error codes are as follows. !! - FS_NO_ERROR: No errors encountered. !! - FS_ARRAY_SIZE_ERROR: Occurs if any of the arrays are not !!      approriately sized. !! - FS_MEMORY_ERROR: Occurs if there is a memory allocation error. ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: p2 = 2.0d-1 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: three = 3.0d0 real ( real64 ), parameter :: p001 = 1.0d-3 real ( real64 ), parameter :: p999 = 0.999d0 ! Local Variables logical :: ok integer ( int32 ) :: iter , i , j , nleft , nright , ns , last , m1 , m2 , n , nsteps , flag real ( real64 ) :: f , delta , d1 , d2 , denom , alpha , cut , eps , cmad , c1 , c9 , r real ( real64 ), allocatable , target , dimension (:) :: rwdef , rsdef real ( real64 ), pointer , dimension (:) :: rw , res class ( errors ), pointer :: errmgr type ( errors ), target :: deferr ! Initialization if ( present ( err )) then errmgr => err else errmgr => deferr end if n = size ( x ) if ( present ( fsmooth )) then f = fsmooth else f = p2 end if if ( present ( nstps )) then nsteps = nstps else nsteps = 2 end if if ( present ( del )) then delta = del else delta = 0.0d0 end if if ( present ( rweights )) then if ( size ( rweights ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"rweights\" , n , & size ( rweights )) return end if rw => rweights else allocate ( rwdef ( n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"lowess\" , flag ) return end if rw => rwdef end if if ( present ( resid )) then if ( size ( resid ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"resid\" , n , & size ( resid )) return end if res => resid else allocate ( rsdef ( n ), stat = flag ) if ( flag /= 0 ) then call report_memory_error ( errmgr , \"lowess\" , flag ) return end if res => rsdef end if ns = max ( min ( int ( f * real ( n ), int32 ), n ), 2 ) eps = epsilon ( eps ) ! Input Checking if ( size ( y ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"y\" , n , size ( y )) return end if if ( size ( ys ) /= n ) then call report_array_size_error ( errmgr , \"lowess\" , \"ys\" , n , size ( ys )) return end if ! Quick Return if ( n < 2 ) then ys = y return end if ! Process do iter = 1 , nsteps + 1 nleft = 1 nright = ns last = 0 i = 1 do do while ( nright < n ) d1 = x ( i ) - x ( nleft ) d2 = x ( nright + 1 ) - x ( i ) if ( d1 <= d2 ) exit nleft = nleft + 1 nright = nright + 1 end do call lowest ( x , y , x ( i ), ys ( i ), nleft , nright , res , iter > 1 , & rw , ok ) if (. not . ok ) ys ( i ) = y ( i ) if ( last < i - 1 ) then denom = x ( i ) - x ( last ) do j = last + 1 , i - 1 alpha = ( x ( j ) - x ( last )) / denom ys ( j ) = alpha * ys ( i ) + ( one - alpha ) * ys ( last ) end do end if last = i cut = x ( last ) + delta do i = last + 1 , n if ( x ( i ) > cut ) exit if ( abs ( x ( i ) - x ( last )) < eps ) then ys ( i ) = ys ( last ) last = i end if end do i = max ( last + 1 , i - 1 ) if ( last >= n ) exit end do res = y - ys if ( iter > nsteps ) exit rw = abs ( res ) call sort ( rw , . true .) m1 = 1 + n / 2 m2 = n - m1 + 1 cmad = three * ( rw ( m1 ) + rw ( m2 )) c9 = p999 * cmad c1 = p001 * cmad do i = 1 , n r = abs ( res ( i )) if ( r <= c1 ) then rw ( i ) = one else if ( r > c9 ) then rw ( i ) = zero else rw ( i ) = ( one - ( r / cmad ) ** 2 ) ** 2 end if end do end do end subroutine ! ****************************************************************************** ! PRIVATE ROUTINES ! ------------------------------------------------------------------------------ ! REF: ! - https://en.wikipedia.org/wiki/Local_regression ! - http://www.aliquote.org/cours/2012_biomed/biblio/Cleveland1979.pdf subroutine lowest ( x , y , xs , ys , nleft , nright , w , userw , rw , ok ) ! Arguments real ( real64 ), intent ( in ), dimension (:) :: x , y , rw ! N ELEMENT real ( real64 ), intent ( in ) :: xs real ( real64 ), intent ( out ) :: ys integer ( int32 ), intent ( in ) :: nleft , nright real ( real64 ), intent ( out ), dimension (:) :: w ! N ELEMENT logical , intent ( in ) :: userw logical , intent ( out ) :: ok ! Parameters real ( real64 ), parameter :: zero = 0.0d0 real ( real64 ), parameter :: one = 1.0d0 real ( real64 ), parameter :: p001 = 1.0d-3 real ( real64 ), parameter :: p999 = 0.999d0 ! Local Variables integer ( int32 ) :: j , n , nrt real ( real64 ) :: range , h , h9 , h1 , a , b , c , r ! Initialization n = size ( x ) range = x ( n ) - x ( 1 ) h = max ( xs - x ( nleft ), x ( nright ) - xs ) h9 = p999 * h h1 = p001 * h a = zero ! Process do j = nleft , n w ( j ) = zero r = abs ( x ( j ) - xs ) if ( r <= h9 ) then if ( r > h1 ) then w ( j ) = ( one - ( r / h ) ** 3 ) ** 3 else w ( j ) = one end if if ( userw ) w ( j ) = rw ( j ) * w ( j ) a = a + w ( j ) else if ( x ( j ) > xs ) then exit end if end do nrt = j - 1 if ( a <= zero ) then ok = . false . else ok = . true . w ( nleft : nrt ) = w ( nleft : nrt ) / a if ( h > zero ) then a = zero do j = nleft , nrt a = a + w ( j ) * x ( j ) end do b = xs - a c = zero do j = nleft , nrt c = c + w ( j ) * ( x ( j ) - a ) ** 2 end do if ( sqrt ( c ) > p001 * range ) then b = b / c do j = nleft , nrt w ( j ) = w ( j ) * ( one + b * ( x ( j ) - a )) end do end if end if ys = zero do j = nleft , nrt ys = ys + w ( j ) * y ( j ) end do end if end subroutine ! ------------------------------------------------------------------------------ end module","tags":"","loc":"sourcefile\\fstats_smoothing.f90.html"}]}
\ No newline at end of file
diff --git a/doc/type/anova_factor.html b/doc/type/anova_factor.html
index 7c47898..d2defe7 100644
--- a/doc/type/anova_factor.html
+++ b/doc/type/anova_factor.html
@@ -114,9 +114,9 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
       </a>
       <div id="vars-0" class="collapse">
         <div class="list-group list-group-flush">
-            <a class="list-group-item" href="../type/anova_factor.html#variable-dof">dof</a>
+            <a class="list-group-item" href="../type/anova_factor.html#variable-dof~7">dof</a>
             <a class="list-group-item" href="../type/anova_factor.html#variable-f_statistic">f_statistic</a>
-            <a class="list-group-item" href="../type/anova_factor.html#variable-probability">probability</a>
+            <a class="list-group-item" href="../type/anova_factor.html#variable-probability~3">probability</a>
             <a class="list-group-item" href="../type/anova_factor.html#variable-sum_of_squares">sum_of_squares</a>
             <a class="list-group-item" href="../type/anova_factor.html#variable-variance">variance</a>
         </div>
@@ -167,7 +167,7 @@ <h2>Components</h2>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-dof"></span>
+              <span class="anchor" id="variable-dof~7"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -201,7 +201,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-probability"></span>
+              <span class="anchor" id="variable-probability~3"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -275,7 +275,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/array_container.html b/doc/type/array_container.html
index f4aff54..a3a2d76 100644
--- a/doc/type/array_container.html
+++ b/doc/type/array_container.html
@@ -114,7 +114,7 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
       </a>
       <div id="vars-0" class="collapse">
         <div class="list-group list-group-flush">
-            <a class="list-group-item" href="../type/array_container.html#variable-x~14">x</a>
+            <a class="list-group-item" href="../type/array_container.html#variable-x~10">x</a>
         </div>
       </div>
     </div>
@@ -164,7 +164,7 @@ <h2>Components</h2>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-x~14"></span>
+              <span class="anchor" id="variable-x~10"></span>
               real(kind=real64),
             </td>
               <td>public,</td>
@@ -204,7 +204,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/binomial_distribution.html b/doc/type/binomial_distribution.html
index 21f3daf..3fbfc2d 100644
--- a/doc/type/binomial_distribution.html
+++ b/doc/type/binomial_distribution.html
@@ -114,7 +114,7 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
       </a>
       <div id="vars-0" class="collapse">
         <div class="list-group list-group-flush">
-            <a class="list-group-item" href="../type/binomial_distribution.html#variable-n~39">n</a>
+            <a class="list-group-item" href="../type/binomial_distribution.html#variable-n~37">n</a>
             <a class="list-group-item" href="../type/binomial_distribution.html#variable-p~16">p</a>
         </div>
       </div>
@@ -181,7 +181,7 @@ <h2>Components</h2>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-n~39"></span>
+              <span class="anchor" id="variable-n~37"></span>
               integer(kind=int32),
             </td>
               <td>public</td>
@@ -630,7 +630,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/bootstrap_regression_statistics.html b/doc/type/bootstrap_regression_statistics.html
index ccfbcfb..5598701 100644
--- a/doc/type/bootstrap_regression_statistics.html
+++ b/doc/type/bootstrap_regression_statistics.html
@@ -116,8 +116,8 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
         <div class="list-group list-group-flush">
