%QDC Quadratic Bayes Normal Classifier (Bayes-Normal-2)
%
% [W,R,S,M] = QDC(A,R,S,M)
% [W,R,S,M] = A*QDC([],R,S,M)
% [W,R,S,M] = A*QDC(R,S,M)
%
% 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
% R Value of regularization parameter R as used
% S Value of regularization parameter S as used
% M Value of regularization parameter M as used
%
% 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
% and sigma^2 is the mean of the K-M smallest eigenvalues.
%
% The use of soft labels is supported. The classification A*W is computed by
% NORMAL_MAP.
%
% If R, S or M is NaN the regularisation parameter is optimised by REGOPTC.
% The best result are usually obtained by R = 0, S = NaN, M = [], or by
% R = 0, S = 0, M = NaN (which is for problems of moderate or low dimensionality
% faster). If no regularisation is supplied a pseudo-inverse of the
% covariance matrix is used in case it is close to singular.
%
% 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 (<a href="http://37steps.com/prtools">PRTools Guide</a>)
% MAPPINGS, DATASETS, REGOPTC, NMC, NMSC, LDC, UDC, QUADRC, NORMAL_MAP
% Copyright: R.P.W. Duin, r.p.w.duin@37steps.com
% Faculty EWI, Delft University of Technology
% P.O. Box 5031, 2600 GA Delft, The Netherlands
% $Id: qdc.m,v 1.8 2010/02/08 15:29:48 duin Exp $
function [w,r,s,dim] = qdc(varargin)
mapname = 'Bayes-Normal-2';
argin = shiftargin(varargin,'scalar');
argin = setdefaults(argin,[],0,0,[]);
if mapping_task(argin,'definition')
w = define_mapping(argin,'untrained',mapname);
elseif mapping_task(argin,'training') % Train a mapping.
[a,r,s,dim] = deal(argin{:});
if any(isnan([r,s,dim])) % optimize regularisation parameters
defs = {0,0,[]};
parmin_max = [1e-8,9.9999e-1;1e-8,9.9999e-1;1,size(a,2)];
[w,r,s,dim] = regoptc(a,mfilename,{r,s,dim},defs,[3 2 1],parmin_max,testc([],'soft'),[1 1 0]);
else % training
islabtype(a,'crisp','soft'); % Assert A has the right labtype.
isvaldfile(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;
dim = round(dim);
if (dim < 1) | (dim > k)
error ('Number of dimensions M should lie in the range [1,K].');
end
[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)
dim = min(rank(F)-1,dim);
[eigvec,eigval] = preig(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 = normal_map(pars,getlab(U),k,c);
w = setcost(w,a);
end
w = setname(w,mapname);
end
return;