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class="menuitem flat-box faa-parent animated-hover" href="/sitemap.xml" title="站点地图" active-action="action-sitemapxml"><i class="fa-duotone fa-sitemap fa-fw"></i> 站点地图</a></li><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://mhuig.instatus.com/" title="Monitors" active-action="action-https:mhuiginstatuscom"><i class="fa-duotone fa-telescope fa-fw"></i> Monitors</a></li><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://ssl.mhuig.top/" title="SSL Status" active-action="action-https:sslmhuigtop"><i class="fa-brands fa-expeditedssl fa-fw"></i> SSL Status</a></li></ul></li><hr><li><a class="menuitem flat-box faa-parent animated-hover" title="工具箱"><i class="fa-duotone fa-tools fa-fw"></i> 工具箱</a><ul class="list-v"><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener 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class="fa-brands fa-github fa-fw"></i> GitHub</a></li></ul></li></ul></div><div class="m_search"><form name="searchform" class="form u-search-form"><i class="icon fa-solid fa-search fa-fw"></i> <input type="text" class="input u-search-input" placeholder="Search..."></form></div><ul class="switcher nav-list-h m-phone"><li><a class="s-search fa-solid fa-search fa-fw" target="_self" title="search"></a></li><li><a class="s-menu fa-solid fa-bars fa-fw" target="_self" title="menu"></a><ul class="menu-phone list-v navigation white-box"><li><a class="menuitem flat-box faa-parent animated-hover" title="索引"><i class="fa-duotone fa-list-alt fa-fw"></i> 索引</a><ul class="list-v"><li><a class="menuitem flat-box faa-parent animated-hover" href="/notes/" title="便签" active-action="action-notes"><i class="fa-duotone fa-books fa-fw"></i> 便签</a></li><li><a class="menuitem flat-box faa-parent animated-hover" href="/categories/" title="分类" active-action="action-categories"><i class="fa-duotone 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fa-user-shield fa-fw"></i> 博客管理</a><ul class="list-v"><li><a class="menuitem flat-box header toggle-mode-btn"><i class="fa-duotone fa-bat fa-fw"></i> 暗黑模式</a></li><li></li><li><a class="menuitem flat-box faa-parent animated-hover" href="/pages/rss/" title="RSS订阅" active-action="action-pagesrss"><i class="fa-duotone fa-rss fa-fw"></i> RSS订阅</a></li><li><a class="menuitem flat-box faa-parent animated-hover" href="/sitemap.xml" title="站点地图" active-action="action-sitemapxml"><i class="fa-duotone fa-sitemap fa-fw"></i> 站点地图</a></li><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://mhuig.instatus.com/" title="Monitors" active-action="action-https:mhuiginstatuscom"><i class="fa-duotone fa-telescope fa-fw"></i> Monitors</a></li><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://ssl.mhuig.top/" title="SSL Status" active-action="action-https:sslmhuigtop"><i class="fa-brands fa-expeditedssl fa-fw"></i> SSL Status</a></li></ul></li><hr><li><a class="menuitem flat-box faa-parent animated-hover" title="工具箱"><i class="fa-duotone fa-tools fa-fw"></i> 工具箱</a><ul class="list-v"><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://rssbox.mhuig.top/" title="RSS Box" active-action="action-https:rssboxmhuigtop"><i class="fa-duotone fa-conveyor-belt-boxes fa-fw"></i> RSS Box</a></li><li><a class="menuitem flat-box faa-parent animated-hover" href="/havefun/Coco/" title="My Cat" active-action="action-havefunCoco"><i class="fa-duotone fa-cat fa-fw"></i> My Cat</a></li><li><a class="menuitem flat-box faa-parent animated-hover" href="/pages/talk/" title="碎言碎语" active-action="action-pagestalk"><i class="fa-duotone fa-comments fa-fw"></i> 碎言碎语</a></li></ul></li><hr><li><a class="menuitem flat-box faa-parent animated-hover" target="_blank" rel="external nofollow noopener noreferrer" href="https://github.com/MHuiG" title="GitHub" active-action="action-https:githubcomMHuiG"><i class="fa-brands fa-github fa-fw"></i> GitHub</a></li></ul></li></ul></li></ul></div></div></div></header><div id="l_body"><div id="l_cover"><div id="none" class="cover-wrapper post featured" style="display:none"><div class="cover-bg lazyload placeholder" data-bg=""></div><div class="cover-body"><div class="top"><p class="title">Magicland</p><p class="subtitle">「看庭前花开花落,望天上云卷云舒」</p></div><div class="bottom"><div class="menu navigation"><div class="list-h"><a target="_blank" rel="external nofollow noopener noreferrer" href="https://github.com/MHuiG" active-action="action-https:githubcomMHuiG"><i class="fa-brands fa-github color-github fa-fw"></i><p>Github</p></a><a href="/pages/friends/" active-action="action-pagesfriends"><i class="fa-duotone fa-link color-friends fa-fw"></i><p>友链</p></a><a href="/pages/about/" active-action="action-pagesabout"><i class="fa-duotone fa-user-tie color-about fa-fw"></i><p>关于</p></a><a href="https://www.travellings.cn/go-by-clouds.html" target="_blank" active-action="action-https:wwwtravellingscngo-by-cloudshtml" rel="external nofollow noopener noreferrer"><i class="fa-duotone fa-subway color-travellings fa-fw"></i><p>Travelling</p></a></div></div></div></div><div id="scroll-down" style="display:none"><i class="fa fa-chevron-down scroll-down-effects"></i></div></div></div><div id="safearea"><div class="body-wrapper"><div id="l_main" class=""><article itemscope="" itemtype="http://schema.org/Article" class="article post white-box reveal md shadow floatable blur article-type-post" id="post" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/p/175a1706/"><span hidden="" itemprop="publisher" itemscope="" itemtype="http://schema.org/Organization"><meta itemprop="name" content="Magicland"></span><span hidden="" itemprop="post" itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="尝试在博客中添加简易文章推荐功能"><meta itemprop="description" content="详解在静态博客中利用TF-IDF算法实现文章推荐功能的过程，包括文本分词、词频-逆文档频率计算、向量空间模型构建及余弦相似度匹配，解决传统标签推荐依赖人工标注的局限，提供技术实现思路。"