Fuzzy c-means clustering models for multivariate time-varying data: Different approaches

The classification of multivariate time-varying data finds application in several fields, such as economics, finance, marketing research, psychometrics, bioinformatics, medicine, signal processing, pattern recognition, etc. In this paper, by considering an exploratory formalization, we propose different unsupervised clustering models for multivariate data time arrays (objects×quantitative variables×times). These models can be classified in two different approaches: the cross sectional and the longitudinal approach. In the first case, after the objects, observed at each time, have been classified, comparison among the classifications made in different time instants will be done. In the second approach, we cluster the time trajectories of the objects; then, we obtain only one classification by comparing the instantaneous and evolutive features of the trajectories of the objects. In particular, in this work, the second approach is analyzed in detail, with reference to the so-called single and double step procedures. Geometric, correlative, instantaneous, evolutive and trend characteristics of the multivariate time arrays are taken into account in the different proposed clustering models. Furthermore, the fuzzy approach, that is particularly suitable in the dynamic classification problem, has been considered. Extensions of a cluster-validity criterion for the proposed fuzzy dynamic clustering models are also suggested. A socio-economic example concludes the paper.