Differences

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--- chain_rules [2025/08/04 13:38] – hkimscil
+++ chain_rules [2026/03/10 22:11] (current) – [e.g.] hkimscil
@@ Line 5: / Line 5: @@
 y & =& f(g(x)) \\
 \frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx}  \\
-&  & \frac {dy}{dt} = f'(t) = f'(g(x)) \\
+&  & \frac {dy}{dt} = f'(t) = f'(g(x)) \;\; \text{and  } \\
-&  & \because{ \frac {dt}{dx} = g'(x) } \\
+&  & \frac {dt}{dx} = g'(x) \\
-&  & \frac {dy}{dx} = f'(g(x)) * g'(x) \\
+\therefore{ \;\; } \frac {dy}{dx} & = & f'(g(x)) * g'(x) \\
 \end{eqnarray*}
+====== E.g ======
 \begin{eqnarray*}
 y & = & (2x^2 + 1)^2 \\
 t & = & 2x^2 + 1 \\
-y & = & t^2 \\b
+y & = & t^2 \\
 t & = & 2x^2 + 1 \\
-\frac{dy}{dt} & = & 2t \\
+\\
-& = & 2 (2x^2 + 1) \\
+&\phantom{=}\, \frac{dy}{dt} & = 2t \\
-& = & (4x^2 + 2) \\
+&\phantom{=}\, & = 2 (2x^2 + 1) \\
-\frac{dt}{dx} & = & 4x \\
+&\phantom{=}\, & = (4x^2 + 2) \\
-\therefore{}
+\\
+&\phantom{=}\, \frac{dt}{dx} & = 4x \\
+\\
 \frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\
 & = & (4x^2 + 2) * 4x \\
@@ Line 26: / Line 29: @@
 ====== e.g. ======
-y.hat = a + 1 * x
+see [[:gradient descent]]
-a = intercept
+\begin{eqnarray*}
-residuals = (y - y.hat)
+\because{ \;\; } \text{predicted value } \; \hat{y} & = & a + b x \\
-d.sum.of.residuals^2 / d.intercept
+\text{and }\;\;  \text{residual} & = & y - \hat{y} \\
-= d.sum.of.residuals^2 / d.sum.of.residuals * d.sum.of.residuals / d.intercept
+\therefore{} \;\; \text{residual}^2 & = & (y - (a + b x)) \\
-= (2 * residual) *
+\therefore{} \sum{\text{residual}^2} & = & \sum{(y - (a + b x))^2} \\
+& = & \text{SSE,  sum of square residuals} \\
+\\
+\dfrac{\text{dSSE}}{\text{da}} & = &   \\
+\end{eqnarray*}
+intercept, a 에 대한 SSR의 미분은
+\begin{eqnarray*}
+\widehat{y} & = & a + b * x \\
+\text{a} & = & \text{intercept} \\
+\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.a}} \\
+& = & (2 * \text{residual}) * \dfrac{ \text{d.Res}} {\text{d.intercept}} \\
+& = & (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*}