DisTools Contents DisTools User Guide
pe_em

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.