PE_EM

Pseudo-Euclidean Linear Embedding

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

Input
D	NxN symmetric dissimilarity matrix (dataset)
ALF	Parameter determining the dimensionality and the mapping (optional, default: Inf)
(0,1)	Fraction of the total (absolute value) preserved variance  Inf - No dimensionality reduction, keeping all dimensions (it's VERY noisy)  'p' - Projection into a Euclidean space based on positive eigenvalues only  'PARp' - Projection into a Euclidean space based on the PAR fraction of  positive eigenvalues; e.g. ALF = '0.9p'  'n' - Projection into a Euclidean space based on negative eigenvalues only  'PARn' - Projection into a (negative) Euclidean space based on the PAR fraction  of negative eigenvalues; e.g. ALF = '0.7n'  'P1pP2n'- Projection into a Euclidean space based on the 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)

Output
W	Linear embedding into a pseudo-Euclidean space

Description

Linear mapping W onto an M-dimensional Pseudo-Euclidean _PE) subspace from a  symmetric, square dissimilarity matrix D such that the dissimilarities are preserved.  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 D*W. The signature of the obtained PE space (numbers of positive and negative  directions) can be found by GETDATA(W,'sig'). The spectrum of the obtained space  can be found by GETDAT(W,'eval').

Reference(s)

1. L. Goldfarb, A unified approach to pattern recognition, Pattern Recognition, vol.17, 575-582, 1984.
2. 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.
3. 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, pcam, getsig, setsig,