></span><div class="headimg-div"> <a class="headimg-a"><img itemprop="image" class="headimg lazyload" src="https://static.mhuig.top/npm/mhgoos@0.0.1663317354137/tf-idf.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663317354137/tf-idf.png" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></a></div><div class="article-meta" id="top"><h1 class="title" itemprop="name headline"> 尝试在博客中添加简易文章推荐功能</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope="" itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：2.7k 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：10 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p>把文档转换成 “向量”，并且尝试用线性代数等数学工具来解决信息检索这类问题，至少可以追溯到 20 世纪 70 年代. 想到在 Hexo 这种静态博客系统，特别是 GitHub Page 这种 Serverless 服务中使用 Word2Vec 等深度学习方法似乎并不现实，所以这里使用的是一个非常简单经典的 TF - IDF 算法.</p><p>TF - IDF 算法原来是搜索引擎中的核心部分，谷歌百度已经使用 TF - IDF 作为内容排名因素很长一段时间.</p><p>现在的搜索引擎一般用如下的算法计算网站页面得分：score (页面得分) = TFIDF 分 * x + 链接分 * y + 用户体验分 * z（其中 x + y+z = 100%），TFIDF 分值百度大约占 40% 左右的权重，谷歌更是达到了 50%.</p><p>这是百度的计算方法： score (页面得分) = 40% 的内容质量相关性（TFIDF）+ 40% 的用户体验分 + 20% 的链接分（域名 + 外链）.</p><p>搜索引擎使用 TF - IDF 来计算内容质量相关性，我们这边也可以用它来计算文章内容相关性，然后实现简易文章推荐功能。</p><p>当然也可以使用文章的 tags 标签最为推荐依据，但是这样过于依赖人工标注数据，顺便最近发现在 Hexo 中分类和标签只能在 <code>source/_posts</code> 文件夹中使用才能渲染出来。</p><div class="story post-story"><h2 id="构建向量空间模型"><a href="#构建向量空间模型" class="headerlink" title="构建向量空间模型"></a>构建向量空间模型</h2><p>将文档转化为向量来表示，这样文档与文档之间就可以定量的去度量他们之间的关系，挖掘文档之间的联系. 上面提到的 Word2Vec 是把单词转化为向量来表示，这样词与词之间就可以定量的去度量他们之间的关系，挖掘词之间的联系.</p><p>比如将词的维度降维到 2 维，用下图的词向量表示词:</p><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1663246894190/1.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663246894190/1.png" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="单词转化为向量"></p><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1663284804787/2.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663284804787/2.png" 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data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(1)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 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title="第一步：数据清洗"></a>第一步：数据清洗</h2><p>简单去除标点符号等干扰信息。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A: 我这里有苹果和西瓜</span><br><span class="line"></span><br><span class="line">句子B: 我喜欢吃西瓜 不喜欢吃苹果</span><br><span class="line"></span><br><span class="line">句子C: 我喜欢吃蔬菜</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第二步：分词"><a href="#第二步：分词" class="headerlink" title="第二步：分词"></a>第二步：分词</h2><p>使用分词工具进行分词。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">[ <span class="string">'我'</span>, <span class="string">'这里'</span>, <span class="string">'有'</span>, <span class="string">'苹果'</span>, <span class="string">'和'</span>, <span class="string">'西瓜'</span> ]</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">[</span><br><span class="line"> <span class="string">'我'</span>, <span class="string">'喜欢'</span>,</span><br><span class="line"> <span class="string">'吃'</span>, <span class="string">'西瓜'</span>,</span><br><span class="line"> <span class="string">'不'</span>, <span class="string">'喜欢'</span>,</span><br><span class="line"> <span class="string">'吃'</span>, <span class="string">'苹果'</span></span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">[ <span class="string">'我'</span>, <span class="string">'喜欢'</span>, <span class="string">'吃'</span>, <span class="string">'蔬菜'</span> ]</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第三步：去重收集所有句子中所有的词"><a href="#第三步：去重收集所有句子中所有的词" class="headerlink" title="第三步：去重收集所有句子中所有的词"></a>第三步：去重收集所有句子中所有的词</h2><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">[</span><br><span class="line"> <span class="string">'我'</span>, <span class="string">'这里'</span>, <span class="string">'有'</span>,</span><br><span class="line"> <span class="string">'苹果'</span>, <span class="string">'和'</span>, <span class="string">'西瓜'</span>,</span><br><span class="line"> <span class="string">'喜欢'</span>, <span class="string">'吃'</span>, <span class="string">'不'</span>,</span><br><span class="line"> <span class="string">'蔬菜'</span></span><br><span class="line">]</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第四步：计算词频"><a href="#第四步：计算词频" class="headerlink" title="第四步：计算词频"></a>第四步：计算词频</h2><p>词频背后的假设是文章的重要程度即文章的相关度与单词在文档中出现的次数成正比。文章的关键词应当比文章中的其他词出现的次数多。