von Mises–Fisher distribution

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In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the -sphere in . If the distribution reduces to the von Mises distribution on the circle.

Definition[edit]

The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector is given by:

where and the normalization constant is equal to

where denotes the modified Bessel function of the first kind at order . If , the normalization constant reduces to

The parameters and are called the mean direction and concentration parameter, respectively. The greater the value of , the higher the concentration of the distribution around the mean direction . The distribution is unimodal for , and is uniform on the sphere for .

The von Mises–Fisher distribution for is also called the Fisher distribution.[1][2] It was first used to model the interaction of electric dipoles in an electric field.[3] Other applications are found in geology, bioinformatics, and text mining.

Relation to normal distribution[edit]

Starting from a normal distribution

the von Mises-Fisher distribution is obtained by expanding

using the fact that and are unit vectors, and recomputing the normalization constant by integrating over the unit sphere.

Estimation of parameters[edit]

A series of N independent measurements are drawn from a von Mises–Fisher distribution. Define

Then[3] the maximum likelihood estimates of and are given by the sufficient statistic

as

and

Thus is the solution to

A simple approximation to is (Sra, 2011)

but a more accurate measure can be obtained by iterating the Newton method a few times

For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as[4]

where

It's then possible to approximate a confidence cone about with semi-vertical angle

where

For example, for a 95% confidence cone, and thus

Generalizations[edit]

The matrix von Mises-Fisher distribution (Also Known as Matrix Langevin Distribution[5]) has the density

supported on the Stiefel manifold of orthonormal p-frames , where is an arbitrary real matrix.[6][7]

See also[edit]

References[edit]

  1. ^ Fisher, R. A. (1953). "Dispersion on a sphere". Proc. Roy. Soc. Lond. A. 217 (1130): 295–305. Bibcode:1953RSPSA.217..295F. doi:10.1098/rspa.1953.0064. S2CID 123166853.
  2. ^ Watson, G. S. (1980). "Distributions on the Circle and on the Sphere". J. Appl. Probab. 19: 265–280. doi:10.2307/3213566. JSTOR 3213566.
  3. ^ a b Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. John Wiley & Sons Ltd. ISBN 978-0-471-95333-3.
  4. ^ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.
  5. ^ Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava (2020). "Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold". Bayesian Analysis. 15 (3): 871–908. doi:10.1214/19-BA1176. ISSN 1936-0975. Retrieved 10 July 2021.
  6. ^ Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
  7. ^ Downs (1972). "Orientational statistics". Biometrika. 59 (3): 665–676. doi:10.1093/biomet/59.3.665.

Further reading[edit]

  • Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
  • Banerjee, A., Dhillon, I. S., Ghosh, J., & Sra, S. (2005). "Clustering on the unit hypersphere using von Mises-Fisher distributions". Journal of Machine Learning Research, 6(Sep), 1345-1382.
  • Sra, S. (2011). "A short note on parameter approximation for von Mises-Fisher distributions: And a fast implementation of I_s(x)". Computational Statistics. 27: 177–190. CiteSeerX 10.1.1.186.1887. doi:10.1007/s00180-011-0232-x. S2CID 3654195.