icaml.m

this is a matlab code for blind source separation using independent component analysis

function [S,A,U,ll,info]=icaML(X,K,par,debug_draw)
% icaML     : ICA by ML (Infomax) with square mixing matrix and no noise.
%
% function [S,A,U,ll,info]=icaML(X,[K],[par]) Independent component analysis (ICA) using
%                                       maximum likelihood, square mixing matrix and 
%                                       no noise [1] (Infomax). Source prior is assumed 
%                                       to be p(s)=1/pi*exp(-ln(cosh(s))). For optimization
%                                       the BFGS algorithm is used [2]. See code for 
%                                       references.
%                                       
%                                       X  : Mixed signals
%                                       K  : Number of source components.
%                                           For K=0 (default) number of sources are equal to number of
%                                           observations.
%                                           For K < number of observations, SVD is used to reduce the
%                                           dimension.
%                                       par:  Vector with 4 elements:
%                                           (1)  :  Expected length of initial step
%                                           Stopping criteria:
%                                           (2)  :  Gradient  ||g||_inf <= par(2) 
%                                           (3)  :  Parameter changes ||dW||_2  <= par(3)*(par(3) + ||W||_2)
%                                           (4)  :  Maximum number of iterations 
%                                           Any illegal element in  opts  is replaced by its
%                                           default value,  [1  1e-4*||g(x0)||_inf  1e-8  100]
%                                       debug_draw : Draw debug information
%                                       
%                                       S  : Estimated source signals with variance
%                                            scaled to one.
%                                       A  : Estimated mixing matrix
%                                       U  : Principal directions of preprocessing PCA. 
%                                            If K (the number of sources) is equal to the number
%                                            of observations then no PCA is performed and U=eye(K).  
%                                       ll : Log likelihood for estimated sources
%                                       info :  Performance information, vector with 6 elements:
%                                           (1:3)  : final values of [ll  ||g||_inf  ||dx||_2] 
%                                           (4:5)  : no. of iteration steps and evaluations of (ll,g)
%                                           (6)    : 1 means stopped by small gradient
%                                                    2 means stopped by small x-step
%                                                    3 means stopped by max number of iterations.
%                                                    4 means stopped by zero step.
%
%
%                                       Eks. Separate with number of sources equal number of
%                                       observations.
%                                       
%                                               [S,A] = icaML(X);
%                                       
%                                       Eks. Separate with number of sources equal to k, using SVD
%                                       as pre-processing.
%                                       
%                                               [S,A,U] = icaML(X,k);
%                                       
% - version 1.5 (Revised 9/9-2003)
% - IMM, Technical University of Denmark

% - version 1.4
% - by Thomas Kolenda 2002 - IMM, Technical University of Denmark

% Revised: 9/9-2003, Mads, [e-mail]
%   * Fixed help message to inform the user about U.
%   * Removed the automatic use of PCA in the quadratic case.

% Bibtex references:
% [1]  
%   @article{Bell95,
%       author =       "A. Bell and T.J. Sejnowski",
%       title =        "An Information-Maximization Approach to Blind Separation and Blind Deconvolution",
%       journal =      "Neural Computation",
%       year =         "1995",
%       volume =       "7",
%       pages =        "1129-1159",
%   }
%
% [2]
%   @techreport{Nielsen01:unopt,
%       author =        "H.B. Nielsen",
%       title =         "UCMINF - an Algorithm for Unconstrained, Nonlinear Optimization ",
%       institution =   "IMM, Technical University of Denmark",
%       number =        "IMM-TEC-0019",
%       year =          "2001",
%       url =           "http://www.imm.dtu.dk/pubdb/views/edoc_download.php/642/ps/imm642.ps",
%   }


% algorithm parameter settings
MaxNrIte = 1000;

try
  debug = debug_draw;
catch
  debug = 0;    
end

if debug==1,fprintf('\n** Start icaML ***************************************\n');tic;end

% Scale X to avoid numerical problems
Xorg = X;
scaleX = max([abs(max(X(:))),abs(min(X(:)))]);
X = X./scaleX;

% set number of source parameters
if nargin<2, K=size(X,1); end
if ((K>0) & (K<size(X,1))),
  [U,X] = callSVD(X,K,debug);
else
  U=eye(size(X,1));
  if debug==1,disp('Don''t use SVD');end
end

% initialize optimize parameters
try 
  ucminf_opt(1) = par(1);
  ucminf_opt(2) = par(2);
  ucminf_opt(3) = par(3);
  ucminf_opt(4) = par(4);
catch
  if debug==1,disp('Use default parameters to optimize');end
  ucminf_opt = [1  1e-4  1e-8  MaxNrIte];
end

% initialize variables
[M,N] = size(X);
W = eye(M);

if debug==1,fprintf('Number of samples %d - Number of sources %d\n',N,M);end

% optimize
if debug==1,fprintf('Optimize ICA ... ');end
par.X=X; par.M=M; par.N=N;
[W,info] = ucminf( 'ica_MLf' , par , W(:) , ucminf_opt );
W = reshape(W,M,M); 

% estimates
A=pinv(W);
S=W*X;

if debug==1,disp('done optimize ICA!');end

% sort components according to energy
Avar=diag(A'*A)/M;
Svar=diag(S*S')/N;
vS=var(S');
sig=Avar.*Svar;
[a,indx]=sort(sig);
S=S(indx(M:-1:1),:);
A=A(:,indx(M:-1:1));

% scale back
A=A.*scaleX;

% log likelihood
if nargout>3
  ll= N*log(abs(det(inv(A)))) - sum(sum(log( cosh(S) ))) - N*M*log(pi);
end

if debug==1,fprintf('** End of icaML ****** time %0.2f sec******************\n\n',toc/60);end



function [f,dW]=ica_MLf(W,par)
% returns the negative log likelihood and its gradient w.r.t. W

X=par.X; M=par.M; N=par.N;
W=reshape(W,M,M);

S=W*X;

% negative log likelihood function
f=-( N*log(abs(det(W))) - sum(sum(log( cosh(S) ))) - N*M*log(pi) );

if nargout>1
  % gradient w.r.t. W
  dW=-(N*inv(W') - tanh(S)*X');
  dW=dW(:);
end



function [U,DV] = callSVD(X,K,draw)
% Reduce dimension with SVD

[M,N] = size(X);

if N>M % Transpose if matrix is flat, to speed up the svd and later transpose back again
    if draw==1,disp('Do Transpose SVD');end
    [V,D,U] = svd(X',0);
else;  
    if draw==1,disp('Do SVD');end
    [U,D,V] = svd(X,0);
end;
  
DV = D(1:K,1:K)*V(:,1:K)';



function  [X, info, perf, D] = ucminf(fun,par, x0, opts, D0)
%UCMINF  BFGS method for unconstrained nonlinear optimization:
% Find  xm = argmin{f(x)} , where  x  is an n-vector and the scalar
% function  F  with gradient  g  (with elements  g(i) = DF/Dx_i )
% must be given by a MATLAB function with with declaration
%           function  [F, g] = fun(x, par)
% par  holds parameters of the function.  It may be dummy.