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.

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

Definition

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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   matrix   must be orthogonal.

Generalization to higher dimensions

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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

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References

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  • Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C. & Hamelryck, T. (2006) Graphical models and directional statistics capture protein structure Archived 2021-05-07 at the Wayback Machine. 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. 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 Archived 2021-05-07 at the Wayback Machine. 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