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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kent distribution ： ウィキペディア英語版 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 $S^\backslash ,$ in $\backslash Bbb^$ . 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, bioinformatics. The probability density function $f(\backslash mathbf)\backslash ,$ of the Kent distribution is given by: :$$ f(\mathbf)=\frac\exp\\cdot\mathbf+\beta()\} where $\backslash mathbf\backslash ,$ is a three-dimensional unit vector and the normalizing constant $\backslash textrm(\backslash kappa,\backslash beta)\backslash ,$ is: $$ c(\kappa,\beta)=2\pi\sum_^\infty\frac)}\beta^\left(\frac\kappa\right)^} I_}(\kappa) Where $I\_v(\backslash kappa)$ is the modified Bessel function. Note that $c(0,0)\; =\; 4\backslash pi$ and $c(\backslash kappa,0)=4\backslash pi\backslash kappa^\backslash sinh(\backslash kappa)$, 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 $\backslash gamma\_1\backslash ,$ is the mean direction, and vectors $\backslash gamma\_2,\backslash gamma\_3\backslash ,$ 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 3×3 matrix $(\backslash gamma\_1,\backslash gamma\_2,\backslash gamma\_3)\backslash ,$ must be orthogonal. ==Generalization to higher dimensions== The Kent distribution can be easily generalized to spheres in higher dimensions. If $x$ is a point on the unit sphere $S^$ in $\backslash mathbb^p$, then the density function of the $p$-dimensional Kent distribution is proportional to : $$ \exp\ + \sum_^p \beta_j (\boldsymbol_j \cdot \mathbf)^2\} Where $\backslash sum\_^p\; \backslash beta\_j\; =0$ and and the vectors $\backslash$ are orthonormal. However the normalization constant becomes very difficult to work with for $p>3$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kent distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.