From ad8fdf1b2bcba28e0f8ce21561afeb0134960ae8 Mon Sep 17 00:00:00 2001
From: scruel <scruelt@hotmail.com>
Date: Thu, 11 Jan 2018 00:45:00 +0800
Subject: [PATCH] =?UTF-8?q?finish=20"6.3=20=E5=86=B3=E7=AD=96=E8=BE=B9?=
 =?UTF-8?q?=E7=95=8C(Decision=20Boundary)"?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 image/20180111_000653.png                     | Bin 0 -> 26926 bytes
 image/20180111_000814.png                     | Bin 0 -> 28642 bytes
 ...fa1f19aaf78265b8a33b_Logistic_function.png | Bin 16623 -> 15790 bytes
 week3.html                                    |   2 +-
 week3.md                                      |  55 ++++++++++++++++--
 5 files changed, 52 insertions(+), 5 deletions(-)
 create mode 100644 image/20180111_000653.png
 create mode 100644 image/20180111_000814.png

diff --git a/image/20180111_000653.png b/image/20180111_000653.png
new file mode 100644
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-<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n40"><a class="md-toc-inner" href="#header-n40">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n62"><a class="md-toc-inner" href="#header-n62">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n65"><a class="md-toc-inner" href="#header-n65">6.4 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n66"><a class="md-toc-inner" href="#header-n66">6.5 Simplified Cost Function and Gradient Descent</a></span><span class="md-toc-item md-toc-h2" data-ref="n67"><a class="md-toc-inner" href="#header-n67">6.6 Advanced Optimization</a></span><span class="md-toc-item md-toc-h2" data-ref="n68"><a class="md-toc-inner" href="#header-n68">6.7 Multiclass Classification_ One-vs-all</a></span><span class="md-toc-item md-toc-h1" data-ref="n69"><a class="md-toc-inner" href="#header-n69">7 Regularization</a></span><span class="md-toc-item md-toc-h2" data-ref="n70"><a class="md-toc-inner" href="#header-n70">7.1 The Problem of Overfitting</a></span><span class="md-toc-item md-toc-h2" data-ref="n71"><a class="md-toc-inner" href="#header-n71">7.2 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n72"><a class="md-toc-inner" href="#header-n72">7.3 Regularized Linear Regression</a></span><span class="md-toc-item md-toc-h2" data-ref="n73"><a class="md-toc-inner" href="#header-n73">7.4 Regularized Logistic Regression</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n3' class='md-header-anchor '></a>6.1 分类(Classification)</h2><p>在分类问题中，预测的结果是离散值（结果是否属于某一类），逻辑回归算法(Logistic Regression)被用于解决这类分类问题。</p><ul><li>垃圾邮件判断</li><li>金融欺诈判断</li><li>肿瘤诊断</li></ul><p>肿瘤诊断问题：</p><p><img src='image/20180109_144040.png' alt='' /></p><p>肿瘤诊断问题是一个<strong>二元分类问题(binary class problems)</strong>，则定义 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E86-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 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class)</strong>，代表恶性肿瘤(&quot;-&quot;)，1 为<strong>正向类(positive class)</strong>，代表良性肿瘤(&quot;+&quot;)。如图，定义最右边的样本为<strong>偏差项</strong>。</p><p>在未加入偏差项时，线性回归算法给出了品红色的拟合直线，若规定</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E88-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E88-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-87">h_\theta(x) \geqslant 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-88-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="E89-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E89-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="E89-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="#E89-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E89-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-88">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-89-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E90-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-89">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-90-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="E91-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E91-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="E91-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="#E91-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E91-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-90">y = 0</script>，即负向类。</p><p>即以 0.5 为<strong>阈值</strong>(threshold)，则我们就可以根据线性回归结果，得到相对正确的分类结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-20-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="E20-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 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="#E20-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-20">y</script>。</p><p>&nbsp;</p><p>接下来加入偏差项，线性回归算法给出了靛青色的拟合直线，如果阈值仍然为 0.5，可以看到算法在某些情况下会给出完全错误的结果。</p><p>不仅如此，线性回归算法的值域为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="1.994ex" viewBox="0 -755.9 759.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E92-MJMATHI-52" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></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="#E92-MJMATHI-52" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-91">R</script>，则当线性回归函数给出诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.342ex" height="2.344ex" viewBox="0 -755.9 10049.8 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E93-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E93-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="E93-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="E93-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E93-MJMAIN-30" x="2002" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-92">h = 10000, h = -10000</script> 等很大/很小(负数)的数值时，结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-93-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E94-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 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68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z"></path><path stroke-width="1" id="E94-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E94-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 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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="#E94-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-2208" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-7B" x="1720" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-30" x="2221" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-2C" x="2721" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-31" x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E94-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-93">y \in \lbrace 0, 1\rbrace</script>，这显得非常怪异。