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week9.html
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<div id='write' class = 'is-node'><h1><a name='header-n0' class='md-header-anchor '></a>第9周</h1><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n0"><a class="md-toc-inner" href="#header-n0">第9周</a></span><span class="md-toc-item md-toc-h2" data-ref="n5"><a class="md-toc-inner" href="#header-n5">十五、异常检测(Anomaly Detection)</a></span><span class="md-toc-item md-toc-h3" data-ref="n6"><a class="md-toc-inner" href="#header-n6">15.1 问题的动机</a></span><span class="md-toc-item md-toc-h3" data-ref="n44"><a class="md-toc-inner" href="#header-n44">15.2 高斯分布</a></span><span class="md-toc-item md-toc-h3" data-ref="n62"><a class="md-toc-inner" href="#header-n62">15.3 算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n95"><a class="md-toc-inner" href="#header-n95">15.4 开发和评价一个异常检测系统</a></span><span class="md-toc-item md-toc-h3" data-ref="n120"><a class="md-toc-inner" href="#header-n120">15.5 异常检测与监督学习对比</a></span><span class="md-toc-item md-toc-h3" data-ref="n145"><a class="md-toc-inner" href="#header-n145">15.6 选择特征</a></span><span class="md-toc-item md-toc-h3" data-ref="n166"><a class="md-toc-inner" href="#header-n166">15.7 多元高斯分布（选修）</a></span><span class="md-toc-item md-toc-h3" data-ref="n236"><a class="md-toc-inner" href="#header-n236">15.8 使用多元高斯分布进行异常检测（可选）</a></span><span class="md-toc-item md-toc-h2" data-ref="n280"><a class="md-toc-inner" href="#header-n280">十六、推荐系统(Recommender Systems)</a></span><span class="md-toc-item md-toc-h3" data-ref="n281"><a class="md-toc-inner" href="#header-n281">16.1 问题形式化</a></span><span class="md-toc-item md-toc-h3" data-ref="n312"><a class="md-toc-inner" href="#header-n312">16.2 基于内容的推荐系统</a></span><span class="md-toc-item md-toc-h3" data-ref="n349"><a class="md-toc-inner" href="#header-n349">16.3 协同过滤</a></span><span class="md-toc-item md-toc-h3" data-ref="n383"><a class="md-toc-inner" href="#header-n383">16.4 协同过滤算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n399"><a class="md-toc-inner" href="#header-n399">16.5 向量化：低秩矩阵分解</a></span><span class="md-toc-item md-toc-h3" data-ref="n470"><a class="md-toc-inner" href="#header-n470">16.6 推行工作上的细节：均值归一化</a></span></p></div><h2><a name='header-n5' class='md-header-anchor '></a>十五、异常检测(Anomaly Detection)</h2><h3><a name='header-n6' class='md-header-anchor '></a>15.1 问题的动机</h3><p>参考文档: 15 - 1 - Problem Motivation (8 min).mkv</p><p>在接下来的一系列视频中，我将向大家介绍异常检测(Anomaly detection)问题。这是机器学习算法的一个常见应用。这种算法的一个有趣之处在于：它虽然主要用于非监督学习问题，但从某些角度看，它又类似于一些监督学习问题。</p><p>什么是异常检测呢？为了解释这个概念，让我举一个例子吧：</p><p>假想你是一个飞机引擎制造商，当你生产的飞机引擎从生产线上流出时，你需要进行QA(质量控制测试)，而作为这个测试的一部分，你测量了飞机引擎的一些特征变量，比如引擎运转时产生的热量，或者引擎的振动等等。</p><p><img src='images/93d6dfe7e5cb8a46923c178171889747.png' alt='' /></p><p>这样一来，你就有了一个数据集，从<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-724-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.663ex" height="2.461ex" viewBox="0 -956.9 1577.2 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E725-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E725-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 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transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-29" x="890" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-724">x^{(1)}</script>到<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-791-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.284ex" height="2.461ex" viewBox="0 -956.9 1844.5 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E792-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E792-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E792-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 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749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E792-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E792-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E792-MJMATHI-6D" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E792-MJMAIN-29" x="1268" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-791">x^{(m)}</script>，如果你生产了m个引擎的话，你将这些数据绘制成图表，看起来就是这个样子：</p><p><img src='images/fe4472adbf6ddd9d9b51d698cc750b68.png' alt='' /></p><p>这里的每个点、每个叉，都是你的无标签数据。这样，异常检测问题可以定义如下：我们假设后来有一天，你有一个新的飞机引擎从生产线上流出，而你的新飞机引擎有特征变量<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-861-Frame" tabindex="-1" 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id="MathJax-Element-861">x_\text{test}</script>。所谓的异常检测问题就是：我们希望知道这个新的飞机引擎是否有某种异常，或者说，我们希望判断这个引擎是否需要进一步测试。因为，如果它看起来像一个正常的引擎，那么我们可以直接将它运送到客户那里，而不需要进一步的测试。</p><p>给定数据集 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-860-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.78ex" height="2.811ex" viewBox="0 -956.9 7224.8 1210.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E861-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-61" x="2494" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-6C" x="3023" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-79" x="3322" y="0"></use></g><g transform="translate(0,-653)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-6E" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-6F" x="600" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-72" x="1086" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-6D" x="1537" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-61" x="2416" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E846-MJMATHI-6C" x="2945" y="0"></use></g></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-845">
if \quad p(x)
\begin{cases}
\leq \varepsilon & anomaly \\\
\> \varepsilon & normal
\end{cases}
