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% function [Lh,Ph,Mu,Pi,LL]=mfa(X,M,K,cyc,tol);
%
% Maximum Likelihood Mixture of Factor Analysis using EM
%
% X - data matrix
% M - number of mixtures (default 1)
% K - number of factors in each mixture (default 2)
% cyc - maximum number of cycles of EM (default 100)
% tol - termination tolerance (prop change in likelihood) (default 0.0001)
%
% Lh - factor loadings
% Ph - diagonal uniquenesses matrix
% Mu - mean vectors
% Pi - priors
% LL - log likelihood curve
%
% Iterates until a proportional change < tol in the log likelihood
% or cyc steps of EM
function [Lh, Ph, Mu, Pi, LL] = mfa(X,M,K,cyc,tol)
if nargin<5 tol=0.0001; end;
if nargin<4 cyc=100; end;
if nargin<3 K=2; end;
if nargin<2 M=1; end;
N=length(X(:,1));
D=length(X(1,:));
tiny=exp(-700);
%rand('state',0);
fprintf('\n');
if (M==1)
[Lh,Ph,LL]=ffa(X,K,cyc,tol);
Mu=mean(X);
Pi=1;
else
if N==1
mX = X;
else
mX=mean(X);
end
cX=cov(X);
scale=det(cX)^(1/D);
randn('state',0);
Lh=randn(D*M,K)*sqrt(scale/K);
Ph=diag(cX)+tiny;
Pi=ones(M,1)/M;
%randn('state',0);
Mu=randn(M,D)*sqrtm(cX)+ones(M,1)*mX;
oldMu=Mu;
I=eye(K);
lik=0;
LL=[];
H=zeros(N,M); % E(w|x)
EZ=zeros(N*M,K);
EZZ=zeros(K*M,K);
XX=zeros(D*M,D);
s=zeros(M,1);
const=(2*pi)^(-D/2);
%%%%%%%%%%%%%%%%%%%%
for i=1:cyc;
%%%% E Step %%%%
Phi=1./Ph;
Phid=diag(Phi);
for k=1:M
Lht=Lh((k-1)*D+1:k*D,:);
LP=Phid*Lht;
MM=Phid-LP*inv(I+Lht'*LP)*LP';
dM=sqrt(det(MM));
Xk=(X-ones(N,1)*Mu(k,:));
XM=Xk*MM;
H(:,k)=const*Pi(k)*dM*exp(-0.5*rsum(XM.*Xk));
EZ((k-1)*N+1:k*N,:)=XM*Lht;
end;
Hsum=rsum(H);
oldlik=lik;
lik=sum(log(Hsum+(Hsum==0)*exp(-744)));
Hzero=(Hsum==0); Nz=sum(Hzero);
H(Hzero,:)=tiny*ones(Nz,M)/M;
Hsum(Hzero)=tiny*ones(Nz,1);
H=rdiv(H,Hsum);
s=csum(H);
s=s+(s==0)*tiny;
s2=sum(s)+tiny;
for k=1:M
kD=(k-1)*D+1:k*D;
Lht=Lh(kD,:);
LP=Phid*Lht;
MM=Phid-LP*inv(I+Lht'*LP)*LP';
Xk=(X-ones(N,1)*Mu(k,:));
XX(kD,:)=rprod(Xk,H(:,k))'*Xk/s(k);
beta=Lht'*MM;
EZZ((k-1)*K+1:k*K,:)=I-beta*Lht +beta*XX(kD,:)*beta';
end;
%%%% log likelihood %%%%
LL=[LL lik];
fprintf('cycle %g \tlog likelihood %g ',i,lik);
if (i<=2)
likbase=lik;
elseif (lik<oldlik)
fprintf(' violation');
elseif ((lik-likbase)<(1 + tol)*(oldlik-likbase)||~isfinite(lik))
break;
end;
fprintf('\n');
%%%% M Step %%%%
% means and covariance structure
Ph=zeros(D,1);
for k=1:M
kD=(k-1)*D+1:k*D;
kK=(k-1)*K+1:k*K;
kN=(k-1)*N+1:k*N;
T0=rprod(X,H(:,k));
T1=T0'*[EZ(kN,:) ones(N,1)];
XH=EZ(kN,:)'*H(:,k);
T2=inv([s(k)*EZZ(kK,:) XH; XH' s(k)]);
T3=T1*T2;
Lh(kD,:)=T3(:,1:K);
Mu(k,:)=T3(:,K+1)';
T4=diag(T0'*X-T3*T1')/s2;
Ph=Ph+T4.*(T4>0);
end;
Phmin=exp(-700);
Ph=Ph.*(Ph>Phmin)+(Ph<=Phmin)*Phmin; % to avoid zero variances
% priors
Pi=s'/s2;
end;
fprintf('\n');
end;
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