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research.html
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<!doctype html>
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</style><title>金融风险管理第二次作业</title>
</head>
<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write' class=''><h1 align="center">市场风险的动态化度量与非线性关系</h1><center>金融20-3班 顾杰 13203427<center></center></center><p><span>本次作业是对西昌电力的市场风险进行度量，用西昌电力的超额收益率作为被解释变量，所选取的解释变量分别为西昌电力昨日收盘价，上证综合指数当日超额收益率及西昌电力当日成交额。</span></p><p><span>参数说明：</span></p><figure><table><thead><tr><th style='text-align:center;' ><span>符号</span></th><th style='text-align:center;' ><span>符号意义</span></th><th style='text-align:center;' ><span>备注</span></th></tr></thead><tbody><tr><td style='text-align:center;' ><em><span>r</span><sub><span>xcdl</span></sub></em></td><td style='text-align:center;' ><span>西昌电力的超额收益率</span></td><td style='text-align:center;' ><span>由西昌电力的当日收益率减去当日一年期国债日利率</span></td></tr><tr><td style='text-align:center;' ><em><span>r</span><sub><span>szzs</span></sub></em></td><td style='text-align:center;' ><span>上证指数的超额收益率</span></td><td style='text-align:center;' ><span>由上证指数的当日收益率减去当日一年期国债日利率</span></td></tr><tr><td style='text-align:center;' ><span>Pre_close</span></td><td style='text-align:center;' ><span>西昌电力的昨日收盘价</span></td><td style='text-align:center;' > </td></tr><tr><td style='text-align:center;' ><span>Amount</span></td><td style='text-align:center;' ><span>西昌电力的成交额</span></td><td style='text-align:center;' > </td></tr></tbody></table></figure><h3 id='1对量与质的数据进行预处理'><span>1、对量与质的数据进行预处理</span></h3><p><span>对数据进行中心标准化，标准化只改变数据在序列中的相对位置，而不改变本身蕴含的信息，标准化对线性回归的拟合效果影响很小。</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n28" cid="n28" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 21.4ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="4.652ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" 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mathvariant="normal">_</mi><mi>c</mi><mi>l</mi><mi>o</mi><mi>s</mi><mi>e</mi><mo>+</mo><mn>0.50</mn><mi>A</mi><mi>m</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi></math></mjx-assistive-mml></mjx-container></div></div><figure><table><thead><tr><th style='text-align:center;' > </th><th style='text-align:center;' ><span>Estimate</span></th><th style='text-align:center;' ><span>Std. Error</span></th><th style='text-align:center;' ><span>t value</span></th><th style='text-align:center;' ><span>Pr(></span><span>|</span><span>t</span><span>|</span><span>)</span></th><th style='text-align:center;' ><span>R</span><sup><span>2</span></sup></th></tr></thead><tbody><tr><td style='text-align:center;' ><em><span>r</span><sub><span>szzs</span></sub></em></td><td style='text-align:center;' ><span>1.05080</span></td><td style='text-align:center;' ><span>0.08535</span></td><td style='text-align:center;' ><span>12.312</span></td><td style='text-align:center;' ><span>< 2e-16 ***</span></td><td style='text-align:center;' ><span>0.4222</span></td></tr><tr><td style='text-align:center;' ><em><span>Pre_close</span></em></td><td style='text-align:center;' ><span>-0.36858</span></td><td style='text-align:center;' ><span>0.11538</span></td><td style='text-align:center;' ><span>-3.195</span></td><td style='text-align:center;' ><span>0.00159 **</span></td><td style='text-align:center;' > </td></tr><tr><td style='text-align:center;' ><em><span>Amount</span></em></td><td style='text-align:center;' ><span>0.49855</span></td><td style='text-align:center;' ><span>0.11521</span></td><td style='text-align:center;' ><span>4.327</span></td><td style='text-align:center;' ><span>2.22e-05 ***</span></td><td style='text-align:center;' > </td></tr></tbody></table></figure><h3 id='2全对数与半对数模型的参数意义'><span>2、全对数与半对数模型的参数意义</span></h3><h4 id='21-因变量为对数形式'><span>2.1 因变量为对数形式</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n64" cid="n64" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 25.915ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.398ex" role="img" focusable="false" 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display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(2)</mtext></mtd><mtd><mi>l</mi><mi>o</mi><mi>g</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><span>公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-41-TEX-N-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 id="MJX-41-TEX-N-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 id="MJX-41-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-41-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="32" xlink:href="#MJX-41-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-41-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(2)</script><span>对x求导得：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n68" cid="n68" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 21.43ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="11.412ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -5.14ex; min-width: 21.43ex;"><defs><path id="MJX-4-TEX-N-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 id="MJX-4-TEX-I-1D466" 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 id="MJX-4-TEX-N-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path id="MJX-4-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-4-TEX-I-1D465" 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 id="MJX-4-TEX-N-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 id="MJX-4-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-4-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-4-TEX-N-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 id="MJX-4-TEX-N-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 id="MJX-4-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.019382,-0.019382) translate(0, -2772)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 2772) matrix(1 0 0 -1 0 0) scale(51.6)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="2658 -2772 1 5044"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1402)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-4-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(225,-686)"><use data-c="1D466" xlink:href="#MJX-4-TEX-I-1D466"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1162.2,0)"><use data-c="2217" xlink:href="#MJX-4-TEX-N-2217"></use></g><g data-mml-node="mfrac" transform="translate(1884.4,0)"><g data-mml-node="mrow" transform="translate(261,676)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-4-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D466" xlink:href="#MJX-4-TEX-I-1D466"></use></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-4-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-4-TEX-I-1D465"></use></g></g><rect width="1292" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3694.2,0)"><use data-c="3D" xlink:href="#MJX-4-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(4750,0)"><use data-c="1D6FD" xlink:href="#MJX-4-TEX-I-1D6FD"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1381)"><g data-mml-node="mtd" transform="translate(124.2,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-4-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-4-TEX-I-1D466"></use></g></g><g data-mml-node="mi" transform="translate(636.5,-686)"><use data-c="1D466" xlink:href="#MJX-4-TEX-I-1D466"></use></g><rect width="1523" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2040.8,0)"><use data-c="3D" xlink:href="#MJX-4-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3096.6,0)"><use data-c="1D6FD" xlink:href="#MJX-4-TEX-I-1D6FD"></use></g><g data-mml-node="mi" transform="translate(3662.6,0)"><use data-c="394" xlink:href="#MJX-4-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(4495.6,0)"><use data-c="1D465" xlink:href="#MJX-4-TEX-I-1D465"></use></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -2772 1 5044"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:3"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-4-TEX-N-28"></use><use data-c="33" xlink:href="#MJX-4-TEX-N-33" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-4-TEX-N-29" transform="translate(889,0)"></use></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3)</mtext></mtd><mtd><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>∗</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>β</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mi>y</mi></mfrac><mo>=</mo><mi>β</mi><mi mathvariant="normal">Δ</mi><mi>x</mi></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div></li><li><p><span>从公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-42-TEX-N-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 id="MJX-42-TEX-N-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 id="MJX-42-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-42-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="33" xlink:href="#MJX-42-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-42-TEX-N-29"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">{(3)}</script><span>可以看出，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.179ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 1405 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-52-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-52-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-52-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-52-TEX-I-1D465"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>x</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta x</script><span>每变化一个单位，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 490 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-54-TEX-I-1D466" 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 id="MJX-54-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-54-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(300.6,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-54-TEX-N-5E"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat y</script><span>相对变化量为</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.456ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4621.