</script></div><p>欺诈检测：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-863-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.487ex" height="2.928ex" viewBox="0 -1007.2 10112.6 1260.5" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E864-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 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284 21 287Z"></path><path stroke-width="1" id="E864-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E864-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E864-MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path></defs><g stroke="currentColor" fill="currentColor" 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transform="translate(1562,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">的</text></g><g transform="translate(2344,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">第</text></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E864-MJMAIN-69" x="3125" y="0"></use><g transform="translate(3403,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">个</text></g><g transform="translate(4185,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">活</text></g><g transform="translate(4966,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">动</text></g><g transform="translate(5748,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">特</text></g><g transform="translate(6529,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(50.259) matrix(1 0 0 -1 0 0)">征</text></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-863">x^{(i)} = \text{用户的第i个活动特征}</script></p><p>模型<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 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xlink:href="#E928-MJMATHI-70" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-29" x="1465" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-927">p(x)</script> 为我们其属于一组数据的可能性</p><p>通过<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-879-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.58ex" height="2.577ex" viewBox="-38.5 -806.1 3694.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E880-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMAIN-29" x="1465" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMAIN-3C" x="2132" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMATHI-3B5" x="3189" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-879">p(x) < \varepsilon</script>检测非正常用户。</p><p>异常检测主要用来识别欺骗。例如在线采集而来的有关用户的数据，一个特征向量中可能会包含如：用户多久登录一次，访问过的页面，在论坛发布的帖子数量，甚至是打字速度等。尝试根据这些特征构建一个模型，可以用这个模型来识别那些不符合该模式的用户。</p><p>再一个例子是检测一个数据中心，特征可能包含：内存使用情况，被访问的磁盘数量，CPU的负载，网络的通信量等。根据这些特征可以构建一个模型，用来判断某些计算机是不是有可能出错了。</p><h3><a name='header-n44' class='md-header-anchor '></a>15.2 高斯分布</h3><p>参考视频: 15 - 2 - Gaussian Distribution (10 min).mkv</p><p>在这个视频中，我将介绍高斯分布，也称为正态分布。回顾高斯分布的基本知识。</p><p>通常如果我们认为变量 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E15-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" 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我们可以利用已有的数据来预测总体中的<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-868-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E869-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E869-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-868">μ</script>和<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-818-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="2.344ex" viewBox="0 -906.7 1025.4 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E819-MJMATHI-3C3" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path stroke-width="1" id="E819-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E819-MJMATHI-3C3" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E819-MJMAIN-32" x="808" y="513"></use></g></svg></span><script type="math/tex" id="MathJax-Element-818">σ^2</script>的计算方法如下：
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831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E870-MJSZ1-2211" d="M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z"></path><path stroke-width="1" 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y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E871-MJMAIN-2212" x="6736" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E871-MJMATHI-3BC" x="7737" y="0"></use><g transform="translate(8340,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E871-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E871-MJMAIN-32" x="550" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-870">\sigma^2=\frac{1}{m}\sum\limits_{i=1}^{m}(x^{(i)}-\mu)^2</script></p><p>高斯分布样例：</p><p><img src='images/fcb35433507a56631dde2b4e543743ee.png' alt='' /></p><p>注：机器学习中对于方差我们通常只除以<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-102-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.04ex" height="1.41ex" viewBox="0 -504.6 878.5 607.1" role="img" 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369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E872-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E872-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 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transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMAIN-31" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMAIN-2F" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMATHI-6D" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-523">1/m</script>还是<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-872-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.177ex" height="2.577ex" viewBox="0 -806.1 4381.9 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E873-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 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-10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E873-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E873-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMAIN-31" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMAIN-2F" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E524-MJMATHI-6D" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-523">1/m</script>这个版本的公式。