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.
Definition The probability density function f ( x ) {\displaystyle f(\mathbf {x} )\,} of the Kent distribution is given by:
f ( x ) = 1 c ( κ , β ) exp { κ γ 1 T x + β [ ( γ 2 T x ) 2 − ( γ 3 T x ) 2 ] } {\displaystyle f(\mathbf {x} )={\frac {1}{{\textrm {c}}(\kappa ,\beta )}}\exp \left\{\kappa {\boldsymbol {\gamma }}_{1}^{T}\mathbf {x} +\beta [({\boldsymbol {\gamma }}_{2}^{T}\mathbf {x} )^{2}-({\boldsymbol {\gamma }}_{3}^{T}\mathbf {x} )^{2}]\right\}}
where x {\displaystyle \mathbf {x} \,} is a three-dimensional unit vector, ( ⋅ ) T {\displaystyle (\cdot )^{T}} denotes the transpose of ( ⋅ ) {\displaystyle (\cdot )} , and the normalizing constant c ( κ , β ) {\displaystyle {\textrm {c}}(\kappa ,\beta )\,} is:
c ( κ , β ) = 2 π ∑ j = 0 ∞ Γ ( j + 1 2 ) Γ ( j + 1 ) β 2 j ( 1 2 κ ) − 2 j − 1 2 I 2 j + 1 2 ( κ ) {\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 )}
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