Kent distribution

Three points sets sampled from the Kent distribution. The mean directions are shown with arrows. The $\kappa\,$ 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 $S^{2}\,$ in $\Bbb{R}^{3}$ . 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 $f(\mathbf{x})\,$ of the Kent distribution is given by:

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

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

$c(\kappa,\beta)=2\pi\sum_{j=0}^\infty\frac{\Gamma(j+\frac{1}{2})}{\Gamma(j+1)}\beta^{2j}(\frac{1}{2}\kappa)^{-2j-\frac{1}{2}}{I}_{2j+\frac{1}{2}}(\kappa)$

Where ${I}_v(\kappa)$ is the modified Bessel function. Note that $c(0,0) = 4\pi$ and $c(\kappa,0)=4\pi\kappa^{-1}sinh(\kappa)$ , the normalizing constant of the Von Mises-Fisher distribution.

The parameter $\kappa\,$ (with $\kappa>0\,$ ) determines the concentration or spread of the distribution, while $\beta\,$ (with $0\leq2\beta<\kappa$ ) determines the ellipticity of the contours of equal probability. The higher the $\kappa\,$ and $\beta\,$ 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 3x3 matrix $(\gamma_{1},\gamma_{2},\gamma_{3})\,$ must be orthogonal.

Generalization to Higher Dimensions [edit]

The Kent distribution can be easily generalized to spheres in higher dimensions. If $x$ is a point on the unit sphere $S^{p-1}$ in $\mathbb{R}^p$ , then the density function of the $p$ -dimensional Kent distribution is proportional to

$\exp\{\kappa \boldsymbol{\gamma}_1\cdot\mathbf{x} + \sum_{j=2}^p \beta_{j} (\boldsymbol{\gamma}_j \cdot \mathbf{x})^2\}$

Where $\sum_{j=2}^p \beta_j =0$ and $0 \le 2|\beta_j| <\kappa$ and the vectors $\{\boldsymbol{\gamma}_j\mid j=1\ldots p\}$ are orthonormal. However the normalization constant becomes very difficult to work with for $p>3$ .

References [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. 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

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham