%QDC Quadratic Bayes Normal Classifier (Bayes-Normal-2)%%   W = QDC(A,R,S)%% INPUT%   A    Dataset%   R,S	 Regularization parameters, 0 <= R,S <= 1 %        (optional; default: no regularization, i.e. R,S = 0)%   M    Dimension of subspace structure in covariance matrix (default: K,%           all dimensions)%% OUTPUT%   W    Quadratic Bayes Normal Classifier mapping%% DESCRIPTION% Computation of the quadratic classifier between the classes of the dataset% A assuming normal densities. R and S (0 <= R,S <= 1) are regularization% parameters used for finding the covariance matrix by% %   G = (1-R-S)*G + R*diag(diag(G)) + S*mean(diag(G))*eye(size(G,1))%% This covariance matrix is then decomposed as G = W*W' + sigma^2 * eye(K),% where W is a K x M matrix containing the M leading principal components.% % The use of soft labels is supported. The classification A*W is computed by% NORMAL_MAP.%% EXAMPLES% See PREX_MCPLOT, PREX_PLOTC.%% REFERENCES% 1. R.O. Duda, P.E. Hart, and D.G. Stork, Pattern classification, 2nd% edition, John Wiley and Sons, New York, 2001. % 2. A. Webb, Statistical Pattern Recognition, John Wiley & Sons, % New York, 2002.%% SEE ALSO% MAPPINGS, DATASETS, NMC, NMSC, LDC, UDC, QUADRC, NORMAL_MAP% Copyright: R.P.W. Duin, duin@ph.tn.tudelft.nl% Faculty of Applied Sciences, Delft University of Technology% P.O. Box 5046, 2600 GA Delft, The Netherlands% $Id: qdc.m,v 1.11 2003/11/22 23:19:26 bob Exp $function w = qdc(a,r,s,dim)	prtrace(mfilename);  if (nargin < 4)    prwarning(4,'subspace dimensionality M not given, assuming K');    dim = [];  end	if (nargin < 3)		prwarning(4,'Regularisation parameter S not given, assuming 0.');		s = 0; 	end	if (nargin < 2)		prwarning(4,'Regularisation parameter R not given, assuming 0.');		r = 0;	end	% No input arguments: return an untrained mapping.	if (nargin < 1) | (isempty(a))		w = mapping(mfilename,{r,s,dim});		w = setname(w,'Bayes-Normal-2');		return	end	islabtype(a,'crisp','soft');	isvaldset(a,2,2); % at least 2 objects per class, 2 classes	[m,k,c] = getsize(a);	% If the subspace dimensionality is not given, set it to all dimensions.	if (isempty(dim)), dim = k; end;  if (dim < 1) | (dim > k)    error ('Number of dimensions M should lie in the range [1,K].');  end	% Assert A has the right labtype.	islabtype(a,'crisp','soft');	[U,G] = meancov(a);	% Calculate means and priors.	pars.mean  = +U;	pars.prior = getprior(a);	% Calculate class covariance matrices.	pars.cov   = zeros(k,k,c);	for j = 1:c		F = G(:,:,j);				% Regularize, if requested.		if (s > 0) | (r > 0) 			F = (1-r-s) * F + r * diag(diag(F)) +s*mean(diag(F))*eye(size(F,1));		end    % If DIM < K, extract the first DIM principal components and estimate    % the noise outside the subspace.    if (dim < k)      [eigvec,eigval] = eig(F); eigval = diag(eigval);      [dummy,ind] = sort(-eigval);      % Estimate sigma^2 as avg. eigenvalue outside subspace.      sigma2 = mean(eigval(ind(dim+1:end)));      % Subspace basis: first DIM eigenvectors * sqrt(eigenvalues).      F = eigvec(:,ind(1:dim)) * diag(eigval(ind(1:dim))) * eigvec(:,ind(1:dim))' + ...            sigma2 * eye(k);    end		pars.cov(:,:,j) = F;	end	w = mapping('normal_map','trained',pars,getlab(U),k,c);	w = setname(w,'Bayes-Normal-2');	w = setcost(w,a);return;