Week 3: Linear Regression - Part 1 Univariate Linear Regression/单变量线性回归 视觉上来说这是一个趋势。 输入： $$ y=f(x; w_0, w_1)=w_0+w_1x $$ $y$: dependent variable $w_0, w_1$: free parameters $x$: independent variable $m$: slope/斜率 $c$: intercept/截距 $$ m=\frac{\Delta y}{\Delta x} $$ $$ \begin{aligned} f(x+\Delta x) &= m(x+\Delta x)+c \\ &= mx+m\cdot\Delta x+c \\ &= f(x)+m\cdot\Delta x \end{aligned} \Rightarrow \begin{aligned} m &=\frac{f(x+\Delta x)-f(x)}{\Delta x}\\ &=\frac{\Delta y}{\Delta x} \end{aligned} $$ Lost Func./Cost Func. Loss Func. = Cost Func. = Loss = Cost = Error. Func. 其是关于 free param. 的函数。 Square Loss/L2 Loss Loss 表示错误，因此必须是非负数 Mean Square Error (MSE): $$ g(w_0, w_1)= \frac{1}{N} \sum_{n=1}^N{ \left( f(x^{(n)}; w_0, w_1) - y^{(n)} \right)^2 } $$ Summarise: What need to do 给出训练集： $$ (x^{(1)}, y^{(1)}), (x^{(2)}, y^{(2)}), \cdots, (x^{(N)}, y^{(N)}) $$ Fit 模型 $$ y=f(x; w_0, w_1)=w_0+w_1x $$ 最小化 Cost Func. $$ g(w_0, w_1)= \frac{1}{N} \sum_{n=1}^N{ \left( f(x^{(n)}; w_0, w_1) - y^{(n)} \right)^2 } $$
