Fuzzy clustering for data time arrays with inlier and outlier time trajectories

In many knowledge discovery and data mining tasks, fuzzy clustering is one of the most common tools for data partitioning. In this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories (time series, sequences) are developed. In particular, by adopting an exploratory approach, based on a geometric-algebraic formulation of the data time array, different kinds of dynamic fuzzy clustering models, based on cross sectional and longitudinal aspects, are suggested. Furthermore, a modified version of the previous clustering models, that can be seen as a generalization of these models, is proposed. By utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories (trajectories with anomalous positions and slopes) are present in the dataset; in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure's domains and are characterized by strange slopes. In these models, the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given. Furthermore, some remarks on dynamic three-way extensions of a few robust fuzzy clustering models for two-way data are suggested. Demonstrative examples are shown and a comparison assessment based on artificial multivariate time-varying data is carried out.