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-45-TEX-N-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 id="MJX-45-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-45-TEX-N-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path id="MJX-45-TEX-N-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 id="MJX-45-TEX-N-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 id="MJX-45-TEX-N-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 id="MJX-45-TEX-N-25" d="M465 605Q428 605 394 614T340 632T319 641Q332 608 332 548Q332 458 293 403T202 347Q145 347 101 402T56 548Q56 637 101 693T202 750Q241 750 272 719Q359 642 464 642Q580 642 650 732Q662 748 668 749Q670 750 673 750Q682 750 688 743T693 726Q178 -47 170 -52Q166 -56 160 -56Q147 -56 142 -45Q137 -36 142 -27Q143 -24 363 304Q469 462 525 546T581 630Q528 605 465 605ZM207 385Q235 385 263 427T292 548Q292 617 267 664T200 712Q193 712 186 709T167 698T147 668T134 615Q132 595 132 548V527Q132 436 165 403Q183 385 203 385H207ZM500 146Q500 234 544 290T647 347Q699 347 737 292T776 146T737 0T646 -56Q590 -56 545 0T500 146ZM651 -18Q679 -18 707 24T736 146Q736 215 711 262T644 309Q637 309 630 306T611 295T591 265T578 212Q577 200 577 146V124Q577 -18 647 -18H651Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-45-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D6FD" xlink:href="#MJX-45-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(1177.2,0)"><use data-c="2217" xlink:href="#MJX-45-TEX-N-2217"></use></g><g data-mml-node="mn" transform="translate(1899.4,0)"><use data-c="31" xlink:href="#MJX-45-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-45-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-45-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" transform="translate(3399.4,0)"><use data-c="29" xlink:href="#MJX-45-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(3788.4,0)"><use data-c="25" xlink:href="#MJX-45-TEX-N-25"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>β</mi><mo>∗</mo><mn>100</mn><mo stretchy="false">)</mo><mi mathvariant="normal">%</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">(\beta*100)\%</script><span>，这时的β也被称为y对于x的半弹性。</span></p></li></ul><p> </p><h4 id='22-自变量为对数形式'><span>2.2 自变量为对数形式</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n73" cid="n73" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 25.096ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.398ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.633ex; min-width: 25.096ex;"><defs><path id="MJX-5-TEX-I-1D466" 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 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556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path id="MJX-5-TEX-N-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 id="MJX-5-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-5-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path><path id="MJX-5-TEX-I-1D45B" 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 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-5-TEX-N-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 id="MJX-5-TEX-I-1D465" 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 id="MJX-5-TEX-N-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 id="MJX-5-TEX-N-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 id="MJX-5-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 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transform="translate(1823.6,0)"><use data-c="1D6FC" xlink:href="#MJX-5-TEX-I-1D6FC"></use></g><g data-mml-node="mo" transform="translate(2685.8,0)"><use data-c="2B" xlink:href="#MJX-5-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(3686,0)"><use data-c="1D6FD" xlink:href="#MJX-5-TEX-I-1D6FD"></use></g><g data-mml-node="mi" transform="translate(4252,0)"><use data-c="1D459" xlink:href="#MJX-5-TEX-I-1D459"></use></g><g data-mml-node="mi" transform="translate(4550,0)"><use data-c="1D45B" xlink:href="#MJX-5-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(5150,0)"><use data-c="28" xlink:href="#MJX-5-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(5539,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-5-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-5-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(6547.6,0)"><use data-c="29" xlink:href="#MJX-5-TEX-N-29"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -780 1 1060"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4" transform="translate(0,720)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-5-TEX-N-28"></use><use data-c="34" xlink:href="#MJX-5-TEX-N-34" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-5-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(4)</mtext></mtd><mtd><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><span>公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-46-TEX-N-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 id="MJX-46-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path id="MJX-46-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-46-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="34" xlink:href="#MJX-46-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-46-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(4)</script><span>对x求导得：</span></li></ul><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n77" cid="n77" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 22.58ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="11.163ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -5.016ex; min-width: 22.58ex;"><defs><path id="MJX-6-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-6-TEX-I-1D466" 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 id="MJX-6-TEX-I-1D465" 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 id="MJX-6-TEX-N-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 id="MJX-6-TEX-I-1D6FD" d="M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z"></path><path id="MJX-6-TEX-N-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path id="MJX-6-TEX-N-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 id="MJX-6-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-6-TEX-N-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 id="MJX-6-TEX-N-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><path id="MJX-6-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.019382,-0.019382) translate(0, -2717)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 2717) matrix(1 0 0 -1 0 0) scale(51.6)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="2912.3 -2717 1 4934"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1347)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(261,676)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-6-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D466" xlink:href="#MJX-6-TEX-I-1D466"></use></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-6-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-6-TEX-I-1D465"></use></g></g><rect width="1292" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1809.8,0)"><use data-c="3D" xlink:href="#MJX-6-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2865.6,0)"><use data-c="1D6FD" xlink:href="#MJX-6-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(3653.8,0)"><use data-c="2217" xlink:href="#MJX-6-TEX-N-2217"></use></g><g data-mml-node="mfrac" transform="translate(4376,0)"><g data-mml-node="mn" transform="translate(474.3,676)"><use data-c="31" xlink:href="#MJX-6-TEX-N-31"></use></g><g data-mml-node="msub" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-6-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-6-TEX-N-31"></use></g></g></g><rect width="1208.