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<h1>
R语言从入门到放弃 (4). 统计回归
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Part 4 of «R语言从入门到放弃»
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#theory" title="Theory">Theory</a><ul><li><a class="toc-href" href="#linear-model" title="linear model">linear model</a></li><li><a class="toc-href" href="#nadaraya-watson-kernel-smoother" title="Nadaraya-Watson kernel smoother">Nadaraya-Watson kernel smoother</a></li><li><a class="toc-href" href="#local-polynomial-smoother" title="Local Polynomial smoother">Local Polynomial smoother</a></li><li><a class="toc-href" href="#smoothing-spline" title="Smoothing Spline">Smoothing Spline</a></li></ul></li><li><a class="toc-href" href="#fit-model_1" title="fit model">fit model</a><ul><li><a class="toc-href" href="#formula" title="formula">formula</a></li><li><a class="toc-href" href="#fit-models" title="fit models">fit models</a></li></ul></li><li><a class="toc-href" href="#predict-fittedresiduals_1" title="predict, fitted/residuals">predict, fitted/residuals</a></li><li><a class="toc-href" href="#bandwidthdf-hat-matrix" title="bandwidth&df: Hat Matrix">bandwidth&df;: Hat Matrix</a></li><li><a class="toc-href" href="#cv-and-hat-matrix" title="CV and Hat Matrix">CV and Hat Matrix</a></li></ul></div>
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<p>R里面的统计函数有很多, 这里只用线性模型<code>lm</code>以及(一维)非参估计最常用的三个smoother: Nadaraya-Watson kernel(<strong>NW, </strong><code>ksmooth</code>), Local Polynomial(<strong>LP, </strong><code>loess</code>), Smoothing Spline(<strong>SS, </strong><code>smooth.spline</code>). 用这三个smoother作为例子, 介绍R里面统计回归的一些用法. </p>
<p>数据的形式是: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image002.png"/></p>
<p>目标是估计函数m(). 例子使用R自带的<code>cars</code>数据集, 它包含两列: 汽车速度speed和刹车距离dist. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">> data(cars)</span></span>
<span class="code-line"><span class="err">> summary(cars)</span></span>
<span class="code-line"><span class="err"> speed dist </span></span>
<span class="code-line"><span class="err"> Min. : 4.0 Min. : 2.00 </span></span>
<span class="code-line"><span class="err"> 1st Qu.:12.0 1st Qu.: 26.00 </span></span>
<span class="code-line"><span class="err"> Median :15.0 Median : 36.00 </span></span>
<span class="code-line"><span class="err"> Mean :15.4 Mean : 42.98 </span></span>
<span class="code-line"><span class="err"> 3rd Qu.:19.0 3rd Qu.: 56.00 </span></span>
<span class="code-line"><span class="err"> Max. :25.0 Max. :120.00 </span></span>
<span class="code-line"><span class="err">> ?cars</span></span>
<span class="code-line"><span class="err">> plot(cars$speed, cars$dist)</span></span>
</pre></div>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image003.png"/></p>
<h1 id="theory">Theory</h1>
<p>首先简单介绍一下这4个smoother的原理: </p>
<h3 id="linear-model">linear model</h3>
<p>认为m是线性形式(包含intercept): </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image008.png"/></p>
<h3 id="nadaraya-watson-kernel-smoother">Nadaraya-Watson kernel smoother</h3>
<p>m_NW 在x处的取值为Yi的加权平均, 权重是按照kernel K()确定的. </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image.png"/></p>
<p>另外m_NW(x)还可以看做是最小化加权的square-error: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image001.png"/></p>
<h3 id="local-polynomial-smoother">Local Polynomial smoother</h3>
<p>m_NW(x)最小化加权sq-err那个表达式里, 可以是用一个<em>常数函数</em>mx来估计在x处的取值, LP将它泛化为p-1阶多项式的形式, m在x附近是多项式形式. m(u)=poly(x-u), 这个多项式的系数为beta(x): </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image004.png"/></p>
<p>最后m_LP在x处的取值为: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image005.png"/></p>
<h3 id="smoothing-spline">Smoothing Spline</h3>
<p>设定m的形式为knot在xi的spline, 加上penalize项: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image006.png"/></p>
<p>另外SS可以看作是bandwidth随x变化的kernel smoother. </p>
<h1 id="fit-model_1">fit model</h1>
<h3 id="formula">formula</h3>
<p><code>lm</code>和<code>loess</code>的文档里都提到参数为formula, 它大概是指示要fit的表达式形式. 这里面的加减号不是算数意义上的加减. 看例子: </p>
<ul>
<li><code>dist ~ speed</code>: 表示dist是speed的函数</li>
<li><code>y ~ .</code> : 表示y是所有其他变量的函数</li>
<li><code>y ~ x1+x2</code>: y 是x1和x2的函数</li>
<li><code>y ~ x - 1</code>: y是x的函数, 且没有intercept项</li>
</ul>
<h3 id="fit-models">fit models</h3>
<p>这几个函数的fit写法各不相同, 有的要提供formula, 有的要提供x和y值, 需要看文档: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">fit.lm</span> <span class="o"><-</span> <span class="nf">lm</span><span class="p">(</span><span class="n">dist</span><span class="o">~</span><span class="n">speed</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">cars</span><span class="p">)</span></span>
