Differences

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--- chain_rules [2026/03/10 08:45] – [e.g.] hkimscil
+++ chain_rules [2026/03/10 22:11] (current) – [e.g.] hkimscil
@@ Line 40: / Line 40: @@
 \end{eqnarray*}
-For SSR 을 intercept 로 미분하면
+intercept, a 에 대한 SSR의 미분은
 \begin{eqnarray*}
 \widehat{y} & = & a + b * x \\
 \text{a} & = & \text{intercept} \\
 \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{y - (a + b * x)} {\text{d.intercept}} \\
@@ Line 52: / Line 53: @@
 & = & -2 * \text{residual} \\
 & = & -2 * (y - (a + b * x)) \\
+& = & -2 * (y - \widehat{y}) \\
 \end{eqnarray*}
-SSR 을 b를 가지고 미분하면
+slope, b 에 대한 SSR의 미분은
-y.hat = a + b * x
+\begin{eqnarray*}
-b = slope
+\widehat{y} & = & a + b * x \\
-d.sum.of.square.res / d.slope
+\text{b} & = & \text{slope} \\
-= d.sum.of.square.res / d.sum.of.res * d.sum.of.res / d.slope
+\text{residual} & = & (y - \widehat{y}) \\
-= d.sum.of.square.res / d.slope
+\text{SSR} & = & \sum  {\text{residual}^2} = \sum{(y - (a + b x))^2}  \\
-= (2 * residual) * d(y - a - bx)
+\dfrac{\text{d.SSR}}{\text{d.a}} & = &
-= (2 * residual) * - x
+\dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.b}} \\
-= - 2 * x * residual
+& = & (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*}