Depth-based methods for clustering of functional data.

Abstract

The problem of detecting clusters is a common issue in the analysis of functional data and some interesting intuitions from approaches relied on depth measures can be considered for construction of basic tools for clustering of curves. Motivated by recent contributions on the problem clustering and alignment of functional data, we also consider the problem of aligning a set of curves when classification procedures are implemented. The variability among curves can be interpreted in terms of two components, phase and amplitude; phase variability, or misalignment, can be eliminated by aligning the curves, according to a similarity index and a warping function. Some approaches address the misalignment as a confounding factor, if it is not suitably taken into account; as opposed to treating phase variability as a nuisance effect, other approaches recognize that both amplitude and phase of curves contain cluster information. The search for suitable transformation of the original data involves the optimization of specific similarities between the warped curves and a natural consequence seems to incorporate the warping step in a clustering approach. Among the similarity indexes considered in the literature on functional data analysis, those defined via statistical depth provide a way to robustly cluster functional data. We implement a procedure exploiting the idea of functional depth, searching both for the set of optimal groups to obtain efficient aligning and clustering curves. We also try to deal with the implications of preprocessing the curves via a warping procedure or alternatively ignoring the misalignment in the further analysis, or explicitly recognizing it as a source of information for clustering. This approach provides an useful tool for analyzing many phenomena and in particular we apply it to seismic curves clustering. Paper supported by the national grant MIUR, PRIN- 2015 program, Prot.20157PRZC4.