这两个版本的公式在理论特性和数学特性上稍有不同，但是在实际使用中，他们的区别甚小，几乎可以忽略不计。</p><h3><a name='header-n62' class='md-header-anchor '></a>15.3 算法</h3><p>参考视频: 15 - 3 - Algorithm (12 min).mkv</p><p>在本节视频中，我将应用高斯分布开发异常检测算法。</p><p>异常检测算法：</p><p>对于给定的数据集 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-873-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.814ex" height="2.811ex" viewBox="0 -956.9 7670 1210.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E874-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 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252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-763-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.385ex" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E879-MJMAIN-32" x="810" y="488"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E879-MJMATHI-6A" x="808" y="-430"></use></g></g></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E879-MJMAIN-29" x="20675" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-878">p(x)=\prod\limits_{j=1}^np(x_j;\mu_j,\sigma_j^2)=\prod\limits_{j=1}^1\frac{1}{\sqrt{2\pi}\sigma_j}exp(-\frac{(x_j-\mu_j)^2}{2\sigma_j^2})</script></p><p>当<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-879-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.58ex" height="2.577ex" viewBox="-38.5 -806.1 3694.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E880-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E880-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMAIN-29" x="1465" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMAIN-3C" x="2132" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E880-MJMATHI-3B5" x="3189" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-879">p(x) < \varepsilon</script>时，为异常。</p><p>下图是一个由两个特征的训练集，以及特征的分布情况：</p><p><img src='images/ba47767a11ba39a23898b9f1a5a57cc5.png' alt='' /></p><p>下面的三维图表表示的是密度估计函数，z轴为根据两个特征的值所估计<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E928-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-70" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-29" x="1465" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-927">p(x)</script>值：</p><p><img src='images/82b90f56570c05966da116c3afe6fc91.jpg' alt='' /></p><p>我们选择一个<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-889-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.083ex" height="1.527ex" viewBox="0 -554.9 466.5 657.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E890-MJMATHI-3B5" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E890-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-889">\varepsilon</script>，将<span 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id="MathJax-Element-883">p(x) > \varepsilon</script>时预测数据为正常数据，否则则为异常。</p><p>在这段视频中，我们介绍了如何拟合<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 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id="MathJax-Element-785">\sigma</script>，然后检测新的样本，确定新样本是否是异常。</p><p>在接下来的课程中，我们将深入研究这一算法，同时更深入地介绍，怎样让算法工作地更加有效。</p><h3><a name='header-n95' class='md-header-anchor '></a>15.4 开发和评价一个异常检测系统</h3><p>参考视频: 15 - 4 - Developing and Evaluating an Anomaly Detection System (13 min). mkv</p><p>异常检测算法是一个非监督学习算法，意味着我们无法根据结果变量 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-490-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.155ex" height="1.877ex" viewBox="0 -504.6 497.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E491-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 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76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E890-MJMATHI-3B5" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-889">\varepsilon</script> 后，针对测试集进行预测，计算异常检验系统的F1值，或者查准率与查全率之比</li></ol><h3><a name='header-n120' class='md-header-anchor '></a>15.5 异常检测与监督学习对比</h3><p>参考视频: 15 - 5 - Anomaly Detection vs. Supervised Learning (8 min).mkv</p><p>之前我们构建的异常检测系统也使用了带标记的数据，与监督学习有些相似，下面的对比有助于选择采用监督学习还是异常检测：</p><p>两者比较：</p><table><thead><tr><th>异常检测</th><th>监督学习</th></tr></thead><tbody><tr><td>非常少量的正向类（异常数据 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-362-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.416ex" height="2.461ex" viewBox="0 -755.9 2332.1 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E363-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 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285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E893-MJMATHI-78" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E893-MJMAIN-3D" x="850" y="0"></use><g transform="translate(1906,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E893-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E893-MJMATHI-63" x="809" y="513"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-892">x=x^c</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-893-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.007ex" height="1.41ex" viewBox="0 -504.6 433.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E894-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E894-MJMATHI-63" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-893">c</script>为 0-1 之间的一个分数，等方法。</p><p><img src='images/0990d6b7a5ab3c0036f42083fe2718c6.jpg' alt='' /></p><p>误差分析：</p><p>一个常见的问题是一些异常的数据可能也会有较高的p(x)值，因而被算法认为是正常的。这种情况下误差分析能够帮助我们，我们可以分析那些被算法错误预测为正常的数据，观察能否找出一些问题。我们可能能从问题中发现我们需要增加一些新的特征，增加这些新特征后获得的新算法能够帮助我们更好地进行异常检测。