- Generalization to higher dimensions
- See also
- References
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 and the normalizing constant 
is:
Where 
is the modified Bessel 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
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
- Directional statistics
- Von Mises–Fisher distribution
- Bivariate von Mises distribution
- Von Mises distribution
- Bingham distribution
References
- 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{{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }}. PLoS Comput Biol 2(9): e131
- Kent, J. T. (1982) [https://www.jstor.org/stable/2984712 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
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