Daniel@0: function [P] = cca(D, P, epochs, Mdist, alpha0, lambda0)
Daniel@0: 
Daniel@0: %CCA Projects data vectors using Curvilinear Component Analysis.
Daniel@0: %
Daniel@0: % P = cca(D, P, epochs, [Dist], [alpha0], [lambda0])
Daniel@0: %
Daniel@0: %  P = cca(D,2,10);           % projects the given data to a plane
Daniel@0: %  P = cca(D,pcaproj(D,2),5); % same, but with PCA initialization
Daniel@0: %  P = cca(D, 2, 10, Dist);   % same, but the given distance matrix is used
Daniel@0: %  
Daniel@0: %  Input and output arguments ([]'s are optional):
Daniel@0: %   D          (matrix) the data matrix, size dlen x dim
Daniel@0: %              (struct) data or map struct            
Daniel@0: %   P          (scalar) output dimension
Daniel@0: %              (matrix) size dlen x odim, the initial projection
Daniel@0: %   epochs     (scalar) training length
Daniel@0: %   [Dist]     (matrix) pairwise distance matrix, size dlen x dlen.
Daniel@0: %                       If the distances in the input space should
Daniel@0: %                       be calculated otherwise than as euclidian
Daniel@0: %                       distances, the distance from each vector
Daniel@0: %                       to each other vector can be given here,
Daniel@0: %                       size dlen x dlen. For example PDIST
Daniel@0: %                       function can be used to calculate the
Daniel@0: %                       distances: Dist = squareform(pdist(D,'mahal'));
Daniel@0: %   [alpha0]   (scalar) initial step size, 0.5 by default
Daniel@0: %   [lambda0]  (scalar) initial radius of influence, 3*max(std(D)) by default
Daniel@0: %  
Daniel@0: %   P          (matrix) size dlen x odim, the projections
Daniel@0: %
Daniel@0: % Unknown values (NaN's) in the data: projections of vectors with
Daniel@0: % unknown components tend to drift towards the center of the
Daniel@0: % projection distribution. Projections of totally unknown vectors are
Daniel@0: % set to unknown (NaN).
Daniel@0: %
Daniel@0: % See also SAMMON, PCAPROJ. 
Daniel@0: 
Daniel@0: % Reference: Demartines, P., Herault, J., "Curvilinear Component
Daniel@0: %   Analysis: a Self-Organizing Neural Network for Nonlinear
Daniel@0: %   Mapping of Data Sets", IEEE Transactions on Neural Networks, 
Daniel@0: %   vol 8, no 1, 1997, pp. 148-154.
Daniel@0: 
Daniel@0: % Contributed to SOM Toolbox 2.0, February 2nd, 2000 by Juha Vesanto
Daniel@0: % Copyright (c) by Juha Vesanto
Daniel@0: % http://www.cis.hut.fi/projects/somtoolbox/
Daniel@0: 
Daniel@0: % juuso 171297 040100
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: %% Check arguments 
Daniel@0: 
Daniel@0: error(nargchk(3, 6, nargin)); % check the number of input arguments
Daniel@0: 
Daniel@0: % input data
Daniel@0: if isstruct(D), 
Daniel@0:   if strcmp(D.type,'som_map'), D = D.codebook; else D = D.data; end
Daniel@0: end
Daniel@0: [noc dim] = size(D);
Daniel@0: noc_x_1  = ones(noc, 1); % used frequently
Daniel@0: me = zeros(1,dim); st = zeros(1,dim);
Daniel@0: for i=1:dim,
Daniel@0:   me(i) = mean(D(find(isfinite(D(:,i))),i));
Daniel@0:   st(i) = std(D(find(isfinite(D(:,i))),i));
Daniel@0: end
Daniel@0: 
Daniel@0: % initial projection
Daniel@0: if prod(size(P))==1, 
Daniel@0:   P = (2*rand(noc,P)-1).*st(noc_x_1,1:P) + me(noc_x_1,1:P); 
Daniel@0: else
Daniel@0:   % replace unknown projections with known values
Daniel@0:   inds = find(isnan(P)); P(inds) = rand(size(inds));
Daniel@0: end
Daniel@0: [dummy odim] = size(P);
Daniel@0: odim_x_1  = ones(odim, 1); % this is used frequently
Daniel@0: 
Daniel@0: % training length
