Differences

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--- chain_rules [2025/08/22 04:04] – hkimscil
+++ chain_rules [2026/03/10 22:11] (current) – [e.g.] hkimscil
@@ Line 31: / Line 31: @@
 see [[:gradient descent]]
 \begin{eqnarray*}
-\hat{y} & = & a + b x \\
+\because{ \;\; } \text{predicted value } \; \hat{y} & = & a + b x \\
-\text{residual} & = & y - \hat{y} \\
+\text{and }\;\;  \text{residual} & = & y - \hat{y} \\
-& = & [y - (a + b x)] \\
+\therefore{} \;\; \text{residual}^2 & = & (y - (a + b x)) \\
+\therefore{} \sum{\text{residual}^2} & = & \sum{(y - (a + b x))^2} \\
+& = & \text{SSE,  sum of square residuals} \\
+\\
+\dfrac{\text{dSSE}}{\text{da}} & = &   \\
 \end{eqnarray*}
-y.hat = a + b * x
+intercept, a 에 대한 SSR의 미분은
-a = intercept
+\begin{eqnarray*}
-residuals = (y - y.hat)
+\widehat{y} & = & a + b * x \\
-d.sum.of.residuals^2 / d.intercept
+\text{a} & = & \text{intercept} \\
-= d.sum.of.residuals^2 / d.sum.of.residuals * d.sum.of.residuals / d.intercept
+\text{residual} & = & (y - \widehat{y}) \\
-= (2 * residual) *  d(y - y.hat)/d.intercept
+\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) *  d(y - a - bx)
+\dfrac{\text{d.SSR}}{\text{d.Res}} * \dfrac{\text{d.Res}}{\text{d.a}} \\
-= (2 * residual) *  -1
+& = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.intercept}} \\
-= -2 * residual
+& = & (2 * \text{residual}) * \dfrac{y - (a + b * x)} {\text{d.intercept}} \\
+& = & 2 * \text{residual} * -1 \\
+& = & -2 * \text{residual} \\
+& = & -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*}
-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