# 1. LOWESS

$\hat{f(x)} = Ave(y_i | x_i \in N_k(x))$

• 没有考虑到不同距离的邻近点应有不同的权重；
• 拟合的曲线不连续（discontinuous），如下图。

$\hat{f(x_0)} = \frac{ \sum_{i=1}^{N} K_{\lambda}(x_0, x_i)y_i }{\sum_{i=1}^{N} K_{\lambda}(x_0, x_i)}$

$K_{\lambda}(x_0, x_i) = D(\frac{|x_0 - x_i|}{\lambda})$

$D(t) = \left \{ { \matrix { {\frac{3}{4} (1-t^2) } & {for |t| < 1} \cr { 0} & {otherwise} \cr } } \right.$

$\lambda = |x_0 - x_{[k]}|$

$\hat{f(x_0)} = \sum_{j=0}^d \beta_j x_0^{j}$

$\min_{\beta} \sum_{i=1}^N K_{\lambda}(x_0, x_i) [y_i - \sum_{j=0}^d \beta_j x_i^j]^2$

$B = \begin{pmatrix} 1 & x_1 & \cdots & x_1^d \\ 1 & x_2 & \cdots & x_2^d \\ \vdots & \vdots & \ddots & \vdots \\ 1 & x_N & \cdots & x_N^d \\ \end{pmatrix}$

$W_{x_0} = \begin{pmatrix} K_{\lambda}(x_0, x_1) & 0 & \cdots & 0 \\ 0 & K_{\lambda}(x_0, x_2) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & K_{\lambda}(x_0, x_N) \\ \end{pmatrix}$

$\Delta = \begin{pmatrix} \beta_0, \beta_1, \cdots, \beta_d \end{pmatrix}^T$

$Y = \begin{pmatrix} y_1, y_2, \cdots, y_N \end{pmatrix}^T$

$\min_{\Delta} (Y-B\Delta)^T W_{x_0} (Y-B\Delta)$

$\Delta = (B^T W_{x_0} B)^{-1} (B^T W_{x_0} Y)$

\begin{aligned} \hat{f(x_0)} &= e(x_0) (B^T W_{x_0} B)^{-1} (B^T W_{x_0} Y) \\ & = \sum_i w_i (x_0) y_i \end{aligned}

# 2. Robust LOWESS

Robust LOWESS是Cleveland [1] 在LOWESS基础上提出来的robust回归方法，能避免outlier对回归的影响。在计算完估计值后，计算残差：

$e_i = y_i - \hat{f(x_i)}$

$\delta_i = B(e_i/6s)$

$B(u) = \left \{ { \matrix { {(1-u^2)^2 } & {for \quad 0 \le u < 1} \cr { 0 } & {for \quad u \ge 1} \cr } } \right.$

# 3. 参考资料

[1] Trevor Hastie, Robert Tibshirani, Jerome H. Friedman. The elements of statistical learning. Springer, Berlin: Springer series in statistics, 2009.
[2] Cleveland, William S. "Robust locally weighted regression and smoothing scatterplots." Journal of the American statistical association 74.368 (1979): 829-836.
[3] peterf, The Local Polynomial Regression Estimator.

posted @ 2017-08-17 17:26  Treant  阅读(6186)  评论(2编辑  收藏  举报