Fix docs

Author: lsbardel

notebooks/_config.yml

@@ -9,8 +9,8 @@ logo: assets/quantflow-light.svg
 # Force re-execution of notebooks on each build.
 # See https://jupyterbook.org/content/execute.html
 execute:
-  #execute_notebooks: "off"
-  execute_notebooks: force
+  execute_notebooks: "off"
+  #execute_notebooks: force
 
 # Define the name of the latex output file for PDF builds
 latex:

notebooks/api/utils/distributions.rst

@@ -4,6 +4,13 @@ Distributions
 
 .. module:: quantflow.utils.distributions
 
+.. autoclass:: Distribution1D
+   :members:
+   :member-order: groupwise
+   :autosummary:
+   :autosummary-nosignatures:
+
+
 .. autoclass:: Exponential
    :members:
    :member-order: groupwise

notebooks/models/poisson.md

@@ -19,16 +19,16 @@ The most common process is the Poisson process.
 
 ## Poisson Process
 
-The Poisson Process $N_t$ with intensity parameter $\lambda > 0$ is a Lévy process with values in $N$ such that each $N_t$ has a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution) with parameter $\lambda t$, that is
+The Poisson Process $n_t$ with intensity parameter $\lambda > 0$ is a Lévy process with values in ${\mathbb N}$ such that each $n_t$ has a [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution) with parameter $\lambda t$, that is
 
 \begin{equation}
-    P\left(N_t=n\right) = \frac{\left(\lambda t\right)^n}{n!}e^{-\lambda t}
+    {\mathbb P}_{n_t}\left(n_t=n\right) = \frac{\left(\lambda t\right)^n}{n!}e^{-\lambda t}
 \end{equation}
 
-The characteristic exponent is given by
+The [](characteristic-exponent) is given by
 
 \begin{equation}
-\phi_{N_t, u} = t \lambda \left(1 - e^{iu}\right)
+\phi_{n_t, u} = t \lambda \left(1 - e^{iu}\right)
 \end{equation}
 
 ```{code-cell} ipython3

notebooks/models/weiner.md

@@ -18,7 +18,7 @@ In this document, we use the term Weiner process $w_t$ to indicate a Brownian mo
 1. $w_t$ is Lévy process
 2. $d w_t = w_{t+dt}-w_t \sim N\left(0, \sigma dt\right)$ where $N$ is the normal distribution
 
-The characteristic exponent of $w$ is
+The [](characteristic-exponent) of $w$ is
 \begin{equation}
     \phi_{w, u} = \frac{\sigma^2 u^2}{2}
 \end{equation}

notebooks/reference/glossary.md

@@ -15,20 +15,30 @@ kernelspec:
 
 ## Characteristic Function
 
-The characteristic function of a random variable $X$ is the Fourier transform of $f_X$, where $f_X$ is the probability density function
-of $X$
+The [characteristic function](../theory/characteristic.md) of a random variable $x$ is the Fourier transform of ${\mathbb P}_x$,
+where ${\mathbb P}_x$ is the distrubution measure of $x$.
+
 \begin{equation}
- \Phi_{X,u} = {\mathbb E}\left[e^{i u X_t}\right] = \int e^{i u x} f_X\left(x\right) dx
+ \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} {\mathbb P}_x\left(d s\right)
 \end{equation}
 
+If $x$ is a continuous random variable, than the characteristic function is the Fourier transform of the PDF $f_x$.
+
+\begin{equation}
+ \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} f_x\left(s\right) ds
+\end{equation}
+
+
 ## Cumulative Distribution Function (CDF)
 
 The cumulative distribution function (CDF), or just distribution function,
-of a real-valued random variable $X$ is the function given by
+of a real-valued random variable $x$ is the function given by
 \begin{equation}
-    F_X(x) = P(X \leq x)
+    F_x(s) = {\mathbb P}_x(x \leq s)
 \end{equation}
 
+where ${\mathbb P}_x$ is the distrubution measure of $x$.
+
 ## Hurst Exponent
 
 The Hurst exponent is a measure of the long-term memory of time series. The Hurst exponent is a measure of the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.
@@ -66,5 +76,5 @@ The [probability density function](https://en.wikipedia.org/wiki/Probability_den
  (PDF), or density, of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. It is related to the CDF by the formula
 
 \begin{equation}
-    F_X(x) = \int_{-\infty}^x f_X(s) ds
+    F_x(x) = \int_{-\infty}^x f_x(s) ds
 \end{equation}

notebooks/theory/characteristic.md

@@ -13,13 +13,14 @@ kernelspec:
 
 # Characteristic Function
 
-The library makes heavy use of characteristic function concept and therefore, it is useful to familiarize with it.
+The library makes heavy use of [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory))
+concept and therefore, it is useful to familiarize with it.
 
 ## Definition
 
-The characteristic function of a random variable $x$ is the Fourier (inverse) transform of $P^x$, where $P^x$ is the distrubution measure of $x$
+The characteristic function of a random variable $x$ is the Fourier (inverse) transform of ${\mathbb P}_x$, where ${\mathbb P}_x$ is the distrubution measure of $x$
 \begin{equation}
- \Phi_{x,u} = {\mathbb E}\left[e^{i u x_t}\right] = \int e^{i u x} P^x\left(dx\right)
+ \Phi_{x,u} = {\mathbb E}\left[e^{i u x}\right] = \int e^{i u s} {\mathbb P}_x\left(ds\right)
 \end{equation}
 
 ## Properties
@@ -30,6 +31,7 @@ The characteristic function of a random variable $x$ is the Fourier (inverse) tr
 * it is continuous
 * characteristic function of a symmetric random variable is real-valued and even
 * moments of $x$ are given by
+
 \begin{equation}
     {\mathbb E}\left[x^n\right] = i^{-n} \left.\frac{\Phi_{x, u}}{d u}\right|_{u=0}
 \end{equation}
@@ -45,10 +47,13 @@ The characteristic function is a great tool for working with linear combination
 \end{equation}
 
 * which means, if $x$ and $y$ are independent, the characteristic function of $x+y$ is the product
+
 \begin{equation}
     \Phi_{x+x,u} = \Phi_{x,u}\Phi_{y,u}
 \end{equation}
+
 * The characteristic function of $ax+b$ is
+
 \begin{equation}
     \Phi_{ax+b,u} = e^{iub}\Phi_{x,au}
 \end{equation}
@@ -62,18 +67,27 @@ There is a one-to-one correspondence between cumulative distribution functions a
 The inversion formula for these distributions is given by
 
 \begin{equation}
-    {\mathbb P}\left(x\right) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-iuk}\Phi_{x, u} du
+    {\mathbb P}_x\left(x=s\right) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-ius}\Phi_{s, u} du
 \end{equation}
 
 ### Discrete distributions
 
-In these distributions, the random variable $x$ takes integer values. For example, the Poisson distribution is discrete.
+In these distributions, the random variable $x$ takes integer values $k$. For example, the Poisson distribution is discrete.
 The inversion formula for these distributions is given by
 
 \begin{equation}
-    {\mathbb P}\left(x=k\right) = \frac{1}{2\pi}\int_{-\pi}^\pi e^{-iuk}\Phi_{x, u} du
+    {\mathbb P}_x\left(x=k\right) = \frac{1}{2\pi}\int_{-\pi}^\pi e^{-iuk}\Phi_{k, u} du
 \end{equation}
 