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0,-1381)"><g data-mml-node="mtd" transform="translate(378.5,0)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-6-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-6-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(1600.8,0)"><use data-c="3D" xlink:href="#MJX-6-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2656.6,0)"><use data-c="1D6FD" xlink:href="#MJX-6-TEX-I-1D6FD"></use></g><g data-mml-node="mfrac" transform="translate(3222.6,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-6-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-6-TEX-I-1D465"></use></g></g><g data-mml-node="msub" transform="translate(418.2,-686)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-6-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-6-TEX-N-31"></use></g></g></g><rect width="1605" height="60" x="120" y="220"></rect></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -2717 1 4934"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:5"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-6-TEX-N-28"></use><use data-c="35" xlink:href="#MJX-6-TEX-N-35" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-6-TEX-N-29" transform="translate(889,0)"></use></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(5)</mtext></mtd><mtd><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>β</mi><mo>∗</mo><mfrac><mn>1</mn><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>=</mo><mi>β</mi><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mfrac></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><span>从公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-47-TEX-N-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 id="MJX-47-TEX-N-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><path id="MJX-47-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-47-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="35" xlink:href="#MJX-47-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-47-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(5)</script><span>可以看出，x相对变化1%，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 490 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-54-TEX-I-1D466" 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 id="MJX-54-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-54-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(300.6,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-54-TEX-N-5E"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat y</script><span> 的值发生为</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.44ex" height="2.034ex" role="img" focusable="false" viewBox="0 -705 3288.4 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-49-TEX-N-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 id="MJX-49-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-49-TEX-N-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 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xlink:href="#MJX-49-TEX-I-1D6FD"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.01</mn><mo>∗</mo><mi>β</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">0.01*\beta</script><span>的变化</span></li></ul><p> </p><h4 id='23-因变量与自变量均为对数形式'><span>2.3 因变量与自变量均为对数形式</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n83" cid="n83" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 28.888ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.398ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.633ex; min-width: 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xlink:href="#MJX-7-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2443.8,0)"><use data-c="3D" xlink:href="#MJX-7-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3499.6,0)"><use data-c="1D6FC" xlink:href="#MJX-7-TEX-I-1D6FC"></use></g><g data-mml-node="mo" transform="translate(4361.8,0)"><use data-c="2B" xlink:href="#MJX-7-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(5362,0)"><use data-c="1D6FD" xlink:href="#MJX-7-TEX-I-1D6FD"></use></g><g data-mml-node="mi" transform="translate(5928,0)"><use data-c="1D459" xlink:href="#MJX-7-TEX-I-1D459"></use></g><g data-mml-node="mi" transform="translate(6226,0)"><use data-c="1D45B" xlink:href="#MJX-7-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(6826,0)"><use data-c="28" xlink:href="#MJX-7-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(7215,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-7-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-7-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(8223.6,0)"><use data-c="29" xlink:href="#MJX-7-TEX-N-29"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -780 1 1060"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:6" transform="translate(0,720)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-7-TEX-N-28"></use><use data-c="36" xlink:href="#MJX-7-TEX-N-36" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-7-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(6)</mtext></mtd><mtd><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><span>公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-50-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 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stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-50-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="36" xlink:href="#MJX-50-TEX-N-36"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-50-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(6)</script><span>同时对x求导：</span></li></ul><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n86" cid="n86" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" 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data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-8-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(225,-686)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1162.2,0)"><use data-c="2217" xlink:href="#MJX-8-TEX-N-2217"></use></g><g data-mml-node="mfrac" transform="translate(1884.4,0)"><g data-mml-node="mrow" transform="translate(261,676)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-8-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-8-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g></g><rect width="1292" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3694.2,0)"><use data-c="3D" xlink:href="#MJX-8-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(4750,0)"><use data-c="1D6FD" xlink:href="#MJX-8-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(5538.2,0)"><use data-c="2217" xlink:href="#MJX-8-TEX-N-2217"></use></g><g data-mml-node="mfrac" transform="translate(6260.4,0)"><g data-mml-node="mn" transform="translate(474.3,676)"><use data-c="31" xlink:href="#MJX-8-TEX-N-31"></use></g><g data-mml-node="msub" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-8-TEX-N-31"></use></g></g></g><rect width="1208.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0,495.3)"><g data-mml-node="mtd" transform="translate(785.7,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(261,676)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g></g><rect width="1605" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2067.2,0)"><use data-c="2217" xlink:href="#MJX-8-TEX-N-2217"></use></g><g data-mml-node="mfrac" transform="translate(2789.4,0)"><g data-mml-node="msub" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-8-TEX-N-31"></use></g></g></g><g data-mml-node="mi" transform="translate(479.3,-686)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g><rect width="1208.