Von Mises–Fisher distribution


In directional statistics, the von Mises–Fisher distribution, is a
probability distribution on the -sphere in. If
the distribution reduces to the von Mises distribution on the circle.
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, also called the Fisher distribution, was first used to model the interaction of electric dipoles in an electric field. Other applications are found in geology, bioinformatics, and text mining.

Relation to normal distribution

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

A series of N independent measurements are drawn from a von Mises–Fisher distribution. Define
Then the maximum likelihood estimates of and are given by the sufficient statistic
as
and
Thus is the solution to
A simple approximation to is
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
where
It's then possible to approximate a confidence cone about with semi-vertical angle
For example, for a 95% confidence cone, and thus

Generalizations

The matrix von Mises-Fisher distribution has the density
supported on the Stiefel manifold of orthonormal p-frames, where is an arbitrary real matrix.