matlab算法大全_算法_matlab_其他

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0"> <span class="_ _0"> </span><span class="ff2">-443-</span></div><div class="t m0 x3 h3 y3 ff2 fs1 fc0 sc1 ls1 ws0">第二十九章<span class="ff3 sc0 ls2"> </span>多元分析<span class="ff1 fs2 sc0 ls0"> </span></div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x5 h4 y5 ff2 fs2 fc0 sc0 ls0 ws0">多元分析（<span class="ff1 ls3 ws1">multivariat<span class="_ _1"></span>e analyses<span class="ff2 ls0 ws0">）<span class="_ _1"></span>是多变量的统计分析方法，是数理统计中应用广</span></span></div><div class="t m0 x4 h5 y6 ff2 fs2 fc0 sc0 ls0 ws0">泛的一个重要分支，<span class="_ _2"></span>其内容庞杂，<span class="_ _2"></span>视角独特，<span class="_ _2"></span>方法多样，<span class="_ _2"></span>深受工程技术人员的青睐和广</div><div class="t m0 x4 h5 y7 ff2 fs2 fc0 sc0 ls0 ws0">泛使用，<span class="_ _3"></span>并在使用中不断完善和创新。<span class="_ _3"></span>由于变量的相关性，<span class="_ _3"></span>不能简单地把每个变量的结</div><div class="t m0 x4 h4 y8 ff2 fs2 fc0 sc0 ls0 ws0">果进行汇总，这是多变量统计分析的基本出发点。<span class="ff1"> </span></div><div class="t m0 x4 h4 y9 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x4 h4 ya ff2 fs2 fc0 sc0 ls0 ws0">§<span class="ff1 ls4 ws2">1 </span>聚类分析<span class="ff1"> </span></div><div class="t m0 x5 h5 yb ff2 fs2 fc0 sc0 ls5 ws0">将认识对象进行分类是人类认识<span class="_ _4"></span>世界的一种重要方法，比<span class="_ _4"></span>如有关世界的时间进程</div><div class="t m0 x4 h5 yc ff2 fs2 fc0 sc0 ls0 ws0">的研究，<span class="_ _2"></span>就形成了历史学，<span class="_ _2"></span>也有关世界空间地域的研究，<span class="_ _2"></span>则形成了地理学。<span class="_ _2"></span>又如在生物</div><div class="t m0 x4 h5 yd ff2 fs2 fc0 sc0 ls0 ws0">学中，<span class="_ _2"></span>为了研究生物的演变，<span class="_ _2"></span>需要对生物进行分类，<span class="_ _2"></span>生物学家根据各种生物的特征，<span class="_ _2"></span>将</div><div class="t m0 x4 h5 ye ff2 fs2 fc0 sc0 ls0 ws0">它们归属于不同的界、<span class="_ _5"></span>门、<span class="_ _5"></span>纲、<span class="_ _5"></span>目、<span class="_ _5"></span>科、<span class="_ _5"></span>属、<span class="_ _5"></span>种之中。<span class="_ _5"></span>事实上，<span class="_ _5"></span>分门别类地对事物进行</div><div class="t m0 x4 h5 yf ff2 fs2 fc0 sc0 ls0 ws0">研究，<span class="_ _3"></span>要远比在一个混杂多变的集合中更清晰、<span class="_ _3"></span>明了和细致，<span class="_ _3"></span>这是因为同一类事物会具</div><div class="t m0 x4 h5 y10 ff2 fs2 fc0 sc0 ls0 ws0">有更多的近似特性。<span class="_ _6"></span>在企业的经营管理中，<span class="_ _6"></span>为了确定其目标市场，<span class="_ _6"></span>首先要进行市场细分。</div><div class="t m0 x4 h5 y11 ff2 fs2 fc0 sc0 ls0 ws0">因为无论一个企业多么庞大和成功，它也无法满足整个市场的各种需求。而市场细分，</div><div class="t m0 x4 h5 y12 ff2 fs2 fc0 sc0 ls0 ws0">可以帮助企业找到适合自己特色，<span class="_ _7"></span>并使企业具有竞争力的分市场，<span class="_ _7"></span>将其作为自己的重点</div><div class="t m0 x4 h4 y13 ff2 fs2 fc0 sc0 ls0 ws0">开发目标。<span class="ff1"> </span></div><div class="t m0 x5 h4 y14 ff2 fs2 fc0 sc0 ls0 ws0">通常，人们可以凭经验和专业知识来实现分类。而聚类分析（<span class="ff1 ls6 ws3">cluster analyses</span>）作</div><div class="t m0 x4 h4 y15 ff2 fs2 fc0 sc0 ls0 ws0">为一种定量方法，将从数据分析的角度，给出一个更准确、细致的分类工具。<span class="ff1"> </span></div><div class="t m0 x5 h4 y16 ff1 fs2 fc0 sc0 ls7 ws4">1.1 <span class="ff2 ls0 ws0">相似性度量<span class="ff1"> </span></span></div><div class="t m0 x5 h4 y17 ff1 fs2 fc0 sc0 ls8 ws5">1.1.1 <span class="ff2 ls0 ws0">样本的相似性度量<span class="ff1"> </span></span></div><div class="t m0 x5 h5 y18 ff2 fs2 fc0 sc0 ls5 ws0">要用数量化的方法对事物进行分<span class="_ _4"></span>类，就必须用数量化的方<span class="_ _4"></span>法描述事物之间的相似</div><div class="t m0 x4 h5 y19 ff2 fs2 fc0 sc0 ls0 ws0">程度。<span class="_ _6"></span>一个事物常常需要用多个变量来刻画。<span class="_ _8"></span>如果对于一群有待分类的样本点需用</div><div class="c x6 y1a w2 h6"><div class="t m0 x7 h7 y1b ff4 fs3 fc0 sc0 ls0 ws0">p</div></div><div class="t m0 x8 h5 y1c ff2 fs2 fc0 sc0 ls0 ws0">个</div><div class="t m0 x4 h5 y1d ff2 fs2 fc0 sc0 ls0 ws0">变量描述，<span class="_ _1"></span>则每个样本点可以看成是</div><div class="t m0 x9 h8 y1e ff4 fs4 fc0 sc0 ls0 ws0">p</div><div class="c xa y1f w3 h9"><div class="t m0 x0 ha y20 ff4 fs5 fc0 sc0 ls0 ws0">R</div></div><div class="t m0 xb h5 y1d ff2 fs2 fc0 sc0 ls0 ws0">空间中的一个点。<span class="_ _1"></span>因此，<span class="_ _1"></span>很自然地想到可以用</div><div class="t m0 x4 h4 y21 ff2 fs2 fc0 sc0 ls0 ws0">距离来度量样本点间的相似程度。<span class="ff1"> </span></div><div class="t m0 x5 h5 y22 ff2 fs2 fc0 sc0 ls0 ws0">记</div><div class="t m0 xc hb y23 ff5 fs6 fc0 sc0 ls0 ws0">Ω<span class="_ _9"> </span><span class="ff2 fs2">是样本点集，距离<span class="_ _a"> </span><span class="ff1 fs7">)<span class="_ _b"></span>,<span class="_ _b"></span>(</span></span></div><div class="c xd y24 w4 hc"><div class="t m0 x0 hd y1b ff5 fs7 fc0 sc0 ls0 ws0">⋅</div></div><div class="c xe y24 w4 hc"><div class="t m0 x0 hd y1b ff5 fs7 fc0 sc0 ls0 ws0">⋅</div></div><div class="t m0 xf he y23 ff4 fs7 fc0 sc0 ls0 ws0">d<span class="_ _c"> </span><span class="ff2 fs2">是</span></div><div class="t m0 x10 hf y25 ff5 fs8 fc0 sc0 ls0 ws0">+</div><div class="t m0 x11 he y23 ff5 fs7 fc0 sc0 ls0 ws0">→<span class="_ _d"></span>Ω<span class="_ _e"></span>×<span class="_ _e"></span>Ω<span class="_ _f"> </span><span class="ff4">R<span class="_ _10"> </span><span class="ff2 fs2">的一个函数，满足条件：<span class="ff1"> </span></span></span></div><div class="t m0 x5 h4 y26 ff1 fs2 fc0 sc0 ls0 ws0">1<span class="ff2">）</span></div><div class="t m0 x12 h10 y27 ff1 fs9 fc0 sc0 ls0 ws0">0<span class="_ _11"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _14"> </span><span class="ff5">≥<span class="_ _15"></span><span class="ff4">y<span class="_ _16"></span>x<span class="_ _e"></span>d<span class="_ _17"> </span><span class="ff2 fs2">，<span class="_ _18"> </span></span><span class="ff5">Ω<span class="_ _15"></span>∈<span class="_ _19"></span><span class="ff4">y<span class="_ _16"></span>x<span class="ff1">,<span class="_ _18"> </span><span class="ff2 fs2">；<span class="ff1"> </span></span></span></span></span></span></span></div><div class="t m0 x5 h4 y28 ff1 fs2 fc0 sc0 ls0 ws0">2<span class="ff2">）</span></div><div class="t m0 x12 h10 y29 ff1 fs9 fc0 sc0 ls0 ws0">0<span class="_ _11"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(</div><div class="c x13 y2a w5 hc"><div class="t m0 x0 h11 y1b ff5 fs9 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x14 h12 y29 ff4 fs9 fc0 sc0 ls0 ws0">y<span class="_ _16"></span>x<span class="_ _e"></span>d<span class="_ _17"> </span><span class="ff2 fs2">当且仅当<span class="_ _1a"> </span></span>y</div><div class="c x15 y2a w6 h6"><div class="t m0 x0 h12 y1b ff4 fs9 fc0 sc0 ls0 ws0">x</div></div><div class="c x16 y2a w5 h6"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x17 h4 y29 ff2 fs2 fc0 sc0 ls0 ws0">；<span class="ff1"> </span></div><div class="t m0 x5 h4 y2b ff1 fs2 fc0 sc0 ls0 ws0">3<span class="ff2">）</span></div><div class="t m0 x18 h10 y2c ff1 fs9 fc0 sc0 ls0 ws0">)<span class="_ _1b"></span>,<span class="_ _1c"></span>(<span class="_ _1d"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _1e"> </span><span class="ff4">x<span class="_ _19"></span>y<span class="_ _1f"></span>d<span class="_ _20"></span>y<span class="_ _16"></span>x<span class="_ _e"></span>d</span></div><div class="c x13 y2d w5 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x19 h5 y2c ff2 fs2 fc0 sc0 ls0 ws0">，</div><div class="c x1a y2d w7 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">Ω</div></div><div class="c x1b y2d w8 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">∈</div></div><div class="t m0 x1c h10 y2c ff4 fs9 fc0 sc0 ls0 ws0">y<span class="_ _16"></span>x<span class="ff1">,<span class="_ _18"> </span><span class="ff2 fs2">；<span class="ff1"> </span></span></span></div><div class="t m0 x5 h4 y2e ff1 fs2 fc0 sc0 ls0 ws0">4<span class="ff2">）</span></div><div class="t m0 x1d h10 y2f ff1 fs9 fc0 sc0 ls0 ws0">)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _21"></span>)<span class="_ _22"></span>,<span class="_ _13"></span>(<span class="_ _1d"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _23"> </span><span class="ff4">y<span class="_ _16"></span>x<span class="_ _e"></span>d<span class="_ _21"></span>z<span class="_ _24"></span>x<span class="_ _e"></span>d<span class="_ _25"></span>y<span class="_ _16"></span>x<span class="_ _e"></span>d<span class="_ _26"> </span><span class="ff5 fs3">+</span></span></div><div class="c x13 y30 w5 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">≤</div></div><div class="t m0 x17 h5 y2f ff2 fs2 fc0 sc0 ls0 ws0">，</div><div class="c x1e y30 w7 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">Ω</div></div><div class="c x1f y30 w8 hc"><div class="t m0 x0 h13 y1b ff5 fs3 fc0 sc0 ls0 ws0">∈</div></div><div class="t m0 x20 h10 y2f ff4 fs9 fc0 sc0 ls0 ws0">z<span class="_ _12"></span>y<span class="_ _16"></span>x<span class="_ _27"> </span><span class="ff1">,<span class="_ _22"></span>,<span class="_ _28"> </span><span class="ff2 fs2">。