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(4515.8,0)"><use data-c="3D" xlink:href="#MJX-8-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(5571.6,0)"><use data-c="1D6FD" xlink:href="#MJX-8-TEX-I-1D6FD"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-2836.9)"><g data-mml-node="mtd" transform="translate(1968,0)"><g data-mml-node="mi"><use data-c="1D6FD" xlink:href="#MJX-8-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(843.8,0)"><use data-c="3D" xlink:href="#MJX-8-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1899.6,0)"><g data-mml-node="mfrac" transform="translate(249,950)"><g data-mml-node="mrow" transform="translate(220,485) scale(0.707)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g></g><g data-mml-node="mi" transform="translate(514.5,-345) scale(0.707)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g><rect width="1135.5" height="60" x="120" y="220"></rect></g><g data-mml-node="mfrac" transform="translate(220,-860.3)"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g></g><g data-mml-node="msub" transform="translate(360.2,-345) scale(0.707)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-8-TEX-N-31"></use></g></g></g><rect width="1193.5" height="60" x="120" y="220"></rect></g><rect width="1633.5" height="60" x="120" y="220"></rect></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -4648.3 1 8796.6"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:7"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-8-TEX-N-28"></use><use data-c="37" xlink:href="#MJX-8-TEX-N-37" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-8-TEX-N-29" transform="translate(889,0)"></use></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(7)</mtext></mtd><mtd><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>∗</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>β</mi><mo>∗</mo><mfrac><mn>1</mn><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac><mo>∗</mo><mfrac><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mi>y</mi></mfrac><mo>=</mo><mi>β</mi></mtd></mtr><mtr><mtd><mi>β</mi><mo>=</mo><mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mi>y</mi></mfrac><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mfrac></mfrac></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><ul><li><span>从公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-51-TEX-N-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 id="MJX-51-TEX-N-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 id="MJX-51-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-51-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="37" xlink:href="#MJX-51-TEX-N-37"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-51-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(7)</script><span>可以看出，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.179ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 1405 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-52-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-52-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-52-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-52-TEX-I-1D465"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>x</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Delta x</script><span>相对变化</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.016ex" height="1.824ex" role="img" focusable="false" viewBox="0 -750 1333 806" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.127ex;"><defs><path id="MJX-53-TEX-N-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 id="MJX-53-TEX-N-25" d="M465 605Q428 605 394 614T340 632T319 641Q332 608 332 548Q332 458 293 403T202 347Q145 347 101 402T56 548Q56 637 101 693T202 750Q241 750 272 719Q359 642 464 642Q580 642 650 732Q662 748 668 749Q670 750 673 750Q682 750 688 743T693 726Q178 -47 170 -52Q166 -56 160 -56Q147 -56 142 -45Q137 -36 142 -27Q143 -24 363 304Q469 462 525 546T581 630Q528 605 465 605ZM207 385Q235 385 263 427T292 548Q292 617 267 664T200 712Q193 712 186 709T167 698T147 668T134 615Q132 595 132 548V527Q132 436 165 403Q183 385 203 385H207ZM500 146Q500 234 544 290T647 347Q699 347 737 292T776 146T737 0T646 -56Q590 -56 545 0T500 146ZM651 -18Q679 -18 707 24T736 146Q736 215 711 262T644 309Q637 309 630 306T611 295T591 265T578 212Q577 200 577 146V124Q577 -18 647 -18H651Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-53-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="25" xlink:href="#MJX-53-TEX-N-25"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi mathvariant="normal">%</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">1\%</script><span>，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 490 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-54-TEX-I-1D466" 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 id="MJX-54-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g 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id='3借助fda方法的参数动态化实现'><span>3、借助FDA方法的参数动态化实现</span></h3><h4 id='31-固定起始点的滚动回归'><span>3.1 固定起始点的滚动回归</span></h4><p><span>将整个2021年交易日划分为12个集合，第一步对第一个集合进行线性回归，然后逐步将后续集合加入回归集合直至所有数据进行一次回归，总计产生12个回归方程。</span></p><h4 id='32-函数化数据分析方法'><span>3.2 函数化数据分析方法</span></h4><ol start='' ><li><span>建立基函数系统</span></li></ol><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n98" cid="n98" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 26.844ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="6.784ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.827ex; min-width: 26.844ex;"><defs><path id="MJX-9-TEX-I-1D453" d="M118 -162Q120 -162 124 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data-c="1D458" xlink:href="#MJX-9-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="3D" xlink:href="#MJX-9-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-9-TEX-N-31"></use></g></g><g data-mml-node="TeXAtom" transform="translate(407.7,1150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-9-TEX-I-1D43E"></use></g></g></g><g data-mml-node="msub" transform="translate(4633.2,0)"><g data-mml-node="mi"><use data-c="1D701" xlink:href="#MJX-9-TEX-I-1D701"></use></g><g data-mml-node="TeXAtom" transform="translate(471,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-9-TEX-I-1D458"></use></g></g></g><g data-mml-node="msub" transform="translate(5522.6,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-9-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-9-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(6570,0)"><use data-c="28" xlink:href="#MJX-9-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6959,0)"><use data-c="1D461" xlink:href="#MJX-9-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(7320,0)"><use data-c="29" xlink:href="#MJX-9-TEX-N-29"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1749.3 1 2998.7"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:8" transform="translate(0,766.4)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-9-TEX-N-28"></use><use data-c="38" xlink:href="#MJX-9-TEX-N-38" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-9-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(8)</mtext></mtd><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>K</mi></mrow></munderover><msub><mi>ζ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><msub><mi>ϕ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>在公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" 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311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path id="MJX-59-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-59-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="38" xlink:href="#MJX-59-TEX-N-38"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-59-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(8)</script><span>中，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.947ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2186.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-57-TEX-I-1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 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749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-57-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-57-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(1047.4,0)"><use data-c="28" xlink:href="#MJX-57-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1436.4,0)"><use data-c="1D461" xlink:href="#MJX-57-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(1797.4,0)"><use data-c="29" xlink:href="#MJX-57-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">ϕ_k (t)</script><span>即是需要建立的基函数系统，目前主流的基函数系统包括傅里叶基函数与样条基函数，本次作业采取死结样条函数构建基函数系统。</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.012ex" height="2.054ex" role="img" focusable="false" viewBox="0 -704 889.4 908" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.462ex;"><defs><path id="MJX-58-TEX-I-1D701" d="M296 643Q298 704 324 704Q342 704 342 687Q342 682 339 664T336 633Q336 623 337 618T338 611Q339 612 341 612Q343 614 354 616T374 618L384 619H394Q471 619 471 586Q467 548 386 546H372Q338 546 320 564L311 558Q235 506 175 398T114 190Q114 171 116 155T125 127T137 104T153 86T171 72T192 61T213 53T235 46T256 39L322 16Q389 -10 389 -80Q389 -119 364 -154T300 -202Q292 -204 274 -204Q247 -204 225 -196Q210 -192 193 -182T172 -167Q167 -159 173 -148Q180 -139 191 -139Q195 -139 221 -153T283 -168Q298 -166 310 -152T322 -117Q322 -91 302 -75T250 -51T183 -29T116 4T65 62T44 160Q44 287 121 410T293 590L302 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data-mml-node="mi"><use data-c="1D701" xlink:href="#MJX-58-TEX-I-1D701"></use></g><g data-mml-node="TeXAtom" transform="translate(471,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-58-TEX-I-1D458"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ζ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">ζ_{k}</script><span>是指基函数的系数，这是在FDA拟合过程中需要估计的参数。