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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 and predicted response ： ウィキペディア英語版 Mean and predicted response In linear regression mean response and predicted response are values of the dependent variable calculated from the regression parameters and a given value of the independent variable. The values of these two responses are the same, but their calculated variances are different. == Straight line regression == In straight line fitting, the model is :$y\_i=\backslash alpha+\backslash beta\; x\_i\; +\backslash epsilon\_i\backslash ,$ where $y\_i$ is the response variable, $x\_i$ is the explanatory variable, ''εi'' is the random error, and $\backslash alpha$ and $\backslash beta$ are parameters. The predicted response value for a given explanatory value, ''xd'', is given by :$\backslash hat\_d=\backslash hat\backslash alpha+\backslash hat\backslash beta\; x\_d\; ,$ while the actual response would be :$y\_d=\backslash alpha+\backslash beta\; x\_d\; +\backslash epsilon\_d\; \backslash ,$ Expressions for the values and variances of $\backslash hat\backslash alpha$ and $\backslash hat\backslash beta$ are given in linear regression. Mean response is an estimate of the mean of the ''y'' population associated with ''xd'', that is $E(y\; |\; x\_d)=\backslash hat\_d\backslash !$. The variance of the mean response is given by :$\backslash text\backslash left(\backslash hat\; +\; \backslash hatx\_d\backslash right)\; =\; \backslash text\backslash left(\backslash hat\backslash right)\; +\; \backslash left(\backslash text\; \backslash hat\backslash right)x\_d^2\; +\; 2\; x\_d\backslash text\backslash left(\backslash hat,\backslash hat\backslash right)\; .$ This expression can be simplified to :$\backslash text\backslash left(\backslash hat\; +\; \backslash hatx\_d\backslash right)\; =\backslash sigma^2\backslash left(\backslash frac\; +\; \backslash frac\backslash right).$ To demonstrate this simplification, one can make use of the identity :$\backslash sum\; (x\_i\; -\; \backslash bar)^2\; =\; \backslash sum\; x\_i^2\; -\; \backslash frac\backslash left(\backslash sum\; x\_i\backslash right)^2\; .$ The predicted response distribution is the predicted distribution of the residuals at the given point ''xd''. So the variance is given by :$\backslash text\backslash left(y\_d\; -\; \backslash left(+\; \backslash hatx\_d\backslash right)\backslash right)\; =\; \backslash text\backslash left(y\_d\backslash right)\; +\; \backslash text\backslash left(\backslash hat\; +\; \backslash hatx\_d\backslash right)\; .$ The second part of this expression was already calculated for the mean response. Since $\backslash text\backslash left(y\_d\backslash right)=\backslash sigma^2$ (a fixed but unknown parameter that can be estimated), the variance of the predicted response is given by :$\backslash text\backslash left(y\_d\; -\; \backslash left(+\; \backslash hatx\_d\backslash right)\backslash right)\; =\; \backslash sigma^2\; +\; \backslash sigma^2\backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; =\; \backslash sigma^2\backslash left(1+\backslash frac\; +\; \backslash frac\backslash right)\; .$ 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 and predicted response」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.