<span class="ff1"> </span></span></span></div><div class="t m0 x5 h5 y31 ff2 fs2 fc0 sc0 ls5 ws0">这一距离的定义是我们所熟知的<span class="_ _4"></span>，它满足正定性，对称性<span class="_ _4"></span>和三角不等式。在聚类</div><div class="t m0 x4 h4 y32 ff2 fs2 fc0 sc0 ls0 ws0">分析中，对于定量变量，最常用的是<span class="_ _29"> </span><span class="ff1 ls9">Minkowski<span class="_"> </span></span>距离<span class="ff1"> </span></div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/bg2.jpg"><div class="t m0 x21 h2 y1 ff1 fs0 fc0 sc0 lsa ws0"> </div><div class="t m0 x21 h2 y2 ff2 fs0 fc0 sc0 ls0 ws0">-444-<span class="ff1"> </span></div><div class="t m0 x22 h4 y33 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x23 h14 y34 ff4 fsa fc0 sc0 ls0 ws0">q</div><div class="t m0 x24 h14 y35 ff4 fsa fc0 sc0 ls0 ws0">p</div><div class="t m0 x25 h14 y36 ff4 fsa fc0 sc0 ls0 ws0">k</div><div class="t m0 x26 h14 y37 ff4 fsa fc0 sc0 ls0 ws0">q</div><div class="t m0 x27 h14 y38 ff4 fsa fc0 sc0 ls0 ws0">k<span class="_ _2a"></span>k<span class="_ _2b"></span>q</div><div class="t m0 x28 h15 y39 ff4 fsb fc0 sc0 ls0 ws0">y<span class="_ _20"></span>x<span class="_ _2c"></span>y<span class="_ _16"></span>x<span class="_ _11"></span>d</div><div class="t m0 x23 h16 y3a ff1 fsa fc0 sc0 ls0 ws0">1</div><div class="t m0 x29 h16 y3b ff1 fsa fc0 sc0 ls0 ws0">1</div><div class="t m0 x2a h17 y39 ff1 fsb fc0 sc0 ls0 ws0">)<span class="_ _12"></span>,<span class="_ _13"></span>(</div><div class="t m0 xe h18 y3c ff5 fsb fc0 sc0 ls0 ws0">⎥</div><div class="t m0 xe h18 y3d ff5 fsb fc0 sc0 ls0 ws0">⎦</div><div class="t m0 xe h18 y3e ff5 fsb fc0 sc0 ls0 ws0">⎤</div><div class="t m0 x2b h18 y3c ff5 fsb fc0 sc0 ls0 ws0">⎢</div><div class="t m0 x2b h18 y3d ff5 fsb fc0 sc0 ls0 ws0">⎣</div><div class="t m0 x2b h18 y3e ff5 fsb fc0 sc0 ls0 ws0">⎡</div><div class="t m0 x18 h18 y39 ff5 fsb fc0 sc0 ls0 ws0">−<span class="_ _2d"></span>=</div><div class="t m0 x12 h19 y3f ff5 fsc fc0 sc0 ls0 ws0">∑</div><div class="t m0 x24 h1a y40 ff5 fsa fc0 sc0 ls0 ws0">=</div><div class="t m0 x2c h17 y33 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _a"> </span><span class="ff1 fsb">0<span class="_ _12"></span><span class="ff5">><span class="_ _12"></span><span class="ff4">q<span class="_ _2e"> </span><span class="ff1 fs2"> </span></span></span></span></div><div class="t m0 x21 h5 y41 ff2 fs2 fc0 sc0 ls0 ws0">当</div><div class="t m0 x2d h10 y42 ff1 fs9 fc0 sc0 ls0 ws0">2<span class="_ _b"></span>,<span class="_ _2f"></span>1<span class="_ _22"></span><span class="ff5 fs3">=<span class="_ _12"></span><span class="ff4 fs9">q<span class="_ _30"> </span><span class="ff2 fs2">或</span></span></span></div><div class="c x2e y43 w9 h1b"><div class="t m0 x0 h1c y1b ff5 fsd fc0 sc0 ls0 ws0">+</div></div><div class="t m0 x2f h1d y42 ff5 fsd fc0 sc0 ls0 ws0">∞<span class="_ _31"></span>→<span class="_ _32"></span><span class="ff4">q<span class="_ _33"> </span><span class="ff2 fs2">时，则分别得到<span class="ff1"> </span></span></span></div><div class="t m0 x21 h4 y44 ff1 fs2 fc0 sc0 lsb ws0"> 1<span class="_ _1f"></span><span class="ff2 ls0">）绝对值距<span class="_ _1"></span>离<span class="ff1"> </span></span></div><div class="t m0 x21 h4 y45 ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x30 h1e y46 ff5 fse fc0 sc0 ls0 ws0">∑</div><div class="t m0 x31 h1f y47 ff5 fsf fc0 sc0 ls0 ws0">=</div><div class="t m0 x32 h20 y45 ff5 fs10 fc0 sc0 ls0 ws0">−<span class="_ _34"></span>=</div><div class="t m0 x31 h21 y48 ff4 fsf fc0 sc0 ls0 ws0">q</div><div class="t m0 x33 h21 y49 ff4 fsf fc0 sc0 ls0 ws0">k</div><div class="t m0 x1c h21 y4a ff4 fsf fc0 sc0 ls0 ws0">k<span class="_ _2a"></span>k</div><div class="t m0 x34 h22 y45 ff4 fs10 fc0 sc0 ls0 ws0">y<span class="_ _20"></span>x<span class="_ _35"></span>y<span class="_ _16"></span>x<span class="_ _32"></span>d</div><div class="t m0 x35 h23 y47 ff1 fsf fc0 sc0 ls0 ws0">1</div><div class="t m0 x36 h23 y4b ff1 fsf fc0 sc0 ls0 ws0">1</div><div class="t m0 x37 h24 y45 ff1 fs10 fc0 sc0 ls0 ws0">)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _36"> </span><span class="ff2 fs2">，<span class="ff1 lsb"> <span class="_ _b"></span> <span class="_ _19"></span> <span class="_ _12"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">1</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></span></div><div class="t m0 x21 h4 y4c ff1 fs2 fc0 sc0 lsb ws0"> 2<span class="_ _1f"></span><span class="ff2 ls0">）欧氏距离<span class="_ _1"></span><span class="ff1"> </span></span></div><div class="t m0 x21 h4 y4d ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x38 h23 y4e ff1 fsf fc0 sc0 ls0 ws0">2</div><div class="t m0 x38 h23 y4f ff1 fsf fc0 sc0 ls0 ws0">1</div><div class="t m0 x39 h23 y50 ff1 fsf fc0 sc0 ls0 ws0">1</div><div class="t m0 x3a h23 y51 ff1 fsf fc0 sc0 ls0 ws0">2</div><div class="t m0 x3b h23 y52 ff1 fsf fc0 sc0 ls0 ws0">2</div><div class="t m0 x3c h24 y53 ff1 fs10 fc0 sc0 ls0 ws0">)<span class="_ _12"></span>,<span class="_ _13"></span>(</div><div class="t m0 x1a h20 y54 ff5 fs10 fc0 sc0 ls0 ws0">⎥</div><div class="t m0 x1a h20 y55 ff5 fs10 fc0 sc0 ls0 ws0">⎦</div><div class="t m0 x1a h20 y56 ff5 fs10 fc0 sc0 ls0 ws0">⎤</div><div class="t m0 x33 h20 y57 ff5 fs10 fc0 sc0 ls0 ws0">⎢</div><div class="t m0 x33 h20 y58 ff5 fs10 fc0 sc0 ls0 ws0">⎣</div><div class="t m0 x33 h20 y59 ff5 fs10 fc0 sc0 ls0 ws0">⎡</div><div class="t m0 x3d h20 y5a ff5 fs10 fc0 sc0 ls0 ws0">−<span class="_ _38"></span>=</div><div class="t m0 x31 h1e y5b ff5 fse fc0 sc0 ls0 ws0">∑</div><div class="t m0 x3e h1f y5c ff5 fsf fc0 sc0 ls0 ws0">=</div><div class="t m0 x3e h21 y5d ff4 fsf fc0 sc0 ls0 ws0">p</div><div class="t m0 x3f h21 y5c ff4 fsf fc0 sc0 ls0 ws0">k</div><div class="t m0 x40 h21 y5e ff4 fsf fc0 sc0 ls0 ws0">k<span class="_ _39"></span>k</div><div class="t m0 x41 h22 y53 ff4 fs10 fc0 sc0 ls0 ws0">y<span class="_ _1d"></span>x<span class="_ _3a"></span>y<span class="_ _16"></span>x<span class="_ _3b"></span>d</div><div class="t m0 x42 h4 y5f ff2 fs2 fc0 sc0 ls0 ws0">，<span class="ff1 lsb"> <span class="_ _b"></span> <span class="_ _19"></span> <span class="_ _12"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">2</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></div><div class="t m0 x21 h4 y60 ff1 fs2 fc0 sc0 lsb ws0"> 3<span class="_ _1f"></span><span class="ff2 ls0">）<span class="ff1 lsc">Chebyshe<span class="_ _1"></span>v<span class="_"> </span><span class="ff2 ls0">距离<span class="ff1"> </span></span></span></span></div><div class="t m0 x21 h4 y61 ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x43 h21 y62 ff4 fsf fc0 sc0 ls0 ws0">k<span class="_ _2a"></span>k</div><div class="t m0 x44 h21 y63 ff4 fsf fc0 sc0 ls0 ws0">p<span class="_ _3c"></span>k</div><div class="t m0 x23 h22 y61 ff4 fs10 fc0 sc0 ls0 ws0">y<span class="_ _20"></span>x<span class="_ _3d"></span>y<span class="_ _16"></span>x<span class="_ _3e"></span>d<span class="_ _3f"> </span><span class="ff5 fs5">−<span class="_ _40"></span>=</span></div><div class="t m0 x39 h25 y64 ff5 fs4 fc0 sc0 ls0 ws0">≤<span class="_ _41"></span>≤</div><div class="t m0 x3b h25 y65 ff5 fs4 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x45 h26 y64 ff1 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x45 h27 y61 ff1 fs5 fc0 sc0 ls0 ws0">max<span class="_ _42"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(<span class="_ _43"> </span><span class="ff2 fs2">。