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

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  • Sra, Suvrit
  • Karp, Dmitrii

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].

Suggested Citation

  • Sra, Suvrit & Karp, Dmitrii, 2013. "The multivariate Watson distribution: Maximum-likelihood estimation and other aspects," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 256-269.
    • Handle: RePEc:eee:jmvana:v:114:y:2013:i:c:p:256-269
    DOI: 10.1016/j.jmva.2012.08.010
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    References listed on IDEAS

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    1. Akihiro Tanabe & Kenji Fukumizu & Shigeyuki Oba & Takashi Takenouchi & Shin Ishii, 2007. "Parameter estimation for von Mises–Fisher distributions," Computational Statistics, Springer, vol. 22(1), pages 145-157, April.
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    Cited by:

    1. Jay Damask, 2019. "A Consistently Oriented Basis for Eigenanalysis," Papers 1912.12983, arXiv.org.
    2. Angela Montanari & Daniela Calò, 2013. "Model-based clustering of probability density functions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(3), pages 301-319, September.
    3. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.

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