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scruel authored 2018-01-25 20:44 . finish week5
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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">4 多变量线性回归(Linear Regression with Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">4.1 多特征(Multiple Features)</a></span><span class="md-toc-item md-toc-h2" data-ref="n38"><a class="md-toc-inner" href="#header-n38">4.2 多变量梯度下降(Gradient Descent for Multiple Variables)</a></span><span class="md-toc-item md-toc-h2" data-ref="n63"><a class="md-toc-inner" href="#header-n63">4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</a></span><span class="md-toc-item md-toc-h2" data-ref="n84"><a class="md-toc-inner" href="#header-n84">4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</a></span><span class="md-toc-item md-toc-h2" data-ref="n123"><a class="md-toc-inner" href="#header-n123">4.5 特征和多项式回归(Features and Polynomial Regression)</a></span><span class="md-toc-item md-toc-h2" data-ref="n138"><a class="md-toc-inner" href="#header-n138">4.6 正规方程(Normal Equation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n200"><a class="md-toc-inner" href="#header-n200">4.7 不可逆性正规方程(Normal Equation Noninvertibility)</a></span><span class="md-toc-item md-toc-h1" data-ref="n232"><a class="md-toc-inner" href="#header-n232">5 Octave/Matlab Tutorial</a></span><span class="md-toc-item md-toc-h2" data-ref="n235"><a class="md-toc-inner" href="#header-n235">5.1 Basic Operations</a></span><span class="md-toc-item md-toc-h2" data-ref="n236"><a class="md-toc-inner" href="#header-n236">5.2 Moving Data Around</a></span><span class="md-toc-item md-toc-h2" data-ref="n237"><a class="md-toc-inner" href="#header-n237">5.3 Computing on Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n238"><a class="md-toc-inner" href="#header-n238">5.4 Plotting Data</a></span><span class="md-toc-item md-toc-h2" data-ref="n239"><a class="md-toc-inner" href="#header-n239">5.5 Control Statements: for, while, if statement</a></span><span class="md-toc-item md-toc-h2" data-ref="n240"><a class="md-toc-inner" href="#header-n240">5.6 向量化(Vectorization)</a></span><span class="md-toc-item md-toc-h2" data-ref="n243"><a class="md-toc-inner" href="#header-n243">5.x 常用函数整理</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>4 多变量线性回归(Linear Regression with Multiple Variables)</h1><h2><a name='header-n3' class='md-header-anchor '></a>4.1 多特征(Multiple Features)</h2><p>对于一个要度量的对象,一般来说会有不同维度的多个特征。比如之前的房屋价格预测例子中,除了房屋的面积大小,可能还有房屋的年限、房屋的层数等等其他特征:</p><p><img src='image/20180107_234509.png' alt='' /></p><p>这里由于特征不再只有一个,引入一些新的记号</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-437-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.395ex" 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id="MathJax-Element-437">n</script>: 特征的总数 </p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-398-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="2.461ex" viewBox="0 -956.9 1467.6 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E413-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 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0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">i</script> 行,也就是第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E73-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">i</script> 个训练实例。</p><p> <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-399-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.409ex" height="3.745ex" viewBox="0 -1107.7 1467.6 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="1" id="E414-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="E414-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="E414-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="1" id="E414-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="E414-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 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id="MathJax-Element-399">{x}_{j}^{\left( i \right)}</script>: 代表样本矩阵中第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E73-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g 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-67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E71-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-71">j</script> 列,也就是第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.802ex" height="1.994ex" viewBox="0 -755.9 345.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E73-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">i</script> 个训练实例的第 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-71-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-11.5 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E71-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E71-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-71">j</script> 个特征。</p></blockquote><p>参照上图,则有 <span 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x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E415-MJMAIN-3D" x="8645" y="0"></use><g transform="translate(9701,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E415-MJMAIN-31"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E415-MJMAIN-34" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E415-MJMAIN-31" x="1001" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E415-MJMAIN-36" x="1501" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-400">{x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416</script></p><p>多变量假设函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-404-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="1.994ex" viewBox="0 -755.9 576.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E419-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></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="#E419-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-404">h</script> 表示为:<span class="MathJax_Preview"></span><span class="MathJax_SVG" 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y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-351">\theta_0</script>,和单特征中一样,我们将其看作基础数值。例如,房价的基础价格。</p><p>参数向量的维度为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-361-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.398ex" height="2.11ex" viewBox="0 -755.9 2323.9 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="1" id="E369-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 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x="1823" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-361">n+1</script> 维特征量向量</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-165-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.384ex" height="1.76ex" viewBox="0 -504.6 1026.4 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E165-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 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y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E303-MJMAIN-2E" x="5149" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E303-MJMAIN-6E" x="5428" y="0"></use></g></g></g></g></svg></span><script type="math/tex" id="MathJax-Element-296">\begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0,1...n}\newline \rbrace\end{align*}</script></p><p>可展开为:</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-411-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.396ex" height="38.531ex" viewBox="0 -8546.1 19975.9 16589.5" role="img" focusable="false" style="vertical-align: -18.682ex;"><defs><path stroke-width="1" id="E426-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 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