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

ABSTRACT

Spherical or directional data arise in many applications of interest. Furthermore, many nondirectional datasets can be usefully re-expressed in the form of directions and analyzed as spherical data. We have proposed a clustering algorithm using mixtures of Poisson-kernel-based densities (PKBD) on the sphere. We prove convergence of the associated generalized EM-algorithm, investigate the identifiability of various forms of PKBD mixture model, and study in detail its performance via simulation and application to real data. Specifically, we discuss a method and simulate data from a variety of PKBD models, then study the performance of the algorithm in terms of adjusted Rand index, macro-precision, and macro-recall. Finally, we compare the PKBD clustering method with other algorithms for clustering data on the sphere. Supplementary materials are available online and provide proofs of the theoretical results and the associated computer code.

KEYWORDS:

• Directional data
• Exit distribution
• Identifiability
• Mixture models
• Poisson kernel
• Rejection sampling

Acknowledgments

Both authors would like to thank the editorial team for suggestions that significantly improved the presentation of this work, and Dr. E. Sofikitou. This article is dedicated to the memory of Professor Bruce G. Lindsay.

Additional information

Funding

The second author would like to acknowledge the Troup Fund, KALEIDA Health Foundation, award number 82114, for supporting this work. The second author also gratefully acknowledges financial support provided by the Department of Biostatistics, University at Buffalo in the form of a start-up package that funded the work of the first author.