Kent distribution

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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, 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.

See also[edit]

References[edit]

  • Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd. ISBN 0-471-95333-4