对于基函数系统来说，最重要的除了确定基函数的形式，更重要的是确定基函数的个数，基函数太少会导致拟合效果差，但是太多则是会出现过拟合的问题，基函数的个数即是公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-59-TEX-N-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 id="MJX-59-TEX-N-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path id="MJX-59-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-59-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="38" xlink:href="#MJX-59-TEX-N-38"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-59-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(8)</script><span>中的</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 889 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-66-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-66-TEX-I-1D43E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">K</script></p><ol start='2' ><li><span>粗糙惩罚函数拟合</span></li></ol><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n103" cid="n103" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>i</mi></msub></mrow></munderover><mrow data-mjx-texclass="ORD"><mo> </mo></mrow><mo stretchy="false">[</mo><msub><mi>y</mi><mi>j</mi></msub><mo>−</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>K</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><mo> </mo></mrow><msub><mi>ζ</mi><mi>k</mi></msub><msub><mi>ϕ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo>+</mo><mi>λ</mi><msub><mo data-mjx-texclass="OP">∫</mo><mi>T</mi></msub><mrow data-mjx-texclass="ORD"><mo> </mo></mrow><mo stretchy="false">[</mo><msup><mi>f</mi><mrow data-mjx-texclass="ORD"><msup><mi></mi><mo data-mjx-alternate="1">″</mo></msup></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mi>d</mi><mi>t</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>粗糙惩罚函数通过在基函数最小二乘法模型的基础上增加一个粗糙惩罚项来控制拟合函数的粗糙程度，在粗糙惩罚平滑中，惩罚项通常由平滑项和粗糙项组成。平滑项的作用是通过对函数的平滑度进行惩罚，来防止过度拟合；而粗糙项的作用则是通过对函数的导数的大小进行惩罚，来防止函数的剧烈波动。公式</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1278 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-61-TEX-N-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 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data-mml-node="mo"><use data-c="28" xlink:href="#MJX-61-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="39" xlink:href="#MJX-61-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="29" xlink:href="#MJX-61-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(9)</script><span>中的</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.027ex;"><defs><path id="MJX-62-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-62-TEX-I-1D706"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">λ</script><span>为平滑参数值，表示模型拟合优度与函数曲线平滑程度之间的比例关系，也是FDA需要确定的参数。</span></p><ol start='3' ><li><span>双参数GCV确定</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 889 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-66-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-66-TEX-I-1D43E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">K</script><span>与</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.027ex;"><defs><path id="MJX-65-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-65-TEX-I-1D706"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda</script></li></ol><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n108" cid="n108" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" 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displaystyle="true"><mlabeledtr><mtd><mtext>(10)</mtext></mtd><mtd><mi>G</mi><mi>C</mi><mi>V</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mi>n</mi><mrow><mi>n</mi><mo>−</mo><mi>d</mi><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>P</mi><mi>E</mi><mi>N</mi><mi>S</mi><mi>S</mi><mi>E</mi><mrow data-mjx-texclass="ORD"><mo></mo></mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>d</mi><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>调整的</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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type="math/tex">\lambda</script><span>与</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 889 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-66-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 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data-c="1D706" xlink:href="#MJX-67-TEX-I-1D706"></use></g><g data-mml-node="mo" transform="translate(3287,0)"><use data-c="2C" xlink:href="#MJX-67-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3731.7,0)"><use data-c="1D43E" xlink:href="#MJX-67-TEX-I-1D43E"></use></g><g data-mml-node="mo" transform="translate(4620.7,0)"><use data-c="29" xlink:href="#MJX-67-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>C</mi><mi>V</mi><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">GCV(λ,K)</script><span>最小即可同时得到最优的最优基函数个数和平滑参数值。</span></p><h4 id='33-参数的函数化拟合效果'><span>3.3 参数的函数化拟合效果</span></h4><ul><li><span>回归参数函数化参数确定</span></li></ul><figure><table><thead><tr><th style='text-align:center;' > </th><th style='text-align:center;' ><span>GCV</span></th><th style='text-align:center;' ><span>log（λ）</span></th><th style='text-align:center;' ><span>K</span></th></tr></thead><tbody><tr><td style='text-align:center;' ><em><span>r</span><sub><span>szzs</span></sub></em></td><td style='text-align:center;' ><span>5.9159*10</span><sup><span>-7</span></sup></td><td style='text-align:center;' ><span>-8.5</span></td><td style='text-align:center;' ><span>11</span></td></tr><tr><td style='text-align:center;' ><em><span>Pre_close</span></em></td><td style='text-align:center;' ><span>1.4859*10</span><sup><span>-5</span></sup></td><td style='text-align:center;' ><span>-9</span></td><td style='text-align:center;' ><span>11</span></td></tr><tr><td style='text-align:center;' ><em><span>Amount</span></em></td><td style='text-align:center;' ><span>4.7877*10</span><sup><span>-6</span></sup></td><td style='text-align:center;' ><span>-9</span></td><td style='text-align:center;' ><span>11</span></td></tr></tbody></table></figure><p><span>从图1，图2，图3</span><em><span>（所提及的图在附录中）</span></em><span>中可以看到，我们所设置的函数化参数较好的拟合了回归所得到的参数。</span></p><p><span>线性回归的斜率项参数是表示自变量x 对因变量y 影响大小的参数，回归系数越大表示x 对y 影响越大。</span></p><ol start='' ><li><span>上证指数收益率</span></li></ol><p><span>上证指数的线性回归参数在前半年呈现上升的趋势，在后半年则是缓慢下降。这表明上证指数收益率对西昌电力超额收益率的影响在上班年逐步增大，在后半年则是逐步减小。回归系数为正表明西昌电力收益率始终与上证指数收益率保持正相关关系，在其他条件没有发生改变的情况下，上证综合指数的超额收益率越高，西昌电力的超额收益率也会随着大盘的拉升而提高，相应的市场风险也会增大。</span></p><ol start='2' ><li><span>西昌电力昨日收盘价</span></li></ol><p><span>西昌电力昨日收盘价的回归系数整体来看都是处于增大的状态，在之间只有仅有的几个月份出现了下跌的趋势。但是与上证指数收益率不同，西昌电力昨日收盘价的回归系数始终为负值，这表明西昌电力收益率与昨日收盘价呈现负相关关系，在其他条件不发生改变的前提下，昨日收盘价越高，西昌电力的收益率越低，相应的市场风险也会降低。</span></p><ol start='3' ><li><span>西昌电力成交量</span></li></ol><p><span>西昌电力成交量的回归系数趋势与上证趋势相反，在前半年迅速下降到后半年缓慢回升。这表明成交量对西昌电力超额收益率的影响在前半年逐步变弱。同时，其回归系数始终为正值，表明西昌电力成交量与西昌电力的超额收益率是正相关关系，在其他条件不发生变化的前提下，成交量数据的提高将会使得西昌电力的收益率提高，同时使得西昌电力的市场风险增大。这种增大的速度在前半年逐步变弱，后半年则是略微有所提高。</span></p><h3 id='4连续估计量的相平面图反映外部冲击的影响'><span>4、连续估计量的相平面图反映外部冲击的影响</span></h3><p><span>相平面图可以反映数据变化的动能与势能,第一象限为加速上升区，第二象限为减速下降区，第三象限为加速下降区，第四象限为减速上升区。并且，越靠近原点速度和加速度的绝对值越小，动能和势能越小，即变化幅度越小和持续时间越短。</span></p><p><span>选取的外部冲击：2021年4月，西昌电力披露年报数据，收入同比增长约20%</span></p><p><span>2021年4月份：西昌电力披露年报，报告期内,公司售电收入同比增长21.28%,主要是居民及一般工商业售电量同比分别增长33.07%、39.94%;同时营业成本同比增长12.38%,主要是售电量增长带动外购电量增长12.25%。</span></p><p><span>外部冲击的影响可以从参数连续估计量的相平面图4，图5，图6</span><em><span>（图见附录1）</span></em><span>来看：</span></p><p><span>受这一消息影响，四月份上证超额收益率对西昌电力超额收益率的影响呈现加速上升的趋势，且变化幅度与持续的时间很长，表明上证指数收益率与西昌电力的超额收益率影响增大；西昌电力昨日收盘价对西昌电力超额收益率的影响也呈现加速上升的趋势，但由于其回归系数值为负，且相平面图上靠近原点，表明这一消息的影响使得昨日收盘价对西昌电力超额收益率的影响变小，但是变小的幅度和持续时间更短；从西昌电力成交量对西昌电力超额收益率的影响则是呈现加速下降的趋势，表明受消息影响，成交量对西昌电力超额收益率的影响在下降，但下降的幅度小，持续时间短。</span></p><p><span>这表明，西昌电力的市场风险受这一消息的影响主要体现在其与上证综合指数这一市场组合的的敏感性上。</span></p><h3 id='5因子的非线性影响--半参数回归的样条光滑估计'><span>5、因子的非线性影响 -半参数回归的样条光滑估计</span></h3><ol start='' ><li><span>非线性影响</span></li></ol><p><span>半参数回归的样条光滑结果见附录图7，图8，图9：</span></p><p><span>本文分别对收益率与三个因子单独进行的样条光滑，结果显示上证收益率对西昌电力收益率的影响是线性的，西昌电力的昨日收盘价和成交量对收益率的影响是非线性的。其中，西昌电力昨日收盘价与西昌电力超额收益率的函数关系是类似与一种S型函数，在标准化收盘价小于-0.5或者是大于1.2时，西昌电力的超额收益率随着昨日收盘价的增加而下降。当西昌电力的标准化收盘价在二者之间，西昌电力的超额收益率基本不会被西昌电力的昨日收盘价影响到。西昌电力的成交量对于西昌电力超额收益率的影响也是一种较为复杂的非线性关系，当标准化西昌电力成交量处在不同区间时，其对西昌电力超额收益率的影响趋势是不一样的，其在前半段表现出的是波动逐步增大的曲线，在后半段则是表现出偏线性的关系。</span></p><ol start='2' ><li><span>整体与局部</span></li></ol><p><span>从解释变量对被解释变量的非线性关系也可以发现，选择不同的角度去看，就会发现两者之间的关系在局部与整体上的表现并不完全一致。</span></p><p><span>从图7来看，上证指数收益率与西昌电力超额收益率之间的关系无论是在整体还是局部都呈现出线性关系，具有一致的趋势。</span></p><p><span>从图9来看，西昌电力超额收益率与西昌电力成交量之间的关系来看，整体上它们之间的关系不是线性的，但是这种非线性的关系并不是存在于全部的的曲线上，显然可以发现，从其拟合结果来看，如果只考虑在标准化成交量小于-1之前，二者之间的关系是一段近似的直线。