</p><p>异常检测误差分析：</p><p><img src='images/f406bc738e5e032be79e52b6facfa48e.png' alt='' /></p><p>我们通常可以通过将一些相关的特征进行组合，来获得一些新的更好的特征（异常数据的该特征值异常地大或小），例如，在检测数据中心的计算机状况的例子中，我们可以用CPU负载与网络通信量的比例作为一个新的特征，如果该值异常地大，便有可能意味着该服务器是陷入了一些问题中。</p><p>在这段视频中，我们介绍了如何选择特征，以及对特征进行一些小小的转换，让数据更像正态分布，然后再把数据输入异常检测算法。同时也介绍了建立特征时，进行的误差分析方法，来捕捉各种异常的可能。希望你通过这些方法，能够了解如何选择好的特征变量，从而帮助你的异常检测算法，捕捉到各种不同的异常情况。</p><h3><a name='header-n166' class='md-header-anchor '></a>15.7 多元高斯分布（选修）</h3><p>参考视频: 15 - 7 - Multivariate Gaussian Distribution (Optional) (14 min).mkv</p><p>假使我们有两个相关的特征，而且这两个特征的值域范围比较宽，这种情况下，一般的高斯分布模型可能不能很好地识别异常数据。其原因在于，一般的高斯分布模型尝试的是去同时抓住两个特征的偏差，因此创造出一个比较大的判定边界。</p><p>下图中是两个相关特征，洋红色的线（根据ε的不同其范围可大可小）是一般的高斯分布模型获得的判定边界，很明显绿色的X所代表的数据点很可能是异常值，但是其<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E928-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E928-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E928-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-70" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-29" x="1465" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-927">p(x)</script>值却仍然在正常范围内。多元高斯分布将创建像图中蓝色曲线所示的判定边界。</p><p><img src='images/598db991a7c930c9021cec5f6ab9beb9.png' alt='' /></p><p>在一般的高斯分布模型中，我们计算 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E928-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E928-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E928-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-70" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-29" x="1465" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-927">p(x)</script> 的方法是：
通过分别计算每个特征对应的几率然后将其累乘起来，在多元高斯分布模型中，我们将构建特征的协方差矩阵，用所有的特征一起来计算 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.398ex" height="2.577ex" viewBox="-38.5 -806.1 1893.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E928-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E928-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E928-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E928-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-70" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-28" x="503" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMATHI-78" x="893" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E928-MJMAIN-29" x="1465" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-927">p(x)</script>。</p><p>我们首先计算所有特征的平均值，然后再计算协方差矩阵：
<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-897-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="52.957ex" height="5.262ex" viewBox="-38.5 -1409.3 22801 2265.7" role="img" focusable="false" style="vertical-align: -1.989ex; margin-left: -0.089ex;"><defs><path stroke-width="1" id="E898-MJMATHI-70" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="1" id="E898-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E898-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 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xlink:href="#E900-MJMATHI-58" x="21014" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E900-MJMAIN-2212" x="22089" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E900-MJMATHI-3BC" x="23090" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E900-MJMAIN-29" x="23693" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-899">\Sigma = \frac{1}{m}\sum_{i=1}^m(x^{(i)}-\mu)(x^{(i)}-\mu)^T=\frac{1}{m}(X-\mu)^T(X-\mu)</script></p><p>注:其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-900-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E901-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 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id="MathJax-Element-902">p(x)=\frac{1}{(2\pi)^{\frac{n}{2}}|\Sigma|^{\frac{1}{2}}}exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)</script>
其中：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-903-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.972ex" height="2.577ex" viewBox="0 -806.1 1279.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E904-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="1" id="E904-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E904-MJMAIN-7C" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E904-MJMAIN-3A3" x="278" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E904-MJMAIN-7C" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-903">|\Sigma|</script>是定矩阵，在 Octave 中用 <code>det(sigma)</code>计算</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-904-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.841ex" height="1.994ex" viewBox="0 -755.9 1223 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E905-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path><path stroke-width="1" id="E905-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E905-MJMAIN-3A3" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E905-MJMAIN-31" x="722" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-904">\Sigma1</script> 是逆矩阵，下面我们来看看协方差矩阵是如何影响模型的：</p><p><img src='images/29df906704d254f18e92a63173dd51e7.jpg' alt='' /></p><p>上图是5个不同的模型，从左往右依次分析：</p><ol start='' ><li>是一个一般的高斯分布模型</li><li>通过协方差矩阵，令特征1拥有较小的偏差，同时保持特征2的偏差</li><li>通过协方差矩阵，令特征2拥有较大的偏差，同时保持特征1的偏差</li><li>通过协方差矩阵，在不改变两个特征的原有偏差的基础上，增加两者之间的正相关性</li><li>通过协方差矩阵，在不改变两个特征的原有偏差的基础上，增加两者之间的负相关性</li></ol><p>多元高斯分布模型与原高斯分布模型的关系：</p><p>可以证明的是，原本的高斯分布模型是多元高斯分布模型的一个子集，即像上图中的第1、2、3，3个例子所示，如果协方差矩阵只在对角线的单位上有非零的值时，即为原本的高斯分布模型了。