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:49.146ex"><svg style="vertical-align:-.566ex;min-width:49.146ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" 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data-latex="\text{(2)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 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class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 1,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 1,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 1,</span><br><span class="line"> <span class="string">'喜欢'</span>: 2,</span><br><span class="line"> <span class="string">'吃'</span>: 2,</span><br><span class="line"> <span class="string">'不'</span>: 1,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 1,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 1,</span><br><span class="line"> <span class="string">'吃'</span>: 1,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 1</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第四步：词频标准化"><a href="#第四步：词频标准化" class="headerlink" title="第四步：词频标准化"></a>第四步：词频标准化</h2><p>文章的篇幅有长有短，为了便于不同文章的比较，进行 "词频" 标准化。词频标准化的目的是把所有的词频在同一维度上分析。</p><p>词频标准化有两种方案：</p><p>方案一：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:50.142ex"><svg style="vertical-align:-2.077ex;min-width:50.142ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{文章的总词数}
"><g data-mml-node="mtable" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{文章的总词数}
" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1418) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="9003.3 -1418 1 2336"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,-8)"><g data-mml-node="mtd"><g data-mml-node="mi" data-latex="词"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">词</text></g><g data-mml-node="mi" data-latex="频" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">频</text></g><g data-mml-node="mo" data-latex="(" transform="translate(2000,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="T" transform="translate(2389,0)"><path data-c="1D447" d="M344 631C344 628 343 621 340 611L208 83C204 68 200 59 197 55 188 44 154 39 94 39 63 39 49 42 49 16 49 5 56 0 69 0 120 0 192 4 235 3L317 2C331 2 386 0 403 0 420 0 428 8 428 24 428 34 416 39 391 39 339 39 309 41 300 46 297 48 295 52 295 58L430 603C433 617 436 626 439 629 443 635 467 638 511 638 566 638 604 634 623 625 642 616 652 595 652 561 652 543 649 517 644 483 642 475 641 469 641 464 641 453 646 447 657 447 666 447 672 456 675 473L702 645C703 650 704 656 704 662 704 672 694 677 673 677L125 677C99 677 97 674 89 655L30 481C27 472 25 466 24 462 24 452 29 447 40 447 48 447 55 455 60 470 85 543 110 588 133 607 159 628 209 638 282 638L321 638C332 638 344 639 344 631Z"></path></g><g data-mml-node="mi" data-latex="F" transform="translate(3093,0)"><path data-c="1D439" d="M199 656C199 646 210 641 231 641 271 641 291 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526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g><g data-mml-node="mo" data-latex="=" transform="translate(4510.8,0)"><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g><g data-mml-node="mfrac" data-latex="\frac{某个词在文章中出现的次数}{文章的总词数}" transform="translate(5566.6,0)"><g data-mml-node="mrow" data-latex="某个词在文章中出现的次数" transform="translate(220,676)"><g data-mml-node="mi" data-latex="某"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">某</text></g><g data-mml-node="mi" data-latex="个" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">个</text></g><g data-mml-node="mi" 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data-mml-node="mi" data-latex="现" transform="translate(8000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">现</text></g><g data-mml-node="mi" data-latex="的" transform="translate(9000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mi" data-latex="次" transform="translate(10000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">次</text></g><g data-mml-node="mi" data-latex="数" transform="translate(11000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g><g data-mml-node="mrow" data-latex="文章的总词数" transform="translate(3220,-710)"><g data-mml-node="mi" data-latex="文"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">文</text></g><g data-mml-node="mi" data-latex="章" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">章</text></g><g data-mml-node="mi" data-latex="的" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mi" data-latex="总" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">总</text></g><g data-mml-node="mi" data-latex="词" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">词</text></g><g data-mml-node="mi" data-latex="数" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g><rect width="12200" height="60" x="120" y="220"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1418 1 2336"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:3" transform="translate(0,740)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(3)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="33" d="M303 353C369 378 431 441 431 526 431 569 410 604 369 631 333 654 292 666 246 666 201 666 162 654 127 631 88 605 68 571 68 528 68 495 90 472 122 472 154 472 176 495 176 527 176 560 157 578 119 580 145 615 186 633 242 633 302 633 332 598 332 527 332 485 324 450 309 421 282 373 245 364 183 364 171 362 165 357 165 348 165 333 172 333 192 333L235 333C310 333 348 280 348 173 348 88 317 14 241 14 176 14 128 36 99 80 134 79 160 105 160 139 160 173 135 198 101 198 62 198 42 178 42 137 42 88 64 49 108 18 147-9 193-22 244-22 301-22 350-3 393 34 436 71 457 117 457 173 457 267 383 332 303 353Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>方案二：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:52.404ex"><svg style="vertical-align:-2.077ex;min-width:52.404ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{该文出现最多的词出现的次数}