<span class="_ _3"></span><span class="ff1 lsd"> <span class="_ _4"></span> <span class="_ _4"></span> <span class="_ _4"></span> <span class="_ _3"></span><span class="ff2 ls0">（<span class="ff1">3</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></span></div><div class="t m0 x21 h4 y66 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">在<span class="_"> </span></span><span class="lse">Minkowsk<span class="_ _1"></span>i<span class="_ _45"> </span><span class="ff2 ls0">距离中，最常用的是欧氏距离，它的主要优点是当坐标轴进行正交</span></span></div><div class="t m0 x21 h5 y67 ff2 fs2 fc0 sc0 ls0 ws0">旋转时，<span class="_ _2"></span>欧氏距离是保持不变的。<span class="_ _2"></span>因此，<span class="_ _2"></span>如果对原坐标系进行平移和旋转变换，<span class="_ _2"></span>则变换</div><div class="t m0 x21 h4 y68 ff2 fs2 fc0 sc0 ls0 ws0">后样本点间的距离和变换前完全相同。<span class="ff1"> </span></div><div class="t m0 x21 h4 y69 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">值得注意的是在采用<span class="_"> </span></span><span class="lsf">Mink<span class="_ _1"></span>owski<span class="_ _45"> </span><span class="ff2 ls0">距离时，一定要采用相同量纲的变量。如果变量</span></span></div><div class="t m0 x21 h5 y6a ff2 fs2 fc0 sc0 ls0 ws0">的量纲不同，<span class="_ _3"></span>测量值变异范围相差悬殊时，<span class="_ _3"></span>建议首先进行数据的标准化处理，<span class="_ _3"></span>然后再计</div><div class="t m0 x21 h4 y6b ff2 fs2 fc0 sc0 ls10 ws0">算距离。在采用<span class="_ _45"> </span><span class="ff1 lse">Minkowski<span class="_ _46"> </span></span>距离时，还应尽可能地避免变量的多重相关性</div><div class="t m0 x21 h4 y6c ff2 fs2 fc0 sc0 ls0 ws0">（<span class="ff1 ls11">multicollinearity</span>）<span class="_ _37"></span><span class="ls5">。多重相关性所造成的信息重叠，会片面强调某些变量的重要性。</span></div><div class="t m0 x21 h4 y6d ff2 fs2 fc0 sc0 ls0 ws0">由于<span class="_ _29"> </span><span class="ff1 lsf">Minkowski<span class="_"> </span></span>距离的这些缺点，一种改进的距离就是马氏距离，定义如下<span class="ff1"> </span></div><div class="t m0 x21 h4 y6e ff1 fs2 fc0 sc0 lsb ws0"> 4<span class="_ _1f"></span><span class="ff2 ls0">）马氏（<span class="ff1 ls12">Ma<span class="_ _1"></span>halanobis<span class="ff2 ls0">）距离<span class="ff1"> </span></span></span></span></div><div class="t m0 x21 h4 y6f ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x46 h28 y70 ff1 fs11 fc0 sc0 ls0 ws0">)<span class="_ _47"></span>(<span class="_ _48"></span>)<span class="_ _47"></span>(<span class="_ _20"></span>)<span class="_ _12"></span>,<span class="_ _13"></span>(</div><div class="t m0 x2c h29 y71 ff1 fs12 fc0 sc0 ls0 ws0">1</div><div class="t m0 x47 h2a y70 ff4 fs11 fc0 sc0 ls0 ws0">y<span class="_ _3e"></span>x<span class="_ _49"></span>y<span class="_ _3e"></span>x<span class="_ _4a"></span>y<span class="_ _16"></span>x<span class="_ _e"></span>d</div><div class="t m0 x15 h2b y71 ff4 fs12 fc0 sc0 ls0 ws0">T</div><div class="t m0 x48 h2c y70 ff5 fs11 fc0 sc0 ls0 ws0">−<span class="_ _4b"></span>Σ<span class="_ _4c"></span>−<span class="_ _4d"></span>=</div><div class="t m0 x49 h2d y71 ff5 fs12 fc0 sc0 ls0 ws0">−</div><div class="t m0 x4a h4 y72 ff1 fs2 fc0 sc0 lsb ws0"> <span class="_ _4e"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">4</span>）<span class="_ _37"></span><span class="ff1"> </span></span></div><div class="t m0 x21 h5 y73 ff2 fs2 fc0 sc0 ls0 ws0">其中</div><div class="t m0 x4b h12 y74 ff4 fs9 fc0 sc0 ls0 ws0">y</div><div class="c x4c y75 w6 h6"><div class="t m0 x0 h12 y76 ff4 fs9 fc0 sc0 ls0 ws0">x</div></div><div class="t m0 x4d h2e y74 ff1 fs3 fc0 sc0 ls0 ws0">,<span class="_ _4f"> </span><span class="ff2 fs2">为来自</span></div><div class="c x4e y75 w2 h6"><div class="t m0 x7 h12 y76 ff4 fs9 fc0 sc0 ls0 ws0">p</div></div><div class="t m0 x4f h5 y74 ff2 fs2 fc0 sc0 ls0 ws0">维总体</div><div class="c x31 y77 wa h2f"><div class="t m0 x0 h12 y78 ff4 fs9 fc0 sc0 ls0 ws0">Z</div></div><div class="t m0 x18 h5 y74 ff2 fs2 fc0 sc0 ls0 ws0">的样本观测值，</div><div class="c x50 y79 wb h30"><div class="t m0 x0 h2c y7a ff5 fs11 fc0 sc0 ls0 ws0">Σ</div></div><div class="t m0 x11 h5 y74 ff2 fs2 fc0 sc0 ls0 ws0">为</div><div class="c x51 y77 wa h2f"><div class="t m0 x0 h12 y78 ff4 fs9 fc0 sc0 ls0 ws0">Z</div></div><div class="t m0 x52 h2c y74 ff2 fs2 fc0 sc0 ls0 ws0">的协方差矩阵，实际中<span class="_ _50"> </span><span class="ff5 fs11">Σ<span class="_ _9"> </span></span>往往是不</div><div class="t m0 x21 h5 y7b ff2 fs2 fc0 sc0 ls0 ws0">知道的，<span class="_ _3"></span>常常需要用样本协方差来估计。<span class="_ _3"></span>马氏距离对一切线性变换是不变的，<span class="_ _3"></span>故不受量</div><div class="t m0 x21 h4 y7c ff2 fs2 fc0 sc0 ls0 ws0">纲的影响。<span class="ff1"> </span></div><div class="t m0 x21 h4 y7d ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">此外，<span class="_ _3"></span>还可采用样本相关系数、<span class="_ _3"></span>夹角余弦和其它关联性度量作为相似性度量。<span class="_ _3"></span>近年</span></div><div class="t m0 x21 h4 y7e ff2 fs2 fc0 sc0 ls0 ws0">来随着数据挖掘研究的深入，这方面的新方法层出不穷。<span class="ff1"> </span></div><div class="t m0 x21 h4 y7f ff1 fs2 fc0 sc0 ls8 ws6"> 1.1.2 <span class="_ _12"></span> <span class="_ _12"></span><span class="ff2 ls0 ws0">类与类间的相似性度量<span class="ff1"> </span></span></div><div class="t m0 x21 h4 y80 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">如果有两个样本类</span></div><div class="t m0 x33 h31 y81 ff1 fs13 fc0 sc0 ls0 ws0">1</div><div class="t m0 x53 h32 y82 ff4 fs14 fc0 sc0 ls0 ws0">G<span class="_ _51"> </span><span class="ff2 fs2">和</span></div><div class="t m0 x54 h31 y81 ff1 fs13 fc0 sc0 ls0 ws0">2</div><div class="t m0 x55 h32 y82 ff4 fs14 fc0 sc0 ls0 ws0">G<span class="_ _52"> </span><span class="ff2 fs2">，我们可以用下面的一系列方法度量它们间的距离：<span class="ff1"> </span></span></div><div class="t m0 x21 h4 y83 ff1 fs2 fc0 sc0 lsb ws0"> 1<span class="_ _1f"></span><span class="ff2 ls0">）最短距离<span class="_ _1"></span>法（<span class="ff1 ls13 ws7">nearest neighbor or si<span class="_ _1"></span>ngle linkage m<span class="_ _1"></span>ethod<span class="ff2 ls0 ws0">）<span class="ff1"> </span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0"> <span class="_ _0"> </span><span class="ff2">-445-</span></div><div class="t m0 x4 h4 y84 ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 xb h33 y85 ff1 fs15 fc0 sc0 ls0 ws0">)}<span class="_ _39"></span>,<span class="_ _16"></span>(<span class="_ _16"></span>{<span class="_ _d"></span>min<span class="_ _53"></span>)<span class="_ _3b"></span>,<span class="_ _54"></span>(</div><div class="t m0 x34 h34 y86 ff1 fs16 fc0 sc0 ls0 ws0">2</div><div class="t m0 x26 h34 y87 ff1 fs16 fc0 sc0 ls0 ws0">1</div><div class="t m0 x45 h35 y88 ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _55"></span>1<span class="_ _56"> </span><span class="ff4">j<span class="_ _e"></span>i</span></div><div class="t m0 x3d h36 y89 ff4 fs17 fc0 sc0 ls0 ws0">G<span class="_ _24"></span>y</div><div class="t m0 x56 h36 y8a ff4 fs17 fc0 sc0 ls0 ws0">G<span class="_ _57"></span>x</div><div class="t m0 x42 h37 y85 ff4 fs15 fc0 sc0 ls0 ws0">y<span class="_ _58"></span>x<span class="_ _e"></span>d<span class="_ _59"></span>G<span class="_ _5a"></span>G<span class="_ _3b"></span>D</div><div class="t m0 x32 h38 y86 ff4 fs16 fc0 sc0 ls0 ws0">j</div><div class="t m0 x3 h38 y87 ff4 fs16 fc0 sc0 ls0 ws0">i</div><div class="t m0 x27 h39 y8b ff5 fs17 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x32 h39 y8c ff5 fs17 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x18 h3a y85 ff5 fs15 fc0 sc0 ls0 ws0">=</div><div class="t m0 x4a h4 y84 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="ff1 lsb"> <span class="_ _16"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span> <span class="_ _12"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">5</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></div><div class="t m0 x4 h4 y8d ff2 fs2 fc0 sc0 ls0 ws0">它的直观意义为两个类中最近两点间的距离。