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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 Bivariate von Mises distribution ： ウィキペディア英語版 Bivariate von Mises distribution In probability theory and statistics, the bivariate von Mises distribution is a probability distribution describing values on a torus. It may be thought of as an analogue on the torus of the bivariate normal distribution. The distribution belongs to the field of directional statistics. The general bivariate von Mises distribution was first proposed by Kanti Mardia in 1975. One of its variants is today used in the field of bioinformatics to formulate a probabilistic model of protein structure in atomic detail. == Definition == The bivariate von Mises distribution is a probability distribution defined on the torus, $S^1\; \backslash times\; S^1$ in $\backslash mathbb^3$. The probability density function of the general bivariate von Mises distribution for the angles $\backslash phi,\; \backslash psi\; \backslash in\; (2\backslash pi)$ is given by〔 :$$ f(\phi, \psi) \propto \exp (\kappa_1 \cos(\phi - \mu) + \kappa_2 \cos(\psi - \nu) + (\cos(\phi-\mu), \sin(\phi-\mu)) \mathbf (\cos(\psi - \nu), \sin(\psi - \nu))^T ), where $\backslash mu$ and $\backslash nu$ are the means for $\backslash phi$ and $\backslash psi$, $\backslash kappa\_1$ and $\backslash kappa\_2$ their concentration and the matrix $\backslash mathbf\; \backslash in\; \backslash mathbb(2,2)$ is related to their correlation. Two commonly used variants of the bivariate von Mises distribution are the sine and cosine variant. The cosine variant of the bivariate von Mises distribution〔 has the probability density function :$$ f(\phi, \psi) = Z_c(\kappa_1, \kappa_2, \kappa_3) \ \exp (\kappa_1 \cos(\phi - \mu) + \kappa_2 \cos(\psi - \nu) - \kappa_3 \cos(\phi - \mu - \psi + \nu) ), where $\backslash mu$ and $\backslash nu$ are the means for $\backslash phi$ and $\backslash psi$, $\backslash kappa\_1$ and $\backslash kappa\_2$ their concentration and $\backslash kappa\_3$ is related to their correlation. $Z\_c$ is the normalization constant. This distribution with $\backslash kappa\_3$=0 has been used for kernel density estimates of the distribution of the protein dihedral angles $\backslash phi$ and $\backslash psi$.〔 The sine variant has the probability density function :$$ f(\phi, \psi) = Z_s(\kappa_1, \kappa_2, \kappa_3) \ \exp (\kappa_1 \cos(\phi - \mu) + \kappa_2 \cos(\psi - \nu) + \kappa_3 \sin(\phi - \mu) \sin(\psi - \nu) ), where the parameters have the same interpretation. 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「Bivariate von Mises 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.