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fit_gmm.m
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function [Priors, Mu, Sigma] = fit_gmm(Xi_ref, Xi_dot_ref, est_options)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Copyright (C) 2018 Learning Algorithms and Systems Laboratory, %
% EPFL, Switzerland %
% Author: Nadia Figueroa %
% email: nadia.figueroafernandez@epfl.ch %
% website: http://lasa.epfl.ch %
% %
% This work was supported by the EU project Cogimon H2020-ICT-23-2014. %
% %
% Permission is granted to copy, distribute, and/or modify this program %
% under the terms of the GNU General Public License, version 2 or any %
% later version published by the Free Software Foundation. %
% %
% This program is distributed in the hope that it will be useful, but %
% WITHOUT ANY WARRANTY; without even the implied warranty of %
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General%
% Public License for more details %
% %
% If you use this code in your research please cite: %
% "A Physically-Consistent Bayesian Non-Parametric Mixture Model for %
% Dynamical System Learning."; N. Figueroa and A. Billard; CoRL 2018 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Parse Options
est_type = est_options.type;
do_plots = est_options.do_plots;
[M,N] = size(Xi_ref);
if est_type == 1
max_gaussians = est_options.maxK;
if isempty(est_options.fixed_K)
fixed_K = 0;
else
fixed_K = est_options.fixed_K;
end
end
if ~isempty(est_options.sub_sample)
sub_sample = est_options.sub_sample;
Xi_ref = Xi_ref(:,1:sub_sample:end);
Xi_dot_ref = Xi_dot_ref(:,1:sub_sample:end);
end
if est_type ~= 1
if isempty(est_options.samplerIter)
if est_type == 0
samplerIter = 20;
end
if est_type == 2
samplerIter = 200;
end
else
samplerIter = est_options.samplerIter;
end
if ~isempty(est_options.l_sensitivity)
l_sensitivity = est_options.l_sensitivity;
else
l_sensitivity = 2;
end
end
switch est_type
case 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% Option1: Non-parametric Clustering with Pos-Vel-cos-sim prior %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute estimate of length-scale
if est_options.estimate_l == 1
[D, mode_hist_D, mean_D] = computePairwiseDistances(Xi_ref',1);
if mode_hist_D == 0
mode_hist_D = mean_D;
end
sigma = sqrt(mode_hist_D/l_sensitivity);
l = 1/(2*sigma^2);
close;
else
l = est_options.length_scale;
end
fprintf('Computed length-scale l=%2.4f \n',l);
% Compute element-wise cosine similarities
S = zeros(length(Xi_dot_ref),length(Xi_dot_ref));
for i=1:length(Xi_dot_ref)
for j=1:length(Xi_dot_ref)
% *** Option 1 ***: Compute Velocity component using tan function
% % Normalize vectors
% xi_dot_i = Xi_dot_ref(:,i)/norm(Xi_dot_ref(:,i));
% xi_dot_j = Xi_dot_ref(:,j)/norm(Xi_dot_ref(:,j));
% % Compute Angle
% s_angle = atan2(xi_dot_i(2),xi_dot_i(1))-atan2(xi_dot_j(2),xi_dot_j(1));
% cos_angle = cos(s_angle);
% *** Option 2 ***: Compute Velocity component using dot product
cos_angle = (Xi_dot_ref(:,i)'*Xi_dot_ref(:,j))/(norm(Xi_dot_ref(:,i))*norm(Xi_dot_ref(:,j)));
% Compute shifted cosine of angle
if isnan(cos_angle)
cos_angle = 0;
end
s = 1 + cos_angle;
% Compute Position component
xi_i = Xi_ref(:,i);
xi_j = Xi_ref(:,j);
% Euclidean pairwise position-kernel
p = exp(-l*norm(xi_i - xi_j));
% Shifted Cosine Similarity of velocity vectors
S(i,j) = p * s;
end
end
% Plot Similarity matrix
if do_plots
if exist('h_sim','var'); delete(h_sim); end;
title_str = 'Physically-Consistent Similarity Confusion Matrix';
h_sim = plotSimilarityConfMatrix(S, title_str);
pause(0);
end
% Setting sampler/model options (i.e. hyper-parameters, alpha, Covariance matrix)
Xi_ref_mean = mean(Xi_ref,2);
options = [];
options.type = 'full'; % Type of Covariance Matrix: 'full' = NIW or 'Diag' = NIG
options.T = samplerIter; % Sampler Iterations
% options.alpha = max(0.5,0.1*(randi(11)-2)); % Concentration parameter
options.alpha = 2; % maximum of similarity function
% Standard Base Distribution Hyper-parameter setting
if strcmp(options.type,'diag')
lambda.alpha_0 = M; % G(sigma_k^-1|alpha_0,beta_0): (degrees of freedom)
lambda.beta_0 = sum(diag(cov(Xi_ref')))/M; % G(sigma_k^-1|alpha_0,beta_0): (precision)
end
if strcmp(options.type,'full')
lambda.nu_0 = M; % IW(Sigma_k|Lambda_0,nu_0): (degrees of freedom)
% lambda.Lambda_0 = eye(M)*sum(diag(cov(Xi_ref')))/M; % IW(Sigma_k|Lambda_0,nu_0): (Scale matrix)
