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| [[Image:Point_sets_from_Kent_distributions_mapped_onto_a_sphere_-_journal.pcbi.0020131.g004.svg|thumb|Three points sets sampled from the Kent distribution. The mean directions are shown with arrows. The <math>\kappa\,</math> parameter is highest for the red set.]]
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| 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]] <math>S^{2}\,</math> in <math>\Bbb{R}^{3}</math> . 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]].
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| The probability density function <math>f(\mathbf{x})\,</math> of the Kent distribution is given by:
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| :<math>
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| f(\mathbf{x})=\frac{1}{\textrm{c}(\kappa,\beta)}\exp\{\kappa\boldsymbol{\gamma}_{1}\cdot\mathbf{x}+\beta[(\boldsymbol{\gamma}_{2}\cdot\mathbf{x})^{2}-(\boldsymbol{\gamma}_{3}\cdot\mathbf{x})^{2}]\}
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| </math>
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| where <math>\mathbf{x}\,</math> is a three-dimensional unit vector and the normalizing constant <math>\textrm{c}(\kappa,\beta)\,</math> is:
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| <math>
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| c(\kappa,\beta)=2\pi\sum_{j=0}^\infty\frac{\Gamma(j+\frac{1}{2})}{\Gamma(j+1)}\beta^{2j}(\frac{1}{2}\kappa)^{-2j-\frac{1}{2}}{I}_{2j+\frac{1}{2}}(\kappa)
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| </math>
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| Where <math>{I}_v(\kappa)</math> is the [[modified Bessel function]]. Note that <math>c(0,0) = 4\pi</math> and <math>c(\kappa,0)=4\pi\kappa^{-1}sinh(\kappa)</math>, the normalizing constant of the [[Von Mises-Fisher distribution]].
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| The parameter <math>\kappa\,</math> (with <math>\kappa>0\,</math> ) determines the concentration or spread of the distribution, while <math>\beta\,</math> (with <math>0\leq2\beta<\kappa</math> ) determines the ellipticity of the contours of equal probability. The higher the <math>\kappa\,</math> and <math>\beta\,</math> parameters, the more concentrated and elliptical the distribution will be, respectively. Vector <math>\gamma_{1}\,</math> is the mean direction, and vectors <math>\gamma_{2},\gamma_{3}\,</math> 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 <math>(\gamma_{1},\gamma_{2},\gamma_{3})\,</math> must be orthogonal.
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| ==Generalization to Higher Dimensions==
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| The Kent distribution can be easily generalized to spheres in higher dimensions. If <math>x</math> is a point on the unit sphere <math>S^{p-1}</math> in <math>\mathbb{R}^p</math>, then the density function of the <math>p</math>-dimensional Kent distribution is proportional to
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| <math>
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| \exp\{\kappa \boldsymbol{\gamma}_1\cdot\mathbf{x} + \sum_{j=2}^p \beta_{j} (\boldsymbol{\gamma}_j \cdot \mathbf{x})^2\}
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| </math>
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| Where <math>\sum_{j=2}^p \beta_j =0</math> and <math>0 \le 2|\beta_j| <\kappa</math> and the vectors <math>\{\boldsymbol{\gamma}_j\mid j=1\ldots p\}</math> are orthonormal. However the normalization constant becomes very difficult to work with for <math>p>3</math>.
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| ==See also==
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| * [[Directional statistics]]
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| * [[Von Mises-Fisher distribution]]
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| * [[Bivariate von Mises distribution]]
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| * [[Von Mises distribution]]
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| * [[Bingham distribution]]
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| ==References==
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| * 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.
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| * 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
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| * 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.
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| * 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.
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| * Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd. ISBN 0-471-95333-4
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| * 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
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| {{ProbDistributions|directional}}
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| [[Category:Probability distributions]]
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| [[Category:Directional statistics]]
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| [[Category:Continuous distributions]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is where her primary income comes from. Ohio is where her home is. I am really fond of to go to karaoke but I've been using on new issues lately.
Here is my page ... free tarot readings - find more,