<span class="ff1"> </span></div><div class="t m0 x57 h4 y8e ff1 fs2 fc0 sc0 ls0 ws0">2<span class="ff2">）最长距离法（</span><span class="ls14 ws8">farthest neighbor or <span class="_ _1"></span>complete linkag<span class="_ _1"></span>e method<span class="ff2 ls0 ws0">）<span class="ff1"> </span></span></span></div><div class="t m0 x4 h4 y8f ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x58 h33 y90 ff1 fs15 fc0 sc0 ls0 ws0">)}<span class="_ _39"></span>,<span class="_ _16"></span>(<span class="_ _16"></span>{<span class="_ _39"></span>max<span class="_ _4d"></span>)<span class="_ _3b"></span>,<span class="_ _54"></span>(</div><div class="t m0 x59 h34 y91 ff1 fs16 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5a h34 y92 ff1 fs16 fc0 sc0 ls0 ws0">1</div><div class="t m0 x45 h35 y93 ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _55"></span>1<span class="_ _5b"> </span><span class="ff4">j<span class="_ _e"></span>i</span></div><div class="t m0 x5b h36 y94 ff4 fs17 fc0 sc0 ls0 ws0">G<span class="_ _24"></span>y</div><div class="t m0 x3d h36 y95 ff4 fs17 fc0 sc0 ls0 ws0">G<span class="_ _57"></span>x</div><div class="t m0 x2 h37 y90 ff4 fs15 fc0 sc0 ls0 ws0">y<span class="_ _58"></span>x<span class="_ _e"></span>d<span class="_ _5c"></span>G<span class="_ _5a"></span>G<span class="_ _3b"></span>D</div><div class="t m0 x27 h38 y91 ff4 fs16 fc0 sc0 ls0 ws0">j</div><div class="t m0 x5c h38 y92 ff4 fs16 fc0 sc0 ls0 ws0">i</div><div class="t m0 xf h39 y96 ff5 fs17 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x27 h39 y97 ff5 fs17 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x18 h3a y90 ff5 fs15 fc0 sc0 ls0 ws0">=</div><div class="t m0 x5d h4 y98 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="ff1 lsb"> <span class="_ _5d"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span> <span class="_ _12"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">6</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></div><div class="t m0 x4 h4 y99 ff2 fs2 fc0 sc0 ls0 ws0">它的直观意义为两个类中最远两点间的距离。<span class="ff1"> </span></div><div class="t m0 x57 h4 y9a ff1 fs2 fc0 sc0 ls0 ws0">3<span class="ff2">）重心法（</span><span class="ls7 ws9">centroid m<span class="_ _1"></span>ethod<span class="ff2 ls0 ws0">）<span class="ff1"> </span></span></span></div><div class="t m0 x4 h3b y9b ff1 fs2 fc0 sc0 lsb ws0"> <span class="_ _5e"> </span><span class="fs14 ls0">)<span class="_ _12"></span>,<span class="_ _1c"></span>(<span class="_ _5f"></span>)<span class="_ _3b"></span>,<span class="_ _54"></span>(</span></div><div class="t m0 x45 h35 y9c ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _55"></span>1</div><div class="t m0 xd h37 y9b ff4 fs15 fc0 sc0 ls0 ws0">y<span class="_ _60"></span>x<span class="_ _e"></span>d<span class="_ _4b"></span>G<span class="_ _5a"></span>G<span class="_ _3b"></span>D<span class="_ _61"> </span><span class="ff5 fs14">=<span class="_ _62"> </span><span class="ff2 fs2">，<span class="ff1 ls15"> <span class="_ _63"> </span> <span class="_ _64"> </span> <span class="_ _64"> </span> <span class="_ _64"> </span> <span class="_ _64"> </span> <span class="_ _64"> </span> <span class="_ _1"></span> <span class="ff2 ls0">（<span class="ff1">7</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></span></span></div><div class="t m0 x4 h5 y9d ff2 fs2 fc0 sc0 ls0 ws0">其中</div><div class="c xc y9e wc h3c"><div class="t m0 x7 h2a y1b ff4 fs11 fc0 sc0 ls0 ws0">y</div></div><div class="c x2d y9e w6 h3c"><div class="t m0 x0 h2a y1b ff4 fs11 fc0 sc0 ls0 ws0">x</div></div><div class="t m0 x5e h28 y9f ff1 fs11 fc0 sc0 ls0 ws0">,<span class="_ _65"> </span><span class="ff2 fs2">分别为</span></div><div class="t m0 x33 h35 ya0 ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _32"></span>1</div><div class="t m0 x5f h37 y9f ff1 fs14 fc0 sc0 ls0 ws0">,<span class="_ _66"></span><span class="ff4 fs15">G<span class="_ _67"></span>G<span class="_ _68"> </span><span class="ff2 fs2">的重心。<span class="ff1"> </span></span></span></div><div class="t m0 x4 h4 ya1 ff1 fs2 fc0 sc0 lsb ws0"> 4<span class="_ _5d"></span><span class="ff2 ls0">）类平均法（<span class="ff1 ls11 wsa">group average method<span class="_ _1"></span><span class="ff2 ls0 ws0">）<span class="ff1"> </span></span></span></span></div><div class="t m0 x4 h4 ya2 ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x32 h3d ya3 ff5 fs18 fc0 sc0 ls16 ws0">∑∑</div><div class="t m0 x60 h39 ya4 ff5 fs17 fc0 sc0 ls17 ws0">∈∈</div><div class="t m0 x61 h2c ya5 ff5 fs11 fc0 sc0 ls0 ws0">=</div><div class="t m0 x59 h3e ya6 ff1 fs19 fc0 sc0 ls18 ws0">12</div><div class="t m0 x5d h28 ya5 ff1 fs11 fc0 sc0 ls0 ws0">)<span class="_ _58"></span>,<span class="_ _16"></span>(</div><div class="t m0 x62 h28 ya7 ff1 fs11 fc0 sc0 ls0 ws0">1</div><div class="t m0 x63 h28 ya5 ff1 fs11 fc0 sc0 ls0 ws0">)<span class="_ _3b"></span>,<span class="_ _58"></span>(</div><div class="t m0 x64 h35 ya8 ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _13"></span>1</div><div class="t m0 x65 h35 ya9 ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _32"></span>1</div><div class="t m0 x15 h36 yaa ff4 fs17 fc0 sc0 ls0 ws0">G<span class="_ _1c"></span><span class="ls19">xG<span class="_ _69"></span><span class="ls0">x</span></span></div><div class="t m0 x66 h36 ya9 ff4 fs17 fc0 sc0 ls0 ws0">j<span class="_ _60"></span>i</div><div class="t m0 x27 h3f ya6 ff4 fs19 fc0 sc0 ls1a ws0">ij</div><div class="t m0 x67 h2a ya5 ff4 fs11 fc0 sc0 ls0 ws0">x<span class="_ _15"></span>x<span class="_ _e"></span>d</div><div class="t m0 x39 h2a yab ff4 fs11 fc0 sc0 ls0 ws0">n<span class="_ _1b"></span>n</div><div class="t m0 x2f h2a ya5 ff4 fs11 fc0 sc0 ls0 ws0">G<span class="_ _67"></span>G<span class="_ _3b"></span>D<span class="_ _6a"> </span><span class="ff2 fs2">，<span class="ff1 lsb"> <span class="_ _6"></span> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">8</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></span></div><div class="t m0 x4 h5 yac ff2 fs2 fc0 sc0 ls0 ws0">它等于</div><div class="t m0 x68 h35 yad ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _32"></span>1</div><div class="t m0 x69 h37 yae ff1 fs14 fc0 sc0 ls0 ws0">,<span class="_ _66"></span><span class="ff4 fs15">G<span class="_ _67"></span>G<span class="_ _68"> </span><span class="ff2 fs2">中两两样本点距离的平均，式中</span></span></div><div class="t m0 x6a h35 yad ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _5d"></span>1</div><div class="t m0 x52 h37 yae ff1 fs14 fc0 sc0 ls0 ws0">,<span class="_ _50"></span><span class="ff4 fs15">n<span class="_ _55"></span>n<span class="_ _a"> </span><span class="ff2 fs2">分别为</span></span></div><div class="t m0 x6b h35 yad ff1 fs17 fc0 sc0 ls0 ws0">2<span class="_ _32"></span>1</div><div class="t m0 x6c h37 yae ff1 fs14 fc0 sc0 ls0 ws0">,<span class="_ _66"></span><span class="ff4 fs15">G<span class="_ _67"></span>G<span class="_ _68"> </span><span class="ff2 fs2">中的样本点个数。<span class="ff1"> </span></span></span></div><div class="t m0 x4 h4 yaf ff1 fs2 fc0 sc0 lsb ws0"> 5<span class="_ _5d"></span><span class="ff2 ls0">）离差平方和法（<span class="ff1 ls1b wsb">sum of squares method</span>）<span class="ff1"> </span></span></div><div class="t m0 x5 h4 yb0 ff2 fs2 fc0 sc0 ls0 ws0">若记<span class="ff1"> </span></div><div class="t m0 x6d h40 yb1 ff5 fs1a fc0 sc0 ls0 ws0">∑</div><div class="t m0 x6e h41 yb2 ff5 fs1b fc0 sc0 ls0 ws0">∈</div><div class="t m0 x28 h42 yb3 ff5 fs1c fc0 sc0 ls0 ws0">−<span class="_ _3d"></span>−<span class="_ _6b"></span>=</div><div class="t m0 x4e h43 yb4 ff1 fs1d fc0 sc0 ls0 ws0">1</div><div class="t m0 xe h44 yb3 ff1 fs1c fc0 sc0 ls0 ws0">)<span class="_ _6c"></span>(<span class="_ _22"></span>)<span class="_ _6c"></span>(</div><div class="t m0 x26 h45 yb5 ff1 fs1b fc0 sc0 ls0 ws0">1<span class="_ _6b"></span>1<span class="_ _6d"></span>1</div><div class="t m0 x36 h46 yb6 ff4 fs1b fc0 sc0 ls0 ws0">G<span class="_ _57"></span>x</div><div class="t m0 x44 h46 yb5 ff4 fs1b fc0 sc0 ls0 ws0">i</div><div class="t m0 x45 h46 yb7 ff4 fs1b fc0 sc0 ls0 ws0">T</div><div class="t m0 x2f h46 yb5 ff4 fs1b fc0 sc0 ls0 ws0">i</div><div class="t m0 x6f h47 yb4 ff4 fs1d fc0 sc0 ls0 ws0">i</div><div class="t m0 x56 h48 yb3 ff4 fs1c fc0 sc0 ls0 ws0">x<span class="_ _6e"></span>x<span class="_ _6f"></span>x<span class="_ _6e"></span>x<span class="_ _38"></span>D</div><div class="t m0 xd h5 yb8 ff2 fs2 fc0 sc0 ls0 ws0">，</div><div class="c x70 yb9 wd h49"><div class="t m0 x0 h3d yba ff5 fs18 fc0 sc0 ls0 ws0">∑</div></div><div class="t m0 x71 h41 yb2 ff5 fs1b fc0 sc0 ls0 ws0">∈</div><div class="t m0 x72 h2c yb8 ff5 fs11 fc0 sc0 ls0 ws0">−<span class="_ _70"></span>−<span class="_ _3d"></span>=</div><div class="t m0 x73 h43 yb4 ff1 fs1d fc0 sc0 ls0 ws0">2</div><div class="t m0 x74 h28 yb8 ff1 fs11 fc0 sc0 ls0 ws0">)<span class="_ _71"></span>(<span class="_ _22"></span>)<span class="_ _71"></span>(</div><div class="t m0 x6c h45 ybb ff1 fs1b fc0 sc0 ls0 ws0">2<span class="_ _72"></span>2<span class="_ _73"></span>2</div><div class="t m0 x5d h46 ybc ff4 fs1b fc0 sc0 ls0 ws0">G<span class="_ _24"></span>x</div><div class="t m0 x75 h46 ybb ff4 fs1b fc0 sc0 ls0 ws0">j</div><div class="t m0 x76 h46 ybd ff4 fs1b fc0 sc0 ls0 ws0">T</div><div class="t m0 x1e h46 ybb ff4 fs1b fc0 sc0 ls0 ws0">j</div><div class="t m0 x77 h47 yb4 ff4 fs1d fc0 sc0 ls0 ws0">j</div><div class="t m0 x78 h2a yb8 ff4 fs11 fc0 sc0 ls0 ws0">x<span class="_ _1d"></span>x<span class="_ _74"></span>x<span class="_ _1d"></span>x<span class="_ _75"></span>D<span class="_ _76"> </span><span class="ff2 fs2">，<span class="ff1"> </span></span></div><div class="t m0 x4e h4a ybe ff5 fs1e fc0 sc0 ls0 ws0">∑</div><div class="t m0 x79 h4b ybf ff5 fs1f fc0 sc0 ls0 ws0">∈</div><div class="t m0 x26 h4c yc0 ff5 fs20 fc0 sc0 ls0 ws0">−<span class="_ _2d"></span>−<span class="_ _77"></span>=</div><div class="t m0 x2b h4d yc1 ff1 fs21 fc0 sc0 ls0 ws0">2<span class="_ _24"></span>1</div><div class="t m0 x7a h4e yc0 ff1 fs20 fc0 sc0 ls0 ws0">)<span class="_ _42"></span>(<span class="_ _1b"></span>)<span class="_ _42"></span>(</div><div class="t m0 x7b h4f yc2 ff1 fs1f fc0 sc0 ls0 ws0">12</div><div class="t m0 x13 h50 yc3 ff4 fs1f fc0 sc0 ls0 ws0">G<span class="_ _60"></span>G<span class="_ _24"></span>x</div><div class="t m0 x27 h50 yc2 ff4 fs1f fc0 sc0 ls0 ws0">k</div><div class="t m0 x39 h50 yc4 ff4 fs1f fc0 sc0 ls0 ws0">T</div><div class="t m0 x25 h50 yc2 ff4 fs1f fc0 sc0 ls0 ws0">k</div><div class="t m0 x36 h51 yc1 ff4 fs21 fc0 sc0 ls0 ws0">k</div><div class="t m0 xd h52 yc0 ff4 fs20 fc0 sc0 ls0 ws0">x<span class="_ _1d"></span>x<span class="_ _21"></span>x<span class="_ _1d"></span>x<span class="_ _78"></span>D</div><div class="t m0 x7c h53 ybf ff6 fs1f fc0 sc0 ls0 ws0">∪</div><div class="t m0 x7d h4 yc5 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="ff1"> </span></div><div class="t m0 x4 h4 yc6 ff2 fs2 fc0 sc0 ls0 ws0">其中<span class="ff1"> </span></div><div class="t m0 x4 h4 yc7 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x7e h54 yc8 ff5 fs22 fc0 sc0 ls0 ws0">∑</div><div class="t m0 x14 h55 yc9 ff5 fs23 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x7f h56 yca ff5 fs24 fc0 sc0 ls0 ws0">=</div><div class="t m0 x4f h57 ycb ff1 fs25 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6f h58 ycc ff1 fs23 fc0 sc0 ls0 ws0">1</div><div class="t m0 x80 h58 ycd ff1 fs23 fc0 sc0 ls0 ws0">1</div><div class="t m0 x81 h59 yce ff1 fs24 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2e h5a yc9 ff4 fs23 fc0 sc0 ls0 ws0">G<span class="_ _1c"></span>x</div><div class="t m0 x2b h5a ycf ff4 fs23 fc0 sc0 ls0 ws0">i</div><div class="t m0 x4e h5b ycb ff4 fs25 fc0 sc0 ls0 ws0">i</div><div class="t m0 x13 h5c yca ff4 fs24 fc0 sc0 ls0 ws0">x</div><div class="t m0 x82 h5c yd0 ff4 fs24 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2d h5c yca ff4 fs24 fc0 sc0 ls0 ws0">x</div><div class="t m0 x25 h5 yd1 ff2 fs2 fc0 sc0 ls0 ws0">，</div><div class="t m0 x5a h3d yd2 ff5 fs18 fc0 sc0 ls0 ws0">∑</div><div class="t m0 x83 h55 yd3 ff5 fs23 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x62 h2c yd1 ff5 fs11 fc0 sc0 ls0 ws0">=</div><div class="t m0 x49 h57 yd4 ff1 fs25 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5c h58 yd5 ff1 fs23 fc0 sc0 ls0 ws0">2</div><div class="t m0 x31 h58 yd6 ff1 fs23 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 h28 yd7 ff1 fs11 fc0 sc0 ls0 ws0">1</div><div class="t m0 x16 h5a yd3 ff4 fs23 fc0 sc0 ls0 ws0">G<span class="_ _24"></span>x</div><div class="t m0 x84 h5a yd8 ff4 fs23 fc0 sc0 ls0 ws0">j</div><div class="t m0 x59 h5b yd4 ff4 fs25 fc0 sc0 ls0 ws0">j</div><div class="t m0 x7a h2a yd1 ff4 fs11 fc0 sc0 ls0 ws0">x</div><div class="t m0 x85 h2a yd9 ff4 fs11 fc0 sc0 ls0 ws0">n</div><div class="t m0 x30 h2a yd1 ff4 fs11 fc0 sc0 ls0 ws0">x<span class="_ _79"> </span><span class="ff2 fs2">，</span></div><div class="t m0 x86 h54 yc8 ff5 fs22 fc0 sc0 ls0 ws0">∑</div><div class="t m0 x87 h55 yc9 ff5 fs23 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x88 h56 yda ff5 fs24 fc0 sc0 ls0 ws0">+</div><div class="t m0 x67 h56 ydb ff5 fs24 fc0 sc0 ls0 ws0">=</div><div class="t m0 x89 h57 ycb ff1 fs25 fc0 sc0 ls0 ws0">2<span class="_ _24"></span>1</div><div class="t m0 x8a h58 ycc ff1 fs23 fc0 sc0 ls0 ws0">2<span class="_ _6e"></span>1</div><div class="t m0 x8b h59 yce ff1 fs24 fc0 sc0 ls0 ws0">1</div><div class="t m0 x8c h5a yc9 ff4 fs23 fc0 sc0 ls0 ws0">G<span class="_ _60"></span>G<span class="_ _24"></span>x</div><div class="t m0 x8d h5a ycf ff4 fs23 fc0 sc0 ls0 ws0">k</div><div class="t m0 x8e h5b ycb ff4 fs25 fc0 sc0 ls0 ws0">k</div><div class="t m0 x8f h5c yca ff4 fs24 fc0 sc0 ls0 ws0">x</div><div class="t m0 x90 h5c yd0 ff4 fs24 fc0 sc0 ls0 ws0">n<span class="_ _5f"></span>n</div><div class="t m0 x9 h5c yca ff4 fs24 fc0 sc0 ls0 ws0">x</div><div class="t m0 x91 h5d yc9 ff6 fs23 fc0 sc0 ls0 ws0">∪</div><div class="t m0 x72 h4 yd1 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x4 h4 ydc ff2 fs2 fc0 sc0 ls0 ws0">则定义<span class="ff1"> </span></div><div class="t m0 x4 h4 ydd ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x1a h58 yde ff1 fs23 fc0 sc0 ls0 ws0">2<span class="_ _7a"></span>1<span class="_ _74"></span>12<span class="_ _7b"></span>2<span class="_ _55"></span>1</div><div class="t m0 x25 h59 ydf ff1 fs24 fc0 sc0 ls0 ws0">)<span class="_ _3b"></span>,<span class="_ _54"></span>(<span class="_ _7c"> </span><span class="ff4">D<span class="_ _7d"></span>D<span class="_ _7b"></span>D<span class="_ _7e"></span>G<span class="_ _67"></span>G<span class="_ _3b"></span>D</span></div><div class="c x41 ye0 we h5e"><div class="t m0 x0 h56 ye1 ff5 fs24 fc0 sc0 ls0 ws0">−</div></div><div class="t m0 x92 h56 ydf ff5 fs24 fc0 sc0 ls0 ws0">−<span class="_ _7f"></span>=<span class="_ _80"> </span><span class="ff1 fs2 lsb"> <span class="_ _12"></span> <span class="_ _19"></span> <span class="_ _12"></span><span class="ff2 ls0">（<span class="ff1">9</span>）<span class="_ _37"></span><span class="ff1"> </span></span></span></div><div class="t m0 x4 h5 ye2 ff2 fs2 fc0 sc0 ls1c ws0">事实上，若</div><div class="t m0 x93 h58 ye3 ff1 fs23 fc0 sc0 ls0 ws0">2<span class="_ _55"></span>1</div><div class="t m0 x94 h59 ye4 ff1 fs24 fc0 sc0 ls0 ws0">,<span class="_ _66"></span><span class="ff4">G<span class="_ _67"></span>G<span class="_ _81"> </span><span class="ff2 fs2 ls1c">内部点与点距离很小，<span class="_ _5"></span>则它们能很好地各自聚为一类，<span class="_ _1"></span>并且这两类</span></span></div><div class="t m0 x4 h5 ye5 ff2 fs2 fc0 sc0 ls0 ws0">又能够充分分离<span class="_ _5"></span>（即</div><div class="t m0 x95 h58 ye6 ff1 fs23 fc0 sc0 ls0 ws0">12</div><div class="t m0 x96 h5c ye7 ff4 fs24 fc0 sc0 ls0 ws0">D<span class="_ _82"> </span><span class="ff2 fs2">很大）<span class="_ _37"></span>，<span class="_ _5"></span>这时必<span class="_ _1"></span>然有</span></div><div class="t m0 x97 h58 ye6 ff1 fs23 fc0 sc0 ls0 ws0">2<span class="_ _25"></span>1<span class="_ _74"></span>12</div><div class="t m0 x98 h5c ye7 ff4 fs24 fc0 sc0 ls0 ws0">D<span class="_ _83"></span>D<span class="_ _84"></span>D<span class="_ _48"></span>D</div><div class="c x99 ye8 we h5e"><div class="t m0 x0 h56 ye9 ff5 fs24 fc0 sc0 ls0 ws0">−</div></div><div class="c x9a ye8 we h5e"><div class="t m0 x0 h56 ye9 ff5 fs24 fc0 sc0 ls0 ws0">−</div></div><div class="c x1e ye8 we h5e"><div class="t m0 x0 h56 ye9 ff5 fs24 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x9b h5 ye7 ff2 fs2 fc0 sc0 ls0 ws0">很大。<span class="_ _5"></span>因此，<span class="_ _5"></span>按定义可</div><div class="t m0 x4 h5 yea ff2 fs2 fc0 sc0 ls0 ws0">以认为，两类</div><div class="t m0 x24 h58 yeb ff1 fs23 fc0 sc0 ls0 ws0">2<span class="_ _32"></span>1</div><div class="t m0 x9c h59 yec ff1 fs24 fc0 sc0 ls0 ws0">,<span class="_ _66"></span><span class="ff4">G<span class="_ _67"></span>G<span class="_ _68"> </span><span class="ff2 fs2">之间的距离很大。离差平方和法最初是由<span class="_ _85"> </span><span class="ff1 ls1d">Wa<span class="_ _86"></span>r<span class="_ _86"></span>d<span class="_ _87"> </span></span>在<span class="_ _85"> </span><span class="ff1 ls1e">1936<span class="_ _85"> </span></span>年提出，</span></span></div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/bg4.jpg"><div class="t m0 x21 h2 y1 ff1 fs0 fc0 sc0 lsa ws0"> </div><div class="t m0 x21 h2 y2 ff2 fs0 fc0 sc0 ls0 ws0">-446-<span class="ff1"> </span></div><div class="t m0 x21 h4 yed ff2 fs2 fc0 sc0 ls0 ws0">后经<span class="_ _29"> </span><span class="ff1 lsc">Orloci<span class="_"> </span></span>等人<span class="_ _29"> </span><span class="ff1 ls1e">1976<span class="_"> </span></span>年发展起来的，故又称为<span class="_ _29"> </span><span class="ff1 ls1d">Wa<span class="_ _86"></span>r<span class="_ _86"></span>d<span class="_ _85"> </span></span>方法。