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<div>[[統計学]]において、'''残差平方和'''（ざんさへいほうわ、{{lang-en-short|residual sum of squares, '''RSS'''}}）は、[[残差]]の平方（二乗）の[[和]]である。残差二乗和、SSR（sum of squared residuals）やSSE（sum of squared errors of prediction）とも呼ばれる。残差平方和はデータと推定モデルとの間の不一致を評価する尺度である。小さいRSSの値はデータに対してモデルがぴったりとフィットしていること示している。<br />
<br />
一般的に、総平方和 = 説明された平方和 + 残差平方和である。<br />
<br />
==説明変数==<br />
単一の説明変数を持つモデルでは、RSSは以下の式で与えられる。<br />
:<math>RSS = \sum_{i=1}^n (y_i - f(x_i))^2, </math><br />
この時''y''<sub>''i''</sub>は''i''番目の変数の値、''x''<sub>''i''</sub>は''i''番目の説明変数の値、<math>f(x_i)</math>は''y''<sub>''i''</sub>（<math>\hat{y_i}</math>とも）の予測値である。標準線形単純[[回帰分析|回帰モデル]]では、 <math>y_i = a+bx_i+\varepsilon_i\,</math>（''a''および''b''は[[係数]]、''y''および''x''はそれぞれ[[従属変数]]および[[独立変数]]、&epsilon;は誤差項）である。残差平方和は&epsilon;<sub>''i''</sub>の[[推定量]]の平方の和であり以下の式で表わされる。<br />
:<math>RSS = \sum_{i=1}^n (\epsilon_i)^2 = \sum_{i=1}^n (y_i - (\alpha + \beta x_i))^2, </math><br />
この時、αは定数項<math>a</math>の推定値、βは回帰係数''b''の推定値である。<br />
<br />
<!--<br />
==OLS残差平方和の行列表現==<br />
''n''個の観測値と''k''個の一般回帰モデル<br />
The general regression model with ''n'' observations and ''k'' explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is <br />
<br />
:<math> y = X \beta + e</math><br />
<br />
where ''y'' is an ''n'' × 1 vector of dependent variable observations, each column of the ''n'' × ''k'' matrix ''X'' is a vector of observations on one of the ''k'' explanators, <math>\beta </math> is a ''k'' × 1 vector of true coefficients, and ''e'' is an ''n''× 1 vector of the true underlying errors. The [[ordinary least squares]] estimator for <math>\beta</math> is<br />
<br />
:<math> \hat \beta = (X^T X)^{-1}X^T y.</math><br />
<br />
The residual vector <math>\hat e</math> is <math>y - X \hat \beta = y - X (X^T X)^{-1}X^T y</math>, so the residual sum of squares <math>\hat e ^T \hat e</math> is, after simplification,<br />
<br />
:<math> RSS = y^T y - y^T X(X^T X)^{-1} X^T y = y^T [I - X(X^T X)^{-1} X^T] y.</math><br />
--><br />
==関連項目==<br />
*[[残差]]<br />
*[[自由度]]<br />
*[[カイ二乗分布]]<br />
<br />
{{DEFAULTSORT:さんさへいほうわ}}<br />
[[Category:回帰分析]]<br />
[[Category:最小二乗法]]<br />
[[Category:統計学]]<br />
[[Category:数学に関する記事]]</div>133.1.90.125 Warning: Cannot modify header information - headers already sent by (output started at /home/users/1/sub.jp-asate/web/wiki/extensions/HeadScript/HeadScript.php:3) in /home/users/1/sub.jp-asate/web/wiki/includes/WebResponse.php on line 46