function [model, GAMMA2, LL] = EM_GMM(Data, model)
% Training of a Gaussian mixture model (GMM) with an expectation-maximization (EM) algorithm.
%
% Writing code takes time. Polishing it and making it available to others takes longer!
% If some parts of the code were useful for your research of for a better understanding
% of the algorithms, please reward the authors by citing the related publications,
% and consider making your own research available in this way.
%
% @article{Calinon15,
% author="Calinon, S.",
% title="A Tutorial on Task-Parameterized Movement Learning and Retrieval",
% journal="Intelligent Service Robotics",
% year="2015"
% }
%
% Copyright (c) 2015 Idiap Research Institute, http://idiap.ch/
% Written by Sylvain Calinon, http://calinon.ch/
%
% This file is part of PbDlib, http://www.idiap.ch/software/pbdlib/
%
% PbDlib is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License version 3 as
% published by the Free Software Foundation.
%
% PbDlib 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.
%
% You should have received a copy of the GNU General Public License
% along with PbDlib. If not, see .
%Parameters of the EM algorithm
nbMinSteps = 5; %Minimum number of iterations allowed
nbMaxSteps = 100; %Maximum number of iterations allowed
maxDiffLL = 1E-4; %Likelihood increase threshold to stop the algorithm
nbData = size(Data,2);
%diagRegularizationFactor = 1E-6; %Regularization term is optional
diagRegularizationFactor = 1E-4; %Regularization term is optional
for nbIter=1:nbMaxSteps
fprintf('.');
%E-step
[L, GAMMA] = computeGamma(Data, model); %See 'computeGamma' function below
GAMMA2 = GAMMA ./ repmat(sum(GAMMA,2),1,nbData);
%M-step
for i=1:model.nbStates
%Update Priors
model.Priors(i) = sum(GAMMA(i,:)) / nbData;
%Update Mu
model.Mu(:,i) = Data * GAMMA2(i,:)';
%Update Sigma
DataTmp = Data - repmat(model.Mu(:,i),1,nbData);
model.Sigma(:,:,i) = DataTmp * diag(GAMMA2(i,:)) * DataTmp' + eye(size(Data,1)) * diagRegularizationFactor;
end
%Compute average log-likelihood
LL(nbIter) = sum(log(sum(L,1))) / nbData;
%Stop the algorithm if EM converged (small change of LL)
if nbIter>nbMinSteps
if LL(nbIter)-LL(nbIter-1)