Kent distribution

The 5-parameter Fisher-Bingham distribution or Kent distribution, named after Ronald Fisher, Christopher Bingham, and John T. Kent, is a probability distribution on the two-dimensional unit sphere $S^\{2\},$ in $Bbb\{R\}^\{3\}$ . It is the analogue on the two-dimensional unit sphere of the bivariate normal distribution with an unconstrained covariance matrix. 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 and bioinformatics.

The probability density function $f(mathbf\{x\}),$ of the Kent distribution is given by:

where $mathbf\{x\},$ is a three-dimensional unit vector and is a normalizing constant.

The parameter $kappa,$ (with $kappa>0,$ ) determines the concentration or spread of the distribution, while (with

ee also

* Directional statistics
* von Mises-Fisher distribution

References

* Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C. & Hamelryck, T. (2006) [http://www.maths.leeds.ac.uk/statistics/workshop/lasr2006/proceedings/hamelryck.pdf 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) [http://compbiol.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pcbi.0020131 Sampling Realistic Protein Conformations Using Local Structural Bias] . "PLoS Comput Biol" 2(9): e131

* Kent, J.T. (1982) [http://links.jstor.org/sici?sici=0035-9246(1982)44%3A1%3C71%3ATFDOTS%3E2.0.CO%3B2-W The Fisher-Bingham distribution on the sphere.] , "J. Royal. Stat. Soc.", 44:71-80.

* Kent, J.T., Hamelryck, T. (2005). [http://www.maths.leeds.ac.uk/statistics/workshop/lasr2005/Proceedings/kent.pdf 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. ISBN 0-471-95333-4

* Peel, D., Whiten, WJ., McLachlan, GJ. (2001) [http://citeseer.ist.psu.edu/235663.html Fitting mixtures of Kent distributions to aid in joint set identification.] "J. Am. Stat. Ass.", 96:56-63