<span class="ff1"> </span></div><div class="t m0 x21 h4 yee ff1 fs2 fc0 sc0 ls7 wsc"> 1.2 <span class="_ _12"></span> <span class="_ _19"></span><span class="ff2 ls0 ws0">系统聚类法<span class="ff1"> </span></span></div><div class="t m0 x21 h4 yef ff1 fs2 fc0 sc0 ls8 ws6"> 1.2.1 <span class="_ _12"></span> <span class="_ _12"></span><span class="ff2 ls0 ws0">系统聚类法的功能与特点<span class="ff1"> </span></span></div><div class="t m0 x21 h4 yf0 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">系统聚类法是聚类分析方法中最常用的一种方法。<span class="_ _37"></span>它的优点在于可以指出由粗到细</span></div><div class="t m0 x21 h4 yf1 ff2 fs2 fc0 sc0 ls0 ws0">的多种分类情况，典型的系统聚类结果可由一个聚类图展示出来。<span class="ff1"> </span></div><div class="t m0 x9d h4 yf2 ff2 fs2 fc0 sc0 ls0 ws0">例如，<span class="_ _3"></span>在平面上有<span class="_ _9"> </span><span class="ff1">7<span class="_ _88"> </span></span>个点</div><div class="t m0 x9 h58 yf3 ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _4d"></span>2<span class="_ _4e"></span>1</div><div class="t m0 x9e h28 yf4 ff1 fs11 fc0 sc0 ls0 ws0">,<span class="_ _15"></span>,<span class="_ _4e"></span>,<span class="_ _33"> </span><span class="ff4">w<span class="_ _71"></span>w<span class="_ _d"></span>w<span class="_ _89"> </span><span class="ff6 fs5"><span class="_ _8a"> </span><span class="ff2 fs2">（如图<span class="_ _29"> </span><span class="ff1">1<span class="_ _3"></span><span class="ff2">（<span class="ff1">a<span class="_ _1"></span><span class="ff2">）<span class="_ _37"></span>）<span class="_ _8b"></span>，<span class="_ _8c"></span>可以用聚类图<span class="_ _8c"></span>（如图<span class="_ _29"> </span><span class="ff1">1<span class="_ _8c"></span><span class="ff2">（<span class="ff1">b</span>）<span class="_ _8b"></span>）</span></span></span></span></span></span></span></span></span></div><div class="t m0 x21 h4 y9 ff2 fs2 fc0 sc0 ls0 ws0">来表示聚类结果。<span class="ff1"> </span></div><div class="t m0 x9f h4 yf5 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x9d h4 yf6 ff1 fs2 fc0 sc0 lsd ws0"> <span class="_ _4"></span> </div><div class="t m0 x2b h2 yf7 ff2 fs0 fc0 sc0 ls0 ws0">图<span class="_ _9"> </span><span class="ff1 wsd">1 </span>聚类方法示意图<span class="ff1"> </span></div><div class="t m0 x9d h28 yf8 ff2 fs2 fc0 sc0 ls0 ws0">记<span class="_ _8d"> </span><span class="ff1 fs11">}<span class="_ _11"></span>,<span class="_ _15"></span>,<span class="_ _55"></span>,<span class="_ _4e"></span>{</span></div><div class="t m0 x45 h58 yf9 ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _4d"></span>2<span class="_ _4e"></span>1</div><div class="t m0 x61 h2a yf8 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _71"></span>w<span class="_ _d"></span>w<span class="_ _89"> </span><span class="ff6 fs5"><span class="_ _8e"></span><span class="ff5 fs11">=<span class="_ _54"></span>Ω<span class="_ _8f"> </span><span class="ff2 fs2">，聚类结果如下：当距离值为</span></span></span></div><div class="t m0 xa0 h58 yf9 ff1 fs23 fc0 sc0 ls0 ws0">5</div><div class="t m0 xa1 h2a yf8 ff4 fs11 fc0 sc0 ls0 ws0">f<span class="_ _10"> </span><span class="ff2 fs2">时，分为一类<span class="ff1"> </span></span></div><div class="t m0 x9d h4 yfa ff1 fs2 fc0 sc0 lsb ws0"> </div><div class="t m0 x71 h28 yfb ff1 fs11 fc0 sc0 ls0 ws0">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _55"></span>,<span class="_ _4e"></span>,<span class="_ _4e"></span>,<span class="_ _55"></span>,<span class="_ _58"></span>{</div><div class="t m0 x46 h58 yfc ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _32"></span>6<span class="_ _55"></span>5<span class="_ _55"></span>4<span class="_ _32"></span>3<span class="_ _55"></span>2<span class="_ _55"></span>1<span class="_ _7d"></span>1</div><div class="t m0 xa2 h2a yfb ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _67"></span>w<span class="_ _67"></span>w<span class="_ _6e"></span>w<span class="_ _67"></span>w<span class="_ _67"></span>w<span class="_ _90"></span>w<span class="_ _42"></span>G<span class="_ _91"> </span><span class="ff5">=<span class="_ _92"> </span><span class="ff2 fs2">；<span class="ff1"> </span></span></span></div><div class="t m0 x9d h5 yfd ff2 fs2 fc0 sc0 ls0 ws0">距离值为</div><div class="t m0 x36 h58 yfe ff1 fs23 fc0 sc0 ls0 ws0">4</div><div class="t m0 x6e h5c yff ff4 fs24 fc0 sc0 ls0 ws0">f<span class="_ _91"> </span><span class="ff2 fs2">分为两类：<span class="ff1"> </span></span></div><div class="t m0 x9d h4 y100 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x45 h28 y101 ff1 fs11 fc0 sc0 ls0 ws0">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _58"></span>{</div><div class="t m0 x30 h58 y102 ff1 fs23 fc0 sc0 ls0 ws0">3<span class="_ _55"></span>2<span class="_ _55"></span>1<span class="_ _7d"></span>1</div><div class="t m0 xa3 h28 y101 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _6e"></span>w<span class="_ _d"></span>w<span class="_ _42"></span>G<span class="_ _91"> </span><span class="ff5">=<span class="_ _93"> </span><span class="ff2 fs2">，<span class="_ _43"> </span></span><span class="ff1">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _4e"></span>,<span class="_ _94"></span>{</span></span></div><div class="t m0 x8b h58 y102 ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _32"></span>6<span class="_ _55"></span>5<span class="_ _32"></span>4<span class="_ _47"></span>2</div><div class="t m0 xa4 h2a y101 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _67"></span>w<span class="_ _67"></span>w<span class="_ _6e"></span>w<span class="_ _7e"></span>G</div><div class="c x5a y103 w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x52 h4 y101 ff2 fs2 fc0 sc0 ls0 ws0">；<span class="ff1"> </span></div><div class="t m0 x9d h5 y104 ff2 fs2 fc0 sc0 ls0 ws0">距离值为</div><div class="t m0 x36 h58 y105 ff1 fs23 fc0 sc0 ls0 ws0">3</div><div class="t m0 x6e h2a y106 ff4 fs11 fc0 sc0 ls0 ws0">f<span class="_ _95"> </span><span class="ff2 fs2">分为三类：<span class="ff1"> </span></span></div><div class="t m0 x9d h4 y107 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x45 h28 y74 ff1 fs11 fc0 sc0 ls0 ws0">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _58"></span>{</div><div class="t m0 x30 h58 y108 ff1 fs23 fc0 sc0 ls0 ws0">3<span class="_ _55"></span>2<span class="_ _55"></span>1<span class="_ _7d"></span>1</div><div class="t m0 xa3 h28 y74 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _6e"></span>w<span class="_ _d"></span>w<span class="_ _42"></span>G<span class="_ _91"> </span><span class="ff5">=<span class="_ _93"> </span><span class="ff2 fs2">，<span class="_ _96"> </span></span><span class="ff1">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _94"></span>{</span></span></div><div class="t m0 x66 h58 y108 ff1 fs23 fc0 sc0 ls0 ws0">6<span class="_ _55"></span>5<span class="_ _32"></span>4<span class="_ _97"></span>2</div><div class="t m0 x58 h2a y74 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _67"></span>w<span class="_ _6e"></span>w<span class="_ _7e"></span>G</div><div class="c x5a y75 w5 h5f"><div class="t m0 x0 h2c y76 ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x1f h28 y74 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _98"> </span><span class="ff1 fs11">}<span class="_ _3e"></span>{</span></div><div class="t m0 x8c h58 y108 ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _47"></span>3</div><div class="t m0 xa5 h2a y74 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _69"></span>G</div><div class="c xa6 y75 w5 h5f"><div class="t m0 x0 h2c y76 ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xa7 h4 y74 ff2 fs2 fc0 sc0 ls0 ws0">；<span class="ff1"> </span></div><div class="t m0 x9d h5 y109 ff2 fs2 fc0 sc0 ls0 ws0">距离值为</div><div class="t m0 x36 h58 y10a ff1 fs23 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6e h5c y10b ff4 fs24 fc0 sc0 ls0 ws0">f<span class="_ _91"> </span><span class="ff2 