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type="math/tex" id="MathJax-Element-61">x_{i1}, \ldots, x_{id}</script>是第<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-62-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -682.0516853480245 345.5 714.103370696049" style="width: 0.811ex; height: 1.622ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use></g></svg></span><script type="math/tex" id="MathJax-Element-62">i</script>个样本<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-63-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -715.0516853480245 523.5 746.103370696049" style="width: 1.158ex; height: 1.737ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-64"></use></g></svg></span><script type="math/tex" id="MathJax-Element-63">d</script>个属性上的取值，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-64-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 834.8053928999522 690.103370696049" style="width: 1.969ex; height: 1.622ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-79"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="693" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-64">y_i</script>是该样本待预测的目标。线性回归模型假设目标<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-65-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 834.8053928999522 690.103370696049" style="width: 1.969ex; height: 1.622ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-79"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="693" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-65">y_i</script>可以被属性间的线性组合描述，即</p><div class="md-section-divider"></div><p data-anchor-id="76r1"><span class="MathJax_Preview"></span><div class="MathJax_SVG_Display" role="textbox" aria-readonly="true" 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transform="translate(3245,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-78"></use><g transform="translate(572,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31" x="345" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2B" x="4738" y="0"></use><g transform="translate(5738,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-3C9"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-32" x="880" y="-213"></use></g><g transform="translate(6815,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-78"></use><g transform="translate(572,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-32" x="345" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2B" x="8308" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2026" x="9308" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2B" x="10703" y="0"></use><g transform="translate(11704,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-3C9"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-64" x="880" y="-213"></use></g><g transform="translate(12797,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-78"></use><g transform="translate(572,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-64" x="345" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2B" x="14306" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-62" x="15307" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="15736" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="16181" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-3D" x="16804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31" x="17861" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="18361" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2026" x="18806" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="20146" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6E" x="20591" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-66">y_i = \omega_1x_{i1} + \omega_2x_{i2} + \ldots + \omega_dx_{id} + b, i=1,\ldots,n</script></p><p data-anchor-id="xieq">例如，在我们将要建模的房价预测问题里，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -463.0516853480245 1208.486940139403 778.0602608392912" style="width: 2.78ex; height: 1.853ex; vertical-align: -0.811ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-78"></use><g transform="translate(572,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6A" x="345" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-67">x_{ij}</script>是描述房子<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-68-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -682.0516853480245 345.5 714.103370696049" style="width: 0.811ex; height: 1.622ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use></g></svg></span><script type="math/tex" id="MathJax-Element-68">i</script>的各种属性（比如房间的个数、周围学校和医院的个数、交通状况等），而 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-69-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 834.8053928999522 690.103370696049" style="width: 1.969ex; height: 1.622ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-79"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="693" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-69">y_i</script>是房屋的价格。