Discriminating between strong and weak structures in three-mode principal component analysis

Recently, a number of model selection heuristics (i.e. DIFFIT, CORCONDIA, the numerical convex hull based heuristic) have been proposed for choosing among Parafac and/or Tucker3 solutions of different complexity for a given three-way three-mode data set. Such heuristics are often validated by means of extensive simulation studies. However, these simulation studies are unrealistic in that it is assumed that the variance in real three-way data can be split into two parts: structural variance, due to a true underlying Parafac or Tucker3 model of low complexity, and random noise. In this paper, we start from the much more reasonable assumption that the variance in any real three-way data set is due to three different sources: (1) a strong Parafac or Tucker3 structure of low complexity, accounting for a considerable amount of variance, (2) a weak Tucker3 structure, capturing less prominent data aspects, and (3) random noise. As such, Parafac and Tucker3 simulation studies are run in which the data are generated by adding a weak Tucker3 structure to a strong Parafac or Tucker3 one and perturbing the resulting data with random noise. The design of these studies is based on the reanalysis of real data sets. In these studies, the performance of the numerical convex hull based model selection method is evaluated with respect to its capability of discriminating strong from weak underlying structures. The results show that in about two-thirds of the simulated cases, the hull heuristic yields a model of the same complexity as the strong underlying structure and thus succeeds in disentangling strong and weak underlying structures. In the vast majority of the remaining third, this heuristic selects a solution that combines the strong structure and (part of) the weak structure.