Kent distribution

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

:$$

f(\mathbf{x}) = \frac{1}{\textrm{c}(\kappa,\beta)}\exp\left\{ \kappa \boldsymbol{\gamma}_1^T \mathbf{x} + \beta[(\boldsymbol{\gamma}_2^T \mathbf{x})^2-(\boldsymbol{\gamma}_3^T \mathbf{x})^2]\right\}

where $\backslash mathbf\{x\}\backslash ,$ is a three-dimensional unit vector, $(\backslash cdot)^T$ denotes the transpose of $(\backslash cdot)$, and the normalizing constant $\backslash textrm\{c\}(\backslash kappa,\backslash beta)\backslash ,$ is:

:$$

c(\kappa,\beta) = 2\pi\sum_{j=0}^\infty \frac{\Gamma(j+\frac{1}{2})}{\Gamma(j+1)}\beta^{2j}\left(\frac{1}{2}\kappa\right)^{-2j-\frac{1}{2}} I_{2j+\frac{1}{2}}(\kappa)

Where $I\_v(\backslash kappa)$ is the modified Bessel function and $\backslash Gamma(\backslash cdot)$ is the gamma function. Note that $c(0,0)\; =\; 4\backslash pi$ and $c(\backslash kappa,0)=4\backslash pi(\backslash kappa^\{-1\})\backslash sinh(\backslash kappa)$, the normalizing constant of the Von Mises–Fisher distribution.

The parameter $\backslash kappa\backslash ,$ (with $\backslash kappa>0\backslash ,$ ) determines the concentration or spread of the distribution, while $\backslash beta\backslash ,$ (with ) determines the ellipticity of the contours of equal probability. The higher the $\backslash kappa\backslash ,$ and $\backslash beta\backslash ,$ parameters, the more concentrated and elliptical the distribution will be, respectively. Vector $\backslash boldsymbol\{\backslash gamma\}\_1\backslash ,$ is the mean direction, and vectors $\backslash boldsymbol\{\backslash gamma\}\_2,\backslash boldsymbol\{\backslash gamma\}\_3\backslash ,$ 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 $3\; \backslash times\; 3$ matrix $(\backslash boldsymbol\{\backslash gamma\}\_1,\backslash boldsymbol\{\backslash gamma\}\_2,\backslash boldsymbol\{\backslash gamma\}\_3)\backslash ,$ must be orthogonal.

The Kent distribution can be easily generalized to spheres in higher dimensions. If $x$ is a point on the unit sphere $S^\{p-1\}$ in $\backslash mathbb\{R\}^p$, then the density function of the $p$-dimensional Kent distribution is proportional to

: $$

\exp \left\{\kappa \boldsymbol{\gamma}_1^T\mathbf{x} + \sum_{j=2}^p \beta_j (\boldsymbol{\gamma}_j^T\mathbf{x})^2 \right\} \ ,

where $\backslash sum\_\{j=2\}^p\; \backslash beta\_j\; =0$ and and the vectors $\backslash \{\; \backslash boldsymbol\{\backslash gamma\}\_j\; \backslash mid\; j=1,\; \backslash ldots,\; p\backslash \}$ are orthonormal. However, the normalization constant becomes very difficult to work with for $p>3$.

• Directional statistics
• Von Mises–Fisher distribution
• Bivariate von Mises distribution
• Von Mises distribution
• Bingham distribution

• 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] {{Webarchive|url=https://web.archive.org/web/20210507110832/http://www1.maths.leeds.ac.uk/statistics/workshop/lasr2006/proceedings/hamelryck.pdf |date=2021-05-07 }}. 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) [https://web.archive.org/web/20171027024924/http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.0020131 Sampling Realistic Protein Conformations Using Local Structural Bias]. PLoS Comput Biol 2(9): e131
• Kent, J. T. (1982) [https://www.jstor.org/stable/2984712 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] {{Webarchive|url=https://web.archive.org/web/20210507102826/http://www1.maths.leeds.ac.uk/statistics/workshop/lasr2005/Proceedings/kent.pdf |date=2021-05-07 }}. 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

{{ProbDistributions|directional}}

Category:Directional statistics

Category:Continuous distributions