Von Mises–Fisher distribution

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Points sampled from three von Mises-Fisher distributions on the sphere (blue: κ = 1, green: κ = 10, red: κ = 100). The mean directions μ are shown with arrows.
Points sampled from three von Mises-Fisher distributions on the sphere (blue: κ = 1, green: κ = 10, red: κ = 100). The mean directions μ are shown with arrows.

The von Mises–Fisher distribution is a probability distribution on the (p − 1)-dimensional sphere in \mathbb{R}^{p}. If p = 2 the distribution reduces to the von Mises distribution on the circle. The distribution belongs to the field of directional statistics.

The probability density function of the von Mises-Fisher distribution for the random p-dimensional unit vector \mathbf{x}\, is given by:

f_{p}(\mathbf{x}; \mu, \kappa)=C_{p}(\kappa)\exp \left( {\kappa \mu^T \mathbf{x} } \right)

where \kappa \ge 0, \left \Vert \mu \right \Vert =1 \, and the normalization constant C_{p}(\kappa)\, is equal to

C_{p}(\kappa)=\frac {\kappa^{p/2-1}} {(2\pi)^{p/2}I_{p/2-1}(\kappa)}. \,

where Iv denotes the modified Bessel function of the first kind and order v.

The parameters \mu\, and \kappa\, are called the mean direction and concentration parameter, respectively. The greater the value of \kappa\,, the higher the concentration of the distribution around the mean direction \mu\,. The distribution is unimodal for \kappa>0\,, and is uniform on the sphere for \kappa=0\,. If p = 3, the distribution is also called the Fisher distribution.

The von Mises-Fisher distribution (for p = 3) was first used to model the interaction of dipoles in an electric field. Other applications are found in geology, bioinformatics and text mining.

[edit] See also

[edit] References

  • Dhillon, I., Sra, S. (2003) Modeling Data using Directional Distributions. Tech. rep., University of Texas, Austin.
  • Fisher, RA, Dispersion on a sphere. (1953) Proc. Roy. Soc. London Ser. A., 217: 295-305
  • Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd. ISBN 0-471-95333-4
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Probability distributions
 
Directional, degenerate, and singular