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Table of Contents
Chain rules
\begin{eqnarray*} y & = & f(t) \\ t & = & g(x) \\ y & =& f(g(x)) \\ \frac {dy}{dx} & = & \frac {dy}{dt} * \frac {dt}{dx} \\ & & \frac {dy}{dt} = f'(t) = f'(g(x)) \;\; \text{and } \\ & & \frac {dt}{dx} = 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 \\ t & = & 2x^2 + 1 \\ \\ &\phantom{=}\, \frac{dy}{dt} & = 2t \\ &\phantom{=}\, & = 2 (2x^2 + 1) \\ &\phantom{=}\, & = (4x^2 + 2) \\ \\ &\phantom{=}\, \frac{dt}{dx} & = 4x \\ \\ \frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\ & = & (4x^2 + 2) * 4x \\ & = & 16x^3 + 8x \\ \end{eqnarray*}
e.g.
see gradient descent
\begin{eqnarray*}
\because{ \;\; } \text{predicted value } \; \hat{y} & = & a + b x \\
\text{and }\;\; \text{residual} & = & y - \hat{y} \\
\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*}
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*}