Kent distribution

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File:10.1371 journal.pcbi.0020131.g004-L.jpg
Three points sets sampled from the FB5 distribution. The mean directions are shown with arrows. The parameter is highest for the red set.

The 5-parameter Fisher-Bingham distribution or Kent distribution, named after Ronald Fisher, Christopher Bingham, and John T. Kent, is a probability distribution on the two-dimensional unit sphere in . It is the analogue on the two-dimensional unit sphere of the bivariate normal distribution with an unconstrained covariance matrix. The distribution belongs to the field of directional statistics.

The probability density function of the Kent distribution is given by:

where is a three-dimensional unit vector and is a normalizing constant.

The parameter (with ) determines the concentration or spread of the distribution, while (with ) determines the ellipticity of the contours of equal probability. The higher the and parameters, the more concentrated and elliptical the distribution will be, respectively. Vector 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 was proposed by John T. Kent in 1982, and is used in geology and bioinformatics.

See also


  • Kent, J.T. (1982) "The Fisher-Bingham distribution on the sphere", J. Royal. Stat. Soc., 44:71-80.
  • Mardia, KVM., Jupp, PE. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd.
  • Peel, D., Whiten, WJ., McLachlan, GJ. (2001) "Fitting mixtures of Kent distributions to aid in joint set identification". J. Am. Stat. Ass., 96:56-63