Generative model-based clustering of directional data
Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining
Clustering on the Unit Hypersphere using von Mises-Fisher Distributions
The Journal of Machine Learning Research
Parameter estimation for von Mises-Fisher distributions
Computational Statistics
Matrix nearness problems in data mining
Matrix nearness problems in data mining
Numerical Methods for Special Functions
Numerical Methods for Special Functions
Handbook of Continued Fractions for Special Functions
Handbook of Continued Fractions for Special Functions
Model-based clustering of probability density functions
Advances in Data Analysis and Classification
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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].