Orthogonal relations decomposition

By relation, PLS is similar to PCR. Both decompose the A-data into a smaller set of variables, i.e., the scores. However, they differ in how they relate the scores to the Y-data. In PCR, the scores from the PCA decomposition of the A -data are regressed onto the Y-data. In contrast, PLS decomposes both the Y- and the A -data into individual score and loading matrices. The orthogonal sets of scores for the X- and Y-data, T, U, respectively, are generated in a way that maximizes their covariance. This is an attractive feature, particularly in situations where not all the major sources of variability in X are correlated to the variability in Y. PLS attempts to find a different set of orthogonal scores for the A -data to give better predictions of the Y-data. Thus, orthogonal vectors may yield a poorer representation of the A-data, while the scores may yield a better prediction of Y than would be possible with PCR. 

A discussion is given of the current decomposition schemes of bond energies and related concepts (exchange (Pauli-)repulsion, polarization, charge transfer). The role of non-orthogonality of fragment orbitals and of kinetic and potential energy for Pauli repulsion and (orbital)polarization is analyzed. 

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