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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 Gaussian copula ： ウィキペディア英語版 Copula (probability theory) In probability theory and statistics, a copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between random variables. Their name comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Sklar's Theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below. ==Mathematical definition== Consider a random vector $(X\_1,X\_2,\backslash dots,X\_d)$. Suppose its margins are continuous, i.e. the marginal CDFs $F\_i(x)\; =\; \backslash mathbb(x)$ are continuous functions. By applying the probability integral transform to each component, the random vector :$(U\_1,U\_2,\backslash dots,U\_d)=\backslash left(F\_1(X\_1),F\_2(X\_2),\backslash dots,F\_d(X\_d)\backslash right)$ has uniformly distributed marginals. The copula of $(X\_1,X\_2,\backslash dots,X\_d)$ is defined as the joint cumulative distribution function of $(U\_1,U\_2,\backslash dots,U\_d)$: :$C(u\_1,u\_2,\backslash dots,u\_d)=\backslash mathbb(u\_1,U\_2\backslash leq\; u\_2,\backslash dots,U\_d\backslash leq\; u\_d)\; .$ The copula ''C'' contains all information on the dependence structure between the components of $(X\_1,X\_2,\backslash dots,X\_d)$ whereas the marginal cumulative distribution functions $F\_i$ contain all information on the marginal distributions. The importance of the above is that the reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample $(U\_1,U\_2,\backslash dots,U\_d)$ from the copula distribution, the required sample can be constructed as :$(X\_1,X\_2,\backslash dots,X\_d)\; =\; \backslash left(F\_1^(U\_1),F\_2^(U\_2),\backslash dots,F\_d^(U\_d)\backslash right).$ The inverses $F\_i^$ are unproblematic as the $F\_i$ were assumed to be continuous. The above formula for the copula function can be rewritten to correspond to this as: :$C(u\_1,u\_2,\backslash dots,u\_d)=\backslash mathbb(F\_1^(u\_1),X\_2\backslash leq\; F\_2^(u\_2),\backslash dots,X\_d\backslash leq\; F\_d^(u\_d))\; .$ 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「Copula (probability theory)」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.