chain_rules
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| chain_rules [2025/08/22 12:33] – [Chain rules] hkimscil | chain_rules [2025/08/22 13:19] (current) – [e.g.] hkimscil | ||
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| Line 11: | Line 11: | ||
| ====== E.g ====== | ====== E.g ====== | ||
| - | |||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| y & = & (2x^2 + 1)^2 \\ | y & = & (2x^2 + 1)^2 \\ | ||
| Line 17: | Line 16: | ||
| y & = & t^2 \\ | y & = & t^2 \\ | ||
| t & = & 2x^2 + 1 \\ | t & = & 2x^2 + 1 \\ | ||
| - | \frac{dy}{dt} & = & 2t \\ | + | \\ |
| - | & = & 2 (2x^2 + 1) \\ | + | & |
| - | & = & (4x^2 + 2) \\ | + | &\phantom{=}\, & = 2 (2x^2 + 1) \\ |
| - | \frac{dt}{dx} & = & 4x \\ | + | &\phantom{=}\, & = (4x^2 + 2) \\ |
| - | \therefore{} | + | \\ |
| + | & | ||
| + | \\ | ||
| \frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\ | \frac{dy}{dx} & = & \frac{dy}{dt}*\frac{dt}{dx} \\ | ||
| & = & (4x^2 + 2) * 4x \\ | & = & (4x^2 + 2) * 4x \\ | ||
| Line 29: | Line 30: | ||
| ====== e.g. ====== | ====== e.g. ====== | ||
| see [[:gradient descent]] | 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, | ||
| + | \\ | ||
| + | \dfrac{\text{dSSE}}{\text{da}} & = & \\ | ||
| + | |||
| + | \end{eqnarray*} | ||
| y.hat = a + b * x | y.hat = a + b * x | ||
chain_rules.1755833622.txt.gz · Last modified: 2025/08/22 12:33 by hkimscil