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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 Mean squared prediction error ： ウィキペディア英語版 Mean squared prediction error In statistics the mean squared prediction error of a smoothing or curve fitting procedure is the expected value of the squared difference between the fitted values $\backslash widehat$ and the (unobservable) function ''g''. If the smoothing procedure has operator matrix ''L'', then :$\backslash operatorname(L)=\backslash operatorname\backslash left(g(x\_i)-\backslash widehat(x\_i)\backslash right)^2\backslash right).$ The MSPE can be decomposed into two terms (just like mean squared error is decomposed into bias and variance); however for MSPE one term is the sum of squared biases of the fitted values and another the sum of variances of the fitted values: :$\backslash operatorname(L)=\backslash sum\_^n\backslash left(\backslash operatorname\backslash left()-g(x\_i)\backslash right)^2+\backslash sum\_^n\backslash operatorname\backslash left().$ Note that knowledge of ''g'' is required in order to calculate MSPE exactly. ==Estimation of MSPE== For the model $y\_i=g(x\_i)+\backslash sigma\backslash varepsilon\_i$ where $\backslash varepsilon\_i\backslash sim\backslash mathcal(0,1)$, one may write :$\backslash operatorname(L)=g\text{'}(I-L)\text{'}(I-L)g+\backslash sigma^2\backslash operatorname\backslash left().$ The first term is equivalent to :$\backslash sum\_^n\backslash left(\backslash operatorname\backslash left()-g(x\_i)\backslash right)^2$ =\operatorname\left()-\sigma^2\operatorname\left(). Thus, :$\backslash operatorname(L)=\backslash operatorname\backslash left()-\backslash sigma^2\backslash left(n-2\backslash operatorname\backslash left()\backslash right).$ If $\backslash sigma^2$ is known or well-estimated by $\backslash widehat^2$, it becomes possible to estimate MSPE by :$\backslash operatorname^n\backslash left(y\_i-\backslash widehat(x\_i)\backslash right)^2-\backslash widehat^2\backslash left(n-2\backslash operatorname\backslash left()\backslash right).$ Colin Mallows advocated this method in the construction of his model selection statistic ''Cp'', which is a normalized version of the estimated MSPE: :$C\_p=\backslash frac(x\_i)\backslash right)^2\}-n+2\backslash operatorname\backslash left().$ where ''p'' comes from the fact that the number of parameters ''p'' estimated for a parametric smoother is given by $p=\backslash operatorname\backslash left()$, and ''C'' is in honor of Cuthbert Daniel. 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「Mean squared prediction error」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.