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learn_MixFHMM.m
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learn_MixFHMM.m
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function mixFHMM = learn_MixFHMM(data, K, R, ...
variance_type, order_constraint, total_EM_tries, max_iter_EM, init_kmeans, threshold, verbose)
%
% The EM algorithm for parameter estimation of the mixture of Hidden Markov
% Models for clustering and segmentation of time series with regime changes
%
% Inputs:
% data: a set of n time series with m observations (dim: [n x m]
% K: number of clusters
% R: number of regimes (states)
% options
%
%
%
% faicel chamroukhi (septembre 2009)
%
%% Please cite the following references for this code
%
% @InProceedings{Chamroukhi-IJCNN-2011,
% author = {F. Chamroukhi and A. Sam\'e and P. Aknin and G. Govaert},
% title = {Model-based clustering with Hidden Markov Model regression for time series with regime changes},
% Booktitle = {Proceedings of the International Joint Conference on Neural Networks (IJCNN), IEEE},
% Pages = {2814--2821},
% Adress = {San Jose, California, USA},
% year = {2011},
% month = {Jul-Aug},
% url = {https://chamroukhi.com/papers/Chamroukhi-ijcnn-2011.pdf}
% }
%
% @PhdThesis{Chamroukhi_PhD_2010,
% author = {Chamroukhi, F.},
% title = {Hidden process regression for curve modeling, classification and tracking},
% school = {Universit\'e de Technologie de Compi\`egne},
% month = {13 december},
% year = {2010},
% type = {Ph.D. Thesis},
% url ={https://chamroukhi.com/papers/FChamroukhi-Thesis.pdf}
% }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
warning off
[n, m] = size(data);%n nbre de signaux (individus); m: nbre de points pour chaque signal
%
Y=reshape(data',[],1);
%
try_EM = 0;
best_loglik = -inf;
cputime_total = [];
while try_EM < total_EM_tries
try_EM = try_EM +1;
fprintf('EM try n° %d\n',try_EM);
time = cputime;
%%%%%%%%%%%%%%%%%%%
% Initialization %
%%%%%%%%%%%%%%%%%%%
mixFHMM = init_MixFHMM(data, K, R,...
variance_type, order_constraint, init_kmeans, try_EM);
% Psi = zeros(nu,1);% vecteur parametre
iter = 0;
converge = 0;
loglik = 0;
prev_loglik=-inf;
stored_loglik=[];
% main algorithm
% % EM %%%%
while ~converge && (iter< max_iter_EM)
%
exp_num_trans_ck = zeros(R,R,n);
exp_num_trans_from_l_ck = zeros(R,n);
%
exp_num_trans = zeros(R,R,n,K);
exp_num_trans_from_l = zeros(R,n,K);
%
w_k_fyi = zeros(n,K);
log_w_k_fyi = zeros(n,K);
%%%%%%%%%%
% E-Step %
%%%%%%%%%%
gamma_ikjr = zeros(n*m,R,K);
for k=1:K
% run a hmm for each sequence
log_fkr_yij =zeros(R,m);
%
Li = zeros(n,1);% to store the loglik for each example
%
mu_kr = mixFHMM.param.mu_kr(:,k);
for i=1:n
y_i = data(i,:);
for r = 1:R
mukr = mu_kr(r);
%sk = sigma_kr(k);
if strcmp(variance_type,'common')
sigma_kr = mixFHMM.param.sigma_k(k);
sk = sigma_kr;
else
sigma_kr = mixFHMM.param.sigma_kr(:,k);
sk = sigma_kr(r);
end
z=((y_i-mukr*ones(1,m)).^2)/sk;
log_fkr_yij(r,:) = -0.5*ones(1,m).*(log(2*pi)+log(sk)) - 0.5*z;% pdf cond à c_i = g et z_i = k de yij
