Poisson Kernel-Based Clustering on the Sphere: Convergence Properties, Identifiability, and a Method of Sampling

TL;DR

This paper introduces a clustering method for spherical data using Poisson kernel-based distributions, proving theoretical properties, proposing an estimation technique, and demonstrating its effectiveness on real and simulated datasets.

Contribution

It establishes identifiability and convergence properties for PKBD mixture models, and introduces a new sampling method and an empirical density distance plot for cluster estimation.

Findings

Proved identifiability of PKBD mixtures

Demonstrated convergence of the EM algorithm for PKBD

Validated methods on real and simulated data

Abstract

Many applications of interest involve data that can be analyzed as unit vectors on a d-dimensional sphere. Specific examples include text mining, in particular clustering of documents, biology, astronomy and medicine among others. Previous work has proposed a clustering method using mixtures of Poisson kernel-based distributions (PKBD) on the sphere. We prove identifiability of mixtures of the aforementioned model, convergence of the associated EM-type algorithm and study its operational characteristics. Furthermore, we propose an empirical densities distance plot for estimating the number of clusters in a PKBD model. Finally, we propose a method to simulate data from Poisson kernel-based densities and exemplify our methods via application on real data sets and simulation experiments.