 ```{code-cell}
 
 ```
+
+(characteristic-exponent)=
+## Characteristic Exponent
+
+The characteristic exponent $\phi_{x,u}$ is defined as
+
+\begin{equation}
+ \Phi_{x,u} = e^{-\phi_{x,u}}
+\end{equation}

notebooks/theory/levy.md

@@ -34,9 +34,10 @@ The independence and stationarity of the increments of the Lévy process imply t
  \Phi_{x_t, u} = {\mathbb E}\left[e^{i u x_t}\right] = e^{-\phi_{x_t, u}} = e^{-t \phi_{x_1,u}}
 \end{equation}
 
-where the **characteristic exponent** $\phi_{x_1,u}$ is given by the [Lévy–Khintchine formula](https://en.wikipedia.org/wiki/L%C3%A9vy_process).
+where the [](characteristic-exponent) $\phi_{x_1,u}$ is given by the [Lévy–Khintchine formula](https://en.wikipedia.org/wiki/L%C3%A9vy_process).
 
-There are several Lévy processes in the literature, including, importantly, the [Poisson process](../models/poisson.md), the compound Poisson process, and the Brownian motion.
+There are several Lévy processes in the literature, including, the [Poisson process](../models/poisson.md), the compound Poisson process
+and the [Brownian motion](../models/weiner.md).
 
 +++
 

quantflow/utils/distributions.py

@@ -24,20 +24,26 @@ def from_variance_and_asymmetry(cls, variance: float, asymmetry: float) -> Self:
         raise NotImplementedError
 
     def asymmetry(self) -> float:
-        """Asymmetry of the distribution"""
+        """Asymmetry of the distribution, 0 for symmetric
+
+        Implemented by distributions that have asymmetry
+        """
         raise NotImplementedError
 
     def set_variance(self, variance: float) -> None:
         """Set the variance of the distribution"""
         raise NotImplementedError
 
     def set_asymmetry(self, asymmetry: float) -> None:
-        """Set the asymmetry of the distribution"""
+        """Set the asymmetry of the distribution
+
+        Implemented by distributions that have asymmetry
+        """
         raise NotImplementedError
 
 
 class Exponential(Distribution1D):
-    r"""A :class:`.Marginal1D` for the `Exponential distribution`_
+    r"""A :class:`.Distribution1D` for the `Exponential distribution`_
 
     The exponential distribution is a continuous probability distribution with PDF
     given by
@@ -90,8 +96,21 @@ def cdf(self, x: FloatArrayLike) -> FloatArrayLike:
 
 
 class Normal(Distribution1D):
+    r"""A :class:`.Distribution1D` for the `Normal distribution`_
+
+    The normal distribution is a continuous probability distribution with PDF
+    given by
+
+    .. math::
+        f(x) = \frac{e^{-\frac{\left(x - \mu\right)^2}{2\sigma^2}}}{\sqrt{2\pi\sigma^2}}
+
+    .. _Normal distribution: https://en.wikipedia.org/wiki/Normal_distribution
+    """
+
     mu: float = Field(default=0, description="mean")
+    r"""The mean :math:`\mu` of the normal distribution"""
     sigma: float = Field(default=1, gt=0, description="standard deviation")
+    r"""The standard deviation :math:`\sigma` of the normal distribution"""
 
     @classmethod
     def from_variance_and_asymmetry(cls, variance: float, asymmetry: float) -> Self:
@@ -188,7 +207,8 @@ def characteristic(self, u: Vector) -> Vector:
         r"""Characteristic function of the double exponential distribution
 
         .. math::
-            \phi(u) = \frac{e^{i u \mu}}{1 - \sigma^2 u^2}
+            \phi(u) = \frac{e^{i u m}}{\left(1 + \frac{i u \kappa}{\lambda}\right)
+                \left(1 - \frac{i u}{\lambda \kappa}\right)}
         """
         den = (1.0 + 1j * u * self.kappa / self.decay) * (
             1.0 - 1j * u / (self.kappa * self.decay)