<span class="code-line"><span class="n">fit.nw</span> <span class="o"><-</span> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">)</span></span>
<span class="code-line"><span class="n">fit.lp</span> <span class="o"><-</span> <span class="nf">loess</span><span class="p">(</span><span class="n">dist</span><span class="o">~</span><span class="n">speed</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">cars</span><span class="p">)</span></span>
<span class="code-line"><span class="n">fit.ss</span> <span class="o"><-</span> <span class="nf">smooth.spline</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">)</span></span>
</pre></div>
<h1 id="predict-fittedresiduals_1">predict, fitted/residuals</h1>
<p>predict的写法也是各不相同, 一般都是用<code>predict</code>函数, 然而这个函数在作用到不同smoother上面, 参数和返回值也都不一样......orz 关于<code>xx</code>smoother的predict函数用法参考?<code>predict.xx</code>. 最奇葩的是NW, 它不能用<code>predict</code>函数, 而要fit的时候在<code>skmooth</code>函数里传入<code>x.points</code>参数... </p>
<p>看predict例子: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">newspeed</span> <span class="o"><-</span> <span class="m">1</span><span class="o">:</span><span class="m">26</span></span>
<span class="code-line"><span class="n">pred.lm</span> <span class="o"><-</span> <span class="nf">predict</span><span class="p">(</span><span class="n">fit.lm</span><span class="p">,</span> <span class="n">newdata</span> <span class="o">=</span> <span class="nf">data.frame</span><span class="p">(</span><span class="n">speed</span> <span class="o">=</span> <span class="n">newspeed</span><span class="p">),</span> <span class="n">interval</span> <span class="o">=</span> <span class="s">"prediction"</span><span class="p">)</span><span class="n">[</span><span class="p">,</span><span class="s">"fit"</span><span class="n">]</span></span>
<span class="code-line"><span class="n">pred.nw</span> <span class="o"><-</span> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">,</span> <span class="n">bandwidth</span><span class="o">=</span><span class="m">2</span><span class="p">,</span> <span class="n">x.points</span><span class="o">=</span><span class="n">newspeed</span><span class="p">)</span><span class="o">$</span><span class="n">y</span></span>
<span class="code-line"><span class="n">pred.lp</span> <span class="o"><-</span> <span class="nf">predict</span><span class="p">(</span><span class="n">fit.lp</span><span class="p">,</span> <span class="n">newdata</span><span class="o">=</span><span class="n">newspeed</span><span class="p">)</span></span>
<span class="code-line"><span class="n">pred.ss</span> <span class="o"><-</span> <span class="nf">predict</span><span class="p">(</span><span class="n">fit.ss</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="n">newspeed</span><span class="p">)</span><span class="o">$</span><span class="n">y</span></span>
</pre></div>
<p>另外, 如果想看smoother在design points(Xi)处的预测值, 可以用<code>fitted</code>函数(NW还是不能用), 例子: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">fitted.lm</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="n">fit.lm</span><span class="p">)</span></span>
<span class="code-line"><span class="n">fitted.nw</span> <span class="o"><-</span> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">,</span> <span class="n">bandwidth</span><span class="o">=</span><span class="m">2</span><span class="p">,</span> <span class="n">x.points</span><span class="o">=</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">)</span><span class="o">$</span><span class="n">y</span></span>
<span class="code-line"><span class="n">fitted.lp</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="n">fit.lp</span><span class="p">)</span></span>
<span class="code-line"><span class="n">fitted.ss</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="n">fit.ss</span><span class="p">)</span></span>
</pre></div>
<p>要看每个点的residual ri=yi-yhat_i, 用<code>residuals</code>函数(NW不行):</p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">resd.lm</span> <span class="o"><-</span> <span class="nf">residuals</span><span class="p">(</span><span class="n">fit.lm</span><span class="p">)</span></span>
<span class="code-line"><span class="n">resd.nw</span> <span class="o"><-</span> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">,</span> <span class="n">bandwidth</span><span class="o">=</span><span class="m">2</span><span class="p">,</span> <span class="n">x.points</span><span class="o">=</span><span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">)</span><span class="o">$</span><span class="n">y</span> <span class="o">-</span> <span class="n">cars</span><span class="o">$</span><span class="n">dist</span> </span>