"><g data-mml-node="mtable" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{该文出现最多的词出现的次数}
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font-family="serif">数</text></g></g><rect width="13200" height="60" x="120" y="220"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1418 1 2336"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4" transform="translate(0,740)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(4)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199 28 163 289 163 289 81C289 63 285 51 278 46 271 41 252 39 219 39L194 39 194 0C223 2 269 3 331 3 393 3 439 2 468 0L468 39 443 39C410 39 391 41 384 46 377 51 373 63 373 81L373 163 471 163 471 202 373 202 373 660C373 670 366 677 353 677M295 553 295 202 67 202Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>一般情况下，第二个标准化方案更适用，因为能够使词频的值相对大点，便于分析。</p><p>有时候常常用<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 500 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" 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transform="translate(3371.2,0)"><path data-c="1D439" d="M199 656C199 646 210 641 231 641 271 641 291 637 291 628 291 625 289 618 286 605L156 82C150 60 140 47 126 42 119 40 101 39 70 39 48 39 38 37 38 16 38 5 44 0 57 0L187 3 335 0C351 0 359 8 359 23 359 40 348 39 322 39 278 39 253 41 247 46 244 47 243 51 243 56L307 321 400 321C446 321 478 319 478 281 478 268 476 252 471 233 470 230 469 226 468 222 468 211 473 206 484 206 489 207 496 214 503 229L557 444C559 452 560 458 560 461 557 470 551 475 544 475 536 475 530 467 526 452 516 414 502 390 486 378 470 366 442 360 402 360L317 360 379 606C387 640 387 641 429 641L559 641C656 641 700 626 700 537 700 517 699 499 697 484 696 473 695 467 695 466 695 455 700 450 710 450 720 450 726 459 728 478L748 649C751 677 744 680 717 680L233 680C210 680 199 679 199 656Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(4122.2,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 这个值来代替原来的 TF 取值。这样的计算保持了一个平衡，既有区分度，但也不至于完全线性增长。</p><p>这里使用比较简单的方案一作为栗子：</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'这里'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'有'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'苹果'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'和'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.125,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0.125,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0.125,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0.25,</span><br><span class="line"> <span class="string">'吃'</span>: 0.25,</span><br><span class="line"> <span class="string">'不'</span>: 0.125,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.25,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0.25,</span><br><span class="line"> <span class="string">'吃'</span>: 0.25,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.25</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第五步：计算逆文档频率"><a href="#第五步：计算逆文档频率" class="headerlink" title="第五步：计算逆文档频率"></a>第五步：计算逆文档频率</h2><p>仅有词频 (TF) 不能比较完整地描述文档的相关度。因为语言的因素，有一些单词可能会比较自然地在很多文档中反复出现，比如这里中的 “我”、“和” 等等。这些词大多起到了链接语句的作用，是保持语言连贯不可或缺的部分。</p><p>很明显，如果有太多文档都涵盖了某个单词，这个单词也就越不重要，或者说是这个单词就越没有信息量。因此，我们需要对词频 (TF) 的值进行修正，而 逆文档频率 (IDF) 的想法是用文档频率 (DF) 的倒数来进行修正。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:57.833ex"><svg style="vertical-align:-2.077ex;min-width:57.833ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
逆文档频率(IDF) = \log(\frac{语料库的文档总数}{包含该词的文档数+1})
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逆文档频率(IDF) = \log(\frac{语料库的文档总数}{包含该词的文档数+1})
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transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>如果一个词越常见，那么分母就越大，逆文档频率就越小越接近 0。分母之所以要加 1，是为了避免分母为 0（即所有文档都不包含该词）。log 表示对得到的值取对数。为什么要用 log 函数？log 函数是单调递增，求 log 是为了归一化，保证逆文档频率不会过大。用 Log 进行变换，也是一个非线性增长的技巧，这样的计算保持了一个平衡，既有区分度，但也不至于完全线性增长。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第六步：计算TF-IDF"><a href="#第六步：计算TF-IDF" class="headerlink" title="第六步：计算TF-IDF"></a>第六步：计算 TF - IDF</h2><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:53.286ex"><svg style="vertical-align:-.566ex;min-width:53.286ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" 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TF-IDF = 词频(TF) \times 逆文档频率(IDF)
"><g data-mml-node="mtable" data-latex="
TF-IDF = 词频(TF) \times 逆文档频率(IDF)
