Bingham distribution

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.

It is widely used in paleomagnetic data analysis, and has been reported as being of use in the field of computer vision.

Its probability density function is given by

$f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle {\frac {1}{2}}};{\textstyle {\frac {n}{2}}};Z)^{-1}\;\cdot \;\exp \left({{\textrm {tr}}\;ZM^{T}\mathbf {x} \mathbf {x} ^{T}M}\right)\;dS^{n-1}$

which may also be written

$f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle {\frac {1}{2}}};{\textstyle {\frac {n}{2}}};Z)^{-1}\;\cdot \;\exp \left({\mathbf {x} ^{T}MZM^{T}\mathbf {x} }\right)\;dS^{n-1}$

where x is an axis, M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, ${}_{1}F_{1}(\cdot ;\cdot ,\cdot )$ is a confluent hypergeometric function of matrix argument.

See also

• Directional statistics
• Kent distribution