Kent distribution

In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere S2 in 3-space R3). It is the analogue on S2 of the bivariate normal distribution with an unconstrained covariance matrix. The Kent distribution was proposed by John T. Kent in 1982, and is used in geology as well as bioinformatics. The probability density function of the Kent distribution is given by: where is a three-dimensional unit vector, denotes the transpose of , and the normalizing constant is: Where is the modified Bessel function and is the gamma function. Note that and , the normalizing constant of the Von Mises–Fisher distribution. 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 3×3 matrix must be orthogonal. The Kent distribution can be easily generalized to spheres in higher dimensions. If is a point on the unit sphere in , then the density function of the -dimensional Kent distribution is proportional to where and and the vectors are orthonormal. However, the normalization constant becomes very difficult to work with for .