A new criterion for simple-structure transformations of core arrays in N-way principal components analysis

Abstract Among the possible (orthogonal) transformations of core arrays in N-way principal components analysis (PCA), the conventional approach of body diagonalization turns out not to provide the simplest structure (in the sense of minimizing the number of significant entries). As an alternative, the maximization of the variance-of-squared core entries is proposed. Both criteria are equivalent in a two-way constellation but may differ markedly for N≥3. Actually, using the variance criterion may provide more insight into the rank structure of the given data, and it is also easily applied to general rectangular core arrays. In order to clarify the relation between body diagonality and variance-of-squares, we prove the following main result of the paper: If some cubic N-way core array can be transformed to exact body diagonality, then the same transformation yields maximum variance-of-squared entries. This result implies the equivalence in the two-way case mentioned above. A solution algorithm is formulated and illustrated with a small numerical example. The application to data examples from environmental chemistry and chromatographic analysis is briefly discussed.