In
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 ...
, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent,
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
, and
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 a
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 unit
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
''S''
2 in
3-space R
3). It is the analogue on ''S''
2 of the bivariate
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with an unconstrained
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
. The Kent distribution was proposed by John T. Kent in 1982, and is used in
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
as well as
bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
.
Definition
The
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 ...
of the Kent distribution is given by:
:
where
is a three-dimensional unit vector,
denotes the transpose of
, and the normalizing constant
is:
:
Where
is the
modified Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
and
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. Note that
and
, the normalizing constant of the
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 ...
.
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 3×3 matrix
must be orthogonal.
Generalization to higher dimensions
The Kent distribution can be easily generalized to spheres in higher dimensions. If
is a point on the unit sphere
in
, then the density function of the
-dimensional Kent distribution is proportional to
:
where
and
and the vectors
are orthonormal. However, the normalization constant becomes very difficult to work with for
.
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 ...
*
Bivariate von Mises distribution
In probability theory and statistics, the bivariate von Mises distribution is a probability distribution describing values on a torus. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belon ...
*
Von Mises distribution
In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wr ...
*
Bingham distribution In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the ''n''-sphere. It is a generalization of the Watson distribution and a special case of the Kent and Fisher-Bing ...
References
* Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C. & Hamelryck, T. (2006
Graphical models and directional statistics capture protein structure In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), ''Interdisciplinary Statistics and Bioinformatics'', pp. 91–94. Leeds, Leeds University Press.
* Hamelryck T, Kent JT, Krogh A (2006
Sampling Realistic Protein Conformations Using Local Structural Bias ''PLoS Comput Biol'' 2(9): e131
* Kent, J. T. (1982
The Fisher–Bingham distribution on the sphere. ''J. Royal. Stat. Soc.'', 44:71–80.
* Kent, J. T., Hamelryck, T. (2005)
Using the Fisher–Bingham distribution in stochastic models for protein structure In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), ''Quantitative Biology, Shape Analysis, and Wavelets'', pp. 57–60. Leeds, Leeds University Press.
* Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd.
* Peel, D., Whiten, WJ., McLachlan, GJ. (2001
''J. Am. Stat. Ass.'', 96:56–63
{{ProbDistributions, directional
Directional statistics
Continuous distributions