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314-247 317-248 318-248Z"></path><path data-c="36" d="M383 504C416 504 432 521 432 555 432 627 378 666 304 666 221 666 155 627 106 548 63 480 42 403 42 316 42 189 65 100 112 47 152 1 198-22 251-22 312-22 362 1 401 47 438 91 457 144 457 205 457 266 439 318 403 361 365 407 316 431 257 431 205 431 165 402 138 346L138 352C138 465 166 561 226 605 252 624 279 633 306 633 342 633 368 623 385 602 351 602 334 583 334 553 334 525 355 504 383 504M344 340C355 317 361 272 361 206 361 141 356 98 345 76 325 35 294 14 251 14 222 14 200 24 184 44 171 60 162 74 158 85 146 116 140 163 140 227 140 255 144 282 151 308 164 355 201 399 256 399 295 399 325 379 344 340Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 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<span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.03596025905647261,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.05068313851352055,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.07192051811294523,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.1013662770270411</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第七步：列出文档向量"><a href="#第七步：列出文档向量" class="headerlink" title="第七步：列出文档向量"></a>第七步：列出文档向量</h2><p>这里准确来说应该是 TF - IDF 特征向量，这是一个稀疏矩阵。下面把它作为描述文章特征的向量。</p><p>说明一点，这里的文章的特征不是由某几个关键词的 TF - IDF 值决定的，而是由所有的词的 TF - IDF 值共同决定的。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.047947012075296815,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0</span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.03596025905647261,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0.05068313851352055,</span><br><span class="line"> 0</span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.07192051811294523,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0.1013662770270411</span><br><span class="line">]</span><br></pre></td></tr></tbody></table></figure><p>可以对文档向量进行标准化，使得这些向量能够不受向量里有效元素多少的影响，也就是不同的文档可能有不同的长度。把向量都标准化为一个单位向量的长度。这个时候再进行点积运算，就相当于在原来的向量上进行余弦相似度的运算。</p><p><mjx-container class="MathJax" 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data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:7" transform="translate(0,578.5)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(7)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="37" d="M475 604C482 613 485 626 485 644L243 644C174 644 135 648 128 657 125 660 122 667 120 676L89 676 55 464 88 464C98 520 106 550 112 555 115 558 146 560 205 560L401 560 295 410C214 295 174 171 174 36 174-3 190-22 223-22 256-22 272-3 272 36L272 87C272 239 296 349 343 416Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p></div><div class="story post-story"><h2 id="第八步：计算余弦相似度"><a href="#第八步：计算余弦相似度" class="headerlink" title="第八步：计算余弦相似度"></a>第八步：计算余弦相似度</h2><p>到这里，如何寻找两篇相似文章的问题转变成了如何计算这两个向量的相似程度的问题。</p><p>什么叫做向量相似？</p><p>一般地，如果两个向量平行或重合，我们认为这两个向量相似度为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></svg></mjx-container>。如果这两个向量垂直，或者说正交，我们认为这两个向量的相似度为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0"><g data-mml-node="mn" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g></g></g></svg></mjx-container>。</p><p>以二维空间为例，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.245ex" height="1.919ex" role="img" focusable="false" viewBox="0 -837 550.5 848"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\vec{a}"><g data-mml-node="TeXAtom" data-latex="\vec{a}" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi" data-latex="a"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158 474 158 468 151 465 137 445 58 421 18 394 18 377 18 368 32 368 60 368 73 372 97 381 132L438 357C443 376 445 387 445 392 445 412 434 422 412 422 391 422 377 410 370 387 349 424 319 442 281 442 216 442 159 409 109 343 63 281 40 217 40 150 40 63 91-11 175-11 218-11 260 12 300 58 311 20 345-11 392-11 461-11 482 70 498 144M341 374C350 353 355 339 355 330 355 326 354 321 353 314L304 122C301 111 294 99 285 87 248 41 212 18 177 18 138 18 118 48 118 107 118 131 124 167 136 215 157 300 187 357 224 388 244 405 263 413 282 413 309 413 329 400 341 374Z"></path></g><g data-mml-node="mo" transform="translate(550.5,26)"><path data-c="20D7" d="M-69 599C-60 602-56 607-56 616-56 625-60 630-69 633-105 645-131 667-147 700-150 707-156 711-163 711-176 711-182 705-182 693-182 678-169 658-143 634L-453 634C-466 634-472 628-472 616-472 604-466 598-453 598L-143 598C-169 574-182 554-182 539-182 527-176 521-163 521-156 521-150 525-147 532-131 565-105 587-69 599Z"></path></g></g></g></g></g></svg></mjx-container> 和<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.058ex" height="2.489ex" role="img" focusable="false" viewBox="0 -1089 467.5 1100"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\vec{b}"><g data-mml-node="TeXAtom" data-latex="\vec{b}" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi" data-latex="b"><path data-c="1D44F" d="M281 445C245 445 209 428 174 394L243 679C241 688 238 694 226 694 193 694 120 685 107 684 92 682 84 675 84 660 84 650 93 645 112 645 131 645 157 646 157 632 157 628 152 608 143 571L63 249C52 207 47 173 47 148 47 62 93-11 175-11 240-11 297 22 347 89 392 151 415 216 415 283 415 370 364 445 281 445M279 415C318 415 337 385 337 326 337 299 331 263 319 218 297 133 268 75 233 44 213 27 194 19 175 19 134 19 114 51 114 115 114 137 119 170 129 214L151 304C154 319 159 330 165 338 204 389 242 415 279 415Z"></path></g><g data-mml-node="mo" transform="translate(467.5,278)"><path data-c="20D7" d="M-69 