Daniel@0: train_len = epochs*noc;
Daniel@0: 
Daniel@0: % random sample order
Daniel@0: rand('state',sum(100*clock));
Daniel@0: sample_inds = ceil(noc*rand(train_len,1));
Daniel@0: 
Daniel@0: % mutual distances
Daniel@0: if nargin<4 | isempty(Mdist) | all(isnan(Mdist(:))),
Daniel@0:   fprintf(2, 'computing mutual distances\r');
Daniel@0:   dim_x_1 = ones(dim,1);
Daniel@0:   for i = 1:noc,
Daniel@0:     x = D(i,:); 
Daniel@0:     Diff = D - x(noc_x_1,:);
Daniel@0:     N = isnan(Diff);
Daniel@0:     Diff(find(N)) = 0; 
Daniel@0:     Mdist(:,i) = sqrt((Diff.^2)*dim_x_1);
Daniel@0:     N = find(sum(N')==dim); %mutual distance unknown
Daniel@0:     if ~isempty(N), Mdist(N,i) = NaN; end
Daniel@0:   end
Daniel@0: else
Daniel@0:   % if the distance matrix is output from PDIST function
Daniel@0:   if size(Mdist,1)==1, Mdist = squareform(Mdist); end
Daniel@0:   if size(Mdist,1)~=noc, 
Daniel@0:     error('Mutual distance matrix size and data set size do not match'); 
Daniel@0:   end
Daniel@0: end
Daniel@0: 
Daniel@0: % alpha and lambda
Daniel@0: if nargin<5 | isempty(alpha0) | isnan(alpha0), alpha0 = 0.5; end
Daniel@0: alpha = potency_curve(alpha0,alpha0/100,train_len);
Daniel@0: 
Daniel@0: if nargin<6 | isempty(lambda0) | isnan(lambda0), lambda0 = max(st)*3; end
Daniel@0: lambda = potency_curve(lambda0,0.01,train_len);
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: %% Action
Daniel@0: 
Daniel@0: k=0; fprintf(2, 'iterating: %d / %d epochs\r',k,epochs);
Daniel@0: 
Daniel@0: for i=1:train_len, 
Daniel@0:   
Daniel@0:   ind = sample_inds(i);     % sample index
Daniel@0:   dx = Mdist(:,ind);        % mutual distances in input space
Daniel@0:   known = find(~isnan(dx)); % known distances
Daniel@0: 
Daniel@0:   if ~isempty(known),
Daniel@0:     % sample vector's projection
Daniel@0:     y = P(ind,:);                 
Daniel@0: 
Daniel@0:     % distances in output space
Daniel@0:     Dy = P(known,:) - y(noc_x_1(known),:); 
Daniel@0:     dy = sqrt((Dy.^2)*odim_x_1);           
Daniel@0:   
Daniel@0:     % relative effect
Daniel@0:     dy(find(dy==0)) = 1;        % to get rid of div-by-zero's
Daniel@0:     fy = exp(-dy/lambda(i)) .* (dx(known) ./ dy - 1);
Daniel@0: 
Daniel@0:     % Note that the function F here is e^(-dy/lambda)) 
Daniel@0:     % instead of the bubble function 1(lambda-dy) used in the 
Daniel@0:     % paper.
Daniel@0:     
Daniel@0:     % Note that here a simplification has been made: the derivatives of the
Daniel@0:     % F function have been ignored in calculating the gradient of error
Daniel@0:     % function w.r.t. to changes in dy.
Daniel@0:     
Daniel@0:     % update
Daniel@0:     P(known,:) = P(known,:) + alpha(i)*fy(:,odim_x_1).*Dy;
Daniel@0:   end
Daniel@0: 
Daniel@0:   % track
Daniel@0:   if rem(i,noc)==0, 
Daniel@0:     k=k+1; fprintf(2, 'iterating: %d / %d epochs\r',k,epochs);
Daniel@0:   end
Daniel@0: 
Daniel@0: end
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: %% clear up
Daniel@0: 
Daniel@0: % calculate error
Daniel@0: error = cca_error(P,Mdist,lambda(train_len));
Daniel@0: fprintf(2,'%d iterations, error %f          \n', epochs, error);
Daniel@0: 
Daniel@0: % set projections of totally unknown vectors as unknown
Daniel@0: unknown = find(sum(isnan(D)')==dim);
Daniel@0: P(unknown,:) = NaN;
Daniel@0: 
Daniel@0: return;
Daniel@0: 
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: %% tips
Daniel@0: 
Daniel@0: % to plot the results, use the code below
Daniel@0: 
Daniel@0: %subplot(2,1,1), 
Daniel@0: %switch(odim), 
Daniel@0: %  case 1, plot(P(:,1),ones(dlen,1),'x')
