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Antipodally symmetric probability distribution on the n-sphere
In
statistics, the Bingham distribution, named after
Christopher Bingham, is an
antipodally symmetric
probability distribution on the
n-sphere.
[1] It is a generalization of the Watson distribution and a special case of the
Kent and Fisher–Bingham distributions.
The Bingham distribution is widely used in
paleomagnetic data analysis,
[2] and has been used in the field of
computer vision.
[3]
[4]
[5]
Its
probability density function is given by
![{\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}={}_{1}F_{1}\left({\tfrac {1}{2}};{\tfrac {n}{2}};Z\right)^{-1}\cdot \exp \left(\operatorname {tr} ZM^{T}\mathbf {x} \mathbf {x} ^{T}M\right)\;dS^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260d027f936957ee570899ab3aad61777ccef1f0)
which may also be written
![{\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}\left({\tfrac {1}{2}};{\tfrac {n}{2}};Z\right)^{-1}\cdot \exp \left(\mathbf {x} ^{T}MZM^{T}\mathbf {x} \right)\;dS^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ae61abfccb2e6bc0a263087d8a859cc34c0c3c)
where x is an axis (i.e., a unit vector), M is an
orthogonal orientation matrix, Z is a diagonal concentration matrix, and
is a confluent
hypergeometric function of matrix argument. The matrices M and Z are the result of
diagonalizing the
positive-definite covariance matrix of the
Gaussian distribution that underlies the Bingham distribution.
See also
References
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Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and
singular | |
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Families | |
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