599C-60 602-56 607-56 616-56 625-60 630-69 633-105 645-131 667-147 700-150 707-156 711-163 711-176 711-182 705-182 693-182 678-169 658-143 634L-453 634C-466 634-472 628-472 616-472 604-466 598-453 598L-143 598C-169 574-182 554-182 539-182 527-176 521-163 521-156 521-150 525-147 532-131 565-105 587-69 599Z"></path></g></g></g></g></g></svg></mjx-container> 是两个向量，我们要计算它们的夹角<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.62ex" role="img" focusable="false" viewBox="0 -705 469 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="θ"><g data-mml-node="mi" data-latex="θ"><path data-c="1D703" d="M164-11C209-11 256 16 302 70 382 163 455 335 455 498 455 539 448 582 434 625 416 678 382 705 333 705 288 705 243 678 196 624 116 532 42 360 42 196 42 94 75-11 164-11M332 675C365 675 382 637 382 560 382 517 372 452 352 367L155 367C180 460 204 529 229 574 266 641 301 675 332 675M145 327 341 327C322 246 299 181 274 132 235 57 199 19 164 19 131 19 115 58 115 135 115 185 125 249 145 327Z"></path></g></g></g></svg></mjx-container> 。余弦定理告诉我们，可以用下面的公式求得：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:29.522ex"><svg style="vertical-align:-1.931ex;min-width:29.522ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.993ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1353.5)"><g data-mml-node="math" data-latex="
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314L304 122C301 111 294 99 285 87 248 41 212 18 177 18 138 18 118 48 118 107 118 131 124 167 136 215 157 300 187 357 224 388 244 405 263 413 282 413 309 413 329 400 341 374Z"></path></g><g data-mml-node="mi" data-latex="b" transform="translate(1029,0)"><path data-c="1D44F" d="M281 445C245 445 209 428 174 394L243 679C241 688 238 694 226 694 193 694 120 685 107 684 92 682 84 675 84 660 84 650 93 645 112 645 131 645 157 646 157 632 157 628 152 608 143 571L63 249C52 207 47 173 47 148 47 62 93-11 175-11 240-11 297 22 347 89 392 151 415 216 415 283 415 370 364 445 281 445M279 415C318 415 337 385 337 326 337 299 331 263 319 218 297 133 268 75 233 44 213 27 194 19 175 19 134 19 114 51 114 115 114 137 119 170 129 214L151 304C154 319 159 330 165 338 204 389 242 415 279 415Z"></path></g></g><rect width="5345.5" height="60" x="120" y="220"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1353.5 1 2206.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:8" transform="translate(0,591.5)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(8)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="38" d="M250 666C201 666 158 650 122 618 86 586 68 546 68 497 68 456 83 419 113 386 120 378 142 361 179 335 88 288 42 227 42 153 42 100 64 57 107 24 147-7 194-22 249-22 305-22 354-4 395 32 436 68 457 115 457 170 457 216 441 257 408 294 396 307 365 330 316 361 393 402 431 454 431 515 431 606 344 666 250 666M379 515C379 463 348 418 286 381L167 459C136 479 120 505 120 536 120 596 185 633 249 633 318 633 379 584 379 515M250 14C168 14 99 73 99 153 99 220 136 274 210 315L328 240C376 209 400 174 400 134 400 62 325 14 250 14Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>假定<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="B"><g data-mml-node="mi" data-latex="B"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659 203 643 223 644 235 644 265 643 283 641 288 640 293 639 295 636 295 631 295 629 294 623 291 613L159 82C154 60 144 47 129 42 122 40 104 39 73 39 52 39 42 31 42 15 42 5 52 0 73 0L426 0C491 0 551 20 608 59 671 102 703 155 703 217 703 296 638 344 567 358 655 379 756 446 756 543M554 644C623 644 658 612 658 547 658 496 638 454 596 420 554 386 507 370 456 370L317 370 377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228 603 176 583 131 542 94 501 57 455 39 402 39L267 39C257 39 250 39 246 40 240 40 237 42 237 45 237 46 237 49 238 53 269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"></path></g></g></g></svg></mjx-container>是两个<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="n"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g></g></g></svg></mjx-container>维向量，我们把这些向量放到<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="n"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g></g></g></svg></mjx-container>维欧几里得空间中讨论，向量之间的距离使用欧几里得距离。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:26.576ex"><svg style="vertical-align:-2.095ex;min-width:26.576ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.321ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1426)"><g data-mml-node="math" data-latex="
\cos \theta=\frac{A \cdot B}{|A| \times |B|}
"><g data-mml-node="mtable" data-latex="
\cos \theta=\frac{A \cdot B}{|A| \times |B|}
