Kent distribution - AbsoluteAstronomy.com

The 5-parameter Fisher-Bingham distribution or Kent distribution, named after Ronald Fisher

Ronald Fisher

Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...

, Christopher Bingham

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.-External links:* faculty web page...

, and John T. Kent, is a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

on the two-dimensional unit sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. It is the analogue on the two-dimensional unit sphere of the bivariate normal distribution with an unconstrained covariance matrix

Covariance matrix

In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...

. The distribution belongs to the field of directional statistics. The Kent distribution was proposed by John T. Kent in 1982, and is used in geology

Geology

Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

and bioinformatics

Bioinformatics

Bioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...

.

The probability density function

of the Kent distribution is given by:

where

is a three-dimensional unit vector and

is a normalizing constant.

The parameter

(with

) determines the concentration or spread of the distribution, while

(with

) determines the ellipticity of the contours of equal probability. The higher the

and

parameters, the more concentrated and elliptical the distribution will be, respectively. Vector

is the mean direction, and vectors

are the major and minor axes. The latter two vectors determine the orientation of the equal probability contours on the sphere, while the first vector determines the common center of the contours. The 3x3 matrix

must be orthogonal.

See also