<span class="code-line"><span class="n">resd.lp</span> <span class="o"><-</span> <span class="nf">residuals</span><span class="p">(</span><span class="n">fit.lp</span><span class="p">)</span></span>
<span class="code-line"><span class="n">resd.ss</span> <span class="o"><-</span> <span class="nf">residuals</span><span class="p">(</span><span class="n">fit.ss</span><span class="p">)</span></span>
</pre></div>
<p>可以画出这几个方法的fit: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">abline(fit.lm, lty=1, col=1) # linear model is just a straigh line</span></span>
<span class="code-line"><span class="err">lines(newspeed, pred.nw, lty=2, col=2)</span></span>
<span class="code-line"><span class="err">lines(newspeed, pred.lp, lty=3, col=3)</span></span>
<span class="code-line"><span class="err">lines(newspeed, pred.ss, lty=4, col=4)</span></span>
<span class="code-line"><span class="err">legend("topleft", c("lm", "nw", "lp", "ss"), lty=1:4, col=1:4)</span></span>
</pre></div>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image009.png"/></p>
<h1 id="bandwidthdf-hat-matrix">bandwidth&df: Hat Matrix</h1>
<p>三个非参估计的smoother都有"带宽"(bandwidth)或者"自由度"(df)的概念, 带宽即NW或LP表达式里的h. </p>
<p>自由度df是带宽的函数, smoother的df可以用它的<strong>hat matrix</strong> S计算出来. </p>
<p>一个smoother的hat matrix S, 是把训练值Y映射到估计值Yhat的矩阵: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image010.png"/></p>
<p>而df则是S的迹: df = tr(S). df的</p>
<p>根据script(P28), S的第j列可以用这个smoother fit一个unit vector来得到: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image012.png"/></p>
<p>所以计算S可以用下面的代码: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">n</span> <span class="o"><-</span> <span class="nf">nrow</span><span class="p">(</span><span class="n">cars</span><span class="p">)</span></span>
<span class="code-line"><span class="n">Snw</span> <span class="o"><-</span> <span class="n">Slp</span> <span class="o"><-</span> <span class="n">Sss</span> <span class="o"><-</span> <span class="nf">matrix</span><span class="p">(</span><span class="m">0</span><span class="p">,</span> <span class="n">nrow</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">ncol</span><span class="o">=</span><span class="n">n</span><span class="p">)</span></span>
<span class="code-line"><span class="n">In</span> <span class="o"><-</span> <span class="nf">diag</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="c1"># identity matrix</span></span>
<span class="code-line"><span class="nf">for</span><span class="p">(</span><span class="n">j</span> <span class="n">in</span> <span class="m">1</span><span class="o">:</span><span class="n">n</span><span class="p">){</span></span>
<span class="code-line"> <span class="n">y</span> <span class="o"><-</span> <span class="n">In[</span><span class="p">,</span><span class="n">j]</span> <span class="c1"># unit vector ej</span></span>
<span class="code-line"> <span class="n">Snw[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">,</span> <span class="n">bandwidth</span> <span class="o">=</span> <span class="m">2</span><span class="p">,</span> <span class="n">x.points</span><span class="o">=</span><span class="n">x</span><span class="p">)</span><span class="o">$</span><span class="n">y</span></span>
<span class="code-line"> <span class="n">Slp[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="nf">loess</span><span class="p">(</span><span class="n">y</span><span class="o">~</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">))</span></span>
<span class="code-line"> <span class="n">Sss[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="nf">smooth.spline</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span></span>
<span class="code-line"><span class="p">}</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="n">df.nw</span> <span class="o"><-</span> <span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Snw</span><span class="p">))</span></span>
<span class="code-line"><span class="n">df.lp</span> <span class="o"><-</span> <span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Slp</span><span class="p">))</span></span>
<span class="code-line"><span class="n">df.ss</span> <span class="o"><-</span> <span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Sss</span><span class="p">))</span></span>
</pre></div>
<p>发现三个非参smoother的自由度不同, 所以上面画图的比较并没有意义, 为了让三者的自由度相同, 可以设定ksmooth/loess/smooth.spline的参数.</p>
