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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 Variance-stabilizing transformation ： ウィキペディア英語版 Variance-stabilizing transformation In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or analysis of variance techniques.〔Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics (2nd Edition)'', CUP. ISBN 0-521-81099-X〕 The aim behind the choice of a variance-stabilizing transformation is to find a simple function ''ƒ'' to apply to values ''x'' in a data set to create new values ''y'' = ''ƒ''(''x'') such that the variability of the values ''y'' is not related to their mean value. For example, suppose that the values x are realizations from different Poisson distributions: i.e. the distributions each have different mean values μ. Then, because for the Poisson distribution the variance is identical to the mean, the variance varies with the mean. However, if the simple variance-stabilizing transformation :$y=\backslash sqrt\; \backslash ,$ is applied, the sampling variance associated with observation will be nearly constant: see Anscombe transform for details and some alternative transformations. While variance-stabilizing transformations are well known for certain parametric families of distributions, such as the Poisson and the binomial distribution, some types of data analysis proceed more empirically: for example by searching among power transformations to find a suitable fixed transformation. Alternatively, if data analysis suggests a functional form for the relation between variance and mean, this can be used to deduce a variance-stabilizing transformation.〔Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9〕 Thus if, for a mean μ, :$\backslash operatorname(X)=g(\backslash mu),\; \backslash ,$ a suitable basis for a variance stabilizing transformation would be :$y=\backslash int^x\; \backslash frac\{\backslash sqrt\{g(v)\}\}\; \backslash ,\; dv,$ where the arbitrary constant of integration can be chosen for convenience. ==Relationship to the delta method== Here, the delta method is presented in a rough way, but it is enough to see the relation with the variance-stabilizing transformations. To see a more formal approach see )=\mu and $Var(X)=\backslash sigma^2$. Define $Y=g(X)$, where $g$ is a regular function. A first order Taylor approximation for $Y=g(x)$ is: : $$ Y=g(X)\approx g(\mu)+g'(\mu)(X-\mu) From the equation above, we obtain: : $$ E()=g(\mu) and $Var()=\backslash sigma^2g\text{'}(\backslash mu)^2$ This approximation method is called delta method. Consider now a random variable $X$ such that $E()=\backslash mu$ and $Var()=kh(\backslash mu)$. Notice the relation between the variance and the mean, which implies, for example, heteroscedasticy in a linear model. Therefore, the goal is to find a function $g$ such that $Y=g(X)$ has a variance independent (at least approximately) of its expectation. Imposing the condition $Var()\backslash approx\; kh(\backslash mu)g\text{'}(\backslash mu)^2=constant$, this equality implies the differential equation: : $$ \frac=\frac=\frac=\frac{\sqrt{h(\mu)}} This last expression appeared for the first time in a M. S. Bartlett paper.〔Bartlett, M.S. (1947) The use of transformations. Biometrics, 3, 39-52. doi:10.2307/3001536〕 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「Variance-stabilizing transformation」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.