<h2><img alt="" height="386" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-800991744b2f89cf2d9e47ee56da2fea.png" width="843"></h2>
<h2>一、因子分析简介</h2>
<h3>1、定义</h3>
<p style="text-indent:33px;">1904年，英国心理学家CharlesSpearman研究了33名学生在古典语、法语和英语三门成绩，三门成绩的相关性系数如下：</p>
<p style="text-align:center;"><img alt="" height="102" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-21133bdac46f2b67113e96be1fc6211b.png" width="355"></p>
<p style="text-indent:33px;">三门成绩的高度相关会不会是由于它们三个成绩的背后有一个共同的因素，来决定这三门成绩的？比如语言能力？</p>
<p style="text-indent:33px;">对于<img alt="p" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-12a879d6031bf6e56869cb07f60403ea.latex">个原始变量<img alt="Y_{1},...,Y_{p}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-6408fe37e3abe1d6b8c5405b3e985270.latex">来说，那些高度相关的变量很可能会遵循一个共同的潜在结构——或可称之为<strong>公共因子 (Common factor）</strong> 。简单的说就是：公共因子是用一个共同的因素来刻画几个高度相关的变量。</p>
<p style="text-indent:33px;">然而，这些“公共因子”通常是无法观测的，故称为潜变量 (latentvariables)。这在心理学、社会学及行为科学等学科中非常常见，比如<br> “智力”和“社会阶层”。</p>
<p style="text-indent:33px;"><strong>因子分析（Factor analysis）</strong>旨在提出<strong>因子模型（Factor model）</strong>来研究如何用几个公共因子，记作<img alt="F_{1},...,F_{m}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-551d35fbede52c0e0b0fa5ea403028ab.latex">，通常<img alt="m<p" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-4aa58dad821acc4ea1c86764a5454ebd.latex">，来刻画原始变量之间的相关性。</p>
<h3>2、正交因子模型</h3>
<p style="text-indent:33px;">Charles Spearman基于学生3门语言成绩的数据提出了单因子模型（Single factor model）：</p>
<p style="text-indent:33px;"><img alt="Y_{1}=l_{1}F+\varepsilon _{1}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-fcfd3b5c14e32b6b1a1d906073672618.latex"></p>
<p style="text-indent:33px;"><img alt="Y_{2}=l_{2}F+\varepsilon _{2}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-1058d2b9e298905f82c7456acef75f35.latex"></p>
<p style="text-indent:33px;"><img alt="Y_{3}=l_{3}F+\varepsilon _{3}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-b60fbeec07cfea9c172f4866a0d31d21.latex"></p>
<p style="text-indent:33px;">其中<img alt="F" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-42685ad271e7dd4b84ea0bac3e3e3f5e.latex">代表公共因子（Common factor），<img alt="\varepsilon _{j}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-a9eeb9494654db71981ea5b0e81552fb.latex">代表特殊因子（Specific factor），即代表<img alt="Y_{j}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-18b39069830b88aa1f1c82cea13274cb.latex">的特殊部分；<img alt="l_{j}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-629567fce59cba7fb60d4ade0e99d630.latex">代表系数/载荷（Loading），即来说明公共因子<img alt="F" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-42685ad271e7dd4b84ea0bac3e3e3f5e.latex">对<img alt="Y_{j}" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-18b39069830b88aa1f1c82cea13274cb.latex">的解释力。 </p>
<p style="text-indent:33px;">当然，大多数时候一个公共因子是不够的，错综复杂的变量可能需要多个公共因子刻画，这就是我们将要学习的正交因子模型（Orthogonal factor model）</p>
<p style="text-indent:33px;">假设可观测随机向量<img alt="y=(Y_1,...,Y_{p})'" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-d4bf520a43bcf106bc22b0fa259ce3a6.latex">的均值为<img alt="\mu" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-66f61b77bbc27510aeb92b9abceb8cac.latex">，协方差矩阵为<img alt="\Sigma" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-a8b0c2500a188ba6d859410b959db675.latex">。正交因子模型假定<img alt="y" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-affb9ae80c2b71c4730e171bbc578d50.latex">线性依赖于<img alt="m" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-fd6c81e40845e2db8dd9249532359ac9.latex">个不可观测公共因子<img alt="f=(F_{1},...,F_{m})'" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-8b5db89f3710f9b916e1d8ea561a63a4.latex">和<img alt="p" class="mathcode" src="htt |
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