statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the Bingham distribution, named after

Christopher Bingham Christopher Bingham is an American statistician who introduced the Bingham distribution. In joint work with C. M. D. Godfrey and John Tukey he introduced complex demodulation into the analysis of time series. The Kent distribution ...

, is an antipodally symmetric

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

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

Kent Kent is a county in South East England and one of the home counties. It borders Greater London to the north-west, Surrey to the west and East Sussex to the south-west, and Essex to the north across the estuary of the River Thames; it faces ...

and Fisher-Bingham distributions. The Bingham distribution is widely used in

paleomagnetic Paleomagnetism (or palaeomagnetismsee ), is the study of magnetic fields recorded in rocks, sediment, or archeological materials. Geophysicists who specialize in paleomagnetism are called ''paleomagnetists.'' Certain magnetic minerals in rock ...

data analysis, and has been reported as being of use in the field of

computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...

. Its

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

is given by :

f(\mathbf\,;\,M,Z)\; dS^ \;=\; _F_(;;Z)^\;\cdot\; \exp\left(\right)\; dS^

which may also be written :

f(\mathbf\,;\,M,Z)\; dS^ \;=\; 
_F_(;;Z)^\;\cdot\; 
\exp\left(\right)\; dS^

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

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

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 In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...

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 fu ...

covariance matrix of the Gaussian distribution that underlies the Bingham distribution.

References

{{ProbDistributions, directional Directional statistics Continuous distributions

See also

References