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szcf-weiya committed Feb 27, 2020
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为什么止步于局部线性拟合处?其实我们可以拟合任意阶 $d$ 的局部多项式拟合,
$$
\underset{\alpha(x_0),\beta_j(x_0),j=1,\ldots,d}{\min}\sum\limits_{i=1}^NK_\lambda(x_0,x_i)\Big[y_i-\alpha(x_0)-\sum\limits_{j=1}^d\beta_j(x_0)x_i^j\Big]\tag{6.11}
\underset{\alpha(x_0),\beta_j(x_0),j=1,\ldots,d}{\min}\sum\limits_{i=1}^NK_\lambda(x_0,x_i)\Big[y_i-\alpha(x_0)-\sum\limits_{j=1}^d\beta_j(x_0)x_i^j\Big]^2\tag{6.11}
$$
解为 $\hat f(x_0)=\hat \alpha(x_0)+\sum_{j=1}^d\hat\beta_j(x_0)x_0^j$.实际上,类似 \eqref{6.10} 的表达式将告诉我们偏差仅仅有 $d+1$ 阶和更高阶的组分([练习 6.2](https://github.com/szcf-weiya/ESL-CN/issues/148)).图 6.5 说明了局部二次回归.局部线性拟合趋向于在真实函数的区域中有偏差,这个现象被称作 **截断山坡 (trimming the hills)****填充山谷 (filling the valleys)**.局部二次回归一般可以纠正这个偏差.

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