             <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-lower_confidence_interval">lower_confidence_interval</a>
             <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-probability~2">probability</a>
-            <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-standard_error">standard_error</a>
-            <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-t_statistic">t_statistic</a>
+            <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-standard_error~2">standard_error</a>
+            <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-t_statistic~2">t_statistic</a>
             <a class="list-group-item" href="../type/bootstrap_regression_statistics.html#variable-upper_confidence_interval">upper_confidence_interval</a>
         </div>
       </div>
@@ -210,7 +210,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error"></span>
+              <span class="anchor" id="variable-standard_error~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -227,7 +227,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-t_statistic"></span>
+              <span class="anchor" id="variable-t_statistic~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -287,7 +287,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/bootstrap_statistics.html b/doc/type/bootstrap_statistics.html
index bec2c0e..23a4a9c 100644
--- a/doc/type/bootstrap_statistics.html
+++ b/doc/type/bootstrap_statistics.html
@@ -117,7 +117,7 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
             <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-bias">bias</a>
             <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-lower_confidence_interval~2">lower_confidence_interval</a>
             <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-population">population</a>
-            <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-standard_error~2">standard_error</a>
+            <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-standard_error~3">standard_error</a>
             <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-statistic_value">statistic_value</a>
             <a class="list-group-item" href="../type/bootstrap_statistics.html#variable-upper_confidence_interval~2">upper_confidence_interval</a>
         </div>
@@ -220,7 +220,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error~2"></span>
+              <span class="anchor" id="variable-standard_error~3"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -294,7 +294,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/chi_squared_distribution.html b/doc/type/chi_squared_distribution.html
index 7d9559a..f58f1b5 100644
--- a/doc/type/chi_squared_distribution.html
+++ b/doc/type/chi_squared_distribution.html
@@ -606,7 +606,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/convergence_info.html b/doc/type/convergence_info.html
index 64419ed..2ea82eb 100644
--- a/doc/type/convergence_info.html
+++ b/doc/type/convergence_info.html
@@ -387,7 +387,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/distribution.html b/doc/type/distribution.html
index aefd10b..40d8bff 100644
--- a/doc/type/distribution.html
+++ b/doc/type/distribution.html
@@ -579,7 +579,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/f_distribution.html b/doc/type/f_distribution.html
index 86193a5..1c3902b 100644
--- a/doc/type/f_distribution.html
+++ b/doc/type/f_distribution.html
@@ -114,8 +114,8 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
       </a>
       <div id="vars-0" class="collapse">
         <div class="list-group list-group-flush">
-            <a class="list-group-item" href="../type/f_distribution.html#variable-d1~3">d1</a>
-            <a class="list-group-item" href="../type/f_distribution.html#variable-d2~3">d2</a>
+            <a class="list-group-item" href="../type/f_distribution.html#variable-d1~2">d1</a>
+            <a class="list-group-item" href="../type/f_distribution.html#variable-d2~2">d2</a>
         </div>
       </div>
     </div>
@@ -180,7 +180,7 @@ <h2>Components</h2>
     <tbody>
         <tr>
             <td>
-              <span class="anchor" id="variable-d1~3"></span>
+              <span class="anchor" id="variable-d1~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -197,7 +197,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-d2~3"></span>
+              <span class="anchor" id="variable-d2~2"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -625,7 +625,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/iteration_controls.html b/doc/type/iteration_controls.html
index 0ca37d3..8dfee51 100644
--- a/doc/type/iteration_controls.html
+++ b/doc/type/iteration_controls.html