</p><p>&nbsp;</p><p>区别于线性回归算法，逻辑回归算法是一个分类算法，<strong>其输出值永远在 0 到 1 之间</strong>，即 <span class="MathJax_Preview"></span><span 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E97-MJMATHI-58" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-96">h_\theta \left( x \right)=\theta^{T}X</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-97-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E98-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></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="#E98-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-97">g</script> 表示逻辑函数(<a href='https://en.wikipedia.org/wiki/Logistic_function' target='_blank'>logistic function</a>)，复合起来，则为线性回归函数。</p><p>一个常用的逻辑函数是 S 形函数，叫做 <a href='https://en.wikipedia.org/wiki/Sigmoid_function' target='_blank'>sigmoid 函数</a>（如下图），其公式为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-98-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.814ex" height="3.395ex" viewBox="0 -956.9 5517.2 1461.5" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E99-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 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type="math/tex" id="MathJax-Element-69">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-101-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="E102-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 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x="3751" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3B" x="4324" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-3B8" x="4769" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-29" x="5238" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-2B" x="5850" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-50" x="6851" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-28" x="7602" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-79" x="7992" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3D" x="8767" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-31" x="9823" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-7C" x="10324" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-78" x="10602" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3B" x="11175" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMATHI-3B8" x="11620" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-29" x="12089" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-3D" x="12757" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E103-MJMAIN-31" x="13813" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-102">\begin{align*}& h_\theta(x) = P(y=1 | x ; \theta) = 1 - P(y=0 | x ; \theta) \newline & P(y = 0 | x;\theta) + P(y = 1 | x ; \theta) = 1\end{align*}</script></p><p>&nbsp;</p><p>以肿瘤诊断为例，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="-1" style="font-size: 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xlink:href="#E104-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E104-MJMAIN-37" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-103">h_\theta \left( x \right)=0.7</script> 表示病人有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-104-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.261ex" height="2.227ex" viewBox="0 -806.1 1834.5 958.9" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E105-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="E105-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 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xlink:href="#E105-MJMAIN-37"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E105-MJMAIN-25" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-104">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n62' class='md-header-anchor '></a>6.3 决策边界(Decision Boundary)</h2><p>&nbsp;</p><h2><a name='header-n65' class='md-header-anchor '></a>6.4 Cost Function</h2><h2><a name='header-n66' class='md-header-anchor '></a>6.5 Simplified Cost Function and Gradient Descent</h2><h2><a name='header-n67' class='md-header-anchor '></a>6.6 Advanced Optimization</h2><h2><a name='header-n68' class='md-header-anchor '></a>6.7 Multiclass Classification_ One-vs-all</h2><h1><a name='header-n69' class='md-header-anchor '></a>7 Regularization</h1><h2><a name='header-n70' class='md-header-anchor '></a>7.1 The Problem of Overfitting</h2><h2><a name='header-n71' class='md-header-anchor '></a>7.2 Cost Function</h2><h2><a name='header-n72' class='md-header-anchor '></a>7.3 Regularized Linear Regression</h2><h2><a name='header-n73' class='md-header-anchor '></a>7.4 Regularized Logistic Regression</h2></div>
