Fuzzy unsupervised classification of multivariate time trajectories with the Shannon entropy regularization

Fuzzy unsupervised clustering models based on entropy regularization are suggested in order to classify time-varying data. In particular, in the proposed models, objective functions, which are the sum of two terms, are minimized. The first term is a dynamic generalization of intra-cluster distance, in a fuzzy framework, that takes into account the instantaneous and/or longitudinal features of the time-varying observations (the so-called multivariate time trajectories); in this way, the within cluster dispersion is minimized (maximize the internal cohesion). The second term represents the Shannon entropy measure as applied to fuzzy partitions (entropy regularization); then, a given measure of entropy is maximized or, equivalently, the converse of the entropy is minimized. Overall, the total functional depending on both the previous aspects is optimized. The dynamic fuzzy entropy clustering models have been applied to a meteorological dataset and an empirical comparison with the instantaneous and/or longitudinal fuzzy C-means clustering models has been made.