The multivariate Watson distribution: Maximum-likelihood estimation and other aspects

Suvrit Sra and Dmitrii Karp

Journal of Multivariate Analysis, 2013, vol. 114, issue C, 256-269

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 ±x are equivalent), for high-dimensions using them can be difficult—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 [K.V. Mardia, P. Jupp, Directional Statistics, second ed., John Wiley & Sons, 2000] or when reasonably accurate [A. Bijral, M. Breitenbach, G.Z. Grudic, Mixture of Watson distributions: a generative model for hyperspherical embeddings, in: Artificial Intelligence and Statistics, AISTATS 2007, 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 et al. [I.S. Dhillon, E.M. Marcotte, U. Roshan, Diametrical clustering for identifying anticorrelated gene clusters, Bioinformatics 19 (13) (2003) 1612–1619].

Keywords: Watson distribution; Kummer function; Confluent hypergeometric; Directional statistics; Diametrical clustering; Special function; Hypergeometric identities (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.jmva.2012.08.010

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