盲源分离的JADE算法的Matlab程序

function B = jadeR(X,m) % B = jadeR(X, m) is an m*n matrix such that Y=B*X are separated sources % extracted from the n*T data matrix X. % If m is omitted, B=jadeR(X) is a square n*n matrix (as many sources as sensors) % % Blind separation of real signals with JADE. Version 1.8. May 2005. % % Usage: % * If X is an nxT data matrix (n sensors, T samples) then % B=jadeR(X) is a nxn separating matrix such that S=B*X is an nxT % matrix of estimated source signals. % * If B=jadeR(X,m), then B has size mxn so that only m sources are % extracted. This is done by restricting the operation of jadeR % to the m first principal components. % * Also, the rows of B are ordered such that the columns of pinv(B) % are in order of decreasing norm; this has the effect that the % `most energetically significant' components appear first in the % rows of S=B*X. % % Quick notes (more at the end of this file) % % o this code is for REAL-valued signals. An implementation of JADE % for both real and complex signals is also available from % http://sig.enst.fr/~cardoso/stuff.html % % o This algorithm differs from the first released implementations of % JADE in that it has been optimized to deal more efficiently % 1) with real signals (as opposed to complex) % 2) with the case when the ICA model does not necessarily hold. % % o There is a practical limit to the number of independent % components that can be extracted with this implementation. Note % that the first step of JADE amounts to a PCA with dimensionality % reduction from n to m (which defaults to n). In practice m % cannot be `very large' (more than 40, 50, 60... depending on % available memory) % % o See more notes, references and revision history at the end of % this file and more stuff on the WEB % http://sig.enst.fr/~cardoso/stuff.html % % o This code is supposed to do a good job! Please report any % problem to cardoso@sig.enst.fr % Copyright : Jean-Francois Cardoso. cardoso@sig.enst.fr verbose = 1 ; % Set to 0 for quiet operation % Finding the number of sources [n,T] = size(X); if nargin==1, m=n ; end; % Number of sources defaults to # of sensors if m>n , fprintf('jade -> Do not ask more sources than sensors here!!!\n'), return,end if verbose, fprintf('jade -> Looking for %d sources\n',m); end ; % to do: add a warning about complex signals % Mean removal %============= if verbose, fprintf('jade -> Removing the mean value\n'); end X = X - mean(X')' * ones(1,T); %%% whitening & projection onto signal subspace % =========================================== if verbose, fprintf('jade -> Whitening the data\n'); end [U,D] = eig((X*X')/T) ; %% An eigen basis for the sample covariance matrix [Ds,k] = sort(diag(D)) ; %% Sort by increasing variances PCs = n:-1:n-m+1 ; %% The m most significant princip. comp. by decreasing variance %% --- PCA ---------------------------------------------------------- B = U(:,k(PCs))' ; % At this stage, B does the PCA on m components %% --- Scaling ------------------------------------------------------ scales = sqrt(Ds(PCs)) ; % The scales of the principal components . B = diag(1./scales)*B ; % Now, B does PCA followed by a rescaling = sphering %% --- Sphering ------------------------------------------------------ X = B*X; %% We have done the easy part: B is a whitening matrix and X is white. clear U D Ds k PCs scales ; %%% NOTE: At this stage, X is a PCA analysis in m components of the real data, except that %%% all its entries now have unit variance. Any further rotation of X will preserve the %%% property that X is a vector of uncorrelated components. It remains to find the %%% rotation matrix such