</p><p>原高斯分布模型和多元高斯分布模型的比较：</p><table><thead><tr><th>原高斯分布模型</th><th>多元高斯分布模型</th></tr></thead><tbody><tr><td>不能捕捉特征之间的相关性 但可以通过将特征进行组合的方法来解决</td><td>自动捕捉特征之间的相关性</td></tr><tr><td>计算代价低，能适应大规模的特征</td><td>计算代价较高 训练集较小时也同样适用</td></tr><tr><td></td><td>必须要有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-905-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.534ex" height="1.644ex" viewBox="0 -605.1 2813.1 707.6" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E906-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E906-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 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17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E907-MJMATHI-6D" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E907-MJMAIN-3E" x="1156" y="0"></use><g transform="translate(2212,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E907-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E907-MJMAIN-30" x="500" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E907-MJMATHI-6E" x="3213" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-906">m>10n</script> 另外特征冗余也会导致协方差矩阵不可逆</td></tr></tbody></table><p>原高斯分布模型被广泛使用着，如果特征之间在某种程度上存在相互关联的情况，我们可以通过构造新新特征的方法来捕捉这些相关性。</p><p>如果训练集不是太大，并且没有太多的特征，我们可以使用多元高斯分布模型。</p><h3><a name='header-n236' class='md-header-anchor '></a>15.8 使用多元高斯分布进行异常检测（可选）</h3><p>参考视频: 15 - 8 - Anomaly Detection using the Multivariate Gaussian Distribution (Optional) (14 min).mkv</p><p>在我们谈到的最后一个视频，关于多元高斯分布，看到的一些建立的各种分布模型，当你改变参数，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>。在这段视频中，让我们用这些想法，并应用它们制定一个不同的异常检测算法。</p><p>要回顾一下多元高斯分布和多元正态分布：</p><p><img src='images/3dbee365617e9264831400e4de247adc.png' alt='' /></p><p>分布有两个参数， <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>。其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script>这一个n维向量和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script> 的协方差矩阵，是一种n乘n的矩阵。而这里的公式<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E15-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E15-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-15">x</script>的概率，如按 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和参数化 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>，和你的变量 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>，你可以得到一个范围的不同分布一样，你知道的，这些都是三个样本，那些我们在以前的视频看过了。</p><p>因此，让我们谈谈参数拟合或参数估计问题：</p><p>我有一组样本<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-917-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg 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xlink:href="#E918-MJMAIN-2E" x="4044" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMAIN-2E" x="4489" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMAIN-2E" x="4935" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMAIN-2C" x="5380" y="0"></use><g transform="translate(5825,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMATHI-6D" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E918-MJMAIN-29" x="1268" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-917">{{{ x^{(1)},x^{(2)},...,x^{(m)}} }}</script>是一个n维向量，我想我的样本来自一个多元高斯分布。我如何尝试估计我的参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 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107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 是你的训练样本的平均值。</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-n253" cid="n253" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" 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227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E847-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E847-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMAIN-31" x="1124" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMATHI-6D" x="582" y="1627"></use></g><g transform="translate(4953,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E847-MJMAIN-29" x="734" y="0"></use></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-846">
\mu=\frac{1}{m}\sum_{i=1}^{m}x^{(i)}
</script></div><p>并设置<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-n255" cid="n255" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-847-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.705ex" height="6.78ex" viewBox="0 -1660.6 13650.6 2919" role="img" focusable="false" style="vertical-align: -2.923ex;"><defs><path stroke-width="1" id="E848-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path><path stroke-width="1" id="E848-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E848-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E848-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E848-MJSZ2-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 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629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-3A3" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-3D" x="1000" y="0"></use><g transform="translate(1778,0)"><g transform="translate(397,0)"><rect stroke="none" width="998" height="60" x="0" y="220"></rect><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-31" x="249" y="676"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-6D" x="60" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-2212" x="6985" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-3BC" x="7986" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-29" x="8589" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-28" x="8979" y="0"></use><g transform="translate(9368,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,412)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-2212" x="11058" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-3BC" x="12059" y="0"></use><g transform="translate(12662,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E848-MJMATHI-54" x="550" y="583"></use></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-847">
Σ=\frac{1}{m}\sum_{i=1}^{m}(x^{(i)}-\mu)(x^{(i)}-\mu)^T
