chain_rules
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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| chain_rules [2026/03/10 08:41] – [e.g.] hkimscil | chain_rules [2026/03/10 22:11] (current) – [e.g.] hkimscil | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| + | intercept, a 에 대한 SSR의 미분은 | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \widehat{y} & = & a + b * x \\ | \widehat{y} & = & a + b * x \\ | ||
| \text{a} & = & \text{intercept} \\ | \text{a} & = & \text{intercept} \\ | ||
| \text{residual} & = & (y - \widehat{y}) \\ | \text{residual} & = & (y - \widehat{y}) \\ | ||
| - | \dfrac{\text{d.[SSR]}^2}{\text{d.a}} & = & | + | \text{SSR} & = & \sum {\text{residual}^2} = \sum{(y - (a + b x))^2} |
| - | \dfrac{\text{d.[SSR]}^2}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.a}} \\ | + | \dfrac{\text{d.SSR}}{\text{d.a}} & = & |
| + | \dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.a}} \\ | ||
| & = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.intercept}} \\ | & = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.intercept}} \\ | ||
| & = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.intercept}} \\ | & = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.intercept}} \\ | ||
| + | & = & 2 * \text{residual} * -1 \\ | ||
| & = & -2 * \text{residual} \\ | & = & -2 * \text{residual} \\ | ||
| & = & -2 * (y - (a + b * x)) \\ | & = & -2 * (y - (a + b * x)) \\ | ||
| + | & = & -2 * (y - \widehat{y}) \\ | ||
| + | \end{eqnarray*} | ||
| + | slope, b 에 대한 SSR의 미분은 | ||
| + | \begin{eqnarray*} | ||
| + | \widehat{y} & = & a + b * x \\ | ||
| + | \text{b} & = & \text{slope} \\ | ||
| + | \text{residual} & = & (y - \widehat{y}) \\ | ||
| + | \text{SSR} & = & \sum {\text{residual}^2} = \sum{(y - (a + b x))^2} | ||
| + | \dfrac{\text{d.SSR}}{\text{d.a}} & = & | ||
| + | \dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.b}} \\ | ||
| + | & = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.b}} \\ | ||
| + | & = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.b}} \\ | ||
| + | & = & 2 * \text{residual} * -x \\ | ||
| + | & = & -2 x * \text{residual} \\ | ||
| + | & = & -2 x * (y - (a + b * x)) \\ | ||
| + | & = & -2 x * (y - \widehat{y}) \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | y.hat = a + b * x | ||
| - | a = intercept | ||
| - | residuals = (y - y.hat) | ||
| - | d.sum.of.residuals^2 / d.intercept | ||
| - | = d.sum.of.residuals^2 / d.sum.of.residuals * d.sum.of.residuals / d.intercept | ||
| - | = (2 * residual) * d(y - y.hat)/ | ||
| - | = (2 * residual) * d(y - (a + bx)) | ||
| - | = (2 * residual) * d(y - a - bx) | ||
| - | = (2 * residual) * -1 | ||
| - | = -2 * residual | ||
| - | y.hat = a + b * x | + | |
| - | b = slope | + | |
| - | d.sum.of.square.res / d.slope | + | |
| - | = d.sum.of.square.res / d.sum.of.res * d.sum.of.res / d.slope | + | |
| - | = d.sum.of.square.res / d.slope | + | |
| - | = (2 * residual) * d(y - a - bx) | + | |
| - | = (2 * residual) * - x | + | |
| - | = - 2 * x * residual | + | |
chain_rules.1773132077.txt.gz · Last modified: by hkimscil