that the entries of X are not only uncorrelated but also `as %%% independent as possible'. This independence is measured by correlations of order %%% higher than 2. We have defined such a measure of independence which %%% 1) is a reasonable approximation of the mutual information %%% 2) can be optimized by a `fast algorithm' %%% This measure of independence also corresponds to the `diagonality' of a set of %%% cumulant matrices. The code below finds the `missing rotation ' as the matrix which %%% best diagonalizes a particular set of cumulant matrices. %%% Estimation of the cumulant matrices. % ==================================== if verbose, fprintf('jade -> Estimating cumulant matrices\n'); end %% Reshaping of the data, hoping to speed up things a little bit... X = X'; dimsymm = (m*(m+1))/2; % Dim. of the space of real symm matrices nbcm = dimsymm ; % number of cumulant matrices CM = zeros(m,m*nbcm); % Storage for cumulant matrices R = eye(m); %% Qij = zeros(m); % Temp for a cum. matrix Xim = zeros(m,1); % Temp Xijm = zeros(m,1); % Temp Uns = ones(1,m); % for convenience %% I am using a symmetry trick to save storage. I should write a short note one of these %% days explaining what is going on here. %% Range = 1:m ; % will index the columns of CM where to store the cum. mats. for im = 1:m Xim = X(:,im) ; Xijm= Xim.*Xim ; %% Note to myself: the -R on next line can be removed: it does not affect %% the joint diagonalization criterion Qij = ((Xijm(:,Uns).*X)' * X)/T - R - 2 * R(:,im)*R(:,im)' ; CM(:,Range) = Qij ; Range = Range + m ; for jm = 1:im-1 Xijm = Xim.*X(:,jm) ; Qij = sqrt(2) *(((Xijm(:,Uns).*X)' * X)/T - R(:,im)*R(:,jm)' - R(:,jm)*R(:,im)') ; CM(:,Range) = Qij ; Range = Range + m ; end ; end; %%%% Now we have nbcm = m(m+1)/2 cumulants matrices stored in a big m x m*nbcm array. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The inefficient code below does the same as above: computing the big CM cumulant matrix. %% It is commented out but you can check that it produces the same result. %% This is supposed to help people understand the (rather obscure) code above. %% See section 4.2 of the Neural Comp paper referenced below. It can be found at %% "http://www.tsi.enst.fr/~cardoso/Papers.PS/neuralcomp_2ppf.ps", %% %% %% %% if 1, %% %% %% Step one: we compute the sample cumulants %% Matcum = zeros(m,m,m,m) ; %% for i1=1:m, %% for i2=1:m, %% for i3=1:m, %% for i4=1:m, %% Matcum(i1,i2,i3,i4) = mean( X(:,i1) .* X(:,i2) .* X(:,i3) .* X(:,i4) ) ... %% - R(i1,i2)*R(i3,i4) ... %% - R(i1,i3)*R(i2,i4) ... %% - R(i1,i4)*R(i2,i3) ; %% end %% end %% end %% end %% %% %% Step 2; We compute a basis of the space of symmetric m*m matrices %% CMM = zeros(m, m, nbcm) ; %% Holds the basis. %% icm = 0 ; %% index to the elements of the basis %% vi = zeros(m,1); %% the ith basis vetor of R^m %% vj = zeros(m,1); %% the jth basis vetor of R^m %% Id = eye (m) ; %% convenience %% for im=1:m, %% vi = Id(:,im) ; %% icm = icm + 1 ; %% CMM(:, :, icm) = vi*vi' ; %% for jm=1:im-1, %% vj = Id(:,jm) ; %% icm = icm + 1 ; %% CMM(:, :, icm) = sqrt(0.5) * (vi*vj'+vj*vi') ; %% end %% end %% %% Now CMM(:,:,i) is the ith element of an orthonormal basis for_ the space of m*m symmetric matrices %% %% %% Step 3. We compute the image of each basis element by the cumulant tensor and store it back into CMM. %% mat = zeros(m) ; %% tmp %% for icm=1:nbcm %% mat = squeeze(CMM(:,:,icm)) ; %% for i1=1:m %% for i2=1:m %% CMM(i1, i2, icm) = sum(sum(squeeze(Matcum(i1,i2,:,:)) .* mat )) ; %% end %% end %% end; %% %% This is doing something like \sum_kl [ Cum(xi,xj,xk,xl) * mat_kl ] %% %% %% Step 4. Now, we can check that CMM a