Kent distribution
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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 Kent distribution was proposed by John T. Kent in 1982, and is used in geology and bioinformatics.
The probability density function 
of the Kent distribution is given by:
![f(\mathbf{x})=\frac{1}{\textrm{c}(\kappa,\beta)}\exp\{\kappa\boldsymbol{\gamma}_{1}\cdot\mathbf{x}+\beta[(\boldsymbol{\gamma}_{2}\cdot\mathbf{x})^{2}-(\boldsymbol{\gamma}_{3}\cdot\mathbf{x})^{2}]\}](../../../../math/3/c/8/3c8eb052912d8d5bcd18249f98b8a62d.png)
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 3x3 matrix 
must be orthogonal.
[edit] See also
[edit] 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. 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
