statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

on the ''n''-sphere. It is a generalization of the Watson distribution and a special case of the

Kent Kent is a Ceremonial counties of England, ceremonial county in South East England. It is bordered by Essex across the Thames Estuary to the north, the Strait of Dover to the south-east, East Sussex to the south-west, Surrey to the west, and Gr ...

and Fisher–Bingham distributions. The Bingham distribution is widely used in paleomagnetic data analysis, and has been used in the field of

computer vision Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...

. Its

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

is given by :

f(\mathbf\,;\,M,Z) \; dS^ = _1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^ \cdot \exp \left( \operatorname Z M^T \mathbf \mathbf^T M \right)\; dS^

which may also be written :

f(\mathbf\,;\,M,Z)\; dS^ \;=\; 
_1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^ \cdot \exp\left( \mathbf^T M Z M^T \mathbf \right)\; dS^

where x is an axis (i.e., a unit vector), ''M'' is an

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

orientation matrix, ''Z'' is a diagonal concentration matrix, and

_F_(\cdot;\cdot,\cdot)

is a confluent hypergeometric function of matrix argument. The matrices ''M'' and ''Z'' are the result of diagonalizing the

positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite ...

covariance matrix of the

Gaussian distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...

that underlies the Bingham distribution.

References

{{ProbDistributions, directional Directional statistics Continuous distributions

See also

References