fkr_yij(r,:) = normpdf(y_i, mukr*ones(1,m), sqrt(sk));
end
% log_fkr_yij = min(log_fkr_yij,log(realmax));
% log_fkr_yij = max(log_fkr_yij ,log(realmin));
% fkr_yij = exp(log_fkr_yij);
% calcul de p(y) : forwards backwards
[gamma_ik, xi_ik, fwd_ik, backw_ik, loglik_i] = forwards_backwards(mixFHMM.param.pi_k(:,k), mixFHMM.param.A_k(:,:,k), fkr_yij);
%
Li(i) = loglik_i; % loglik of the ith curve
%
gamma_ikjr((i-1)*m+1:i*m,:,k) = gamma_ik';%[n*m K G]
% xi_ikjrl(:,:,(i-1)*(m-1)+1:i*(m-1),g) = xi_ik;%[KxK n*m G]
%
exp_num_trans_ck(:,:,i) = sum(xi_ik,3); % [K K n]
exp_num_trans_from_l_ck(:,i) = gamma_ik(:,1);%[K x n]
%
end
exp_num_trans_from_l(:,:,k) = exp_num_trans_from_l_ck;%[K n G]
exp_num_trans(:,:,:,k) = exp_num_trans_ck;%[K K n G]
% for the MAP partition: the numerator of the cluster post
% probabilities
num_log_post_prob(:,k) = log(mixFHMM.param.w_k(k)) + Li;
% for computing the global loglik
w_k_fyi(:,k) = mixFHMM.param.w_k(k)*exp(Li);%[nx1]
log_w_k_fyi(:,k) = log(mixFHMM.param.w_k(k)) + Li;
end
log_w_k_fyi = min(log_w_k_fyi,log(realmax));
log_w_k_fyi = max(log_w_k_fyi,log(realmin));
tau_ik = exp(log_w_k_fyi)./(sum(exp(log_w_k_fyi),2)*ones(1,K));
% % log-likelihood
loglik = sum(log(sum(exp(log_w_k_fyi),2)));
%%%%%%%%%%
% M-Step %
%%%%%%%%%%
% Maximization of Q1 w.r.t w_k
mixFHMM.param.w_k = sum(tau_ik,1)'/n;
for k=1:K
if strcmp(variance_type,'common'), s=0; end
weights_cluster_k = tau_ik(:,k);
% Maximization of Q2 w.r.t \pi^g
exp_num_trans_k_from_l = (ones(R,1)*weights_cluster_k').*exp_num_trans_from_l(:,:,k);%[K x n]
mixFHMM.param.pi_k(:,k) = (1/sum(tau_ik(:,k)))*sum(exp_num_trans_k_from_l,2);% sum over i
% Maximization of Q3 w.r.t A^g
for r=1:R
if n==1
exp_num_trans_k(r,:,:) = (ones(R,1)*weights_cluster_k)'.*squeeze(exp_num_trans(r,:,:,k));
else
%exp_num_trans_k(k,:,:,g)
exp_num_trans_k(r,:,:) = (ones(R,1)*weights_cluster_k').*squeeze(exp_num_trans(r,:,:,k));
end
end
if n==1
temp = exp_num_trans_k;
else
temp = sum(exp_num_trans_k,3);%sum over i
end
mixFHMM.param.A_k(:,:,k) = mk_stochastic(temp);
% if HMM with order constraints
if order_constraint
mixFHMM.param.A_k(:,:,k) = mk_stochastic(mixFHMM.stats.mask.*mixFHMM.param.A_k(:,:,k));
end
% Maximisation de Q4 par rapport aux muk et sigmak
% each sequence i (m observations) is first weighted by the cluster weights
weights_cluster_k = repmat((tau_ik(:,k))',m,1);
weights_cluster_k = weights_cluster_k(:);
% secondly, the m observations of each sequance are weighted by the
% wights of each segment k (post prob of the segments for each
% cluster g)
gamma_ijk = gamma_ikjr(:,:,k);% [n*m K]
nm_kr=sum(gamma_ijk,1);% cardinal nbr of the segments k,k=1,...,K within each cluster g, at iteration q
sigma_kr = zeros(R,1);
for r=1:R
nmkr = nm_kr(r);%cardinal nbr of segment k for the cluster g
% % Maximization w.r.t muk
weights_seg_k = gamma_ijk(:,r);
mu_kr(r) = (1/sum(weights_cluster_k.*weights_seg_k))*sum((weights_cluster_k.*weights_seg_k).*Y);
% % Maximization w.r.t sigmak :
z = sqrt(weights_cluster_k.*weights_seg_k).*(Y-ones(n*m,1)*mu_kr(r));
if strcmp(variance_type,'common')
