PE_EM

Embedding to a Pseudo-Euclidean (PE) space

      W = PE_EM(D,ALF,P)
      W = D*PE_EM([],ALF,P)
      X = C*W

Input
D	NxN symmetric dissimilarity matrix (dataset)
ALF	Determines dimensionality of the mapping (optional, default: Inf)
(0,1)	Fraction of the total (absolute value) preserved variance  Inf - No reduction (default)  'p' - Mapping to Euclidean space using positive eigenvalues only  'PARp' - Projection to Euclidean space based on the PAR fraction of  positive eigenvalues; e.g. ALF = '0.9p' (the positive  space)  'n' - Mapping to Euclidean space using negative eigenvalues only  'PARn' - Projection to Euclidean space based on the PAR fraction  of negative eigenvalues; e.g. ALF = '0.7n' (the negative  space)  'P1pP2n'- Mapping to Euclidean space using P1 positive eigenvalues  and P2 negative eigenvalues;  e.g. ALF = '0.7p0.1n', ALF = '7p2n'
1	.. N - Number of dimensions in total
[P1	P2] - P1 dimensions or preserved fraction of variance in the  positive subspace and P2 dimensions or preserved fraction  of variance in the negative subspace;  e.g. ALF = [5 10], ALF = [0.9 0.1]
P	Integer between 0 and N specifying which object is mapped at the  origin; 0 stands for the mean; (optional, default: 0)
C	Dataset, dissimilarity matrix to be mapped to PE space.

Output
W	Mapping to PE space
X	Pseudo-Euclidean dataset, see PE_DATASET

Description

Mapping W onto an M-dimensional Pseudo-Euclidean _PE) subspace from a  symmetric, square dissimilarity matrix D such that the dissimilarities  are preserved. W is linear in D.^2.  M is determined by ALF. E.g., the subspace is found such that at least a  fraction ALF of the total variance is preserved for ALF in (0,1). The  resulting X is found by C*W in which C is a dissimilarity matrix  referring to the same objects as D (same number columns). In case D equal  C, the resulting X = D*W is a PE-DATASET such that PE_DISTM(X) approximates the the original D.^2. In case W = PE_EM(D) this is exact.

X is an object of class PE_DATASET which is a child of PRDATASET.It  stores the positive and negative subspaces of the PE_space. See  PE_DATASET for more information.

Reference(s)

1. L. Goldfarb, A unified approach to pattern recognition, Pattern Recognition, vol.17, 575-582, 1984.
2. E. Pekalska and R.P.W. Duin, The Dissimilarity Representation for Pattern Recognition, Foundations and Applications, World Scientific, Singapore, 2005, 1-607.
3. E. Pekalska, P. Paclik and R.P.W. Duin, A Generalized Kernel Approach to Dissimilarity-based Classification, Journal of Machine Learning Research, vol.2, no.2, 175-211, 2002.
4. E. Pekalska and R.P.W. Duin, Beyond traditional kernels: classification % in two dissimilarity-based representation spaces, IEEE Trans. on Systems, Man Cybernetics, vol. 38, no. 6, 2008, 729-744.

See also

mappings, datasets, pe_dataset, pe2complex, pcam, getsig, setsig,