<p>控制带宽, Lp的参数为<code>span</code>, SS的参数为<code>spar</code>; 而指定想要的自由度则分别是<code>enp.target</code>和<code>df</code>. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="nf">cat</span><span class="p">(</span><span class="s">"let all 3 np smoother use the same df="</span><span class="p">,</span> <span class="n">df.nw</span><span class="p">)</span></span>
<span class="code-line"><span class="n">Slp</span> <span class="o"><-</span> <span class="n">Sss</span> <span class="o"><-</span> <span class="nf">matrix</span><span class="p">(</span><span class="m">0</span><span class="p">,</span> <span class="n">nrow</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">ncol</span><span class="o">=</span><span class="n">n</span><span class="p">)</span></span>
<span class="code-line"><span class="n">In</span> <span class="o"><-</span> <span class="nf">diag</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="c1"># identity matrix</span></span>
<span class="code-line"><span class="nf">for</span><span class="p">(</span><span class="n">j</span> <span class="n">in</span> <span class="m">1</span><span class="o">:</span><span class="n">n</span><span class="p">){</span></span>
<span class="code-line"> <span class="n">y</span> <span class="o"><-</span> <span class="n">In[</span><span class="p">,</span><span class="n">j]</span></span>
<span class="code-line"> <span class="n">Slp[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="nf">loess</span><span class="p">(</span><span class="n">y</span><span class="o">~</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">enp.target</span><span class="o">=</span><span class="n">df.nw</span><span class="p">))</span></span>
<span class="code-line"> <span class="n">Sss[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">fitted</span><span class="p">(</span><span class="nf">smooth.spline</span><span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">df</span><span class="o">=</span><span class="n">df.nw</span><span class="p">))</span></span>
<span class="code-line"><span class="p">}</span></span>
<span class="code-line"><span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Slp</span><span class="p">))</span></span>
<span class="code-line"><span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Sss</span><span class="p">))</span></span>
</pre></div>
<p>发现SS的df参数使用以后控制的非常接近NW的df了, 不过lp的df还是不够接近, 用span来控制应该更准确一些, 为了找到合适的span, 用以下代码来寻找使得df=df.nw的span取值: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">dflp</span> <span class="o"><-</span> <span class="nf">function</span><span class="p">(</span><span class="n">span</span><span class="p">,</span> <span class="n">val</span><span class="p">){</span></span>
<span class="code-line"> <span class="nf">for</span><span class="p">(</span><span class="n">j</span> <span class="n">in</span> <span class="m">1</span><span class="o">:</span><span class="n">n</span><span class="p">)</span></span>
<span class="code-line"> <span class="n">Slp[</span><span class="p">,</span><span class="n">j]</span> <span class="o"><-</span> <span class="nf">loess</span><span class="p">(</span><span class="n">In[</span><span class="p">,</span><span class="n">j]</span> <span class="o">~</span> <span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">span</span> <span class="o">=</span> <span class="n">span</span><span class="p">)</span><span class="o">$</span><span class="n">fitted</span></span>
<span class="code-line"> <span class="nf">return</span><span class="p">(</span><span class="nf">sum</span><span class="p">(</span><span class="nf">diag</span><span class="p">(</span><span class="n">Slp</span><span class="p">))</span> <span class="o">-</span> <span class="n">val</span><span class="p">)</span></span>
<span class="code-line"><span class="p">}</span></span>
<span class="code-line"><span class="n">chosen_span</span> <span class="o"><-</span> <span class="nf">uniroot</span><span class="p">(</span><span class="n">dflp</span><span class="p">,</span> <span class="nf">c</span><span class="p">(</span><span class="m">0.2</span><span class="p">,</span> <span class="m">0.5</span><span class="p">),</span> <span class="n">val</span> <span class="o">=</span> <span class="n">df.nw</span><span class="p">)</span><span class="o">$</span><span class="n">root</span></span>
</pre></div>
<p>如果不用这个循环计算的话, 可以用<code>sfsmisc</code>包里的<code>hatMat</code>函数: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image013.png"/> </p>
<p>参数<code>trace</code>取TRUE的话, 直接返回hat matrix的迹, 否则返回整个hat matrix.