</span></p><div style="page-break-after: always;"></div><h3 id='附录1-作业中的图'><span>附录1 作业中的图</span></h3><p><img src="C:\Users\20392\Desktop\BETA1.png" alt="BETA1" style="zoom: 33%;" /></p><p><img src="C:\Users\20392\Desktop\BETA2.png" alt="BETA2" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\BEAT3.png" alt="BEAT3" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\相位1.png" alt="相位1" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\相位2.png" alt="相位2" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\相位3.png" alt="相位3" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\样条平滑1.png" alt="样条平滑1" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\样条平滑2.png" alt="样条平滑2" style="zoom:33%;" /></p><p><img src="C:\Users\20392\Desktop\样条平滑3.png" alt="样条平滑3" style="zoom:33%;" /></p><div style="page-break-after: always;"></div><h3 id='附录2-代码'><span>附录2 代码</span></h3><pre class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" spellcheck="false" lang="R" style="break-inside: unset;"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="r"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.51796px; left: 43.9836px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35.9959px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">options</span>(<span class="cm-variable">warn</span><span class="cm-operator">=-</span><span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">lubridate</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">xts</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">quantmod</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">Tushare</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">TSA</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">fUnitRoots</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">urca</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">fBasics</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">forecast</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">FinTS</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">fda</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">xlsx</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">splines</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">Matrix</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">MASS</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">rf</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">read.xlsx</span>(<span class="cm-string">"D:</span><span class="cm-string-2">\\</span><span class="cm-string">R</span><span class="cm-string-2">\\</span><span class="cm-string">政府债券利率历史数据.xlsx"</span>,<span class="cm-variable">sheetIndex</span> <span class="cm-operator">=</span> <span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">rf</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">X1</span>年期国债收益率<span class="cm-operator cm-arrow"><-</span><span class="cm-variable">as.numeric</span>(<span class="cm-variable">rf</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">X1</span>年期国债收益率)<span class="cm-operator">/</span><span class="cm-number">365</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">index0</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">as.Date</span>(<span class="cm-variable">ymd</span>(<span class="cm-variable">rf</span><span class="cm-operator cm-dollar">$</span>日期))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">wufengxian</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xts</span>(<span class="cm-variable">rf</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">X1</span>年期国债收益率,<span class="cm-variable">order.by</span> <span class="cm-operator">=</span> <span class="cm-variable">index0</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">szzs</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">read.csv</span>(<span class="cm-string">"D:</span><span class="cm-string-2">\\</span><span class="cm-string">R</span><span class="cm-string-2">\\</span><span class="cm-string">2017-2022上证指数.csv"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">index1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">as.Date</span>(<span class="cm-variable">ymd</span>(<span class="cm-variable">szzs</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">date</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">szzs_close</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xts</span>(<span class="cm-variable">szzs</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">close</span>,<span class="cm-variable">order.by</span> <span class="cm-operator">=</span> <span class="cm-variable">index1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">szzs_return</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">dailyReturn</span>(<span class="cm-variable">szzs_close</span>,<span class="cm-variable">tyoe</span><span class="cm-operator">=</span><span class="cm-string">"arithmetic"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_x1</span><span class="cm-operator cm-arrow"><-</span>(<span class="cm-variable">window</span>(<span class="cm-variable">szzs_return</span>,<span class="cm-variable">start</span> <span class="cm-operator">=</span> <span class="cm-string">'2021-01-01'</span>)<span class="cm-operator">-</span><span class="cm-variable">wufengxian</span>)<span class="cm-operator">*</span><span class="cm-number">100</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">api</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">pro_api</span>(<span class="cm-variable">token</span><span class="cm-operator">=</span><span class="cm-string">"731ee2eafa47ae6b8c98993e8b30001b51bd3085cd69d63870e745c9"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xcdl</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">api</span>(<span class="cm-variable">api_name</span> <span class="cm-operator">=</span><span class="cm-string">'daily'</span>,<span class="cm-variable">ts_code</span> <span class="cm-operator">=</span><span class="cm-string">'601318.SH'</span>,<span class="cm-variable">start_date</span><span class="cm-operator">=</span><span class="cm-string">'20210101'</span>,<span class="cm-variable">end_date</span><span class="cm-operator">=</span><span class="cm-string">'20220101'</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">index2</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">as.Date</span>(<span class="cm-variable">ymd</span>(<span class="cm-variable">xcdl</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">trade_date</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xcdl_return</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xts</span>(<span class="cm-variable">xcdl</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">pct_chg</span>,<span class="cm-variable">order.by</span> <span class="cm-operator">=</span> <span class="cm-variable">index2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_y</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xcdl_return</span><span class="cm-operator">-</span><span class="cm-variable">wufengxian</span><span class="cm-operator">*</span><span class="cm-number">100</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xcdl1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">read.csv</span>(<span class="cm-string">"D:</span><span class="cm-string-2">\\</span><span class="cm-string">R</span><span class="cm-string-2">\\</span><span class="cm-string">xcdl.csv"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">index3</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">as.Date</span>(<span class="cm-variable">ymd</span>(<span class="cm-variable">xcdl1</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">date</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_x2</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xts</span>(<span class="cm-variable">xcdl</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">pre_close</span>,<span class="cm-variable">order.by</span> <span class="cm-operator">=</span> <span class="cm-variable">index2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_x3</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">xts</span>(<span class="cm-variable">xcdl</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">amount</span>,<span class="cm-variable">order.by</span> <span class="cm-operator">=</span> <span class="cm-variable">index2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#### 数据的处理 ####</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_x3</span><span class="cm-operator cm-arrow"><-</span>(<span class="cm-variable">model_x3</span><span class="cm-operator">-</span><span class="cm-variable">mean</span>(<span class="cm-variable">model_x3</span>))<span class="cm-operator">/</span><span class="cm-variable">sd</span>(<span class="cm-variable">model_x3</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">model_x2</span><span class="cm-operator cm-arrow"><-</span>(<span class="cm-variable">model_x2</span><span class="cm-operator">-</span><span class="cm-variable">mean</span>(<span class="cm-variable">model_x2</span>))<span class="cm-operator">/</span><span class="cm-variable">sd</span>(<span class="cm-variable">model_x2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">fit</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">lm</span>(<span class="cm-variable">model_y</span><span class="cm-operator">~</span><span class="cm-variable">model_x1</span><span class="cm-operator">+</span><span class="cm-variable">model_x2</span><span class="cm-operator">+</span><span