Von Mises–Fisher distribution

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 distributionbelongs 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} } ight)

where $kappa\; ge\; 0,\; left\; Vert\; mu\; ight\; Vert\; =1\; ,$ and the normalization constant $C\_\{p\}(kappa),$ is equal to

: $C\_\{p\}(kappa)=frac\; \{kappa^\{p/2-1\; \{(2pi)^\{p/2\}I\_\{p/2-1\}(kappa)\}.\; ,$

where $I\_\{v\}$ 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.

ee also

* Directional statistics
* Kent distribution, a related distribution on the two-dimensional unit sphere
* von Mises distribution, von Mises–Fisher distribution where p=2, the one-dimensional unit circle

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