需要把要计算的smoother包装成一个pred.sm函数传入, 这个函数接受x和y, 返回fitted数值. 例子: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">hatMat(cars$speed, T, </span></span>
<span class="code-line"><span class="err"> pred.sm = function(x,y) </span></span>
<span class="code-line"><span class="err"> ksmooth(x, y, kernel="normal", bandwidth=2, x.points=x)$y )</span></span>
<span class="code-line"><span class="err">hatMat(cars$speed, T, </span></span>
<span class="code-line"><span class="err"> pred.sm = function(x,y) fitted(loess(y~x, span=chosen_span)) )</span></span>
<span class="code-line"><span class="err">hatMat(cars$speed, T, </span></span>
<span class="code-line"><span class="err"> pred.sm = function(x,y) fitted(smooth.spline(x, y, df=df.nw)) )</span></span>
</pre></div>
<h1 id="cv-and-hat-matrix">CV and Hat Matrix</h1>
<p>为了预测smoother的performance, 用loo CV来估计MSE(mean sq err)的值. </p>
<p>loo CV可以用下面这个通用函数得到(注意看对于参数的要求): </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">##</span><span class="s1">' Calculates the LOO CV score for given data and regression prediction function</span></span>
<span class="code-line"><span class="s1">##'</span><span class="w"></span></span>
<span class="code-line"><span class="err">##</span><span class="s1">' @param reg.data: regression data; data.frame with columns '</span><span class="n">x</span><span class="s1">', '</span><span class="n">y</span><span class="s1">'</span></span>
<span class="code-line"><span class="s1">##'</span><span class="w"> </span><span class="nv">@param</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="nl">fcn</span><span class="p">:</span><span class="w"> </span><span class="n">regr</span><span class="p">.</span><span class="n">prediction</span><span class="w"> </span><span class="k">function</span><span class="p">;</span><span class="w"> </span><span class="nl">arguments</span><span class="p">:</span><span class="w"></span></span>
<span class="code-line"><span class="err">##</span><span class="s1">' reg.x: regression x-values</span></span>
<span class="code-line"><span class="s1">##'</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="nl">y</span><span class="p">:</span><span class="w"> </span><span class="n">regression</span><span class="w"> </span><span class="n">y</span><span class="o">-</span><span class="k">values</span><span class="w"></span></span>
<span class="code-line"><span class="err">##</span><span class="s1">' x: x-value(s) of evaluation point(s)</span></span>
<span class="code-line"><span class="s1">##'</span><span class="w"> </span><span class="k">value</span><span class="err">:</span><span class="w"> </span><span class="n">prediction</span><span class="w"> </span><span class="k">at</span><span class="w"> </span><span class="n">point</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="n">x</span><span class="w"></span></span>
<span class="code-line"><span class="err">##'</span><span class="w"> </span><span class="nv">@return</span><span class="w"> </span><span class="n">LOOCV</span><span class="w"> </span><span class="n">score</span><span class="w"></span></span>
<span class="code-line"><span class="n">loocv</span><span class="w"> </span><span class="o"><-</span><span class="w"> </span><span class="k">function</span><span class="p">(</span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="n">fcn</span><span class="p">)</span><span class="err">{</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="err">##</span><span class="w"> </span><span class="n">Help</span><span class="w"> </span><span class="k">function</span><span class="w"> </span><span class="k">to</span><span class="w"> </span><span class="n">calculate</span><span class="w"> </span><span class="n">leave</span><span class="o">-</span><span class="n">one</span><span class="o">-</span><span class="k">out</span><span class="w"> </span><span class="n">regression</span><span class="w"> </span><span class="k">values</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">loo</span><span class="p">.</span><span class="n">reg</span><span class="p">.</span><span class="k">value</span><span class="w"> </span><span class="o"><-</span><span class="w"> </span><span class="k">function</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="n">fcn</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="p">(</span><span class="n">reg</span><span class="p">.