Daniel@0: %  case 2, plot(P(:,1),P(:,2),'x'); 
Daniel@0: %  otherwise, plot3(P(:,1),P(:,2),P(:,3),'x'); rotate3d on
Daniel@0: %end
Daniel@0: %subplot(2,1,2), dydxplot(P,Mdist);
Daniel@0: 
Daniel@0: % to a project a new point x in the input space to the output space
Daniel@0: % do the following:
Daniel@0: 
Daniel@0: % Diff = D - x(noc_x_1,:); Diff(find(isnan(Diff))) = 0; 
Daniel@0: % dx = sqrt((Diff.^2)*dim_x_1);
Daniel@0: % p = project_point(P,x,dx); % this function can be found from below
Daniel@0: % tlen = size(p,1);
Daniel@0: % plot(P(:,1),P(:,2),'bx',p(tlen,1),p(tlen,2),'ro',p(:,1),p(:,2),'r-')
Daniel@0: 
Daniel@0: % similar trick can be made to the other direction
Daniel@0: 
Daniel@0: 
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: %% subfunctions
Daniel@0: 
Daniel@0: function vals = potency_curve(v0,vn,l)
Daniel@0: 
Daniel@0:   % curve that decreases from v0 to vn with a rate that is 
Daniel@0:   % somewhere between linear and 1/t
Daniel@0:   vals = v0 * (vn/v0).^([0:(l-1)]/(l-1));
Daniel@0: 
Daniel@0: 
Daniel@0: function error = cca_error(P,Mdist,lambda)
Daniel@0: 
Daniel@0:   [noc odim] = size(P);
Daniel@0:   noc_x_1 = ones(noc,1);
Daniel@0:   odim_x_1 = ones(odim,1);
Daniel@0: 
Daniel@0:   error = 0;
Daniel@0:   for i=1:noc,
Daniel@0:     known = find(~isnan(Mdist(:,i)));
Daniel@0:     if ~isempty(known),   
Daniel@0:       y = P(i,:);                 
Daniel@0:       Dy = P(known,:) - y(noc_x_1(known),:);
Daniel@0:       dy = sqrt((Dy.^2)*odim_x_1);
Daniel@0:       fy = exp(-dy/lambda);
Daniel@0:       error = error + sum(((Mdist(known,i) - dy).^2).*fy);
Daniel@0:     end
Daniel@0:   end
Daniel@0:   error = error/2;
Daniel@0: 
Daniel@0: 
Daniel@0: function [] = dydxplot(P,Mdist)
Daniel@0: 
Daniel@0:   [noc odim] = size(P);
Daniel@0:   noc_x_1 = ones(noc,1);
Daniel@0:   odim_x_1 = ones(odim,1);
Daniel@0:   Pdist = zeros(noc,noc);
Daniel@0:     
Daniel@0:   for i=1:noc,
Daniel@0:     y = P(i,:);                 
Daniel@0:     Dy = P - y(noc_x_1,:);
Daniel@0:     Pdist(:,i) = sqrt((Dy.^2)*odim_x_1);
Daniel@0:   end
Daniel@0: 
Daniel@0:   Pdist = tril(Pdist,-1); 
Daniel@0:   inds = find(Pdist > 0); 
Daniel@0:   n = length(inds);
Daniel@0:   plot(Pdist(inds),Mdist(inds),'.');
Daniel@0:   xlabel('dy'), ylabel('dx')
Daniel@0: 
Daniel@0: 
Daniel@0: function p = project_point(P,x,dx)
Daniel@0: 
Daniel@0:   [noc odim] = size(P);
Daniel@0:   noc_x_1 = ones(noc,1);
Daniel@0:   odim_x_1 = ones(odim,1);
Daniel@0: 
Daniel@0:   % initial projection
Daniel@0:   [dummy,i] = min(dx);
Daniel@0:   y = P(i,:)+rand(1,odim)*norm(P(i,:))/20;
Daniel@0:  
Daniel@0:   % lambda 
Daniel@0:   lambda = norm(std(P));
Daniel@0: 
Daniel@0:   % termination
Daniel@0:   eps = 1e-3; i_max = noc*10;
Daniel@0:   
Daniel@0:   i=1; p(i,:) = y; 
Daniel@0:   ready = 0;
Daniel@0:   while ~ready,
Daniel@0: 
Daniel@0:     % mutual distances
Daniel@0:     Dy = P - y(noc_x_1,:);        % differences in output space
Daniel@0:     dy = sqrt((Dy.^2)*odim_x_1);  % distances in output space
Daniel@0:     f = exp(-dy/lambda);
Daniel@0:   
Daniel@0:     fprintf(2,'iteration %d, error %g \r',i,sum(((dx - dy).^2).*f));
Daniel@0: 
Daniel@0:     % all the other vectors push the projected one
Daniel@0:     fy = f .* (dx ./ dy - 1) / sum(f);
Daniel@0:   
Daniel@0:     % update    
Daniel@0:     step = - sum(fy(:,odim_x_1).*Dy);
Daniel@0:     y = y + step;
Daniel@0:   
Daniel@0:     i=i+1;
Daniel@0:     p(i,:) = y;   
Daniel@0:     ready = (norm(step)/norm(y) < eps | i > i_max);
Daniel@0: 
Daniel@0:   end
Daniel@0:   fprintf(2,'\n');
Daniel@0:      
Daniel@0: