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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 Box–Cox transformation ： ウィキペディア英語版 Power transform In statistics, the power transform corresponds to a family of functions that are applied to create a monotonic transformation of data using power functions. This is a useful data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association such as the Pearson correlation between variables and for other data stabilization procedures. ==Definition== The power transformation is defined as a continuously varying function, with respect to the power parameter ''λ'', in a piece-wise function form that makes it continuous at the point of singularity (''λ'' = 0). For data vectors (''y''1,..., ''y''''n'') in which each ''y''''i'' > 0, the power transform is : $y\_i^\; =$ \begin \dfrac} , &\text \lambda \neq 0 \\() \operatorname(y)\ln , &\text \lambda = 0 \end where : $\backslash operatorname(y)\; =\; (y\_1\backslash cdots\; y\_n)^\; \backslash ,$ is the geometric mean of the observations ''y''1, ..., ''y''''n''. The inclusion of the (''λ'' − 1)th power of the geometric mean in the denominator simplifies the scientific interpretation of any equation involving $y\_i^$, because the units of measurement do not change as ''λ'' changes. Box and Cox (1964) introduced the geometric mean into this transformation by first including the Jacobian of rescaled power transformation : $\backslash dfrac$. with the likelihood. This Jacobian is as follows: : $J(\backslash lambda;\; y\_1,\; ...,\; y\_n)\; =\; \backslash prod\_^n\; |d\; y\_i^\; /\; dy|$ = \prod_^n y_i^ = \operatorname(y)^ This allows the normal log likelihood at its maximum to be written as follows: :$$ \log ( \mathcal (\hat\mu,\hat\sigma)) = (-n/2)(\log(2\pi\hat\sigma^2) +1) + n(\lambda-1) \log(\operatorname(y)) = (-n/2)(\log(2\pi\hat\sigma^2 / \operatorname(y)^) + 1). From here, absorbing $\backslash operatorname(y)^$ into the expression for $\backslash hat\backslash sigma^2$ produces an expression that establishes that minimizing the sum of squares of residuals from $y\_i^$ is equivalent to maximizing the sum of the normal log likelihood of deviations from $(y^\backslash lambda-1)/\backslash lambda$ and the log of the Jacobian of the transformation. The value at ''Y'' = 1 for any ''λ'' is 0, and the derivative with respect to ''Y'' there is 1 for any ''λ''. Sometimes ''Y'' is a version of some other variable scaled to give ''Y'' = 1 at some sort of average value. The transformation is a power transformation, but done in such a way as to make it continuous with the parameter ''λ'' at ''λ'' = 0. It has proved popular in regression analysis, including econometrics. Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter. : \operatorname(y+\alpha)\ln(y_i + \alpha)& \text \lambda=0,\end which holds if ''y''''i'' + α > 0 for all ''i''. If τ(''Y'', λ, α) follows a truncated normal distribution, then ''Y'' is said to follow a Box–Cox distribution. Bickel and Doksum eliminated the need to use a truncated distribution by extending the range of the transformation to all ''y'', as follows: :$\backslash tau(y\_i;\backslash lambda,\; \backslash alpha)\; =\; \backslash begin$ \dfrac} & \text \lambda\neq 0, \\ \\ \operatorname(y+\alpha)\operatorname(y+\alpha)\ln(y_i + \alpha)& \text \lambda=0,\end, where sgn(.) is the Sign function. This change in definition has little practical import as long as $\backslash alpha$ is less than $\backslash operatorname(y\_i)$, which it usually is. Bickel and Doksum also proved that the parameter estimates are consistent and asymptotically normal under appropriate regularity conditions, though the standard Cramér–Rao lower bound can substantially underestimate the variance when parameter values are small relative to the noise variance.〔 However, this problem of underestimating the variance may not be a substantive problem in many applications. 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「Power transform」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.