</script></div><p>这其实只是当我们使用PCA算法时候，有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script> 时写出来。所以你只需插入上述两个公式，这会给你你估计的参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和你估计的参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>。所以，这里给出的数据集是你如何估计 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-928-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script> 和 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>。让我们以这种方法而只需将其插入到异常检测算法。那么，我们如何把所有这一切共同开发一个异常检测算法？</p><p><img src='images/d1a228f2bec262f2206379ed844c7f4a.png' alt='' /></p><p>首先，我们把我们的训练集，和我们的拟合模型，我们计算<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-927-Frame" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.402ex" height="1.877ex" viewBox="0 -504.6 603.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E929-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E929-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-928">\mu</script>和描述的一样<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>。</p><p><img src='images/015cee3a224dde6da0181215cf91a23d.png' alt='' /></p><p>如图，该分布在中央最多，越到外面的圈的范围越小。</p><p>并在该点是出路这里的概率非常低。</p><p>原始模型与多元高斯模型的关系如图：</p><p>其中：协方差矩阵<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-930-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.678ex" height="1.994ex" viewBox="0 -755.9 722.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E931-MJMAIN-3A3" d="M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E931-MJMAIN-3A3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-930">\Sigma</script>为：</p><p><img src='images/7104dd2548f1251e4c423e059d1d2594.png' alt='' /></p><p>原始模型和多元高斯分布比较如图：</p><p><img src='images/f4585239738f2b5149608879fa166889.png' alt='' /></p><h2><a name='header-n280' class='md-header-anchor '></a>十六、推荐系统(Recommender Systems)</h2><h3><a name='header-n281' class='md-header-anchor '></a>16.1 问题形式化</h3><p>参考视频: 16 - 1 - Problem Formulation (8 min).mkv</p><p>在接下来的视频中，我想讲一下推荐系统。我想讲推荐系统有两个原因：</p><p>第一、仅仅因为它是机器学习中的一个重要的应用。在过去几年，我偶尔访问硅谷不同的技术公司，我常和工作在这儿致力于机器学习应用的人们聊天，我常问他们，最重要的机器学习的应用是什么，或者，你最想改进的机器学习应用有哪些。我最常听到的答案是推荐系统。现在，在硅谷有很多团体试图建立很好的推荐系统。因此，如果你考虑网站像亚马逊，或网飞公司或易趣，或iTunes Genius，有很多的网站或系统试图推荐新产品给用户。如，亚马逊推荐新书给你，网飞公司试图推荐新电影给你，等等。这些推荐系统，根据浏览你过去买过什么书，或过去评价过什么电影来判断。这些系统会带来很大一部分收入，比如为亚马逊和像网飞这样的公司。因此，对推荐系统性能的改善，将对这些企业的有实质性和直接的影响。</p><p>推荐系统是个有趣的问题，在学术机器学习中因此，我们可以去参加一个学术机器学习会议，推荐系统问题实际上受到很少的关注，或者，至少在学术界它占了很小的份额。但是，如果你看正在发生的事情，许多有能力构建这些系统的科技企业，他们似乎在很多企业中占据很高的优先级。这是我为什么在这节课讨论它的原因之一。</p><p>我想讨论推荐系统地第二个原因是：这个班视频的最后几集我想讨论机器学习中的一些大思想，并和大家分享。这节课我们也看到了，对机器学习来说，特征是很重要的，你所选择的特征，将对你学习算法的性能有很大的影响。因此，在机器学习中有一种大思想，它针对一些问题，可能并不是所有的问题，而是一些问题，有算法可以为你自动学习一套好的特征。因此，不要试图手动设计，而手写代码这是目前为止我们常干的。有一些设置，你可以有一个算法，仅仅学习其使用的特征，推荐系统就是类型设置的一个例子。还有很多其它的，但是通过推荐系统，我们将领略一小部分特征学习的思想，至少，你将能够了解到这方面的一个例子，我认为，机器学习中的大思想也是这样。因此，让我们开始讨论推荐系统问题形式化。</p><p>我们从一个例子开始定义推荐系统的问题。</p><p>假使我们是一个电影供应商，我们有 5 部电影和 4 个用户，我们要求用户为电影打分。</p><p><img src='images/c2822f2c28b343d7e6ade5bd40f3a1fc.png' alt='' /></p><p>前三部电影是爱情片，后两部则是动作片，我们可以看出Alice和Bob似乎更倾向与爱情片， 而 Carol 和 Dave 似乎更倾向与动作片。并且没有一个用户给所有的电影都打过分。我们希望构建一个算法来预测他们每个人可能会给他们没看过的电影打多少分，并以此作为推荐的依据。</p><p>下面引入一些标记：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-931-Frame" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.95ex" height="1.994ex" viewBox="0 -504.6 1270.2 858.4" role="img" focusable="false" style="vertical-align: -0.822ex;"><defs><path stroke-width="1" id="E937-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E937-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E937-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E937-MJMATHI-6A" x="1242" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-936">m_j</script>代表用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script> 评过分的电影的总数</p><h3><a name='header-n312' class='md-header-anchor '></a>16.2 基于内容的推荐系统</h3><p>参考视频: 16 - 2 - Content Based Recommendations (15 min).mkv</p><p>在一个基于内容的推荐系统算法中，我们假设对于我们希望推荐的东西有一些数据，这些数据是有关这些东西的特征。</p><p>在我们的例子中，我们可以假设每部电影都有两个特征，如<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-220-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.644ex" viewBox="0 -504.6 1026.4 707.6" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E220-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E220-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E220-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E220-MJMAIN-31" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-220">x_1</script>代表电影的浪漫程度，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-221-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.644ex" viewBox="0 -504.6 1026.4 707.6" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E221-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E221-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E221-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E221-MJMAIN-32" x="809" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-221">x_2</script> 代表电影的动作程度。</p><p><img src='images/747c1fd6bff694c6034da1911aa3314b.png' alt='' /></p><p>则每部电影都有一个特征向量，如<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-724-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.663ex" height="2.461ex" viewBox="0 -956.9 1577.2 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E725-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E725-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E725-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E725-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E725-MJMAIN-29" x="890" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-724">x^{(1)}</script>是第一部电影的特征向量为[0.9 0]。</p><p>下面我们要基于这些特征来构建一个推荐系统算法。