</p><p data-anchor-id="xdaz">初看起来，这个假设实在过于简单了，变量间的真实关系很难是线性的。但由于线性回归模型有形式简单和易于建模分析的优点，它在实际问题中得到了大量的应用。很多经典的统计学习、机器学习书籍[<a href="#参考文献">2,3,4</a>]也选择对线性模型独立成章重点讲解。</p><div class="md-section-divider"></div><h2 data-anchor-id="a18j" id="效果展示">效果展示</h2><p data-anchor-id="f50x">我们使用从<a href="https://archive.ics.uci.edu/ml/datasets/Housing" target="_blank">UCI Housing Data Set</a>获得的波士顿房价数据集进行模型的训练和预测。下面的散点图展示了使用模型对部分房屋价格进行的预测。其中，每个点的横坐标表示同一类房屋真实价格的中位数，纵坐标表示线性回归模型根据特征预测的结果，当二者值完全相等的时候就会落在虚线上。所以模型预测得越准确，则点离虚线越近。</p><p align="center" data-anchor-id="4jfr"> <img src="https://raw.githubusercontent.com/PaddlePaddle/book/develop/fit_a_line/image/predictions.png" width="400"><br> 图1. 预测值 V.S. 真实值 </p><div class="md-section-divider"></div><h2 data-anchor-id="739z" id="模型概览">模型概览</h2><div class="md-section-divider"></div><h3 data-anchor-id="tzb9" id="模型定义">模型定义</h3><p data-anchor-id="0i1k">在波士顿房价数据集中，和房屋相关的值共有14个：前13个用来描述房屋相关的各种信息，即模型中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -463.0516853480245 916.8053928999522 641.5886520702876" style="width: 2.085ex; height: 1.506ex; vertical-align: -0.463ex; margin: 1px 0px;"><g stroke="black" 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transform="translate(828,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-33" x="500" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2B" x="14945" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-62" x="15946" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-72">\hat{Y} = \omega_1X_{1} + \omega_2X_{2} + \ldots + \omega_{13}X_{13} + b</script></p><p data-anchor-id="ch16"><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -943.8583648847041 763.5 963.9100502327285" style="width: 1.737ex; height: 2.201ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-5E" x="249" y="228"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">\hat{Y}</script> 表示模型的预测结果，用来和真实值<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-74-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -704.0516853480245 763.5 724.103370696049" style="width: 1.737ex; height: 1.737ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use></g></svg></span><script type="math/tex" id="MathJax-Element-74">Y</script>区分。模型要学习的参数即：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-75-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -715.0516853480245 5610.887498618268 931.103370696049" style="width: 13.089ex; height: 2.201ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-3C9"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31" x="880" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="1076" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2026" x="1521" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="2860" y="0"></use><g transform="translate(3305,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-3C9"></use><g transform="translate(622,-150)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-33" x="500" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2C" x="4736" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-62" x="5181" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-75">\omega_1, \ldots, \omega_{13}, b</script>。</p><p data-anchor-id="9qug">建立模型后，我们需要给模型一个优化目标，使得学到的参数能够让预测值<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-76-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -943.8583648847041 763.5 963.9100502327285" style="width: 1.737ex; height: 2.201ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-5E" x="249" y="228"></use></g></svg></span><script type="math/tex" id="MathJax-Element-76">\hat{Y}</script>尽可能地接近真实值<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-77-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -704.0516853480245 763.5 724.103370696049" style="width: 1.737ex; height: 1.737ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use></g></svg></span><script type="math/tex" id="MathJax-Element-77">Y</script>。这里我们引入损失函数（<a href="https://en.wikipedia.org/wiki/Loss_function" target="_blank">Loss Function</a>，或Cost Function）这个概念。 输入任意一个数据样本的目标值<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-78-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 834.8053928999522 690.103370696049" style="width: 1.969ex; height: 1.622ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-79"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="693" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-78">y_{i}</script>和模型给出的预测值<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-79-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -703.8583648847041 834.8053928999522 929.9100502327285" style="width: 1.969ex; height: 2.201ex; vertical-align: -0.579ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-79"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="693" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-5E" x="167" y="-12"></use></g></svg></span><script type="math/tex" id="MathJax-Element-79">\hat{y_{i}}</script>，损失函数输出一个非负的实值。这个实质通常用来反映模型误差的大小。</p><p data-anchor-id="e3lc">对于线性回归模型来讲，最常见的损失函数就是均方误差（Mean Squared Error， <a href="https://en.wikipedia.org/wiki/Mean_squared_error" target="_blank">MSE</a>）了，它的形式是：</p><div class="md-section-divider"></div><p data-anchor-id="zib5"><span class="MathJax_Preview"></span><div class="MathJax_SVG_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-80-Frame" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -1585.0070961948516 10841.351063117105 2802.850285968542" style="width: 25.135ex; height: 6.486ex; vertical-align: -2.896ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-4D"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-53" x="1051" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-45" x="1697" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-3D" x="2739" y="0"></use><g transform="translate(3915,0)"><rect stroke="none" width="720" height="60" x="0" y="220"></rect><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31" x="110" y="676"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6E" x="60" y="-686"></use></g><g transform="translate(4922,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJSZ2-2211"></use><g transform="translate(147,-1090)"><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-31" x="1124" y="0"></use></g><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6E" x="721" y="1627"></use></g><g transform="translate(6533,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-28"></use><g transform="translate(389,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="822" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-5E" x="212" y="228"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-2212" x="1537" y="0"></use><g transform="translate(2538,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-59"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="822" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-29" x="3464" y="0"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-32" x="5449" y="920"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-80">MSE=\frac{1}{n}\sum_{i=1}^{n}{(\hat{Y_i}-Y_i)}^2</script></p><p data-anchor-id="etto">即对于一个大小为<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-81-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 600.5 496.10337069604896" style="width: 1.39ex; height: 1.158ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6E"></use></g></svg></span><script type="math/tex" id="MathJax-Element-81">n</script>的测试集，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-82-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -726.0516853480245 2461.5 769.103370696049" style="width: 5.676ex; height: 1.737ex; vertical-align: -0.232ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-4D"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-53" x="1051" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-45" x="1697" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-82">MSE</script>是<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-83-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 600.5 496.10337069604896" style="width: 1.39ex; height: 1.158ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-6E"></use></g></svg></span><script type="math/tex" id="MathJax-Element-83">n</script>个数据预测结果误差平方的均值。</p><div class="md-section-divider"></div><h3 data-anchor-id="m9rr" id="训练过程">训练过程</h3><p data-anchor-id="hj08">定义好模型结构之后，我们要通过以下几个步骤进行模型训练 <br> 1. 初始化参数，其中包括权重<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-84-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -464.0516853480245 966.8053928999522 642.5886520702876" style="width: 2.201ex; height: 1.506ex; vertical-align: -0.463ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-3C9"></use><use transform="scale(0.7071067811865476)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-69" x="880" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-84">\omega_i</script>和偏置<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-85-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -715.0516853480245 429.5 747.103370696049" style="width: 1.042ex; height: 1.737ex; vertical-align: -0.116ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMATHI-62"></use></g></svg></span><script type="math/tex" id="MathJax-Element-85">b</script>，对其进行初始化（如0均值，1方差）。 <br> 2. 网络正向传播计算网络输出和损失函数。 <br> 3. 根据损失函数进行反向误差传播 （<a href="https://en.wikipedia.org/wiki/Backpropagation" target="_blank">backpropagation</a>），将网络误差从输出层依次向前传递, 并更新网络中的参数。 <br> 4. 重复2~3步骤，直至网络训练误差达到规定的程度或训练轮次达到设定值。