@@ -381,7 +381,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/lm_solver_options.html b/doc/type/lm_solver_options.html
index f157e82..322449e 100644
--- a/doc/type/lm_solver_options.html
+++ b/doc/type/lm_solver_options.html
@@ -117,7 +117,7 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
             <a class="list-group-item" href="../type/lm_solver_options.html#variable-damping_decrease_factor">damping_decrease_factor</a>
             <a class="list-group-item" href="../type/lm_solver_options.html#variable-damping_increase_factor">damping_increase_factor</a>
             <a class="list-group-item" href="../type/lm_solver_options.html#variable-finite_difference_step_size">finite_difference_step_size</a>
-            <a class="list-group-item" href="../type/lm_solver_options.html#variable-method~4">method</a>
+            <a class="list-group-item" href="../type/lm_solver_options.html#variable-method">method</a>
         </div>
       </div>
     </div>
@@ -229,7 +229,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-method~4"></span>
+              <span class="anchor" id="variable-method"></span>
               integer(kind=int32),
             </td>
               <td>public</td>
@@ -328,7 +328,7 @@ <h4>Arguments</h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/normal_distribution.html b/doc/type/normal_distribution.html
index ff99ed7..b86b29c 100644
--- a/doc/type/normal_distribution.html
+++ b/doc/type/normal_distribution.html
@@ -679,7 +679,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/regression_statistics.html b/doc/type/regression_statistics.html
index 9a3ed7a..89ae1d1 100644
--- a/doc/type/regression_statistics.html
+++ b/doc/type/regression_statistics.html
@@ -115,9 +115,9 @@ <h4 class="card-header bg-primary text-white">Variables</h4>
       <div id="vars-0" class="collapse">
         <div class="list-group list-group-flush">
             <a class="list-group-item" href="../type/regression_statistics.html#variable-confidence_interval">confidence_interval</a>
-            <a class="list-group-item" href="../type/regression_statistics.html#variable-probability~3">probability</a>
-            <a class="list-group-item" href="../type/regression_statistics.html#variable-standard_error~3">standard_error</a>
-            <a class="list-group-item" href="../type/regression_statistics.html#variable-t_statistic~2">t_statistic</a>
+            <a class="list-group-item" href="../type/regression_statistics.html#variable-probability">probability</a>
+            <a class="list-group-item" href="../type/regression_statistics.html#variable-standard_error">standard_error</a>
+            <a class="list-group-item" href="../type/regression_statistics.html#variable-t_statistic">t_statistic</a>
         </div>
       </div>
     </div>
@@ -187,7 +187,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-probability~3"></span>
+              <span class="anchor" id="variable-probability"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -212,7 +212,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-standard_error~3"></span>
+              <span class="anchor" id="variable-standard_error"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -232,7 +232,7 @@ <h2>Components</h2>
         </tr>
         <tr>
             <td>
-              <span class="anchor" id="variable-t_statistic~2"></span>
+              <span class="anchor" id="variable-t_statistic"></span>
               real(kind=real64),
             </td>
               <td>public</td>
@@ -275,7 +275,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/single_factor_anova_table.html b/doc/type/single_factor_anova_table.html
index 78efa02..0a9ed0d 100644
--- a/doc/type/single_factor_anova_table.html
+++ b/doc/type/single_factor_anova_table.html
@@ -293,7 +293,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/t_distribution.html b/doc/type/t_distribution.html
index ec72ef3..8eaace4 100644
--- a/doc/type/t_distribution.html
+++ b/doc/type/t_distribution.html
@@ -608,7 +608,7 @@ <h4>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>
diff --git a/doc/type/two_factor_anova_table.html b/doc/type/two_factor_anova_table.html
index b09355f..af91a6a 100644
--- a/doc/type/two_factor_anova_table.html
+++ b/doc/type/two_factor_anova_table.html
@@ -329,7 +329,7 @@ <h2>Components</h2>
             <p class="text-end">
               Documentation generated by
               <a href="https://github.com/Fortran-FOSS-Programmers/ford">FORD</a>
- on 2024-04-16 20:51              </p>
+ on 2024-04-17 05:57              </p>
           </div>
         </div>
         <br>