+<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n216"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n216">6 逻辑回归(Logistic Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n217"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n217">6.1 分类(Classification)</a></span><span class="md-toc-item md-toc-h2" data-ref="n254"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n254">6.2 假设函数表示(Hypothesis Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n278"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n278">6.3 决策边界(Decision Boundary)</a></span><span class="md-toc-item md-toc-h2" data-ref="n291"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n291">6.4 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n292"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n292">6.5 Simplified Cost Function and Gradient Descent</a></span><span class="md-toc-item md-toc-h2" data-ref="n293"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n293">6.6 Advanced Optimization</a></span><span class="md-toc-item md-toc-h2" data-ref="n294"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n294">6.7 Multiclass Classification_ One-vs-all</a></span><span class="md-toc-item md-toc-h1" data-ref="n295"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n295">7 Regularization</a></span><span class="md-toc-item md-toc-h2" data-ref="n296"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n296">7.1 The Problem of Overfitting</a></span><span class="md-toc-item md-toc-h2" data-ref="n297"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n297">7.2 Cost Function</a></span><span class="md-toc-item md-toc-h2" data-ref="n298"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n298">7.3 Regularized Linear Regression</a></span><span class="md-toc-item md-toc-h2" data-ref="n299"><a class="md-toc-inner" style="cursor: pointer;" href="#header-n299">7.4 Regularized Logistic Regression</a></span></p></div><h1><a name='header-n216' class='md-header-anchor '></a>6 逻辑回归(Logistic Regression)</h1><h2><a name='header-n217' class='md-header-anchor '></a>6.1 分类(Classification)</h2><p>在分类问题中，预测的结果是离散值（结果是否属于某一类），逻辑回归算法(Logistic Regression)被用于解决这类分类问题。</p><ul><li>垃圾邮件判断</li><li>金融欺诈判断</li><li>肿瘤诊断</li></ul><p>肿瘤诊断问题：</p><p><img src='image/20180109_144040.png' alt='' /></p><p>肿瘤诊断问题是一个<strong>二元分类问题(binary class problems)</strong>，则定义 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: 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xlink:href="#E4-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJAMS-2A7E" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E4-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-3">h_\theta(x) \geqslant 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-4-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="E5-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E5-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="E5-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="#E5-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E5-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E5-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-4">y = 1</script>，即正向类；</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.551ex" height="2.577ex" viewBox="0 -806.1 4973.5 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E6-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E6-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="E6-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="E6-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="E6-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="E6-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path stroke-width="1" id="E6-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E6-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E6-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></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="#E6-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E6-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-5">h_\theta(x) \lt 0.5</script> ，预测为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-6-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="E7-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E7-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="E7-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="#E7-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E7-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E7-MJMAIN-30" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-6">y = 0</script>，即负向类。</p><p>即以 0.5 为<strong>阈值</strong>(threshold)，则我们就可以根据线性回归结果，得到相对正确的分类结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-7-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="E8-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 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="#E8-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-7">y</script>。</p><p>&nbsp;</p><p>接下来加入偏差项，线性回归算法给出了靛青色的拟合直线，如果阈值仍然为 0.5，可以看到算法在某些情况下会给出完全错误的结果。</p><p>不仅如此，线性回归算法的值域为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.764ex" height="1.994ex" viewBox="0 -755.9 759.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E9-MJMATHI-52" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></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="#E9-MJMATHI-52" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-8">R</script>，则当线性回归函数给出诸如 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.342ex" height="2.344ex" viewBox="0 -755.9 10049.8 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E10-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E10-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="E10-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="E10-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 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xlink:href="#E10-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-30" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-30" x="1501" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E10-MJMAIN-30" x="2002" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-9">h = 10000, h = -10000</script> 等很大/很小(负数)的数值时，结果 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.68ex" height="2.577ex" viewBox="0 -806.1 4167.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E11-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 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x="3166" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E11-MJMAIN-7D" x="3667" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-10">y \in \lbrace 0, 1\rbrace</script>，这显得非常怪异。</p><p>&nbsp;</p><p>区别于线性回归算法，逻辑回归算法是一个分类算法，<strong>其输出值永远在 0 到 1 之间</strong>，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.348ex" height="2.577ex" viewBox="0 -806.1 4024.