class="cm-variable">model_x3</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">summary</span>(<span class="cm-variable">fit</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#### 参数的动态化 ####</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">source</span>(<span class="cm-string">"curve_fda.R"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">cofe</span><span class="cm-operator">=</span><span class="cm-variable">matrix</span>(<span class="cm-number">0</span>,<span class="cm-number">12</span>,<span class="cm-number">3</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> (<span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-number">23</span><span class="cm-operator">:</span><span class="cm-variable">length</span>(<span class="cm-variable">model_y</span>)){</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">model_new_y</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">model_y</span>[<span class="cm-number">1</span><span class="cm-operator">:</span><span class="cm-variable">i</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">model_new_x1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">model_x1</span>[<span class="cm-number">1</span><span class="cm-operator">:</span><span class="cm-variable">i</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">model_new_x2</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">model_x2</span>[<span class="cm-number">1</span><span class="cm-operator">:</span><span class="cm-variable">i</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">model_new_x3</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">model_x3</span>[<span class="cm-number">1</span><span class="cm-operator">:</span><span class="cm-variable">i</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">lms</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">lm</span>(<span class="cm-variable">model_new_y</span><span class="cm-operator">~</span><span class="cm-number">0</span><span class="cm-operator">+</span><span class="cm-variable">model_new_x1</span><span class="cm-operator">+</span><span class="cm-variable">model_new_x2</span><span class="cm-operator">+</span><span class="cm-variable">model_new_x3</span>) </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">49</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">50</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> (<span class="cm-variable">a</span> <span class="cm-keyword">in</span> <span class="cm-number">1</span><span class="cm-operator">:</span><span class="cm-number">3</span>){</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">51</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">cofe</span>[(<span class="cm-variable">i</span><span class="cm-operator">-</span><span class="cm-number">3</span>)<span class="cm-operator">/</span><span class="cm-number">20</span>,<span class="cm-variable">a</span>]<span class="cm-operator">=</span><span class="cm-variable">lms</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">coefficients</span>[<span class="cm-variable">a</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">52</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">53</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">i</span><span class="cm-operator">=</span><span class="cm-variable">i</span><span class="cm-operator">+</span><span class="cm-number">20</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">54</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">55</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">curve_beta1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">curve_fda</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">1</span>],<span class="cm-number">8</span>,<span class="cm-number">2021</span>,<span class="cm-number">2022</span>,<span class="cm-number">12</span>,<span class="cm-operator">-</span><span class="cm-number">10</span>,<span class="cm-number">3</span>,<span class="cm-number">0.5</span>,<span class="cm-string">"β1的函数化参数"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">56</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">curve_beta2</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">curve_fda</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">2</span>],<span class="cm-number">8</span>,<span class="cm-number">2021</span>,<span class="cm-number">2022</span>,<span class="cm-number">12</span>,<span class="cm-operator">-</span><span class="cm-number">15</span>,<span class="cm-operator">-</span><span class="cm-number">5</span>,<span class="cm-number">0.5</span>,<span class="cm-string">"β2的函数化参数"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">57</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">curve_beta3</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">curve_fda</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">3</span>],<span class="cm-number">8</span>,<span class="cm-number">2021</span>,<span class="cm-number">2022</span>,<span class="cm-number">12</span>,<span class="cm-operator">-</span><span class="cm-number">10</span>,<span class="cm-number">3</span>,<span class="cm-number">0.5</span>,<span class="cm-string">"β3的函数化参数"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">58</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">date1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">seq</span>(<span class="cm-number">2021</span>,<span class="cm-number">2022</span>,<span class="cm-variable">length.out</span><span class="cm-operator">=</span><span class="cm-number">12</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">59</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plotfit.fd</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">1</span>],<span class="cm-variable">date1</span>,<span class="cm-variable">curve_beta1</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">60</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图1：上证综合指数收益率线性回归参数的FDA拟合"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">61</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plotfit.fd</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">2</span>],<span class="cm-variable">date1</span>,<span class="cm-variable">curve_beta2</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">62</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图2：西昌电力昨日收盘价线性回归参数的FDA拟合"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">63</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plotfit.fd</span>(<span class="cm-variable">cofe</span>[,<span class="cm-number">3</span>],<span class="cm-variable">date1</span>,<span class="cm-variable">curve_beta3</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">64</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图3：西昌电力成交量线性回归参数的FDA拟合"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">65</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#### 外部冲击的影响 ####</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">66</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#### 参数动态化相位分析</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">67</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xiangwei</span><span class="cm-operator cm-arrow"><-</span><span class="cm-keyword">function</span>(<span class="cm-variable">fdpar</span>,<span class="cm-variable">mains</span>){</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">68</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">R_amount.mean</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">mean.fd</span>(<span class="cm-variable">fdpar</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">69</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">R_amountmean</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">R_amount.mean</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">70</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">DR_amount</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">deriv.fd</span>(<span class="cm-variable">fdpar</span>,<span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">71</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">R_amount.mean1</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">deriv.fd</span>(<span class="cm-variable">R_amount.mean</span>,<span class="cm-number">1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">72</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">AR_amount</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">deriv.fd</span>(<span class="cm-variable">fdpar</span>,<span class="cm-number">2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">73</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">R_amount.mean2</span><span class="cm-operator cm-arrow"><-</span><span class="cm-variable">deriv.fd</span>(<span class="cm-variable">R_amount.mean</span>,<span