The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects

TL;DR

This paper addresses the challenges of maximum-likelihood estimation for multivariate Watson distributions in high dimensions, proposing new accurate approximations and exploring their application in mixture modeling and connection to clustering algorithms.

Contribution

We derive theoretically justified, numerically accurate approximations for maximum-likelihood estimation of Watson distributions and reveal a novel link to diametrical clustering.

Findings

New approximations for ML estimation are accurate and computationally efficient.

Established a connection between Watson mixture models and diametrical clustering.

Enhanced understanding of high-dimensional Watson distribution modeling.

Abstract

This paper studies fundamental aspects of modelling data using multivariate Watson distributions. Although these distributions are natural for modelling axially symmetric data (i.e., unit vectors where $\pm \x$ are equivalent), for high-dimensions using them can be difficult. Why so? Largely because for Watson distributions even basic tasks such as maximum-likelihood are numerically challenging. To tackle the numerical difficulties some approximations have been derived---but these are either grossly inaccurate in high-dimensions (\emph{Directional Statistics}, Mardia & Jupp. 2000) or when reasonably accurate (\emph{J. Machine Learning Research, W. & C.P., v2}, Bijral \emph{et al.}, 2007, pp. 35--42), they lack theoretical justification. We derive new approximations to the maximum-likelihood estimates; our approximations are theoretically well-defined, numerically accurate, and easy to compute. We build on our parameter estimation and discuss mixture-modelling with Watson distributions; here we uncover a hitherto unknown connection to the "diametrical clustering" algorithm of Dhillon \emph{et al.} (\emph{Bioinformatics}, 19(13), 2003, pp. 1612--1619).