残差平方和とデータの各平方和

残差平方和とデータの各平方和

$$\begin{array}{rcl}{S}_e& =& \sum _{i=1}^n{e}_i^2\\ & =& \sum _{i=1}^n{(y_i-\widehat{y_i})}^2\dots \widehat{y_i}=\hat{\alpha }+\hat{\beta }{x}_i\\ & =& \left(\sum _{i=1}^n{y}_i^2\right)-2n\overline{y}\hat{\alpha }-2\left(\sum _{i=1}^n{x}_i{y}_i\right)\hat{\beta }+n{\hat{\alpha }}^2+2n\overline{x}\hat{\alpha }\hat{\beta }+\left(\sum _{i=1}^n{x}_i^2\right){\hat{\beta }}^2\dots \mathrm{残}\mathrm{差}\mathrm{平}\mathrm{方}\mathrm{和}\mathrm{の}\mathrm{展}\mathrm{開}\mathrm{し}\mathrm{た}\mathrm{式}\\ & =& \left(\sum _{i=1}^n{y}_i^2\right)-2n\overline{y}(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x})-2\left(\sum _{i=1}^n{x}_i{y}_i\right)\frac{S_{xy}}{S_{xx}}+n{(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x})}^2+2n\overline{x}(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x})\frac{S_{xy}}{S_{xx}}+\left(\sum _{i=1}^n{x}_i^2\right){\left(\frac{S_{xy}}{S_{xx}}\right)}^2\dots \hat{\alpha }=\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x},\hat{\beta }=\frac{S_{xy}}{S_{xx}}\\ & =& \left(\sum _{i=1}^n{y}_i^2\right)-2n{\overline{y}}^2+2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}-2\frac{S_{xy}}{S_{xx}}\left(\sum _{i=1}^n{x}_i{y}_i\right)+n{\overline{y}}^2-2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}+n{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{\overline{x}}^2+2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}-2n{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{\overline{x}}^2+\left(\sum _{i=1}^n{x}_i^2\right){\left(\frac{S_{xy}}{S_{xx}}\right)}^2\\ & =& \{\left(\sum _{i=1}^n{y}_i^2\right)-2n{\overline{y}}^2+n{\overline{y}}^2\}+\{2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}-2\frac{S_{xy}}{S_{xx}}\left(\sum _{i=1}^n{x}_i{y}_i\right)-2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}+2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}\}+\{n{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{\overline{x}}^2-2n{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{\overline{x}}^2+\left(\sum _{i=1}^n{x}_i^2\right){\left(\frac{S_{xy}}{S_{xx}}\right)}^2\}\\ & =& \{\left(\sum _{i=1}^n{y}_i^2\right)-n{\overline{y}}^2\}+\{2n\frac{S_{xy}}{S_{xx}}\overline{x}\overline{y}-2\frac{S_{xy}}{S_{xx}}\left(\sum _{i=1}^n{x}_i{y}_i\right)\}+\{-n{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{\overline{x}}^2+\left(\sum _{i=1}^n{x}_i^2\right){\left(\frac{S_{xy}}{S_{xx}}\right)}^2\}\\ & =& \{\left(\sum _{i=1}^n{y}_i^2\right)-n{\overline{y}}^2\}-2\frac{S_{xy}}{S_{xx}}\{\left(\sum _{i=1}^n{x}_i{y}_i\right)-n\overline{x}\overline{y}\}+{\left(\frac{S_{xy}}{S_{xx}}\right)}^2\{\left(\sum _{i=1}^n{x}_i^2\right)-n{\overline{x}}^2\}\dots \sum _{i=1}^n{y}_i^2-n{\overline{y}}^2=S_{yy},\sum _{i=1}^n{x}_i^2-n{\overline{x}}^2=S_{xx},\sum _{i=1}^n{x}_i{y}_i-n\overline{x}\overline{y}=S_{xy}\\ & =& S_{yy}-2\frac{S_{xy}}{S_{xx}}{S}_{xy}+{\left(\frac{S_{xy}}{S_{xx}}\right)}^2{S}_{xx}\\ & =& S_{yy}-2\frac{S_{xy}^2}{S_{xx}}+\frac{S_{xy}^2}{S_{xx}}\\ & =& S_{yy}-\frac{S_{xy}^2}{S_{xx}}\end{array}$$ よって，残差平方和$S_e$はデータの各平方和($S_{xx},S_{yy},S_{xy}$)より求めることができる．