Mattia Ciollaro (Carnegie Mellon University)

An inferential theory of clustering for functional data

Recently, it has been shown that Morse theory can be exploited to de- velop a sound inferential background for clustering: one can rigorously define both population and empirical clusters by means of the gradient flows asso- ciated to the population density p and the estimated density pˆ. In this framework, clusters are well-defined entities corresponding to the basins of attraction of the density’s critical points. The population parameter of inter- est corresponds to the collection of population clusters associated to p, and a natural estimator is given by the collection of empirical clusters associated to pˆ. While this framework is already well-developed for finite-dimensional random vectors, very little is known about the extent to which this theory generalizes to infinite-dimensional random variables (functional data). We discuss the challenges that functional data pose for the development of a clus- tering inferential paradigm based on Morse theory, and we describe how these challenges can be addressed. We focus on smooth random curves, but the theory that we develop can be further extended to more complex functional data types. If time allows, we will also discuss how population clusters of functional data can be estimated in practice and how to assess the statistical significance of the estimated clusters.