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
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. It was first used to model the interaction of electric dipoles in an electric field. Other applications are found in geology, bioinformatics, and text mining.
Note on the normalization constant
In the textbook, Directional Statistics by Mardia and Jupp, the normalization constant given for the Von Mises Fisher probability density is apparently different from the one given here: . In that book, the normalization constant is specified as:
where is the gamma function. This is resolved by noting that Mardia and Jupp give the density "with respect to the uniform distribution", while the density here is specified in the usual way, with respect to Lebesgue measure. The density (w.r.t. Lebesgue measure) of the uniform distribution is the reciprocal of the surface area of the (p-1)-sphere, so that the uniform density function is given by the constant:
It then follows that:
While the value for was derived above via the surface area, the same result may be obtained by setting in the above formula for . This can be done by noting that the series expansion for divided by has but one non-zero term at . (To evaluate that term, one needs to use the definition .)
Support
The support of the Von Mises–Fisher distribution is the hypersphere, or more specifically, the -sphere, denoted as
This is a -dimensional manifold embedded in -dimensional Euclidean space, .
Relation to normal distribution
Starting from a normal distribution with isotropic covariance and mean of length , whose density function is:
the Von Mises–Fisher distribution is obtained by conditioning on . By expanding
and using the fact that the first two right-hand-side terms are fixed, the Von Mises-Fisher density, is recovered by recomputing the normalization constant by integrating over the unit sphere. If , we get the uniform distribution, with density .
More succinctly, the restriction of any isotropic multivariate normal density to the unit hypersphere, gives a Von Mises-Fisher density, up to normalization.
This construction can be generalized by starting with a normal distribution with a general covariance matrix, in which case conditioning on gives the Fisher-Bingham distribution.
Estimation of parameters
Mean direction
A series of N independent unit vectors are drawn from a von Mises–Fisher distribution. The maximum likelihood estimates of the mean direction is simply the normalized arithmetic mean, a sufficient statistic:
Concentration parameter
Use the modified Bessel function of the first kind to define
Then:
Thus is the solution to
A simple approximation to is (Sra, 2011)
A more accurate inversion can be obtained by iterating the Newton method a few times
Standard error
For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as:
where
It is then possible to approximate a a spherical confidence interval (a confidence cone) about with semi-vertical angle: where For example, for a 95% confidence cone, and thus
Expected value
The expected value of the Von Mises–Fisher distribution is not on the unit hypersphere, but instead has a length of less than one. This length is given by as defined above. For a Von Mises–Fisher distribution with mean direction and concentration , the expected value is: . For , the expected value is at the origin. For finite , the length of the expected value, is strictly between zero and one and is a monotonic rising function of .