Kent distribution

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Three points sets sampled from the Kent distribution. The mean directions are shown with arrows. The parameter is highest for the red set.

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

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

where is a three-dimensional unit vector, denotes the transpose of , and the normalizing constant is:

Where is the modified Bessel function and is the gamma function. Note that and , the normalizing constant of the Von Mises–Fisher distribution.

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 3×3 matrix must be orthogonal.

Generalization to higher dimensions[edit]

The Kent distribution can be easily generalized to spheres in higher dimensions. If is a point on the unit sphere in , then the density function of the -dimensional Kent distribution is proportional to

where and and the vectors are orthonormal. However, the normalization constant becomes very difficult to work with for .

See also[edit]


  • Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C. & Hamelryck, T. (2006) Graphical models and directional statistics capture protein structure. In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Interdisciplinary Statistics and Bioinformatics, pp. 91–94. Leeds, Leeds University Press.
  • Hamelryck T, Kent JT, Krogh A (2006) Sampling Realistic Protein Conformations Using Local Structural Bias[permanent dead link]. PLoS Comput Biol 2(9): e131
  • Kent, J. T. (1982) The Fisher–Bingham distribution on the sphere., J. Royal. Stat. Soc., 44:71–80.
  • Kent, J. T., Hamelryck, T. (2005). Using the Fisher–Bingham distribution in stochastic models for protein structure. In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Quantitative Biology, Shape Analysis, and Wavelets, pp. 57–60. Leeds, Leeds University Press.
  • Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd. ISBN 0-471-95333-4
  • 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