Clusterwise Parafac to identify heterogeneity in three-way data

In many research areas, the Parafac model is adopted to disclose the underlying structure of three-way three-mode data. In this model, a set of latent variables, called components, that captures the complex interaction between the elements of the three modes is sought. An important assumption of this model is that these components are the same for all elements of the three modes. In many cases, however, it makes sense to assume that the components may differ (i.e., qualitative differences in underlying component structure) across groups of elements of one of the modes. Therefore, in this paper, we present Clusterwise Parafac. In this new model, the elements of one of the three modes are assigned to a limited number of mutually exclusive clusters and, simultaneously, the data within each cluster are modeled with Parafac. As such, elements that belong to the same cluster are assumed to be governed by the same components, whereas elements that are assigned to different clusters have a different underlying component structure. To evaluate the performance of the new Clusterwise Parafac strategy, an extensive simulation study is conducted. Moreover, the strategy is applied to sensory profiling data regarding different cheeses.