Additive Models 到 BackFitting

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原文出处：http://zhfuzh.blog.163.com/blog/static/14553938720128995950696/

对原文的一些注释·

这里因为是一个自己选取的smoothing函数（任意指定），但是必须要使的它在样本上的均值为零。也就是该维特征的期望为0。

例子：
Figure 8.1: Estimated (solid lines) versus true additive component functions (circles at the input values)

$\includegraphics[width=1.4\defpicwidth]{SPMdemob1.ps}$

Consider a regression problem with four input variables

. When

is small, it is difficult to obtain a precise nonparametric kernel estimate due to the curse of dimensionality. Let us take a sample of

regression values. We use explanatory variables that are independent and uniformly distributed on

and responses generated from the additive model

$\displaystyle Y= \sum_{\alpha=1}^4 g_\alpha (X_\alpha) + \varepsilon,\quad\varepsilon \sim N(0,1).$

The component functions are chosen as

$\begin{displaymath}\begin{array}{ll} g_1 (X_1) = -\sin (2X_1), &\quad g_2 (X_2)......_3 , &\quad g_4(X_4) = \exp(-X_4)-E\{\exp(-X_4)\}.\end{array}\end{displaymath}$

自己指定的smooth函数

In Figure 8.1 we have plotted the true functions (at the corresponding observations $X_{i\alpha}$ ) and the estimated curves. We used backfitting with univariate local linear smoothers and set the bandwidth to

for each dimension (using the Quartic kernel). We see that even for this small sample size the estimator gives rather precise results.

Next, we turn to a real data example demonstrating the use of additive regression estimators in practice and manifesting that even for a high dimensional data set the backfitting estimator works reasonably well.

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