</p><div class="md-section-divider"></div><h2 data-anchor-id="wozw" id="数据准备">数据准备</h2><p data-anchor-id="ytgk">执行以下命令来准备数据:</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="jh2a"><ol class="linenums"><li class="L0"><code class="language-bash"><span class="pln">cd data </span><span class="pun">&&</span><span class="pln"> python prepare_data</span><span class="pun">.</span><span class="pln">py</span></code></li></ol></pre><div class="md-section-divider"></div><div class="md-section-divider"></div><div class="md-section-divider"></div><p align="center" data-anchor-id="b8vw"> <img src="https://raw.githubusercontent.com/PaddlePaddle/book/develop/fit_a_line/image/ranges.png" width="550"><br> 图2. 各维属性的取值范围 </p><div class="md-section-divider"></div><h4 data-anchor-id="ye0u" id="整理训练集与测试集">整理训练集与测试集</h4><p data-anchor-id="m87z">我们将数据集分割为两份：一份用于调整模型的参数，即进行模型的训练，模型在这份数据集上的误差被称为<strong>训练误差</strong>；另外一份被用来测试，模型在这份数据集上的误差被称为<strong>测试误差</strong>。我们训练模型的目的是为了通过从训练数据中找到规律来预测未知的新数据，所以测试误差是更能反映模型表现的指标。分割数据的比例要考虑到两个因素：更多的训练数据会降低参数估计的方差，从而得到更可信的模型；而更多的测试数据会降低测试误差的方差，从而得到更可信的测试误差。一种常见的分割比例为<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" role="textbox" aria-readonly="true" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 -687.0516853480245 1835.0555555555557 730.103370696049" style="width: 4.286ex; height: 1.737ex; vertical-align: -0.232ex; margin: 1px 0px;"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-38"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-3A" x="778" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#MJMAIN-32" x="1334" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-86">8:2</script>，感兴趣的读者朋友们也可以尝试不同的设置来观察这两种误差的变化。</p><p data-anchor-id="gqtf">执行如下命令可以分割数据集，并将训练集和测试集的地址分别写入train.list 和 test.list两个文件中，供PaddlePaddle读取。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="g3k3"><ol class="linenums"><li class="L0"><code class="language-python"><span class="pln">python prepare_data</span><span class="pun">.</span><span class="pln">py </span><span class="pun">-</span><span class="pln">r </span><span class="lit">0.8</span><span class="pln"> </span><span class="com">#默认使用8:2的比例进行分割</span></code></li></ol></pre><p data-anchor-id="4qk8">在更复杂的模型训练过程中，我们往往还会多使用一种数据集：验证集。因为复杂的模型中常常还有一些超参数（<a href="https://en.wikipedia.org/wiki/Hyperparameter_optimization" target="_blank">Hyperparameter</a>）需要调节，所以我们会尝试多种超参数的组合来分别训练多个模型，然后对比它们在验证集上的表现选择相对最好的一组超参数，最后才使用这组参数下训练的模型在测试集上评估测试误差。由于本章训练的模型比较简单，我们暂且忽略掉这个过程。</p><div class="md-section-divider"></div><h3 data-anchor-id="qt61" id="提供数据给paddlepaddle">提供数据给PaddlePaddle</h3><p data-anchor-id="s85w">准备好数据之后，我们使用一个Python data provider来为PaddlePaddle的训练过程提供数据。一个 data provider 就是一个Python函数，它会被PaddlePaddle的训练过程调用。在这个例子里，只需要读取已经保存好的数据，然后一行一行地返回给PaddlePaddle的训练进程即可。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="yknv"><ol class="linenums"><li class="L0"><code class="language-python"><span class="kwd">from</span><span class="pln"> paddle</span><span class="pun">.</span><span class="pln">trainer</span><span class="pun">.</span><span class="typ">PyDataProvider2</span><span class="pln"> </span><span class="kwd">import</span><span class="pln"> </span><span class="pun">*</span></code></li><li class="L1"><code class="language-python"><span class="kwd">import</span><span class="pln"> numpy </span><span class="kwd">as</span><span class="pln"> np</span></code></li><li class="L2"><code class="language-python"><span class="com">#定义数据的类型和维度</span></code></li><li class="L3"><code class="language-python"><span class="lit">@provider</span><span class="pun">(</span><span class="pln">input_types</span><span class="pun">=[</span><span class="pln">dense_vector</span><span class="pun">(</span><span class="lit">13</span><span class="pun">),</span><span class="pln"> dense_vector</span><span class="pun">(</span><span class="lit">1</span><span class="pun">)])</span></code></li><li class="L4"><code class="language-python"><span class="kwd">def</span><span class="pln"> process</span><span class="pun">(</span><span class="pln">settings</span><span class="pun">,</span><span class="pln"> input_file</span><span class="pun">):</span></code></li><li class="L5"><code class="language-python"><span class="pln"> data </span><span class="pun">=</span><span class="pln"> np</span><span class="pun">.</span><span class="pln">load</span><span class="pun">(</span><span class="pln">input_file</span><span class="pun">.</span><span class="pln">strip</span><span class="pun">())</span></code></li><li class="L6"><code class="language-python"><span class="pln"> </span><span class="kwd">for</span><span class="pln"> row </span><span class="kwd">in</span><span class="pln"> data</span><span class="pun">:</span></code></li><li class="L7"><code class="language-python"><span class="pln"> </span><span class="kwd">yield</span><span class="pln"> row</span><span class="pun">[:-</span><span class="lit">1</span><span class="pun">].