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E12-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E12-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="1" id="E12-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="E12-MJMAIN-30" d="M96 585Q152 666 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transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E14-MJMATHI-58" x="4928" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-13">h_\theta \left( x \right)=\theta^{T}X</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.116ex" height="1.877ex" viewBox="0 -504.6 480.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E15-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></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-67" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-14">g</script> 表示逻辑函数(<a href='https://en.wikipedia.org/wiki/Logistic_function' target='_blank'>logistic function</a>)，复合起来，则称为逻辑回归函数。</p><p>逻辑函数是 S 形函数，会将所有实数映射到 <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="5.168ex" height="2.577ex" viewBox="0 -806.1 2225.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E16-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="E16-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E16-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 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y="572"></use><g transform="translate(59,-376)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E17-MJMAIN-31" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E17-MJMAIN-2B" x="500" y="0"></use><g transform="translate(904,0)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E17-MJMATHI-65" x="0" y="0"></use><g transform="translate(329,204)"><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E17-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E17-MJMATHI-7A" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-16">g\left( z \right)=\frac{1}{1+{{e}^{-z}}}</script>。 </p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>应用 sigmoid 函数，则逻辑回归模型：<span 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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="#E20-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-19">x</script> 以及参数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" 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id="MathJax-Element-20">\theta</script>，计算得出”输出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-21-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="E22-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E22-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="E22-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="#E22-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-3D" x="775" y="0"></use><use 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y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-3D" x="2804" y="0"></use><g transform="translate(3860,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E24-MJMAIN-37" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-23">h_\theta \left( x \right)=0.7</script> 表示病人有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.261ex" height="2.227ex" viewBox="0 -806.1 1834.5 958.9" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E25-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path stroke-width="1" id="E25-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E25-MJMAIN-30" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E25-MJMAIN-25" x="1001" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-24">70\%</script> 的概率得了恶性肿瘤。</p><h2><a name='header-n278' class='md-header-anchor '></a>6.3 决策边界(Decision Boundary)</h2><p>决策边界的概念可以帮助我们更好地理解逻辑回归模型。</p><p>在逻辑回归中，有假设函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-188-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.171ex" height="3.044ex" viewBox="0 -906.7 10407.1 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E249-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E249-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="E249-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 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x="5866" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJMATHI-67" x="6922" y="0"></use><g transform="translate(7569,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJMATHI-54" x="663" y="513"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJMATHI-58" x="1526" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E249-MJSZ1-29" x="2378" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-188">h_\theta \left( x \right)=g(z)=g\left(\theta^{T}X \right)</script>。</p><p>为了得出分类的结果，这里和前面一样，规定以 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="-1" style="font-size: 100%; display: 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511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E28-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E28-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="E28-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="E28-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-15,0)"><g transform="translate(0,651)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2265" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-31" x="8361" y="0"></use></g><g transform="translate(0,-700)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3C" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-35" x="779" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-2192" x="5251" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMATHI-79" x="6529" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-3D" x="7304" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E28-MJMAIN-30" x="8361" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-27">\begin{align*}& h_\theta(x) \geq 0.5 \rightarrow y = 1 \newline& h_\theta(x) < 0.5 \rightarrow y = 0 \newline\end{align*}</script></p><p>回忆一下逻辑函数的图像：</p><p><img src='image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png' alt='sigmoid function' /></p><p>观察可得当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.084ex" height="2.577ex" viewBox="0 -806.1 4341.