fs2">分为四类：<span class="ff1"> </span></span></div><div class="t m0 x9d h4 y10c ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x45 h28 y29 ff1 fs11 fc0 sc0 ls0 ws0">}<span class="_ _11"></span>,<span class="_ _55"></span>,<span class="_ _58"></span>{</div><div class="t m0 x30 h58 y10d ff1 fs23 fc0 sc0 ls0 ws0">3<span class="_ _55"></span>2<span class="_ _55"></span>1<span class="_ _7d"></span>1</div><div class="t m0 xa3 h28 y29 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _6e"></span>w<span class="_ _d"></span>w<span class="_ _42"></span>G<span class="_ _91"> </span><span class="ff5">=<span class="_ _93"> </span><span class="ff2 fs2">，<span class="_ _99"> </span></span><span class="ff1">}<span class="_ _11"></span>,<span class="_ _94"></span>{</span></span></div><div class="t m0 xa8 h58 y10d ff1 fs23 fc0 sc0 ls0 ws0">5<span class="_ _55"></span>4<span class="_ _97"></span>2</div><div class="t m0 xa h2a y29 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _6e"></span>w<span class="_ _7e"></span>G</div><div class="c x5a y2a w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x46 h28 y29 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _98"> </span><span class="ff1 fs11">}<span class="_ _9a"></span>{</span></div><div class="t m0 xa9 h58 y10d ff1 fs23 fc0 sc0 ls0 ws0">6<span class="_ _47"></span>3</div><div class="t m0 xaa h2a y29 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _69"></span>G</div><div class="c xab y2a w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xac h28 y29 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _9b"> </span><span class="ff1 fs11">}<span class="_ _3e"></span>{</span></div><div class="t m0 xad h58 y10d ff1 fs23 fc0 sc0 ls0 ws0">7<span class="_ _97"></span>4</div><div class="t m0 xae h2a y29 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _7e"></span>G</div><div class="c x98 y2a w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xaf h4 y29 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x9d h5 y10e ff2 fs2 fc0 sc0 ls0 ws0">距离值为</div><div class="t m0 x36 h58 ye3 ff1 fs23 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6e h5c ye4 ff4 fs24 fc0 sc0 ls0 ws0">f<span class="_ _95"> </span><span class="ff2 fs2">分为六类：<span class="ff1"> </span></span></div><div class="t m0 x9d h4 y10f ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x63 h28 y2f ff1 fs11 fc0 sc0 ls0 ws0">}<span class="_ _11"></span>,<span class="_ _94"></span>{</div><div class="t m0 x37 h58 y110 ff1 fs23 fc0 sc0 ls0 ws0">5<span class="_ _55"></span>4<span class="_ _47"></span>1</div><div class="t m0 x13 h2a y2f ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _6e"></span>w<span class="_ _42"></span>G<span class="_ _10"> </span><span class="ff5">=<span class="_ _9c"> </span><span class="ff2 fs2">，<span class="_ _62"> </span><span class="ff1 fs26">}<span class="_ _3b"></span>{</span></span></span></div><div class="t m0 x59 h60 y110 ff1 fs27 fc0 sc0 ls0 ws0">1<span class="_ _83"></span>2</div><div class="t m0 x56 h28 y2f ff4 fs26 fc0 sc0 ls0 ws0">w<span class="_ _7e"></span>G<span class="_ _9d"> </span><span class="ff5">=<span class="_ _14"> </span><span class="ff2 fs2">，<span class="_ _9e"> </span><span class="ff1 fs11">}<span class="_ _3e"></span>{</span></span></span></div><div class="t m0 x5d h60 y110 ff1 fs27 fc0 sc0 ls0 ws0">2<span class="_ _47"></span>3</div><div class="t m0 x70 h2a y2f ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _69"></span>G</div><div class="c x2 y30 w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xb0 h28 y2f ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _9e"> </span><span class="ff1 fs11">}<span class="_ _9f"></span>{</span></div><div class="t m0 xa1 h60 y110 ff1 fs27 fc0 sc0 ls0 ws0">3<span class="_ _97"></span>4</div><div class="t m0 xb1 h2a y2f ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _7e"></span>G</div><div class="c x6a y30 w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xb2 h28 y2f ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _9e"> </span><span class="ff1 fs11">}<span class="_ _9a"></span>{</span></div><div class="t m0 xb3 h60 y110 ff1 fs27 fc0 sc0 ls0 ws0">6<span class="_ _97"></span>5</div><div class="t m0 xb4 h2a y2f ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _69"></span>G</div><div class="c x74 y30 w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xb5 h28 y2f ff2 fs2 fc0 sc0 ls0 ws0">，<span class="_ _61"> </span><span class="ff1 fs11">}<span class="_ _3e"></span>{</span></div><div class="t m0 xb6 h60 y110 ff1 fs27 fc0 sc0 ls0 ws0">7<span class="_ _97"></span>6</div><div class="t m0 xb7 h2a y2f ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _7e"></span>G<span class="_ _9d"> </span><span class="ff5">=<span class="_ _a0"> </span><span class="ff1 fs2"> </span></span></div><div class="t m0 x21 h5 y111 ff2 fs2 fc0 sc0 ls0 ws0">距离小于</div><div class="t m0 x7f h60 yeb ff1 fs27 fc0 sc0 ls0 ws0">1</div><div class="t m0 x7b h61 yec ff4 fs26 fc0 sc0 ls0 ws0">f<span class="_ _95"> </span><span class="ff2 fs2">分为七类，每一个点自成一类。<span class="ff1"> </span></span></div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>

<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6280691d9b6e2b6d55893c9d/bg5.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0"> <span class="_ _0"> </span><span class="ff2">-447-</span></div><div class="t m0 x4 h28 y112 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">怎样才能生成这样的聚类图呢？步骤如下：设<span class="_ _8d"> </span></span><span class="fs11">}<span class="_ _9f"></span>,<span class="_ _54"></span>,<span class="_ _55"></span>,<span class="_ _58"></span>{</span></div><div class="t m0 xb8 h60 y113 ff1 fs27 fc0 sc0 ls0 ws0">7<span class="_ _4d"></span>2<span class="_ _4e"></span>1</div><div class="t m0 xb9 h2a y112 ff4 fs11 fc0 sc0 ls0 ws0">w<span class="_ _71"></span>w<span class="_ _d"></span>w<span class="_ _89"> </span><span class="ff6 fs5"></span></div><div class="c x76 y114 w5 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">=</div></div><div class="c x6a y114 w7 h5f"><div class="t m0 x0 h2c y1b ff5 fs11 fc0 sc0 ls0 ws0">Ω</div></div><div class="t m0 xba h4 y112 ff2 fs2 fc0 sc0 ls0 ws0">，<span class="ff1"> </span></div><div class="t m0 x4 h62 y115 ff1 fs2 fc0 sc0 lsb ws0"> 1<span class="_ _5d"></span><span class="ff2 ls0">）计算<span class="_ _50"> </span><span class="ff4 fs28">n<span class="_ _29"> </span></span>个样本点两两之间的距离</span></div><div class="t m0 xbb h63 y116 ff1 fs29 fc0 sc0 ls0 ws0">}<span class="_ _9f"></span>{</div><div class="t m0 xbc h64 y117 ff4 fs2a fc0 sc0 ls0 ws0">ij</div><div class="t m0 x70 h65 y116 ff4 fs29 fc0 sc0 ls0 ws0">d</div><div class="t m0 xbd h5 y115 ff2 fs2 fc0 sc0 ls0 ws0">，记为矩阵</div><div class="t m0 xbe h64 y117 ff4 fs2a fc0 sc0 ls0 ws0">n<span class="_ _a1"></span>n<span class="_ _57"></span>ij</div><div class="t m0 x74 h65 y116 ff4 fs29 fc0 sc0 ls0 ws0">d<span class="_ _a2"></span>D</div><div class="t m0 xb9 h66 y117 ff5 fs2a fc0 sc0 ls0 ws0">×</div><div class="c xbf y118 w5 h67"><div class="t m0 x0 h68 y1b ff5 fs29 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 xc0 h63 y116 ff1 fs29 fc0 sc0 ls0 ws0">)<span class="_ _32"></span>(</div><div class="t m0 xc1 h4 y115 ff2 fs2 fc0 sc0 ls0 ws0">；<span class="ff1"> </span></div><div class="t m0 x4 h4 y119 ff1 fs2 fc0 sc0 lsb ws0"> 2<span class="_ _5d"></span><span class="ff2 ls0">）首先构造</span></div><div class="t m0 x2b h62 y11a ff4 fs28 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc2 h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">个类，每一个类中只包含一个样本点，每一类的平台高度均为零；<span class="ff1"> </span></div><div class="t m0 x4 h4 y7 ff1 fs2 fc0 sc0 lsb ws0"> 3<span class="_ _5d"></span><span class="ff2 ls0">）合并距离最近的两类为新类，并<span class="_ _4"></span>且以这两类间的距离值作为聚类图中的平台高</span></div><div class="t m0 x4 h4 y8 ff2 fs2 fc0 sc0 ls0 ws0">度；<span class="ff1"> </span></div><div class="t m0 x5 h4 y9 ff1 fs2 fc0 sc0 ls0 ws0">4<span class="ff2">）计算新类与当前各类的距离，若类的个数已经等于<span class="_ _29"> </span></span>1<span class="ff2">，<span class="_ _1"></span>转入步骤<span class="_ _29"> </span><span class="ff1">5</span>）<span class="_ _8b"></span>，否则，回</span></div><div class="t m0 x4 h4 ya ff2 fs2 fc0 sc0 ls0 ws0">到步骤<span class="_ _29"> </span><span class="ff1">3</span>）<span class="_ _8b"></span>；<span class="ff1"> </span></div><div class="t m0 x5 h4 yb ff1 fs2 fc0 sc0 ls0 ws0">5<span class="ff2">）画聚类图；</span> </div><div class="t m0 x5 h4 yc ff1 fs2 fc0 sc0 ls0 ws0">6<span class="ff2">）决定类的个数和类。