假设我们采用线性回归模型，我们可以针对每一个用户都训练一个线性回归模型，如<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-937-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.424ex" height="2.694ex" viewBox="0 -1057.4 1474.2 1160" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E938-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E938-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E938-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E938-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E938-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,432)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E938-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E938-MJMAIN-31" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E938-MJMAIN-29" x="890" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-937">{{\theta }^{(1)}}</script>是第一个用户的模型的参数。
于是，我们有：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-938-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.28ex" height="2.461ex" viewBox="0 -956.9 1412 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E939-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E939-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E939-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E939-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E939-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E939-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E939-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E939-MJMAIN-29" x="802" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-938">\theta^{(j)}</script>用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script> 的参数向量</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-685-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="2.461ex" viewBox="0 -956.9 1467.6 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E686-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E686-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E686-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E686-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E686-MJMAIN-29" x="734" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-685">x^{(i)}</script>电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script> 的特征向量</p><p>对于用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script> 和电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script>，我们预测评分为：<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-939-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.887ex" height="2.928ex" viewBox="0 -956.9 4256.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E940-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E940-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E940-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E940-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="1" id="E940-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="1" id="E940-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E940-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-28" x="0" y="0"></use><g transform="translate(389,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMATHI-6A" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-29" x="802" y="0"></use></g></g><g transform="translate(1801,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMATHI-54" x="550" y="513"></use></g><g transform="translate(2789,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E940-MJMAIN-29" x="734" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-939">(\theta^{(j)})^T x^{(i)}</script></p><p>代价函数</p><p>针对用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 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\min_{\theta (j)}\frac{1}{2}\sum_{i:r(i,j)=1}\left((\theta^{(j)})^Tx^{i}-y^{(i,j)}\right)^2+\frac{\lambda}{2}\left(\theta_{k}^{(j)}\right)^2
</script></div><p>其中 i:r(i,j)表示我们只计算那些用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 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\min_{\theta^{(1)},...,\theta^{(n_u)}} \frac{1}{2}\sum_{j=1}^{n_u}\sum_{i:r(i,j)=1}\left((\theta^{(j)})^Tx^{(i)}-y^{(i,j)}\right)^2+\frac{\lambda}{2}\sum_{j=1}^{n_u}\sum_{k=1}^{n}(\theta_k^{(j)})^2
</script></div><p>如果我们要用梯度下降法来求解最优解，我们计算代价函数的偏导数后得到梯度下降的更新公式为：</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-n345" cid="n345" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-850-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="56.965ex" height="6.079ex" viewBox="0 -1107.7 24526.4 2617.5" role="img" focusable="false" style="vertical-align: -3.507ex;"><defs><path stroke-width="1" id="E851-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 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</script></div><p></p><h3><a name='header-n349' class='md-header-anchor '></a>16.3 协同过滤</h3><p>参考视频: 16 - 3 - Collaborative Filtering (10 min).mkv</p><p>在之前的基于内容的推荐系统中，对于每一部电影，我们都掌握了可用的特征，使用这些特征训练出了每一个用户的参数。相反地，如果我们拥有用户的参数，我们可以学习得出电影的特征。</p><div contenteditable="false" class="mathjax-block md-end-block" id="mathjax-n354" cid="n354" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-852-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="60.489ex" height="7.363ex" viewBox="0 -1660.6 26043.9 3170.3" role="img" focusable="false" style="vertical-align: -3.507ex;"><defs><path stroke-width="1" id="E853-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 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\mathop{min}\limits_{x^{(1)},...,x^{(n_m)}}\frac{1}{2}\sum_{i=1}^{n_m}\sum_{j{r(i,j)=1}}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)})^2+\frac{\lambda}{2}\sum_{i=1}^{n_m}\sum_{k=1}^{n}(x_k^{(i)})^2
</script></div><p>但是如果我们既没有用户的参数，也没有电影的特征，这两种方法都不可行了。协同过滤算法可以同时学习这两者。</p><p>我们的优化目标便改为同时针对<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E15-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 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\theta_k^{(i)}:=\theta_k^{(i)}-\alpha\left(\sum_{i:r(i,j)=1}((\theta^{(j)})^Tx^{(i)}-y^{(i,j)}x_k^{(i)}+\lambda \theta_k^{(j)}\right)
</script></div><p></p><p>注：在协同过滤从算法中，我们通常不使用方差项，如果需要的话，算法会自动学得。