</span><span class="pln">tolist</span><span class="pun">(),</span><span class="pln"> row</span><span class="pun">[-</span><span class="lit">1</span><span class="pun">:].</span><span class="pln">tolist</span><span class="pun">()</span></code></li></ol></pre><div class="md-section-divider"></div><h2 data-anchor-id="9xee" id="模型配置说明">模型配置说明</h2><div class="md-section-divider"></div><h3 data-anchor-id="jcsa" id="数据定义">数据定义</h3><p data-anchor-id="rq5t">首先，通过 <code>define_py_data_sources2</code> 来配置PaddlePaddle从上面的<code>dataprovider.py</code>里读入训练数据和测试数据。 PaddlePaddle接受从命令行读入的配置信息，例如这里我们传入一个名为<code>is_predict</code>的变量来控制模型在训练和测试时的不同结构。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="v6zd"><ol class="linenums"><li class="L0"><code class="language-python"><span class="kwd">from</span><span class="pln"> paddle</span><span class="pun">.</span><span class="pln">trainer_config_helpers </span><span class="kwd">import</span><span class="pln"> </span><span class="pun">*</span></code></li><li class="L1"><code class="language-python"></code></li><li class="L2"><code class="language-python"><span class="pln">is_predict </span><span class="pun">=</span><span class="pln"> get_config_arg</span><span class="pun">(</span><span class="str">'is_predict'</span><span class="pun">,</span><span class="pln"> bool</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">False</span><span class="pun">)</span></code></li><li class="L3"><code class="language-python"></code></li><li class="L4"><code class="language-python"><span class="pln">define_py_data_sources2</span><span class="pun">(</span></code></li><li class="L5"><code class="language-python"><span class="pln"> train_list</span><span class="pun">=</span><span class="str">'data/train.list'</span><span class="pun">,</span></code></li><li class="L6"><code class="language-python"><span class="pln"> test_list</span><span class="pun">=</span><span class="str">'data/test.list'</span><span class="pun">,</span></code></li><li class="L7"><code class="language-python"><span class="pln"> module</span><span class="pun">=</span><span class="str">'dataprovider'</span><span class="pun">,</span></code></li><li class="L8"><code class="language-python"><span class="pln"> obj</span><span class="pun">=</span><span class="str">'process'</span><span class="pun">)</span></code></li></ol></pre><div class="md-section-divider"></div><h3 data-anchor-id="0tzy" id="算法配置">算法配置</h3><p data-anchor-id="0n6j">接着，指定模型优化算法的细节。由于线性回归模型比较简单，我们只要设置基本的<code>batch_size</code>即可，它指定每次更新参数的时候使用多少条数据计算梯度信息。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="klfv"><ol class="linenums"><li class="L0"><code class="language-python"><span class="pln">settings</span><span class="pun">(</span><span class="pln">batch_size</span><span class="pun">=</span><span class="lit">2</span><span class="pun">)</span></code></li></ol></pre><div class="md-section-divider"></div><h3 data-anchor-id="omgp" id="网络结构">网络结构</h3><p data-anchor-id="8a14">最后，使用<code>fc_layer</code>和<code>LinearActivation</code>来表示线性回归的模型本身。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="v0zt"><ol class="linenums"><li class="L0"><code class="language-python"><span class="com">#输入数据，13维的房屋信息</span></code></li><li class="L1"><code class="language-python"><span class="pln">x </span><span class="pun">=</span><span class="pln"> data_layer</span><span class="pun">(</span><span class="pln">name</span><span class="pun">=</span><span class="str">'x'</span><span class="pun">,</span><span class="pln"> size</span><span class="pun">=</span><span class="lit">13</span><span class="pun">)</span></code></li><li class="L2"><code class="language-python"></code></li><li class="L3"><code class="language-python"><span class="pln">y_predict </span><span class="pun">=</span><span class="pln"> fc_layer</span><span class="pun">(</span></code></li><li class="L4"><code class="language-python"><span class="pln"> input</span><span class="pun">=</span><span class="pln">x</span><span class="pun">,</span></code></li><li class="L5"><code class="language-python"><span class="pln"> param_attr</span><span class="pun">=</span><span class="typ">ParamAttr</span><span class="pun">(</span><span class="pln">name</span><span class="pun">=</span><span class="str">'w'</span><span class="pun">),</span></code></li><li class="L6"><code class="language-python"><span class="pln"> size</span><span class="pun">=</span><span class="lit">1</span><span class="pun">,</span></code></li><li class="L7"><code class="language-python"><span class="pln"> act</span><span class="pun">=</span><span class="typ">LinearActivation</span><span class="pun">(),</span></code></li><li class="L8"><code class="language-python"><span