Kent distribution

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

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

where ${\displaystyle \mathbf {x} \,}$ is a three-dimensional unit vector, ${\displaystyle (\cdot )^{T}}$ denotes the transpose of ${\displaystyle (\cdot )}$, and the normalizing constant ${\displaystyle {\textrm {c}}(\kappa ,\beta )\,}$ is:

${\displaystyle c(\kappa ,\beta )=2\pi \sum _{j=0}^{\infty }{\frac {\Gamma (j+{\frac {1}{2}})}{\Gamma (j+1)}}\beta ^{2j}\left({\frac {1}{2}}\kappa \right)^{-2j-{\frac {1}{2}}}I_{2j+{\frac {1}{2}}}(\kappa )}$

Where ${\displaystyle I_{v}(\kappa )}$ is the modified Bessel function and ${\displaystyle \Gamma (\cdot )}$ is the gamma function. Note that ${\displaystyle c(0,0)=4\pi }$ and ${\displaystyle c(\kappa ,0)=4\pi (\kappa ^{-1})\sinh(\kappa )}$, the normalizing constant of the Von Mises–Fisher distribution.

The parameter ${\displaystyle \kappa \,}$ (with ${\displaystyle \kappa >0\,}$ ) determines the concentration or spread of the distribution, while ${\displaystyle \beta \,}$ (with ${\displaystyle 0\leq 2\beta <\kappa }$ ) determines the ellipticity of the contours of equal probability. The higher the ${\displaystyle \kappa \,}$ and ${\displaystyle \beta \,}$ parameters, the more concentrated and elliptical the distribution will be, respectively. Vector ${\displaystyle \gamma _{1}\,}$ is the mean direction, and vectors ${\displaystyle \gamma _{2},\gamma _{3}\,}$ 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 3×3 matrix ${\displaystyle (\gamma _{1},\gamma _{2},\gamma _{3})\,}$ must be orthogonal.

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

${\displaystyle \exp \left\{\kappa {\boldsymbol {\gamma }}_{1}^{T}\cdot \mathbf {x} +\sum _{j=2}^{p}\beta _{j}({\boldsymbol {\gamma }}_{j}^{T}\cdot \mathbf {x} )^{2}\right\}\ ,}$

where ${\displaystyle \sum _{j=2}^{p}\beta _{j}=0}$ and ${\displaystyle 0\leq 2|\beta _{j}|<\kappa }$ and the vectors ${\displaystyle \{{\boldsymbol {\gamma }}_{j}\mid j=1,\ldots ,p\}}$ are orthonormal. However, the normalization constant becomes very difficult to work with for ${\displaystyle p>3}$.

• Directional statistics
• Von Mises–Fisher distribution
• Bivariate von Mises distribution
• Von Mises distribution
• Bingham distribution

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• 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