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43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container>是<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="16.005ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7074 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="[A1, A2, ..., An] "><g data-mml-node="mo" data-latex="["><path data-c="5B" d="M233-202 159-202 159 702 233 702C248 702 256 710 256 726 256 742 248 750 233 750L114 750 114-250 233-250C248-250 256-242 256-226 256-214 245-202 233-202Z"></path></g><g data-mml-node="mi" data-latex="A" transform="translate(278,0)"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mn" data-latex="1" transform="translate(1028,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(1528,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 165-1 174 8L175 0C175-63 154-117 112-160 105-168 101-174 101-178 101-188 105-193 114-193 123-193 135-181 152-158 186-110 203-57 203 0 203 53 185 106 139 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82C154 60 144 47 129 42 122 40 104 39 73 39 52 39 42 31 42 15 42 5 52 0 73 0L426 0C491 0 551 20 608 59 671 102 703 155 703 217 703 296 638 344 567 358 655 379 756 446 756 543M554 644C623 644 658 612 658 547 658 496 638 454 596 420 554 386 507 370 456 370L317 370 377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228 603 176 583 131 542 94 501 57 455 39 402 39L267 39C257 39 250 39 246 40 240 40 237 42 237 45 237 46 237 49 238 53 269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"></path></g><g data-mml-node="mn" data-latex="1" transform="translate(1037,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(1537,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 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data-latex="." transform="translate(4130,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106 110 106 86 82 86 53 86 24 110 0 139 0 168 0 192 24 192 53Z"></path></g><g data-mml-node="mo" data-latex="." transform="translate(4574.7,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106 110 106 86 82 86 53 86 24 110 0 139 0 168 0 192 24 192 53Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(5019.3,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 165-1 174 8L175 0C175-63 154-117 112-160 105-168 101-174 101-178 101-188 105-193 114-193 123-193 135-181 152-158 186-110 203-57 203 0 203 53 185 106 139 106Z"></path></g><g data-mml-node="mi" data-latex="B" transform="translate(5464,0)"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659 203 643 223 644 235 644 265 643 283 641 288 640 293 639 295 636 295 631 295 629 294 623 291 613L159 82C154 60 144 47 129 42 122 40 104 39 73 39 52 39 42 31 42 15 42 5 52 0 73 0L426 0C491 0 551 20 608 59 671 102 703 155 703 217 703 296 638 344 567 358 655 379 756 446 756 543M554 644C623 644 658 612 658 547 658 496 638 454 596 420 554 386 507 370 456 370L317 370 377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228 603 176 583 131 542 94 501 57 455 39 402 39L267 39C257 39 250 39 246 40 240 40 237 42 237 45 237 46 237 49 238 53 269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"></path></g><g data-mml-node="mi" data-latex="n" transform="translate(6223,0)"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g><g data-mml-node="mo" data-latex="]" transform="translate(6823,0)"><path data-c="5D" d="M45-250 164-250 164 750 45 750C30 750 22 742 22 726 22 710 30 702 45 702L119 702 119-202 45-202C30-202 22-210 22-226 22-242 30-250 45-250Z"></path></g></g></g></svg></mjx-container> ，则<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container>与<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="B"><g data-mml-node="mi" data-latex="B"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659 203 643 223 644 235 644 265 643 283 641 288 640 293 639 295 636 295 631 295 629 294 623 291 613L159 82C154 60 144 47 129 42 122 40 104 39 73 39 52 39 42 31 42 15 42 5 52 0 73 0L426 0C491 0 551 20 608 59 671 102 703 155 703 217 703 296 638 344 567 358 655 379 756 446 756 543M554 644C623 644 658 612 658 547 658 496 638 454 596 420 554 386 507 370 456 370L317 370 377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228 603 176 583 131 542 94 501 57 455 39 402 39L267 39C257 39 250 39 246 40 240 40 237 42 237 45 237 46 237 49 238 53 269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"></path></g></g></g></svg></mjx-container>的夹角<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.62ex" role="img" focusable="false" viewBox="0 -705 469 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="θ"><g data-mml-node="mi" data-latex="θ"><path data-c="1D703" d="M164-11C209-11 256 16 302 70 382 163 455 335 455 498 455 539 448 582 434 625 416 678 382 705 333 705 288 705 243 678 196 624 116 532 42 360 42 196 42 94 75-11 164-11M332 675C365 675 382 637 382 560 382 517 372 452 352 367L155 367C180 460 204 529 229 574 266 641 301 675 332 675M145 327 341 327C322 246 299 181 274 132 235 57 199 19 164 19 131 19 115 58 115 135 115 185 125 249 145 327Z"></path></g></g></g></svg></mjx-container>的余弦等于：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:44.014ex"><svg style="vertical-align:-3.24ex;min-width:44.014ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="7.61ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1931.9)"><g data-mml-node="math" data-latex="
\cos\theta=\frac{\sum_{i=1}^{n}(A_i \times B_i)}{\sqrt{\sum_{i=1}^{n}A_i^2} \times \sqrt{\sum_{i=1}^{n}B_i^2} }
"><g data-mml-node="mtable" data-latex="
\cos\theta=\frac{\sum_{i=1}^{n}(A_i \times B_i)}{\sqrt{\sum_{i=1}^{n}A_i^2} \times \sqrt{\sum_{i=1}^{n}B_i^2} }
