Bingham distribution
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In
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.


See also

*
Directional statistics Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes (lines through the origin in R''n'') or rotations in R''n''. M ...
*
von Mises–Fisher distribution In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the (p-1)-sphere in \mathbb^. If p=2 the distribution reduces to the von Mises distribution on the ci ...
*
Kent distribution In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere ''S''2 in ...


References

{{ProbDistributions, directional Directional statistics Continuous distributions