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In [[statistics]], the '''Bingham distribution''', named after [[Christopher Bingham]], is an [[antipodal symmetry|antipodally symmetric]] [[probability distribution]] on the [[hypersphere|''n''-sphere]].<ref>Bingham, Ch. (1974) "An antipodally symmetric distribution on the sphere". ''Annals of Statistics'', 2(6):1201–1225.</ref>  
 
It is widely used in [[paleomagnetic]] data analysis,<ref>Onstott, T.C. (1980) "Application of the Bingham distribution function in paleomagnetic studies". ''Journal of Geophysical Research'', 85:1500–1510.</ref> and has been reported as being of use in the field of [[computer vision]].<ref>{{cite web|title=Belief Propagation with Directional Statistics for Solving the Shape-from-Shading Problem|url=http://link.springer.com/chapter/10.1007%2F978-3-540-88690-7_58|publisher=Springer|date=2008|accessdate=November 29, 2013}}</ref><ref>{{cite web|title=Better robot vision: A neglected statistical tool could help robots better understand the objects in the world around them.|url=http://web.mit.edu/newsoffice/2013/better-robot-vision-1007.html|publisher=MIT News|date=October 7, 2013|accessdate=October 7, 2013}}</ref>
 
Its [[probability density function]] is given by
 
: <math>
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}\; Z M^{T}\mathbf{x} \mathbf{x}^{T}M}\right)\; dS^{n-1}
</math>
which may also be written
: <math>
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} M Z M^{T}\mathbf{x} }\right)\; dS^{n-1}
</math>
 
where '''x''' is an axis, ''M'' is an [[orthogonal matrix|orthogonal]] orientation matrix, ''Z'' is a diagonal concentration matrix,  
<math>{}_{1}F_{1}(\cdot;\cdot,\cdot)</math> is a [[confluent hypergeometric function]] of matrix argument.
 
==See also==
* [[Directional statistics]]
* [[Kent distribution]]
 
==References==
{{reflist}}
 
{{ProbDistributions|directional}}
 
[[Category:Probability distributions]]
[[Category:Directional statistics]]
[[Category:Continuous distributions]]

Revision as of 19:56, 8 September 2013

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.[1]

It is widely used in paleomagnetic data analysis,[2] and has been reported as being of use in the field of computer vision.[3][4]

Its probability density function is given by

f(x;M,Z)dSn1=1F1(12;n2;Z)1exp(trZMTxxTM)dSn1

which may also be written

f(x;M,Z)dSn1=1F1(12;n2;Z)1exp(xTMZMTx)dSn1

where x is an axis, M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, 1F1(;,) is a confluent hypergeometric function of matrix argument.

See also

References

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  1. Bingham, Ch. (1974) "An antipodally symmetric distribution on the sphere". Annals of Statistics, 2(6):1201–1225.
  2. Onstott, T.C. (1980) "Application of the Bingham distribution function in paleomagnetic studies". Journal of Geophysical Research, 85:1500–1510.
  3. Template:Cite web
  4. Template:Cite web