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E123-MJMATHI-67" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path stroke-width="1" id="E123-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="E123-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E123-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="E123-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="1" id="E123-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E123-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E123-MJMAIN-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></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="#E123-MJMATHI-67" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-28" x="480" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMATHI-7A" x="870" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-29" x="1338" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-2265" x="2005" y="0"></use><g transform="translate(3062,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E123-MJMAIN-35" x="779" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-113">g(z) \geq 0.5</script> 时，有 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.349ex" height="2.227ex" viewBox="0 -755.9 2303.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E204-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E204-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="1" id="E204-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="#E204-MJMATHI-7A" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-2265" x="746" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E204-MJMAIN-30" x="1802" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-153">z \geq 0</script>，即 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-211-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.721ex" height="2.577ex" viewBox="0 -906.7 3754.7 1109.7" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E285-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="E285-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="E285-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="1" id="E285-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path stroke-width="1" id="E285-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="#E285-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E285-MJMATHI-54" x="663" y="513"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E285-MJMATHI-58" x="1067" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E285-MJMAIN-2265" x="2197" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E285-MJMAIN-30" x="3254" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-211">\theta^TX \geq 0</script>。</p><p>同线性回归模型的不同点在于： <span 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E605-MJMATHI-7A" x="10505" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E605-MJMAIN-29" x="10973" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E605-MJMAIN-3D" x="11641" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E605-MJMAIN-30" x="12697" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-433">\begin{align*}z \to +\infty, e^{-\infty} \to 0 \Rightarrow g(z)=1 \newline z \to -\infty, e^{\infty}\to \infty \Rightarrow g(z)=0 \end{align*}</script></p><p>直观一点来个例子，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-224-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="29.162ex" height="2.577ex" viewBox="0 -806.1 12555.8 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E299-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E299-MJMAIN-29" x="7658" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-224">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\right)</script> 是下图模型的假设函数。</p><p><img src='image/20180111_000814.png' alt='' /></p><p>根据上面的讨论，要进行分类，那么只要 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-250-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.144ex" height="2.461ex" viewBox="0 -806.1 9103.5 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E343-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="E343-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E343-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="1" id="E343-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="E343-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="E343-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><path stroke-width="1" id="E343-MJMAIN-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></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="#E343-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-30" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-2B" x="1145" y="0"></use><g transform="translate(2146,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(3069,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-2B" x="4318" y="0"></use><g transform="translate(5319,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6242,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-2265" x="7546" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E343-MJMAIN-30" x="8602" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-250"> {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\geq0</script> 时，就预测 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-4-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="E5-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E5-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="E5-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="#E5-MJMATHI-79" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E5-MJMAIN-3D" x="775" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E5-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-4">y = 1</script>，即预测为正向类。