第5講回帰分析

重相関係数

\begin{aligned} R & = \frac{\sum_{i = 1}^{n} (y_{i} - \bar{y}) ({\hat{y}}_{i} - \bar{\hat{y}})}{\sqrt{\sum_{i = 1}^{n} (y_{i} - \bar{y})^{2}} \sqrt{\sum_{i = 1}^{n} ({\hat{y}}_{i} - \bar{\hat{y}})^{2}}} \\ = \frac{S_{y \hat{y}}}{\sqrt{S_{y y}} \sqrt{S_{\hat{y} \hat{y}}}} \\ = \frac{S_{\hat{y} \hat{y}}}{\sqrt{S_{y y}} \sqrt{S_{\hat{y} \hat{y}}}} \\ = \frac{\sqrt{S_{\hat{y} \hat{y}}}}{\sqrt{S_{y y}}} \\ = \sqrt{\frac{S_{\hat{y} \hat{y}}}{S_{y y}}} \\ = \sqrt{\frac{S_{R}}{S_{y y}}} \end{aligned}

$\begin{align} R &= \dfrac{\ds \sum_{i=1}^{n}(y_i - \bar{y})(\hat{y}_i - \bar{\hat{y}})}{\ds \sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}\sqrt{\sum_{i=1}^{n}(\hat{y}_i - \bar{\hat{y}})^2}} \\ &=\frac{S_{y\hat{y}}}{\sqrt{S_{yy}}\sqrt{S_{\hat{y}\hat{y}}}} \\ &= \frac{S_{\hat{y}\hat{y}}}{\sqrt{S_{yy}}\sqrt{S_{\hat{y}\hat{y}}}} \\ &= \frac{\sqrt{S_{\hat{y}\hat{y}}}}{\sqrt{S_{yy}}} \\ &= \sqrt{\frac{S_{\hat{y}\hat{y}}}{S_{yy}}} \\ &= \sqrt{\frac{S_{R}}{S_{yy}}} \end{align}$

第5講 回帰分析

重相関係数

第5講回帰分析