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transform="translate(0,1137.1)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(10)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z" transform="translate(389,0)"></path><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z" transform="translate(889,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(1389,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 1.0000000000000002,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子B'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子B'</span>: 1,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.33484380220099325</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子C'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子C'</span>: 1</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第九步：余弦相似度按照降序排序"><a href="#第九步：余弦相似度按照降序排序" class="headerlink" title="第九步：余弦相似度按照降序排序"></a>第九步：余弦相似度按照降序排序</h2><p>两两计算向量夹角的余弦值，然后按降序排列，取排在最前面的几个文章，就是相似度较高的文章，作为推荐结果。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 1.0000000000000002,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子B'</span>: {</span><br><span class="line"> <span class="string">'句子B'</span>: 1,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子C'</span>: {</span><br><span class="line"> <span class="string">'句子C'</span>: 1,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p>相同句子的余弦值是<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></svg></mjx-container>.余弦值越接近<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></svg></mjx-container>，就表明夹角越接近<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0"><g data-mml-node="mn" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g></g></g></svg></mjx-container>度，也就是两个向量越相似.</p></div><div class="story post-story"><h2 id="第十步：收集推荐结果"><a href="#第十步：收集推荐结果" class="headerlink" title="第十步：收集推荐结果"></a>第十步：收集推荐结果</h2><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: [ <span class="string">'句子B'</span>, <span class="string">'句子C'</span> ],</span><br><span class="line"> <span class="string">'句子B'</span>: [ <span class="string">'句子C'</span>, <span class="string">'句子A'</span> ],</span><br><span class="line"> <span class="string">'句子C'</span>: [ <span class="string">'句子B'</span>, <span class="string">'句子A'</span> ]</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p>即：</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">A-B</span><br><span class="line">B-C</span><br><span class="line">C-B</span><br></pre></td></tr></tbody></table></figure><p>A 和 C 之间没什么关系。</p></div><div class="story post-story"><h2 id="源码实现"><a href="#源码实现" class="headerlink" title="源码实现"></a>源码实现</h2><div class="tag link"><a class="link-card" title="Volantis主题实现文章推荐" target="_blank" rel="external nofollow noopener noreferrer" 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class="toc-item toc-level-2"><a class="toc-link" href="#%E6%9E%84%E5%BB%BA%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4%E6%A8%A1%E5%9E%8B"><span class="toc-text">构建向量空间模型</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%B8%BE%E4%B8%AA%E6%A0%97%E5%AD%90"><span class="toc-text">举个栗子</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%80%E6%AD%A5%EF%BC%9A%E6%95%B0%E6%8D%AE%E6%B8%85%E6%B4%97"><span class="toc-text">第一步：数据清洗</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%BA%8C%E6%AD%A5%EF%BC%9A%E5%88%86%E8%AF%8D"><span class="toc-text">第二步：分词</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%89%E6%AD%A5%EF%BC%9A%E5%8E%BB%E9%87%8D%E6%94%B6%E9%9B%86%E6%89%80%E6%9C%89%E5%8F%A5%E5%AD%90%E4%B8%AD%E6%89%80%E6%9C%89%E7%9A%84%E8%AF%8D"><span class="toc-text">第三步：去重收集所有句子中所有的词</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%9B%9B%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E8%AF%8D%E9%A2%91"><span class="toc-text">第四步：计算词频</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%9B%9B%E6%AD%A5%EF%BC%9A%E8%AF%8D%E9%A2%91%E6%A0%87%E5%87%86%E5%8C%96"><span class="toc-text">第四步：词频标准化</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%BA%94%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E9%80%86%E6%96%87%E6%A1%A3%E9%A2%91%E7%8E%87"><span class="toc-text">第五步：计算逆文档频率</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%85%AD%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97TF-IDF"><span class="toc-text">第六步：计算 TF - IDF</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%83%E6%AD%A5%EF%BC%9A%E5%88%97%E5%87%BA%E6%96%87%E6%A1%A3%E5%90%91%E9%87%8F"><span class="toc-text">第七步：列出文档向量</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%85%AB%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E4%BD%99%E5%BC%A6%E7%9B%B8%E4%BC%BC%E5%BA%A6"><span class="toc-text">第八步：计算余弦相似度</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B9%9D%E6%AD%A5%EF%BC%9A%E4%BD%99%E5%BC%A6%E7%9B%B8%E4%BC%BC%E5%BA%A6%E6%8C%89%E7%85%A7%E9%99%8D%E5%BA%8F%E6%8E%92%E5%BA%8F"><span class="toc-text">第九步：余弦相似度按照降序排序</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%8D%81%E6%AD%A5%EF%BC%9A%E6%94%B6%E9%9B%86%E6%8E%A8%E8%8D%90%E7%BB%93%E6%9E%9C"><span class="toc-text">第十步：收集推荐结果</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%BA%90%E7%A0%81%E5%AE%9E%E7%8E%B0"><span class="toc-text">源码实现</span></a></li></ol></div></section></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div 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