%FISHERC Fisher's Least Square Linear Classifier% %   W = FISHERC(A)% % INPUT%   A  Dataset%% OUTPUT%   W  Fisher's linear classifier %% DESCRIPTION  % Finds the linear discriminant function between the classes in the % dataset A by minimizing the errors in the least square sense. This % is a multi-class implementation using the one-against-all strategy.%% This classifier also works for soft and  target labels.%% For high dimensional datasets or small sample size situations, the % Pseudo-Fisher procedure is used, which is based on a pseudo-inverse.%% This classifier, like all other non-density based classifiers, does not% use the prior probabilities stored in the dataset A. Consequently, it% is just for two-class problems and equal class prior probabilities % equivalent to LDC, which assumes normal densities with equal covariance% matrices.%% Note that A*(KLMS([],N)*NMC) performs a very similar operation, but uses% the prior probabilities to estimate the mean class covariance matrix used% in the pre-whitening operation performed by KLMS. The reduced% dimensionality N controls some regularization.% % REFERENCES% 1. R.O. Duda, P.E. Hart, and D.G. Stork, Pattern classification, 2nd ed.% John Wiley and Sons, New York, 2001.% 2. A. Webb, Statistical Pattern Recognition, Wiley, New York, 2002.% 3. S. Raudys and R.P.W. Duin, On expected classification error of the% Fisher linear classifier with pseudo-inverse covariance matrix, Pattern% Recognition Letters, vol. 19, no. 5-6, 1998, 385-392.% SEE ALSO% MAPPINGS, DATASETS, TESTC, LDC, NMC, FISHERM% Copyright: R.P.W. Duin, r.p.w.duin@prtools.org% Faculty EWI, Delft University of Technology% P.O. Box 5031, 2600 GA Delft, The Netherlands% $Id: fisherc.m,v 1.12 2005/11/08 22:26:24 duin Exp $function W = fisherc(a)	prtrace(mfilename);		% No input arguments, return an untrained mapping.	if (nargin < 1) | (isempty(a))		W = mapping(mfilename); 		W = setname(W,'Fisher');		return;	end	isvaldset(a,1,2); % at least 1 object per class, 2 classes		[m,k,c] = getsize(a);	if c > 2		W = mclassc(a,fisherc);		return	end	y = gettargets(a);	u = mean(a);    	% Shift A to the origin. This is not significant, just increases accuracy.	% A is extended by ones(m,1), a trick to incorporate a free weight in the 	% hyperplane definition. 	b = [+a-repmat(u,m,1), ones(m,1)]; 	if (rank(b) <= k)		% This causes Fisherc to be the Pseudo-Fisher Classifier		prwarning(1,'The dimensionality is too large. Pseudo-Fisher is trained instead.');  		v = pinv(b)*y;                 	else		% Fisher is identical to the Min-Square-Error solution.				v = b\y;               	end	offset = v(k+1,:) - u*v(1:k,:); 	% Free weight. 	W = affine(v(1:k,:),offset,a,getlablist(a),k,c);	% Normalize the weights for good posterior probabilities. 	W = cnormc(W,a);										W = setname(W,'Fisher');return;