</p><p>如果取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-303-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.001ex" height="9.114ex" viewBox="0 -2213.4 4736.6 3924.2" role="img" focusable="false" style="vertical-align: -3.974ex;"><defs><path stroke-width="1" id="E446-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="E446-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="E446-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E446-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E446-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E446-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="E446-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E446-MJSZ4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path stroke-width="1" id="E446-MJSZ4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path stroke-width="1" id="E446-MJSZ4-23A2" d="M319 0V602H403V0H319Z"></path><path stroke-width="1" id="E446-MJSZ4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path stroke-width="1" id="E446-MJSZ4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path stroke-width="1" id="E446-MJSZ4-23A5" d="M263 0V602H347V0H263Z"></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="#E446-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><g transform="translate(0,2150)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJSZ4-23A1" x="0" y="-1155"></use><g transform="translate(0,-2048.5066225165565) scale(1,0.49337748344370863)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJSZ4-23A3" x="0" y="-3155"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><g transform="translate(0,1350)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJMAIN-33" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJMAIN-31" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E446-MJMAIN-31" x="389" y="-1450"></use></g></g><g transform="translate(2265,2150)"><use 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120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="1" id="E477-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="E477-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E477-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 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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="E477-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="E477-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 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463Z"></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="#E304-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMAIN-31" x="809" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMAIN-2B" x="1248" y="0"></use><g transform="translate(2249,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMAIN-2265" x="3553" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E304-MJMAIN-33" x="4609" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-228">{x_1}+{x_2} \geq 3</script> 时，易绘制图中的品红色直线即<strong>决策边界</strong>，为正向类（以红叉标注的数据）给出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-21-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="E22-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 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E22-MJMAIN-31" x="1831" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-21">y=1</script> 的分类预测结果。</p><p>&nbsp;</p><p>上面讨论了逻辑回归模型中线性拟合的例子，下面则是一个多项式拟合的例子，和线性回归中讨论的其实没有多大区别。</p><p>为了拟合下图数据，建模多项式假设函数：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-222-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="44.221ex" height="3.044ex" viewBox="0 -906.7 19039.3 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="1" id="E297-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-29" x="962" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-3D" x="2804" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-67" x="3860" y="0"></use><g transform="translate(4507,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-2B" x="1604" y="0"></use><g transform="translate(2604,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(3528,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-31" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-2B" x="4776" y="0"></use><g transform="translate(5777,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(6701,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-32" x="809" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-2B" x="7949" y="0"></use><g transform="translate(8950,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(9873,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-2B" x="11122" y="0"></use><g transform="translate(12123,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-34" x="663" y="-213"></use></g><g transform="translate(13046,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-32" x="809" y="488"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJMAIN-32" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E297-MJSZ1-29" x="14072" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-222">{h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}+{\theta_{3}}x_{1}^{2}+{\theta_{4}}x_{2}^{2} \right)</script></p><p>这里取 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-415-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.001ex" height="15.651ex" viewBox="0 -3620.7 4736.6 6738.7" role="img" focusable="false" style="vertical-align: -7.242ex;"><defs><path stroke-width="1" id="E584-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="E584-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="E584-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="1" id="E584-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="1" id="E584-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="E584-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E584-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="1" id="E584-MJSZ4-23A1" d="M319 -645V1154H666V1070H403V-645H319Z"></path><path stroke-width="1" id="E584-MJSZ4-23A3" d="M319 -644V1155H403V-560H666V-644H319Z"></path><path stroke-width="1" id="E584-MJSZ4-23A2" d="M319 0V602H403V0H319Z"></path><path stroke-width="1" id="E584-MJSZ4-23A4" d="M0 1070V1154H347V-645H263V1070H0Z"></path><path