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<h1 id="偏相关系数计算"><a href="#偏相关系数计算" class="headerlink" title="偏相关系数计算"></a>偏相关系数计算</h1><p>参考<br><em><a target="_blank" rel="noopener" href="https://book.douban.com/subject/5418944/">陈彦光编著. 地理数学方法及其应用. 2008, 北京大学城市与环境学院.</a></em><br><em><a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Partial_correlation">维基百科</a></em></p>
<p>简单相关系数旨在反映变量之间两两线性关系，但实际上，每一个简单相关系数不可能绝对不包括其他因素的相关成分。为了克服简单相关系数的间接相关信息，有人设计了另一种检验指标，称为偏相关系数（ partial correlation coefficient）。偏相关系数旨在排除其它因素的影响，单纯反映某个自变量与因变量之间的密切程度。</p>
<p>当自变量较多时，利用公式计算偏相关系数相当麻烦，比较便捷的方式是借助简单相关系数构成的相关矩阵进行运算，计算公式如下：</p>
<p>$R_{x_{j} y}=\frac{-c_{j y}}{\sqrt{c_{j j} c_{y y}}}$</p>
<p>这里$R_{x_{j} y}$ 为第 <em>j</em> 个自变量与因变量 <em>y</em> 的偏相关系数， <em>c</em> 为相关系数矩阵的逆矩阵中对应的元素。 </p>
<p>下面是python实现</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># -*- coding: utf-8 -*-</span></span><br><span class="line"><span class="string">"""</span></span><br><span class="line"><span class="string">Created on Mon Dec 20 16:53:39 2021</span></span><br><span class="line"><span class="string">modified: https://gist.github.com/fabianp/9396204419c7b638d38f</span></span><br><span class="line"><span class="string">@author: pan</span></span><br><span class="line"><span class="string">"""</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"><span class="keyword">from</span> numpy.linalg <span class="keyword">import</span> inv</span><br><span class="line"><span class="keyword">from</span> osgeo <span class="keyword">import</span> gdal, gdal_array</span><br><span class="line"><span class="keyword">import</span> os, time,glob</span><br><span class="line"><span class="keyword">from</span> sklearn <span class="keyword">import</span> linear_model</span><br><span class="line"><span class="keyword">from</span> sklearn <span class="keyword">import</span> preprocessing</span><br><span class="line"><span class="keyword">from</span> matplotlib <span class="keyword">import</span> pyplot <span class="keyword">as</span> plt</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">partial_corr</span>(<span class="params">C</span>):</span></span><br><span class="line"> <span class="string">"""</span></span><br><span class="line"><span class="string"> Returns the sample linear partial correlation coefficients between pairs of variables in C, controlling </span></span><br><span class="line"><span class="string"> for the remaining variables in C.</span></span><br><span class="line"><span class="string"> Parameters</span></span><br><span class="line"><span class="string"> ----------</span></span><br><span class="line"><span class="string"> C : array-like, shape (n, p)</span></span><br><span class="line"><span class="string"> Array with the different variables. Each column of C is taken as a variable</span></span><br><span class="line"><span class="string"> Returns</span></span><br><span class="line"><span class="string"> -------</span></span><br><span class="line"><span class="string"> P_corr : array-like, shape (p, p)</span></span><br><span class="line"><span class="string"> P_corr[i, j] contains the partial correlation of C[:, i] and C[:, j] controlling</span></span><br><span class="line"><span class="string"> for the remaining variables in C.</span></span><br><span class="line"><span class="string"> """</span></span><br><span class="line"> </span><br><span class="line"> C = np.asarray(C)</span><br><span class="line"> p = C.shape[<span class="number">1</span>]</span><br><span class="line"> P_corr = np.zeros((p, p)) <span class="comment"># sample linear partial correlation coefficients</span></span><br><span class="line"> </span><br><span class="line"> corr = np.corrcoef(C,rowvar=<span class="literal">False</span>) <span class="comment"># Pearson product-moment correlation coefficients.</span></span><br><span class="line"> corr_inv = inv(corr) <span class="comment"># the (multiplicative) inverse of a matrix.</span></span><br><span class="line"> </span><br><span class="line"> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(p):</span><br><span class="line"> P_corr[i, i] = <span class="number">1</span></span><br><span class="line"> <span class="keyword">for</span> j <span class="keyword">in</span> <span class="built_in">range</span>(i+<span class="number">1</span>, p):</span><br><span class="line"> pcorr_ij = -corr_inv[i,j]/(np.sqrt(corr_inv[i,i]*corr_inv[j,j]))</span><br><span class="line"> P_corr[i,j]=pcorr_ij</span><br><span class="line"> P_corr[j,i]=pcorr_ij</span><br><span class="line"> </span><br><span class="line"> <span class="keyword">return</span> P_corr</span><br></pre></td></tr></table></figure>
</div></article></div><div class="card"><article class="card-content article" role="article"><div class="article-meta is-size-7 is-uppercase level is-mobile"><div class="level-left"><span class="level-item">Posted <time dateTime="2021-07-15T07:31:57.000Z" title="2021/7/15 下午3:31:57">2021-07-15</time></span><span class="level-item">Updated <time dateTime="2021-07-15T07:56:14.588Z" title="2021/7/15 下午3:56:14">2021-07-15</time></span><span class="level-item"><a class="link-muted" href="/categories/%E5%85%B6%E5%AE%83/">其它</a></span><span class="level-item">2 minutes read (About 276 words)</span></div></div><h1 class="title is-3 is-size-4-mobile"><a class="link-muted" href="/2021/07/15/%E6%96%87%E7%8C%AE%E7%AE%A1%E7%90%86%E4%B8%8E%E4%BF%A1%E6%81%AF%E5%88%86%E6%9E%90%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0/">文献管理与信息分析学习笔记</a></h1><div class="content"><p>关于文献管理-信息分析-科学知识图谱学习中的一些笔记，帮助我们提升科研的效率。</p></div><a class="article-more button is-small is-size-7" 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