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Encyclopedia > Kent distribution

The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. The Kent distribution is the analogue on the unit sphere of the bivariate normal distribution with an unconstrained covariance matrix. The distribution belongs to the field of directional statistics. In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A sphere (< Greek ÏƒÏ†Î±Î¯ÏÎ±) is a perfectly symmetrical geometrical object. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

The probability density function $f(mathbf{x}),$ of the Kent distribution is given by:

$f(mathbf{x})=frac{1}{textrm{c}(kappa,beta)}exp{kappaboldsymbol{gamma}_{1}cdotmathbf{x}+beta[(boldsymbol{gamma}_{2}cdotmathbf{x})^{2}-(boldsymbol{gamma}_{3}cdotmathbf{x})^{2}]}$

where $mathbf{x},$ is a three-dimensional unit vector and $textrm{c}(kappa,beta),$ is a normalizing constant.

The parameter $kappa,$ (with $kappa>0,$ ) determines the concentration or spread of the distribution, while (with $0leq2beta ) determines the ellipticity of the contours of equal probability. The higher the $kappa,$ and parameters, the more concentrated and elliptical the distribution will be, respectively. Vector $gamma_{1},$ is the mean direction, and vectors $gamma_{2},gamma_{3},$ 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 is used in geology and bioinformatics. The Blue Marble: The famous photo of the Earth taken en route to the Moon by Apollo 17s Harrison Schmitt on December 7, 1972. ... Making sense of the huge amounts of DNA data (pictured) produced by gene sequencing projects is just one of the tasks faced by bioinformatics. ...

## References

• Kent, JT. (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.), Proceedings of the 24th LASR Workshop, pp. 57-60. Leeds, Leeds University Press.
• 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
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