2016/12/21 局部回归中不同的窗宽结果 顿武变不重要了 如果目的是估计m,则当P一 3.稳健回归LOWESS Step1:Defining the window width locally weighted scatterplot smoothe 基本思想： e 16 ne 16，r MAD -{sMD-(i-mdn( the welgt -时2016/12/21 3 与参数模型不同，局部多项式估计拟合的复杂性是 由带宽来控制的, 通常 是较小的，故而选择 的问 题就变得不重要了．如果目的是估计 ，则当 是奇数，局部多项式拟合自动修正边界偏倚．进一 步，则当 是奇数，与 阶拟合相比较， 阶 拟合包含了一个多余常数，但没有增加 估计的 方差。不过这个参数创造了一个降低偏倚的机会， 特别是在边界区域．另一方面，带宽 的选择在多 项式拟合中起着重要作用．太大的带宽引起过渡平 滑，产生过大的建模偏倚，而太小的带宽会导致不 足平滑，获得受干扰的估计。 p p v m p v  p v  p 1 p v m h h 局部回归中不同的窗宽结果 14 3.稳健回归LOWESS locally weighted scatterplot smoother • 基本思想： 局部线性估计 稳健的权重平滑 （残差大的减小权重） 15 MAD=median(|ri-median(ri)|) MAD 16 #Step1 #Defining the window width plot(TIME, LIBERAL, xlab="Time (in days)", ylab="Liberal Support", type='n', main="Defining the Window Width") ord <- order(TIME) Lib=LIBERAL time <- TIME[ord] pre <- LIBERAL[ord] x0 <- time[10] diffs <- abs(time - x0) which.diff <- sort(diffs)[16] abline(v=c(x0-which.diff, x0+which.diff), lty=2) abline(v=x0) points(time[diffs > which.diff], Lib[diffs > which.diff], pch=16, cex=2, col=gray(.75)) points(time[diffs <= which.diff], Lib[diffs <= which.diff],cex=2) x.n <- time[diffs <= which.diff] y.n <- Lib[diffs <= which.diff] text(locator(1), "Window Width") 17 #Step 2 #Applying the Tricube Weight #Tricube function tricube <- function(z) { ifelse (abs(z) < 1, (1 - (abs(z))^3)^3, 0) } #Bisquare weight bisquare <- function(z) { ifelse (abs(z) < 1, (1 - (abs(z))^2)^2, 0) } plot(range(TIME), c(0,1), xlab="Time (in days)", ylab="Tricube Weight", type='n', main="The Tricube Weight") abline(v=c(x0-which.diff, x0+which.diff), lty=2) abline(v=x0) xwts <- seq(x0-which.diff, x0+which.diff, len=250) lines(xwts, tricube((xwts-x0)/which.diff), lty=1, lwd=2) points(x.n, tricube((x.n - x0)/which.diff), cex=2) #Step 3 #The local polynomial plot(TIME, LIBERAL, xlab="Time (in days)", ylab="Liberal Support", type='n', main="Local Linear Regression") abline(v=c(x0-which.diff, x0+which.diff), lty=2) abline(v=x0) points(x.n, y.n, cex=2) mod <- lm(y.n ~ x.n, weights=tricube((x.n-x0)/which.diff)) reg.line(mod, lwd=2, col=1) points(x0, predict(mod, data.frame(x.n=x0)), pch=16, cex=1.8) text(locator(1), "Fitted Value of Y at Focal X") 18