%NLFISHERM Non-linear Fisher Mapping according to Marco Loog % % W = NLFISHERM(A,N) % % INPUT % A Dataset % N Number of dimensions (optional; default: MIN(K,C)-1, where % K is the dimensionality of A and C is the number of classes) % % OUTPUT % W Non-linear Fisher mapping % % DESCRIPTION % Finds a mapping of the labeled dataset A to a N-dimensional linear % subspace emphasizing the class separability for neighboring classes. % % REFERENCES % 1. R. Duin, M. Loog and R. Haeb-Umbach, Multi-Class Linear Feature % Extraction by Nonlinear PCA, in: ICPR15, 15th Int. Conf. on Pattern % Recognition, vol.2, IEEE Computer Society Press, 2000, 398-401. % 2. M. Loog, R.P.W. Duin and R. Haeb-Umbach, Multiclass Linear Dimension % Reduction by Weighted Pairwise Fisher Criteria, IEEE Trans. on % Pattern Analysis and Machine Intelligence, vol.23, no.7, 2001, 762-766. % % SEE ALSO % MAPPINGS, DATASETS, FISHERM, KLM, PCA % Copyright: M. Loog, R.P.W. Duin, duin@ph.tn.tudelft.nl % Faculty of Applied Physics, Delft University of Technology % P.O. Box 5046, 2600 GA Delft, The Netherlands % $Id: nlfisherm.m,v 1.9 2003/11/22 23:24:36 bob Exp $ function W = nlfisherm(a,n) prtrace(mfilename); if (nargin < 2) n = []; end % No input data, an untrained mapping returned. if (nargin < 1) | (isempty(a)) W = mapping('nlfisherm',n); W = setname(W,'Non-linear Fisher mapping'); return; end islabtype(a,'crisp'); isvaldset(a,1,2); % at least 2 objects per class, 2 classes [m,k,c] = getsize(a); prior = getprior(a); if (isempty(n)) n = min(k,c)-1; prwarning(4,'Dimensionality N not supplied, assuming MIN(K,C)-1.'); end if (n >= m) | (n >= c) error('Dataset too small or has too few classes for demanded output dimensionality.') end % Non-linear Fisher mapping is determined by the eigenvectors of CW^{-1}*CB, % where CW is the within-scatter, understood as the averaged covariance % matrix weighted by the prior probabilities, and CB is the between-scatter, % modified in a nonlinear way. % To simplify the computations, CW can be set to the identity matrix. w = klms(a); % A is changed such that CW = I and the mean of A is shifted to the origin. b = a*w; k = size(b,2); u = +meancov(b); d = +distm(u); % D is the Mahalanobis distance between the classes. % Compute the weights E to be used in the modified between-scatter matrix G % E should diminish the influence of large distances D. e = 0.5*erf(sqrt(d)/(2*sqrt(2))); G = zeros(k,k); for j = 1:c for i=j+1:c % Marco-Loog Mapping G = G + prior(i)*prior(j)*e(i,j)*(u(j,:)-u(i,:))'*(u(j,:)-u(i,:))/d(i,j); G = (G + G')/2; % Avoid a numerical inaccuracy: cov. matrix should be symmetric! end end % Perform the eigendecomposition of the modified between-scatter matrix. [F,V] = eig(G); [v,I] = sort(-diag(V)); I = I(1:n); rot = F(:,I); off = -mean(b*F(:,I)); % After non-linear transformations, NLFISHERM is stored as an affine (linear) map. W = affine(rot,off,a); W = setname(w*W,'Non-linear Fisher mapping'); return;
