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The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects
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).Comment: 24 pages; extensively updated numerical result
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