Title: The area of the convex hull of sampled curves: A robust functional statistical depth measure
Authors: Guillaume Staerman - Telecom Paris, Institut Polytechnique de Paris (France) [presenting]
Pavlo Mozharovskyi - Telecom Paris, Institut Polytechnique de Paris (France)
Stephan Clemencon - Telecom ParisTech (France)
Abstract: With the increasing industrial digitalization contemporary data are often present in the form of temporal series or functions. Out of existing statistical tools for functional data analysis, statistical data depth distinguishes by its non-parametric nature and robustness. Having undergone theoretical and computational developments in the recent decades, it has proven to be of particular use in functional spaces. Nevertheless, most of the existing functional depths share a common feature of treating evaluations for different arguments independently of each other, and by that may possess certain insensitivity to the shape changes. We propose a notion of functional depth based on the area of the convex hull of the functions' graphs. This approach allows for capturing gradual departures from centrality, even beyond the envelope of the data, but additionally provides a possibility to simultaneously consider functional evaluations for multiple arguments. We discuss the practical relevance of commonly imposed axioms on functional depths and their satisfaction by the proposed notion, and construct an efficient estimation algorithm. An extension to generic geometric transformations is also suggested, as well as a generalization to multivariate functional data. Simulation and real data studies demonstrate exploratory properties of the developed depth function. In particular, its application for functional anomaly detection is advantageous.