Embedding to a Pseudo-Euclidean (PE) space
W = PE_EM(D,ALF,P)
W = D*PE_EM(,ALF,P)
X = C*W
| 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.|
| W|| Mapping to PE space|
| X|| Pseudo-Euclidean dataset, see PE_DATASET |
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.
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.
mappings, datasets, pe_dataset, pe2complex, pcam, getsig, setsig,
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