Integral nonlinearity

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

$f(\mathbf{x};M,Z)d{S}^{n-1}={}_1{F}_1(\frac{1}{2};\frac{n}{2};Z{)}^{-1}\cdot \exp \left(\text{<mrow data-mjx-texclass="ORD">tr</mrow>}Z{M}^T\mathbf{x}{\mathbf{x}}^{T}M\right)d{S}^{n-1}$

which may also be written

$f(\mathbf{x};M,Z)d{S}^{n-1}={}_1{F}_1(\frac{1}{2};\frac{n}{2};Z{)}^{-1}\cdot \exp \left({\mathbf{x}}^{T}MZ{M}^T\mathbf{x}\right)d{S}^{n-1}$

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

References

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