stroke-width="1" id="E584-MJSZ4-23A6" d="M263 -560V1155H347V-644H0V-560H263Z"></path><path stroke-width="1" id="E584-MJSZ4-23A5" d="M263 0V602H347V0H263Z"></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="#E584-MJMATHI-3B8" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-3D" x="747" y="0"></use><g transform="translate(1803,0)"><g transform="translate(0,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A1" x="0" y="-1155"></use><g transform="translate(0,-4843.8708609271525) scale(1,5.129139072847682)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A2"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A3" x="0" y="-5955"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><g transform="translate(0,2750)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-2212" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-31" x="778" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-30" x="389" y="1350"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-30" x="389" y="-50"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-31" x="389" y="-1450"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJMAIN-31" x="389" y="-2850"></use></g></g><g transform="translate(2265,3550)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A4" x="0" y="-1155"></use><g transform="translate(0,-4843.8708609271525) scale(1,5.129139072847682)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A5"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E584-MJSZ4-23A6" x="0" y="-5955"></use></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-415">\theta = \begin{bmatrix} -1\\0\\0\\1\\1\end{bmatrix}</script>，决策边界对应了一个在原点处的单位圆，如此便可给出分类结果了，如图中品红色曲线：</p><p>&nbsp;</p><p><img src='image/20180111_000653.png' alt='' /></p><p>当然，通过一些更为复杂的多项式，还能拟合那些图像显得非常怪异的数据，使得决策边界像碗状、爱心状等等。</p><p>&nbsp;</p><p>简单来说，决策边界就是<strong>分类的分界线</strong>，分类现在实际就由 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-115-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.088ex" height="1.41ex" viewBox="0 -504.6 468.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E126-MJMATHI-7A" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></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="#E126-MJMATHI-7A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-115">z</script> (中的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E21-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></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="#E21-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-20">\theta</script>)决定啦。</p><h2><a name='header-n291' class='md-header-anchor '></a>6.4 代价函数(Cost Function)</h2><p>&nbsp;</p><h2><a name='header-n292' class='md-header-anchor '></a>6.5 Simplified Cost Function and Gradient Descent</h2><h2><a name='header-n293' class='md-header-anchor '></a>6.6 Advanced Optimization</h2><h2><a name='header-n294' class='md-header-anchor '></a>6.7 Multiclass Classification_ One-vs-all</h2><h1><a name='header-n295' class='md-header-anchor '></a>7 Regularization</h1><h2><a name='header-n296' class='md-header-anchor '></a>7.1 The Problem of Overfitting</h2><h2><a name='header-n297' class='md-header-anchor '></a>7.2 Cost Function</h2><h2><a name='header-n298' class='md-header-anchor '></a>7.3 Regularized Linear Regression</h2><h2><a name='header-n299' class='md-header-anchor '></a>7.4 Regularized Logistic Regression</h2></div>
 </body>
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diff --git a/week3.md b/week3.md
index 121b66b..15f59c5 100644
--- a/week3.md
+++ b/week3.md
@@ -38,11 +38,13 @@ $h_\theta(x) \lt 0.5$ ，预测为 $y = 0$，即负向类。
 
 为了使 $h \in \left(0, 1\right)$，引入逻辑回归模型，定义假设函数
 $$
-h_\theta \left( x \right)=g\left(\theta^{T}X \right)
+h_\theta \left( x \right)=g(z)=g\left(\theta^{T}X \right)
 $$
-对比线性回归函数 $h_\theta \left( x \right)=\theta^{T}X$，$g$ 表示逻辑函数([logistic function][1])，复合起来，则为线性回归函数。
+对比线性回归函数 $h_\theta \left( x \right)=\theta^{T}X$，$g$ 表示逻辑函数([logistic function][1])，复合起来，则称为逻辑回归函数。
 
-一个常用的逻辑函数是 S 形函数，叫做 [sigmoid 函数][2]（如下图），其公式为 $g\left( z \right)=\frac{1}{1+{{e}^{-z}}}$。 
+逻辑函数是 S 形函数，会将所有实数映射到 $(0, 1)$ 范围。
+
+[sigmoid 函数][2]（如下图）是逻辑函数的特殊情况，其公式为 $g\left( z \right)=\frac{1}{1+{{e}^{-z}}}$。 
 
 ![sigmoid function](image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png)
 
@@ -61,9 +63,54 @@ $\begin{align*}& h_\theta(x) = P(y=1 | x ; \theta) = 1 - P(y=0 | x ; \theta) \ne
 
 ## 6.3 决策边界(Decision Boundary)
 
+决策边界的概念可以帮助我们更好地理解逻辑回归模型。
+
+在逻辑回归中，有假设函数 $h_\theta \left( x \right)=g(z)=g\left(\theta^{T}X \right)$。
+
+为了得出分类的结果，这里和前面一样，规定以 $0.5$ 为阈值：
+
+
+$\begin{align*}& h_\theta(x) \geq 0.5 \rightarrow y = 1 \newline& h_\theta(x) < 0.5 \rightarrow y = 0 \newline\end{align*}$
+
+回忆一下逻辑函数的图像：
+
+![sigmoid function](image/2413fbec8ff9fa1f19aaf78265b8a33b_Logistic_function.png)
+
+观察可得当 $g(z) \geq 0.5$ 时，有 $z \geq 0$，即 $\theta^TX \geq 0$。
+
+同线性回归模型的不同点在于： $\begin{align*}z \to +\infty, e^{-\infty} \to 0 \Rightarrow g(z)=1 \newline z \to -\infty, e^{\infty}\to \infty \Rightarrow g(z)=0 \end{align*}$
+
+直观一点来个例子，${h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\right)$ 是下图模型的假设函数。
+
+![](image/20180111_000814.png)
+
+根据上面的讨论，要进行分类，那么只要 $ {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}\geq0$ 时，就预测 $y = 1$，即预测为正向类。
+
+如果取 $\theta = \begin{bmatrix} -3\\1\\1\end{bmatrix}$，则有 $z = -3+{x_1}+{x_2}$，当 $z \geq 0$ 即 ${x_1}+{x_2} \geq 3$ 时，易绘制图中的品红色直线即**决策边界**，为正向类（以红叉标注的数据）给出 $y=1$ 的分类预测结果。
+
+
+
+上面讨论了逻辑回归模型中线性拟合的例子，下面则是一个多项式拟合的例子，和线性回归中讨论的其实没有多大区别。
+
+为了拟合下图数据，建模多项式假设函数：
+
+${h_\theta}\left( x \right)=g\left( {\theta_0}+{\theta_1}{x_1}+{\theta_{2}}{x_{2}}+{\theta_{3}}x_{1}^{2}+{\theta_{4}}x_{2}^{2} \right)$
+
+这里取 $\theta = \begin{bmatrix} -1\\0\\0\\1\\1\end{bmatrix}$，决策边界对应了一个在原点处的单位圆，如此便可给出分类结果了，如图中品红色曲线：
+
+
+
+![](image/20180111_000653.png)
+
+当然，通过一些更为复杂的多项式，还能拟合那些图像显得非常怪异的数据，使得决策边界像碗状、爱心状等等。
+
+
+
+简单来说，决策边界就是**分类的分界线**，分类现在实际就由 $z$ (中的 $\theta$)决定啦。
+
+## 6.4 代价函数(Cost Function)
 
 
-## 6.4 Cost Function
 
 ## 6.5 Simplified Cost Function and Gradient Descent