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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 Von Mises–Fisher distribution ： ウィキペディア英語版 Von Mises–Fisher distribution In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises, is a probability distribution on the $(p-1)$-dimensional sphere in $\backslash mathbb^$. If $p=2$ the distribution reduces to the von Mises distribution on the circle. The probability density function of the von Mises–Fisher distribution for the random ''p''-dimensional unit vector $\backslash mathbf\backslash ,$ is given by: :$$ f_(\mathbf; \mu, \kappa)=C_(\kappa)\exp \left( \right) where $\backslash kappa\; \backslash ge\; 0,\; \backslash left\; \backslash Vert\; \backslash mu\; \backslash right\; \backslash Vert\; =1\; \backslash ,$ and the normalization constant $C\_(\backslash kappa)\backslash ,$ is equal to : $$ C_(\kappa)=\frac I_(\kappa)}. \, where $I\_$ denotes the modified Bessel function of the first kind at order $v$. If $p=3$, the normalization constant reduces to : $$ C_(\kappa)=\frac =\frac )}. \, The parameters $\backslash mu\backslash ,$ and $\backslash kappa\backslash ,$ are called the ''mean direction'' and ''concentration parameter'', respectively. The greater the value of $\backslash kappa\backslash ,$, the higher the concentration of the distribution around the mean direction $\backslash mu\backslash ,$. The distribution is unimodal for $\backslash kappa>0\backslash ,$, and is uniform on the sphere for $\backslash kappa=0\backslash ,$. The von Mises–Fisher distribution for $p=3$, also called the Fisher distribution, was first used to model the interaction of electric dipoles in an electric field (Mardia, 2000). Other applications are found in geology, bioinformatics, and text mining. ==Estimation of parameters== A series of ''N'' independent measurements $x\_i$ are drawn from a von Mises–Fisher distribution. Define : $$ A_(\kappa)=\frac . \, Then (Sra, 2011) the maximum likelihood estimates of $\backslash mu\backslash ,$ and $\backslash kappa\backslash ,$ are given by :$$ \mu = \frac , :$$ \kappa = A_p^(\bar) . Thus $\backslash kappa\backslash ,$ is the solution to :$$ A_p(\kappa) = \frac = \bar . A simple approximation to $\backslash kappa$ is :$$ \hat = \frac^2)} , but a more accurate measure can be obtained by iterating the Newton method a few times :$$ \hat_1 = \hat - \frac})} , :$$ \hat_2 = \hat_1 - \frac}A_p(\hat_1)} . For ''N'' ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as :$\backslash hat\; =\; \backslash left(\backslash frac\backslash right)^$ where :$d\; =\; 1\; -\; \backslash frac\backslash sum\_i^N\; (\backslash mu^Tx\_i)^2$ It's then possible to approximate a $100(1-\backslash alpha)\backslash \%$ confidence cone about $\backslash mu$ with semi-vertical angle :$q\; =\; \backslash arcsin(e\_\backslash alpha^\backslash hat)$ , where $e\_\backslash alpha\; =\; -\backslash ln(\backslash alpha).$ For example, for a 95% confidence cone, $\backslash alpha\; =\; 0.05,\; e\_\backslash alpha\; =\; -\backslash ln(0.05)\; =\; 2.996,$ and thus $q\; =\; \backslash arcsin(1.731\backslash hat).$ 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「Von Mises–Fisher 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.