lambda.Lambda_0 = diag(diag(cov(Xi_ref'))); % IW(Sigma_k|Lambda_0,nu_0): (Scale matrix)
end
lambda.mu_0 = Xi_ref_mean; % hyper for N(mu_k|mu_0,kappa_0)
lambda.kappa_0 = 1; % hyper for N(mu_k|mu_0,kappa_0)
% Run Collapsed Gibbs Sampler
options.lambda = lambda;
options.verbose = 1;
[Psi Psi_Stats] = run_ddCRP_sampler(Xi_ref, S, options);
est_labels = Psi.Z_C';
% Extract Learnt cluster parameters
unique_labels = unique(est_labels);
est_K = length(unique_labels);
Priors = zeros(1, est_K);
singletons = zeros(1, est_K);
for k=1:est_K
assigned_k = sum(est_labels==unique_labels(k));
Priors(k) = assigned_k/N;
singletons(k) = assigned_k < 2;
end
Mu = Psi.Theta.Mu;
Sigma = Psi.Theta.Sigma;
if any(singletons)
[~, est_labels] = my_gmm_cluster(Xi_ref, Priors, Mu, Sigma, 'hard', []);
unique_labels = unique(est_labels);
est_K = length(unique_labels);
singleton_idx = find(singletons == 1);
Mu(:,singleton_idx) = [];
Sigma(:,:,singleton_idx) = [];
Priors = [];
for k=1:est_K
assigned_k = sum(est_labels==unique_labels(k));
Priors(k) = assigned_k/N;
end
end
%% Re-estimate GMM parameters, needed for >2D data
if M > 2
Mu_k = Mu; Sigma_k = Sigma;
for k=1:length(unique_labels)
cluster_points = Xi_ref(:,est_labels == unique_labels(k));
if ~isempty(cluster_points)
[ V_k, L_k, Mu_k(:,k) ] = my_pca( cluster_points );
Sigma_k(:,:,k) = V_k*L_k*V_k';
end
end
rel_dilation_fact = 0.15;
Sigma_k = adjust_Covariances(Priors, Sigma_k, 1, rel_dilation_fact);
Mu = Mu_k;
Sigma = Sigma_k;
end
if do_plots
if exist('h1b','var') && isvalid(h1b), delete(h1b);end
stats_options = [];
stats_options.dataset = 'Sampler Stats';
stats_options.true_labels = [];
stats_options.Psi = Psi;
[ h1b ] = plotSamplerStats( Psi_Stats, stats_options );
end
case 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Option2: Cluster Trajectories with GMM-EM + BIC Model Selection %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if fixed_K == 0
em_type = 'matlab'; repetitions = 10;
[bic_scores, k] = fit_gmm_bic([Xi_ref],max_gaussians, repetitions, em_type, do_plots);
else
k = fixed_K;
end
% Train GMM with Optimal k
warning('off', 'all'); % there are a lot of really annoying warnings when fitting GMMs
%fit a GMM to our data
GMM_full = fitgmdist([Xi_ref]', k, 'Start', 'plus', 'CovarianceType','full', 'Regularize', .000001, 'Replicates', 10);
warning('on', 'all');
% Extract Model Parameters
Priors = GMM_full.ComponentProportion;
Mu = transpose(GMM_full.mu);
Sigma = GMM_full.Sigma;
case 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Option3: Cluster Trajectories with Chinese Restaurant Process MM sampler (CRP-GMM) %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CRP-GMM (Frank-Wood's implementation) -- faster
[class_id, mean_record, covariance_record, K_record, lP_record, alpha_record] = sampler(Xi_ref, samplerIter);
[val , Maxiter] = max(lP_record);
est_labels = class_id(:,Maxiter);
% Visualization and plotting options
if do_plots
figure('Color',[1 1 1])
subplot(2,1,1)
semilogx(1:samplerIter, lP_record'); hold on;
semilogx(Maxiter,lP_record(Maxiter),'ko','MarkerSize',10);
grid on
xlabel('Gibbs Iteration','Interpreter','LaTex','Fontsize',20); ylabel('LogPr','Interpreter','LaTex','Fontsize',20)
xlim([1 samplerIter])
legend({'$p(Z|Y, \alpha, \lambda)$'},'Interpreter','LaTex','Fontsize',14)
title(sprintf('CRP-GMM Sampling results, optimal K=%d at iter=%d', length(unique(est_labels)), Maxiter), 'Interpreter','LaTex','Fontsize',20)
subplot(2,1,2)
stairs(K_record, 'LineWidth',2);
set(gca, 'XScale', 'log')
xlim([1 samplerIter])
xlabel('Gibbs Iteration','Interpreter','LaTex','Fontsize',20); ylabel('$\Psi$ = Estimated K','Interpreter','LaTex','Fontsize',20);
end
% Extract Learnt cluster parameters
unique_labels = unique(est_labels);
est_K = length(unique_labels);
Priors = zeros(1, est_K);
singletons = zeros(1, est_K);
for k=1:est_K
assigned_k = sum(est_labels==unique_labels(k));
Priors(k) = assigned_k/N;
singletons(k) = assigned_k < 2;
end
Mu = mean_record {Maxiter};
Sigma = covariance_record{Maxiter};
% Remove Singleton Clusters
if any(singletons)
[~, est_labels] = my_gmm_cluster(Xi_ref, Priors, Mu, Sigma, 'hard', []);
unique_labels = unique(est_labels);
est_K = length(unique_labels);
Mu = Mu(:,unique_labels);
Sigma = Sigma(:,:,unique_labels);
Priors = [];
for k=1:est_K
assigned_k = sum(est_labels==unique_labels(k));
Priors(k) = assigned_k/N;
end
end
end
end