</span> </div><div class="t m0 x5 h5 yd ff2 fs2 fc0 sc0 ls5 ws0">显而易见，这种系统归类过程与<span class="_ _4"></span>计算类和类之间的距离有<span class="_ _4"></span>关，采用不同的距离定</div><div class="t m0 x4 h4 ye ff2 fs2 fc0 sc0 ls0 ws0">义，有可能得出不同的聚类结果。<span class="ff1"> </span></div><div class="t m0 x5 h4 yf ff1 fs2 fc0 sc0 ls8 ws5">1.2.2 <span class="ff2 ls0 ws0">最短距离法与最长距离法<span class="ff1"> </span></span></div><div class="t m0 x4 h4 y10 ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">如果使用最短距离法来测量类与类之间的距离，<span class="_ _37"></span>即称其为系统聚类法中的最短距离</span></div><div class="t m0 x4 h4 y11 ff2 fs2 fc0 sc0 ls0 ws0">法<span class="_ _5"></span>（又称最近邻法）<span class="_ _a3"></span>，<span class="_ _a3"></span>最先由<span class="_ _29"> </span><span class="ff1 ls1f">Florek<span class="_"> </span></span>等人<span class="_ _29"> </span><span class="ff1 ls1e">1951<span class="_ _88"> </span></span><span class="ls20">年和<span class="_ _29"> </span><span class="ff1 ls11">Sneath1957<span class="_"> </span></span></span>年引入。<span class="_ _a3"></span>下面举例说明</div><div class="t m0 x4 h4 y12 ff2 fs2 fc0 sc0 ls0 ws0">最短距离法的计算步骤。<span class="ff1"> </span></div><div class="t m0 x4 h4 y11b ff1 fs2 fc0 sc0 ls0 ws0"> <span class="_ _44"> </span><span class="ff2">例<span class="_ _50"> </span></span><span class="ls4 wse">1 </span><span class="ff2">设有<span class="_ _50"> </span></span>5<span class="_ _50"></span><span class="ff2">个销售员</span></div><div class="t m0 xbc h60 yf9 ff1 fs27 fc0 sc0 ls0 ws0">5<span class="_ _55"></span>4<span class="_ _55"></span>3<span class="_ _55"></span>2<span class="_ _4e"></span>1</div><div class="t m0 xb h63 yf8 ff1 fs29 fc0 sc0 ls0 ws0">,<span class="_ _4e"></span>,<span class="_ _58"></span>,<span class="_ _55"></span>,<span class="_ _a4"> </span><span class="ff4">w<span class="_ _67"></span>w<span class="_ _67"></span>w<span class="_ _67"></span>w<span class="_ _3e"></span>w<span class="_ _a5"> </span><span class="ff2 fs2">，<span class="_ _8b"></span>他们的销售业绩由二维变量<span class="_ _a6"> </span><span class="ff1 fs26">)<span class="_ _54"></span>,<span class="_ _19"></span>(</span></span></span></div><div class="t m0 xc3 h60 yf9 ff1 fs27 fc0 sc0 ls0 ws0">2<span class="_ _e"></span>1</div><div class="t m0 xc4 h61 yf8 ff4 fs26 fc0 sc0 ls0 ws0">v<span class="_ _54"></span>v<span class="_ _a7"> </span><span class="ff2 fs2">描述，</span></div><div class="t m0 x4 h4 y11c ff2 fs2 fc0 sc0 ls0 ws0">见表<span class="_ _29"> </span><span class="ff1">1</span>。<span class="ff1"> </span></div><div class="t m0 x54 h2 y11d ff2 fs0 fc0 sc0 ls0 ws0">表<span class="_ _9"> </span><span class="ff1 wsd">1 </span>销售员业绩表<span class="ff1"> </span></div><div class="t m0 x14 h2 y11e ff2 fs0 fc0 sc0 ls0 ws0">销售员<span class="ff1"> </span></div><div class="t m0 x1c h60 y11f ff1 fs27 fc0 sc0 ls0 ws0">1</div><div class="t m0 x59 h61 y120 ff4 fs26 fc0 sc0 ls0 ws0">v<span class="_ _51"> </span><span class="ff2 fs0">（销售量）百件<span class="ff1"> </span></span></div><div class="t m0 x87 h60 y11f ff1 fs27 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa6 h61 y120 ff4 fs26 fc0 sc0 ls0 ws0">v<span class="_ _52"> </span><span class="ff2 fs0">（回收款项）万元<span class="ff1"> </span></span></div><div class="t m0 xc5 h60 y121 ff1 fs27 fc0 sc0 ls0 ws0">1</div><div class="t m0 x4f h61 y122 ff4 fs26 fc0 sc0 ls0 ws0">w<span class="_ _51"> </span><span class="ff1 fs0"> </span></div><div class="t m0 x42 h2 y123 ff1 fs0 fc0 sc0 ls0 wsf">1 0 </div><div class="t m0 xc5 h60 y124 ff1 fs27 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4f h61 y125 ff4 fs26 fc0 sc0 ls0 ws0">w<span class="_ _45"> </span><span class="ff1 fs0"> </span></div><div class="t m0 x42 h2 y126 ff1 fs0 fc0 sc0 ls0 wsf">1 1 </div><div class="t m0 xc5 h60 y127 ff1 fs27 fc0 sc0 ls0 ws0">3</div><div class="t m0 x4f h65 y128 ff4 fs29 fc0 sc0 ls0 ws0">w<span class="_ _45"> </span><span class="ff1 fs0"> </span></div><div class="t m0 x42 h2 y129 ff1 fs0 fc0 sc0 ls0 wsf">3 2 </div><div class="t m0 xc5 h60 y12a ff1 fs27 fc0 sc0 ls0 ws0">4</div><div class="t m0 x4f h61 y12b ff4 fs26 fc0 sc0 ls0 ws0">w<span class="_ _45"> </span><span class="ff1 fs0"> </span></div><div class="t m0 x42 h2 y12c ff1 fs0 fc0 sc0 ls0 wsf">4 3 </div><div class="t m0 xc5 h60 y12d ff1 fs27 fc0 sc0 ls0 ws0">5</div><div class="t m0 x4f h65 y12e ff4 fs29 fc0 sc0 ls0 ws0">w<span class="_ _45"> </span><span class="ff1 fs0"> </span></div><div class="t m0 x42 h2 y12f ff1 fs0 fc0 sc0 ls0 wsf">2 5 </div><div class="t m0 x4 h4 y130 ff1 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x5 h63 y131 ff2 fs2 fc0 sc0 ls5 ws0">记销售员<span class="_ _a5"> </span><span class="ff1 fs29 ls0">)<span class="_ _a8"></span>5<span class="_ _a9"></span>,<span class="_ _a9"></span>4<span class="_ _b"></span>,<span class="_ _aa"></span>3<span class="_ _a9"></span>,<span class="_ _a9"></span>2<span class="_ _b"></span>,<span class="_ _2f"></span>1<span class="_ _5a"></span>(<span class="_ _91"> </span><span class="ff5">=<span class="_ _ab"></span><span class="ff4">i<span class="_ _5d"></span>w</span></span></span></div><div class="t m0 x2f h5a y132 ff4 fs23 fc0 sc0 ls0 ws0">i</div><div class="t m0 x83 h63 y131 ff2 fs2 fc0 sc0 ls5 ws0">的销售业绩为<span class="_ _ac"> </span><span class="ff1 fs29 ls0">)<span class="_ _3b"></span>,<span class="_ _5d"></span>(</span></div><div class="t m0 xc6 h5a y132 ff1 fs27 fc0 sc0 ls0 ws0">2<span class="_ _55"></span>1<span class="_ _65"> </span><span class="ff4 fs23">i<span class="_ _e"></span>i</span></div><div class="t m0 xc7 h65 y131 ff4 fs29 fc0 sc0 ls0 ws0">v<span class="_ _94"></span>v<span class="_ _ad"> </span><span class="ff2 fs2 ls5">。如果使用绝对值距离来测量点</span></div><div class="t m0 x4 h4 y133 ff2 fs2 fc0 sc0 ls0 ws0">与点之间的距离，使用最短距离法来测量类与类之间的距离，即<span class="ff1"> </span></div><div class="t m0 x2b h3d y134 ff5 fs18 fc0 sc0 ls0 ws0">∑</div><div class="t m0 xc8 h55 y135 ff5 fs23 fc0 sc0 ls0 ws0">=</div><div class="t m0 x3e h68 y136 ff5 fs29 fc0 sc0 ls0 ws0">−<span class="_ _ae"></span>=</div><div class="t m0 x12 h60 y137 ff1 fs27 fc0 sc0 ls0 ws0">2</div><div class="t m0 xc2 h60 y138 ff1 fs27 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2a h63 y136 ff1 fs29 fc0 sc0 ls0 ws0">)<span class="_ _3b"></span>,<span class="_ _5d"></span>(</div><div class="t m0 x65 h5a y135 ff4 fs23 fc0 sc0 ls0 ws0">k</div><div class="t m0 x3 h5a y139 ff4 fs23 fc0 sc0 ls0 ws0">jk<span class="_ _31"></span>ik<span class="_ _ae"></span>j<span class="_ _15"></span>i</div><div class="t m0 x64 h63 y136 ff4 fs29 fc0 sc0 ls0 ws0">v<span class="_ _af"></span>v<span class="_ _2d"></span>w<span class="_ _d"></span>w<span class="_ _58"></span>d<span class="_ _b0"> </span><span class="ff2 fs2">，<span class="_ _b1"> </span></span><span class="ff1">)}<span class="_ _5f"></span>,<span class="_ _1f"></span>(<span class="_ _60"></span>{<span class="_ _90"></span>min<span class="_ _4b"></span>)<span class="_ _3b"></span>,<span class="_ _94"></span>(</span></div><div class="t m0 xc9 h5a y13a ff4 fs23 fc0 sc0 ls0 ws0">j<span class="_ _54"></span>i</div><div class="t m0 x52 h5a y13b ff4 fs23 fc0 sc0 ls0 ws0">G<span class="_ _16"></span>w</div><div class="t m0 xca h5a y13c ff4 fs23 fc0 sc0 ls0 ws0">G<span class="_ _24"></span>w</div><div class="t m0 x70 h5a y13a ff4 fs23 fc0 sc0 ls0 ws0">q<span class="_ _94"></span>p</div><div class="t m0 x8d h65 y136 ff4 fs29 fc0 sc0 ls0 ws0">w<span class="_ _3e"></span>w<span class="_ _4e"></span>d<span class="_ _b2"></span>G<span class="_ _5f"></span>G<span class="_ _3b"></span>D</div><div class="t m0 xc7 h5b y13d ff4 fs25 fc0 sc0 ls0 ws0">q<span class="_ _13"></span>j</div><div class="t m0 xc7 h5b y13e ff4 fs25 fc0 sc0 ls0 ws0">p<span class="_ _57"></span>i</div><div class="t m0 xcb h55 y13f ff5 fs23 fc0 sc0 ls0 ws0">∈</div><div class="t m0 xab h55 y140 ff5 fs23 fc0 sc0 ls0 ws0">∈</div><div class="c xbb y141 w5 h5f"><div class="t m0 x0 h68 y1b ff5 fs29 fc0 sc0 ls0 ws0">=</div></div><div class="t m0 x97 h4 y136 ff1 fs2 fc0 sc0 ls0 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>