Kent distribution

In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere S2 in 3-space R3).

It is the analogue on S2 of the bivariate normal distribution with an unconstrained covariance matrix.

The Kent distribution was proposed by John T. Kent in 1982, and is used in geology as well as bioinformatics.

The probability density function

is a three-dimensional unit vector,

denotes the transpose of

, and the normalizing constant

( κ , β )

is the modified Bessel function and

is the gamma function.

c ( κ , 0 ) = 4 π (

) sinh ⁡ ( κ )

, the normalizing constant of the Von Mises–Fisher distribution.

) determines the concentration or spread of the distribution, while

0 ≤ 2 β < κ

) determines the ellipticity of the contours of equal probability.

parameters, the more concentrated and elliptical the distribution will be, respectively.

is the mean direction, and vectors

are the major and minor axes.

The latter two vectors determine the orientation of the equal probability contours on the sphere, while the first vector determines the common center of the contours.

The Kent distribution can be easily generalized to spheres in higher dimensions.

is a point on the unit sphere

, then the density function of the

-dimensional Kent distribution is proportional to where

However, the normalization constant becomes very difficult to work with for