Nonparametric Clustering of Functional Data Using Pseudo-Densities

TL;DR

This paper introduces a nonparametric clustering method for smooth functional data based on pseudo-density gradients, providing a consistent algorithm applicable to both infinite-dimensional curves and discretized noisy data.

Contribution

It develops a novel clustering approach using pseudo-density gradients that is theoretically sound under weak assumptions and applicable to real-world discretized functional data.

Findings

Clustering is well-defined at both population and sample levels.

The proposed algorithm reliably identifies significant local modes.

The method is consistent even with noisy, discretized observations.

Abstract

We study nonparametric clustering of smooth random curves on the basis of the L2 gradient flow associated to a pseudo-density functional and we show that the clustering is well-defined both at the population and at the sample level. We provide an algorithm to mark significant local modes, which are associated to informative sample clusters, and we derive its consistency properties. Our theory is developed under weak assumptions, which essentially reduce to the integrability of the random curves, and does not require to project the random curves on a finite-dimensional subspace. However, if the underlying probability distribution is supported on a finite-dimensional subspace, we show that the pseudo-density and the expectation of a kernel density estimator induce the same gradient flow, and therefore the same clustering. Although our theory is developed for smooth curves that belong to an infinite-dimensional functional space, we also provide consistent procedures that can be used with real data (discretized and noisy observations).