</span><span class="n">fcn</span><span class="p">(</span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="err">$</span><span class="n">x</span><span class="o">[</span><span class="n">-i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="err">$</span><span class="n">y</span><span class="o">[</span><span class="n">-i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="err">$</span><span class="n">x</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">))</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="err">##</span><span class="w"> </span><span class="n">Calculate</span><span class="w"> </span><span class="n">LOO</span><span class="w"> </span><span class="n">regression</span><span class="w"> </span><span class="k">values</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="n">the</span><span class="w"> </span><span class="n">help</span><span class="w"> </span><span class="k">function</span><span class="w"> </span><span class="n">above</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o"><-</span><span class="w"> </span><span class="n">nrow</span><span class="p">(</span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="n">loo</span><span class="p">.</span><span class="k">values</span><span class="w"> </span><span class="o"><-</span><span class="w"> </span><span class="n">sapply</span><span class="p">(</span><span class="mi">1</span><span class="err">:</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">loo</span><span class="p">.</span><span class="n">reg</span><span class="p">.</span><span class="k">value</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="p">,</span><span class="w"> </span><span class="n">reg</span><span class="p">.</span><span class="n">fcn</span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="w"> </span><span class="err">##</span><span class="w"> </span><span class="n">Calculate</span><span class="w"> </span><span class="ow">and</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">MSE</span><span class="w"></span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="p">(</span><span class="w"> </span><span class="n">mean</span><span class="p">(</span><span class="w"> </span><span class="p">(</span><span class="n">loo</span><span class="p">.</span><span class="k">values</span><span class="o">-</span><span class="n">reg</span><span class="p">.</span><span class="k">data</span><span class="err">$</span><span class="n">y</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">)</span><span class="w"></span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p>比如, 为了计算NW的CV数值, 需要这样: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">regfcn.nw</span> <span class="o"><-</span> <span class="nf">function</span><span class="p">(</span><span class="n">regx</span><span class="p">,</span> <span class="n">regy</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span></span>
<span class="code-line"> <span class="nf">ksmooth</span><span class="p">(</span><span class="n">regx</span><span class="p">,</span> <span class="n">regy</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s">"normal"</span><span class="p">,</span> <span class="n">bandwidth</span><span class="o">=</span><span class="m">2</span><span class="p">,</span> <span class="n">x.points</span><span class="o">=</span><span class="n">x</span><span class="p">)</span><span class="o">$</span><span class="n">y</span></span>
<span class="code-line"><span class="nf">loocv</span><span class="p">(</span><span class="nf">data.frame</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">cars</span><span class="o">$</span><span class="n">speed</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="n">cars</span><span class="o">$</span><span class="n">dist</span><span class="p">),</span> <span class="n">regfcn.nw</span><span class="p">)</span></span>
</pre></div>
<p>不过, 如果得到了hat Matrix S, 根据公式4.5, loo CV可以这样一次计算出来: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image014.png"/></p>
<p>试一下: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="c1"># compute CV using shortcut euqation</span></span>
<span class="code-line"><span class="n">yhat.nw</span> <span class="o"><-</span> <span class="n">Snw</span> <span class="o">%*%</span> <span class="n">cars</span><span class="o">$</span><span class="n">y</span> <span class="c1"># or use regfcn.nw(cars$speed, cars$dist, cars$speed)</span></span>
<span class="code-line"><span class="nf">mean</span><span class="p">(</span> <span class="p">(</span> <span class="p">(</span><span class="n">cars</span><span class="o">$</span><span class="n">y</span><span class="o">-</span><span class="n">yhat.nw</span><span class="p">)</span><span class="o">/</span> <span class="p">(</span><span class="m">1</span><span class="o">-</span><span class="nf">diag</span><span class="p">(</span><span class="n">Snw</span><span class="p">))</span> <span class="p">)</span><span class="n">^2</span> <span class="p">)</span></span>
</pre></div>
<p>得到的结果和之前用loocv一样, 都是253.9128 !~ </p>
<p>或者只用df, 计算generalized CV, 公式为: </p>
<p><img alt="" class="img-responsive" src="../images/Rnotes-4-regression/pasted_image015.png"/></p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">#compute GCV</span></span>
<span class="code-line"><span class="err">mean( (cars$dist-yhat.nw)^2 ) / ( 1 - df.nw/n )^2</span></span>
</pre></div>
<p>得到gcv=269.3911, 和looCV也比较接近. </p>
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