class="pln"> bias_attr</span><span class="pun">=</span><span class="typ">ParamAttr</span><span class="pun">(</span><span class="pln">name</span><span class="pun">=</span><span class="str">'b'</span><span class="pun">))</span></code></li><li class="L9"><code class="language-python"></code></li><li class="L0"><code class="language-python"><span class="kwd">if</span><span class="pln"> </span><span class="kwd">not</span><span class="pln"> is_predict</span><span class="pun">:</span><span class="pln"> </span><span class="com">#训练时，我们使用MSE，即regression_cost作为损失函数</span></code></li><li class="L1"><code class="language-python"><span class="pln"> y </span><span class="pun">=</span><span class="pln"> data_layer</span><span class="pun">(</span><span class="pln">name</span><span class="pun">=</span><span class="str">'y'</span><span class="pun">,</span><span class="pln"> size</span><span class="pun">=</span><span class="lit">1</span><span class="pun">)</span></code></li><li class="L2"><code class="language-python"><span class="pln"> cost </span><span class="pun">=</span><span class="pln"> regression_cost</span><span class="pun">(</span><span class="pln">input</span><span class="pun">=</span><span class="pln">y_predict</span><span class="pun">,</span><span class="pln"> label</span><span class="pun">=</span><span class="pln">y</span><span class="pun">)</span></code></li><li class="L3"><code class="language-python"><span class="pln"> outputs</span><span class="pun">(</span><span class="pln">cost</span><span class="pun">)</span><span class="pln"> </span><span class="com">#训练时输出MSE来监控损失的变化</span></code></li><li class="L4"><code class="language-python"><span class="kwd">else</span><span class="pun">:</span><span class="pln"> </span><span class="com">#测试时，输出预测值</span></code></li><li class="L5"><code class="language-python"><span class="pln"> outputs</span><span class="pun">(</span><span class="pln">y_predict</span><span class="pun">)</span></code></li></ol></pre><div class="md-section-divider"></div><h2 data-anchor-id="ak2p" id="训练模型">训练模型</h2><p data-anchor-id="e18b">在对应代码的根目录下执行PaddlePaddle的命令行训练程序。这里指定模型配置文件为<code>trainer_config.py</code>，训练30轮，结果保存在<code>output</code>路径下。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="k7nw"><ol class="linenums"><li class="L0"><code class="language-bash"><span class="pun">./</span><span class="pln">train</span><span class="pun">.</span><span class="pln">sh</span></code></li></ol></pre><div class="md-section-divider"></div><h2 data-anchor-id="tjol" id="应用模型">应用模型</h2><p data-anchor-id="z7d8">现在来看下如何使用已经训练好的模型进行预测。</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="ew76"><ol class="linenums"><li class="L0"><code class="language-bash"><span class="pln">python predict</span><span class="pun">.</span><span class="pln">py</span></code></li></ol></pre><p data-anchor-id="1bsn">这里默认使用<code>output/pass-00029</code>中保存的模型进行预测，并将数据中的房价与预测结果进行对比，结果保存在 <code>predictions.png</code>中。 <br> 如果你想使用别的模型或者其它的数据进行预测，只要传入新的路径即可：</p><div class="md-section-divider"></div><pre class="prettyprint linenums prettyprinted" data-anchor-id="9q38"><ol class="linenums"><li class="L0"><code class="language-bash"><span class="pln">python predict</span><span class="pun">.</span><span class="pln">py </span><span class="pun">-</span><span class="pln">m output</span><span class="pun">/</span><span class="pln">pass</span><span class="pun">-</span><span class="lit">00020</span><span class="pln"> </span><span class="pun">-</span><span class="pln">t data</span><span class="pun">/</span><span class="pln">housing</span><span class="pun">.</span><span class="pln">test</span><span class="pun">.</span><span class="pln">npy</span></code></li></ol></pre><div class="md-section-divider"></div><h2 data-anchor-id="p7c3" id="总结">总结</h2><p data-anchor-id="3chd">在这章里，我们借助波士顿房价这一数据集，介绍了线性回归模型的基本概念，以及如何使用PaddlePaddle实现训练和测试的过程。很多的模型和技巧都是从简单的线性回归模型演化而来，因此弄清楚线性模型的原理和局限非常重要。</p><div class="md-section-divider"></div><h2 data-anchor-id="r5es" id="参考文献">参考文献</h2><ol data-anchor-id="2umn"> <li><a href="https://en.wikipedia.org/wiki/Linear_regression" target="_blank">https://en.wikipedia.org/wiki/Linear_regression</a></li> <li>Friedman J, Hastie T, Tibshirani R. The elements of statistical learning[M]. Springer, Berlin: Springer series in statistics, 2001.</li> <li>Murphy K P. Machine learning: a probabilistic perspective[M]. MIT press, 2012.</li> <li>Bishop C M. Pattern recognition[J]. Machine Learning, 2006, 128.</li> </ol><p data-anchor-id="x9ai"><br> <br> <img src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png" alt="知识共享许可协议"> <br> 本教程由<a href="http://book.paddlepaddle.org" target="_blank">PaddlePaddle</a>创作，采用<a href="http://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">知识共享 署名-非商业性使用-相同方式共享 4.0 国际 许可协议</a>进行许可。</p></div> </body> </html>
