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function [model, LL] = EM_tensorGMM(Data, model)
% Training of a task-parameterized Gaussian mixture model (GMM) with an expectation-maximization (EM) algorithm.
% The approach allows the modulation of the centers and covariance matrices of the Gaussians with respect to
% external parameters represented in the form of candidate coordinate systems.
% Author: Sylvain Calinon, 2014
% This source code is given for free! In exchange, I would be grateful if you cite
% the following reference in any academic publication that uses this code or part of it:
% @inproceedings{Calinon14ICRA,
% author="Calinon, S. and Bruno, D. and Caldwell, D. G.",
% title="A task-parameterized probabilistic model with minimal intervention control",
% booktitle="Proc. {IEEE} Intl Conf. on Robotics and Automation ({ICRA})",
% year="2014",
% month="May-June",
% address="Hong Kong, China",
% pages=""
% }
%Parameters of the EM algorithm
nbMinSteps = 5; %Minimum number of iterations allowed
nbMaxSteps = 200; %Maximum number of iterations allowed
maxDiffLL = 1E-4; %Likelihood increase threshold to stop the algorithm
nbData = size(Data,3);
diagRegularizationFactor = 1E-6;
for nbIter=1:nbMaxSteps
[L, GAMMA, GAMMA0] = computeGamma(Data, model); %See 'computeGamma' function below
GAMMA2 = GAMMA ./ repmat(sum(GAMMA,2),1,nbData);
for i=1:model.nbStates
%Update Priors
model.Priors(i) = sum(sum(GAMMA(i,:))) / nbData;
for m=1:model.nbFrames
%Matricization/flattening of tensor
DataMat(:,:) = Data(:,m,:);
%Update Mu
model.Mu(:,m,i) = DataMat * GAMMA2(i,:)';
%Update Sigma (regularization term is optional)
DataTmp = DataMat - repmat(model.Mu(:,m,i),1,nbData);
model.Sigma(:,:,m,i) = DataTmp * diag(GAMMA2(i,:)) * DataTmp' + eye(model.nbVar) * diagRegularizationFactor;
%Compute average log-likelihood
LL(nbIter) = sum(log(sum(L,1))) / size(L,2);
%Stop the algorithm if EM converged (small change of LL)
if nbIter>nbMinSteps
if LL(nbIter)-LL(nbIter-1)<maxDiffLL || nbIter==nbMaxSteps-1
disp(['EM converged after ' num2str(nbIter) ' iterations.']);
disp(['The maximum number of ' num2str(nbMaxSteps) ' EM iterations has been reached.']);
function [L, GAMMA, GAMMA0] = computeGamma(Data, model)
nbData = size(Data, 3);
L = ones(model.nbStates, nbData);
GAMMA0 = zeros(model.nbStates, model.nbFrames, nbData);
for m=1:model.nbFrames
DataMat(:,:) = Data(:,m,:); %Matricization/flattening of tensor
for i=1:model.nbStates
%GAMMA0(i,m,:) = model.Priors(i) * gaussPDF(DataMat, model.Mu(:,m,i), model.Sigma(:,:,m,i));
GAMMA0(i,m,:) = gaussPDF(DataMat, model.Mu(:,m,i), model.Sigma(:,:,m,i));
L(i,:) = L(i,:) .* squeeze(GAMMA0(i,m,:))';
for i=1:model.nbStates
L(i,:) = model.Priors(i) * L(i,:);
GAMMA = L ./ repmat(sum(L,1)+realmin,size(L,1),1);