📂Probability Distribution

Kent Distribution

Definition ¹

Concentration $\kappa > 0$ and $\beta \in \mathbb{R}$, Mean $\gamma_{1} \in S^{p-1}$, Major Axis $\gamma_{2} \in S^{p-1}$, Minor Axis $\gamma_{3} \in S^{p-1}$ are characterized by the following probability density function for the multivariate distribution $\text{FB}_{5} \left( \left( \gamma_{1} , \gamma_{2} , \gamma_{3} \right) , \kappa , \beta \right)$, known as the Kent Distribution. $$ f \left( \mathbf{x} \right) = {{ 1 } \over { c \left( \kappa , \nu \right) }} \exp \left( \kappa \gamma^{T} \mathbf{x} + \beta \left[ \left( \gamma_{2}^{T} \mathbf{x} \right)^{2} - \left( \gamma_{3}^{T} \mathbf{x} \right)^{2} \right] \right) \qquad , \mathbf{x} \in S^{p-1} $$

Especially when $0 \le \beta < \kappa / 2$, this distribution is oval on the sphere, and $c \left( \kappa , \nu \right) > 0$ is the normalizing constant such that $\int_{S^{p-1}} f(\mathbf{x}) d \mathbf{x} = 1$. $$ c \left( \kappa , \beta \right) = 2 \pi \sum_{j=0}^{\infty} {{ \Gamma \left( j + {{ 1 } \over { 2 }} \right) } \over { \Gamma \left( j+1 \right) }} \beta^{2j} \left( {{ 2 } \over { \kappa }} \right)^{2j + {{ 1 } \over { 2 }}} I_{2j + {{ 1 } \over { 2 }}} \left( \kappa \right) $$

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• $S^{p-1} \subset \mathbb{R}^{p}$ is the unit sphere.
• $\gamma ^{T}$ is the transpose of the vector $\gamma$.
• $\Gamma$ is the gamma function.
• $I_{\nu}$ denotes the modified Bessel function of the first kind of order $\nu$, the reason for using such complex functions is referred to in the post Why Modified Bessel Functions Appear in Directional Statistics.

Description

The Kent distribution draws oval-shaped contours on the sphere similar to the geometric meaning given by the non-trivial covariance matrix of the multivariate normal distribution, i.e., it seems like one could just draw an ellipse on the plane and project it onto a sphere, but as introduced in the definition, modeling with complex formulas is necessary to avoid distortion on the sphere.

Eccentricity $2 \beta / \kappa$ defined by the distribution’s parameters indicates how different the contours are from a circle.