s = s + z'*z;
ngm = sum(sum((weights_cluster_k*ones(1,R)).*gamma_ijk));
sigma_k = s/ngm;
else
ngmk = sum(weights_cluster_k.*weights_seg_k);
sigma_kr(r)= z'*z/(ngmk);
end
end
mixFHMM.param.mu_kr(:,k) = mu_kr;
if strcmp(variance_type,'common')
mixFHMM.param.sigma_k(k) = sigma_k;
else
mixFHMM.param.sigma_kr(:,k) = sigma_kr;
end
end
iter=iter+1;
if prev_loglik-loglik > 1e-3, fprintf(1, '!!!!! EM log-lik is decreasing from %6.4f to %6.4f!\n', prev_loglik, loglik);end
if verbose
fprintf(1,'EM : Iteration : %d log-likelihood : %f \n', iter,loglik);
end
converge = abs((loglik-prev_loglik)/prev_loglik) <= threshold;
prev_loglik = loglik;
stored_loglik = [stored_loglik loglik];
end % end of EM loop
cputime_total = [cputime_total cputime-time];
mixFHMM.param = mixFHMM.param;
if strcmp(variance_type,'common')
mixFHMM.stats.Psi = [mixFHMM.param.w_k(:); mixFHMM.param.A_k(:); mixFHMM.param.pi_k(:); mixFHMM.param.mu_kr(:); mixFHMM.param.sigma_k(:)];
else
mixFHMM.stats.Psi = [mixFHMM.param.w_k(:); mixFHMM.param.A_k(:); mixFHMM.param.pi_k(:); mixFHMM.param.mu_kr(:); mixFHMM.param.sigma_kr(:)];
end
mixFHMM.stats.tau_ik = tau_ik;
mixFHMM.stats.gamma_ikjr = gamma_ikjr;
mixFHMM.stats.loglik = loglik;
mixFHMM.stats.stored_loglik = stored_loglik;
mixFHMM.stats.log_w_k_fyi = log_w_k_fyi;
if mixFHMM.stats.loglik > best_loglik
best_loglik = mixFHMM.stats.loglik;
best_mixFHMM.stats = mixFHMM.stats;
end
if try_EM>=1, fprintf('log-lik at convergence: %f \n', mixFHMM.stats.loglik); end
end
mixFHMM.stats.loglik = best_loglik;
%
if try_EM>1, fprintf('log-lik max: %f \n', mixFHMM.stats.loglik); end
mixFHMM.stats = best_mixFHMM.stats;
% Finding the curve partition by using the MAP rule
[klas, Cik] = MAP(mixFHMM.stats.tau_ik);% MAP partition of the n sequences
mixFHMM.stats.klas = klas;
% cas ou on prend la moyenne des gamma ijkr
smoothed = zeros(m,K);
mean_curves = zeros(m,R,K);
%mean_gamma_ijk = zeros(m,R,K);
for k=1:K
weighted_segments = sum(gamma_ikjr(:,:,k).*(Y*ones(1,R)),2);
%weighted_segments = sum(gamma_ikjr(:,:,g).*(ones(n*m,1)*mixFHMM.param.mu_kr(:,k)'),2);
%
weighted_segments = reshape(weighted_segments,m,n);
weighted_clusters = (ones(m,1)*mixFHMM.stats.tau_ik(:,k)').* weighted_segments;
smoothed(:,k) = (1/sum(mixFHMM.stats.tau_ik(:,k)))*sum(weighted_clusters,2);
end
mixFHMM.stats.smoothed = smoothed;
mixFHMM.stats.mean_curves = mean_curves;
%mixFHMM.stats.mean_gamma_ijk = mean_gamma_ijk;
mixFHMM.stats.cputime = mean(cputime_total);
% % Segmentation of each cluster using the MAP rule
% for k=1:K
% [segments_k Zjk] = MAP(mixFHMM.stats.mean_gamma_ijk(:,:,k));%MAP segmentation of each cluster of sequences
% mixFHMM.stats.segments(:,k) = segments_k;
% end
nu = length(mixFHMM.stats.Psi);
% BIC AIC et ICL*
mixFHMM.stats.BIC = mixFHMM.stats.loglik - (nu*log(n)/2);%n*m/2!
mixFHMM.stats.AIC = mixFHMM.stats.loglik - nu;
% ICL*
% Compute the comp-log-lik
cik_log_w_k_fyi = (Cik).*(mixFHMM.stats.log_w_k_fyi);
comp_loglik = sum(sum(cik_log_w_k_fyi,2));
mixFHMM.stats.ICL1 = comp_loglik - nu*log(n)/2;%n*m/2!