协同过滤算法使用步骤如下：</p><ol start='' ><li>初始 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-940-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.961ex" height="2.811ex" viewBox="0 -956.9 15913.9 1210.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E941-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-29" x="890" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-2C" x="11738" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-2E" x="12183" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-2E" x="12628" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-2E" x="13073" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-2C" x="13518" y="0"></use><g transform="translate(13964,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMATHI-3B8" x="0" y="0"></use><g transform="translate(469,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMATHI-6E" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMATHI-75" x="989" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E941-MJMAIN-29" x="1562" y="0"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-940">x^{(1)},x^{(1)},...x^{(nm)},\ \theta^{(1)},\theta^{(2)},...,\theta^{(nu)}</script>为一些随机小值</li><li>使用梯度下降算法最小化代价函数</li><li>在训练完算法后，我们预测<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-941-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.887ex" height="2.928ex" viewBox="0 -956.9 4256.8 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E942-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 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id="MathJax-Element-941">(\theta^{(j)})^Tx^{(i)}</script>为用户 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 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302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script> 的评分</li></ol><p>通过这个学习过程获得的特征矩阵包含了有关电影的重要数据，这些数据不总是人能读懂的，但是我们可以用这些数据作为给用户推荐电影的依据。</p><p>例如，如果一位用户正在观看电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-685-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="2.461ex" viewBox="0 -956.9 1467.6 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E686-MJMATHI-78" d="M52 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</script></div><p></p><h3><a name='header-n399' class='md-header-anchor '></a>16.5 向量化：低秩矩阵分解</h3><p>参考视频: 16 - 5 - Vectorization_ Low Rank Matrix Factorization (8 min).mkv</p><p>在上几节视频中，我们谈到了协同过滤算法，本节视频中我将会讲到有关该算法的向量化实现，以及说说有关该算法你可以做的其他事情。</p><p>举例子：</p><ol start='' ><li>当给出一件产品时，你能否找到与之相关的其它产品。</li><li>一位用户最近看上一件产品，有没有其它相关的产品，你可以推荐给他。</li></ol><p>我将要做的是：实现一种选择的方法，写出协同过滤算法的预测情况。</p><p>我们有关于五部电影的数据集，我将要做的是，将这些用户的电影评分，进行分组并存到一个矩阵中。</p><p>我们有五部电影，以及四位用户，那么 这个矩阵 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-953-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.773ex" height="1.877ex" viewBox="0 -755.9 763.5 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="1" id="E954-MJMATHI-59" d="M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E954-MJMATHI-59" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-953">Y</script> 就是一个5行4列的矩阵，它将这些电影的用户评分数据都存在矩阵里：</p><table><thead><tr><th><strong>Movie</strong></th><th><strong>Alice (1)</strong></th><th><strong>Bob (2)</strong></th><th><strong>Carol (3)</strong></th><th><strong>Dave (4)</strong></th></tr></thead><tbody><tr><td>Love at last</td><td>5</td><td>5</td><td>0</td><td>0</td></tr><tr><td>Romance forever</td><td>5</td><td>?</td><td>?</td><td>0</td></tr><tr><td>Cute puppies of love</td><td>?</td><td>4</td><td>0</td><td>?</td></tr><tr><td>Nonstop car chases</td><td>0</td><td>0</td><td>5</td><td>4</td></tr><tr><td>Swords vs. karate</td><td>0</td><td>0</td><td>5</td><td>?</td></tr></tbody></table><p><img src='images/42a92e07b32b593bb826f8f6bc4d9eb3.png' alt='' /></p><p>推出评分：</p><p><img src='images/c905a6f02e201a4767d869b3791e8aeb.png' alt='' /></p><p>找到相关影片：</p><p><img src='images/0a8b49da1ab852f2996a02afcaca2322.png' alt='' /></p><p>现在既然你已经对特征参数向量进行了学习，那么我们就会有一个很方便的方法来度量两部电影之间的相似性。例如说：电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script>和电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script> 在某种程度上有相似，至少在某种意义上，某些人喜欢电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script>，或许更有可能也对电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script> 感兴趣。总结一下，当用户在看某部电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script> 的时候，如果你想找5部与电影非常相似的电影，为了能给用户推荐5部新电影，你需要做的是找出电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E116-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E116-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-116">j</script>，在这些不同的电影中与我们要找的电影 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E117-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E117-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-117">i</script> 的距离最小，这样你就能给你的用户推荐几部不同的电影了。</p><p>通过这个方法，希望你能知道，如何进行一个向量化的计算来对所有的用户和所有的电影进行评分计算。同时希望你也能掌握，通过学习特征参数，来找到相关电影和产品的方法。</p><h3><a name='header-n470' class='md-header-anchor '></a>16.6 推行工作上的细节：均值归一化</h3><p>参考视频: 16 - 6 - Implementational Detail_ Mean Normalization (9 min).mkv</p><p>让我们来看下面的用户评分数据：</p><p><img src='images/54b1f7c3131aed24f9834d62a6835642.png' alt='' /></p><p>如果我们新增一个用户 Eve，并且 Eve 没有为任何电影评分，那么我们以什么为依据为Eve推荐电影呢？</p><p>我们首先需要对结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-952-Frame" tabindex="-1" style="font-size: 100%; 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y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMAIN-2B" x="4479" y="0"></use><g transform="translate(5479,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E955-MJMATHI-69" x="853" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-954">(\theta^{(j)})^T x^{(i)}+\mu_i</script>，对于Eve，我们的新模型会认为她给每部电影的评分都是该电影的平均分。</p></div>
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