%FISHERM Optimal discrimination linear mapping (Fisher mapping, LDA)
%
% W = FISHERM(A,N,ALF)
%
% INPUT
% A Dataset
% N Number of dimensions to map to, N < C, where C is the number of classes
% (default: min(C,K)-1, where K is the number of features in A)
% ALF Preserved variance in the pre-whitening step
%
% OUTPUT
% W Fisher mapping
%
% DESCRIPTION
% Finds a mapping of the labeled dataset A onto an N-dimensional linear
% subspace such that it maximizes the the between scatter over the within
% scatter (also called the Fisher mapping [1 or LDA]). Note that N should be
% less than the number of classes in A. If supplied, ALF determines the
% preserved variance in the prewhitening step (i.e. removal of insignificant
% eigenvectors in the within-scatter, the EFLD procedure [2]), see KLMS.
%
% The resulting mapping is not orthogonal. It may be orthogonalised by ORTH.
%
% REFERENCES
% [1] K. Fukunaga, Introduction to statistical pattern recognition, 2nd
% ed., Academic Press, New York, 1990.
% [2] C. Liu and H. Wechsler, Robust Coding Schemes for Indexing and Retrieval
% from Large Face Databases, IEEE Transactions on Image Processing, vol. 9,
% no. 1, 2000, 132-136.
%
% SEE ALSO (<a href="http://37steps.com/prtools">PRTools Guide</a>)
% MAPPINGS, DATASETS, NLFISHERM, KLM, PCAM, KLMS, ORTH
% 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: fisherm.m,v 1.4 2007/04/21 23:05:59 duin Exp $
function w = fisherm(a,n,alf)
if (nargin < 3)
alf = [];
prwarning (4, 'no pre-whitening reduction specified, so take all features');
end
if (nargin < 2)
prwarning (4, 'number of dimensions to map to not specified, assuming min(C,K)-1');
n = [];
end
% If no arguments are specified, return an untrained mapping.
if (nargin < 1) | (isempty(a))
w = prmapping('fisherm',{n,alf});
w = setname(w,'Fisher mapping');
return
end
islabtype(a,'crisp','soft');
a = testdatasize(a);
isvaldset(a,2,2); % at least 2 objects per class, 2 classes
[m,k,c] = getsize(a);
% If N is not given, set it.
%DXD Question: why do we subtract here 1 from k? Should it not be
% the next one??
%if (isempty(n)), n = min(k,c)-1; end
if (isempty(n)), n = min(k,c-1); end
if (n >= m) | (n >= c) | (n > k)
error('The dataset is too small or has too few classes for the requested dimensionality')
end
% A Fisher mapping is determined by the eigenvectors of inv(W)*B, where W
% is the within-scatter matrix, i.e. the average covariance matrix
% weighted by the prior probabilities, and B is the between-scatter
% matrix.
% To simplify computations, W can be set to the identity matrix if A is
% centered and sphered by KLMS.
a = setprior(a,getprior(a)); % set priors to avoid unnecessary warnings
v = klms(a,alf); a = a*v;
% Calculate eigenvectors of B.
u = pcam(meancov(a),n);
if (n == 0)
%DXD: Careful here: I wanted to make the option n==0 behave the
% same as in pcam(x,0) and bhatm(x,0). Unfortunately, it was
% already defined. So I overrule it now!!!! Please correct
% if you don't agree!
w = u;
return
%w = v; % Only one feature is available (k=1)
else
w = v*u; % Construct final mapping.
end
w = setname(w,'Fisher mapping');
return