class="cm-number">2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">74</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">phaseplanePlot</span>(<span class="cm-variable">c</span>(<span class="cm-number">2021</span>,<span class="cm-number">2022</span>),<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-variable">mains</span>,<span class="cm-variable">R_amount.mean</span>,<span class="cm-variable">labels</span><span class="cm-operator">=</span><span class="cm-builtin">list</span>(<span class="cm-variable">evalarg</span><span class="cm-operator">=</span><span class="cm-variable">seq</span>(<span class="cm-number">2021</span>,<span class="cm-number">2022</span>,<span class="cm-variable">length</span><span class="cm-operator">=</span><span class="cm-number">12</span>),<span class="cm-variable">labels</span><span class="cm-operator">=</span><span class="cm-variable">c</span>(<span class="cm-string">"1"</span>,<span class="cm-string">"2"</span>,<span class="cm-string">"3"</span>,<span class="cm-string">"4"</span>,<span class="cm-string">"5"</span>,<span class="cm-string">"6"</span>,<span class="cm-string">"7"</span>,<span class="cm-string">"8"</span>,<span class="cm-string">"9"</span>,<span class="cm-string">"10"</span>,<span class="cm-string">"11"</span>,<span class="cm-string">"12"</span>),<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"6"</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">75</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">}</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">76</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xiangwei</span>(<span class="cm-variable">fdpar</span><span class="cm-operator">=</span><span class="cm-variable">curve_beta1</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>,<span class="cm-variable">mains</span><span class="cm-operator">=</span><span class="cm-string">"图4：上证综合指数收益率回归参数相平面图"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">77</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xiangwei</span>(<span class="cm-variable">fdpar</span><span class="cm-operator">=</span><span class="cm-variable">curve_beta2</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>,<span class="cm-variable">mains</span><span class="cm-operator">=</span><span class="cm-string">"图5：西昌电力昨日收盘价回归参数相平面图"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">78</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">xiangwei</span>(<span class="cm-variable">fdpar</span><span class="cm-operator">=</span><span class="cm-variable">curve_beta3</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">fd</span>,<span class="cm-variable">mains</span><span class="cm-operator">=</span><span class="cm-string">"图6：西昌电力成交量回归参数相平面图"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">79</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#### 半参数模型 ####</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">80</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">x3</span><span class="cm-operator">=</span><span class="cm-variable">as.vector</span>(<span class="cm-variable">model_x3</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">81</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">x2</span><span class="cm-operator">=</span><span class="cm-variable">as.vector</span>(<span class="cm-variable">model_x2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">82</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">x1</span><span class="cm-operator">=</span><span class="cm-variable">as.vector</span>(<span class="cm-variable">model_x1</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">83</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">y</span><span class="cm-operator">=</span><span class="cm-variable">as.vector</span>(<span class="cm-variable">model_y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">84</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">library</span>(<span class="cm-variable">splines</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">85</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plot</span>(<span class="cm-variable">x1</span>,<span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">86</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">res3</span> <span class="cm-operator cm-arrow"><-</span> <span class="cm-variable">smooth.spline</span>(<span class="cm-variable">x1</span>, <span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">87</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">lines</span>(<span class="cm-variable">spline</span>(<span class="cm-variable">res3</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">x</span>, <span class="cm-variable">res3</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">y</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-string">"red"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">88</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">legend</span>(<span class="cm-string">"top"</span>, <span class="cm-variable">lwd</span><span class="cm-operator">=</span><span class="cm-variable">c</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>,<span class="cm-number">1</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"red"</span>),<span class="cm-variable">legend</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"样条平滑结果"</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">89</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图7：西昌电力超额收益率与上证收益率非线性关系的样条光滑估计"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">90</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plot</span>(<span class="cm-variable">x2</span>,<span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">91</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">res2</span> <span class="cm-operator cm-arrow"><-</span> <span class="cm-variable">smooth.spline</span>(<span class="cm-variable">x2</span>, <span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">92</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">lines</span>(<span class="cm-variable">spline</span>(<span class="cm-variable">res2</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">x</span>, <span class="cm-variable">res2</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">y</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-string">"red"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">93</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">legend</span>(<span class="cm-string">"top"</span>, <span class="cm-variable">lwd</span><span class="cm-operator">=</span><span class="cm-variable">c</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>,<span class="cm-number">1</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"red"</span>),<span class="cm-variable">legend</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"样条平滑结果"</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">94</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图8：西昌电力超额收益率与昨日收盘价非线性关系的样条光滑估计"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">95</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plot</span>(<span class="cm-variable">x3</span>,<span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">96</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">res</span> <span class="cm-operator cm-arrow"><-</span> <span class="cm-variable">smooth.spline</span>(<span class="cm-variable">x3</span>, <span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">97</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">lines</span>(<span class="cm-variable">spline</span>(<span class="cm-variable">res</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">x</span>, <span class="cm-variable">res</span><span class="cm-operator cm-dollar">$</span><span class="cm-variable">y</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-string">"red"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">98</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">legend</span>(<span class="cm-string">"top"</span>, <span class="cm-variable">lwd</span><span class="cm-operator">=</span><span class="cm-variable">c</span>(<span class="cm-number">2</span>,<span class="cm-number">1</span>,<span class="cm-number">1</span>), <span class="cm-variable">col</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"red"</span>),<span class="cm-variable">legend</span><span class="cm-operator">=</span><span class="cm-variable">c</span>( <span class="cm-string">"样条平滑结果"</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35.9959px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">99</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">title</span>(<span class="cm-variable">main</span><span class="cm-operator">=</span><span class="cm-string">"图9：西昌电力超额收益率与成交量非线性关系的样条光滑估计"</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 2304px;"></div><div class="CodeMirror-gutters" style="height: 2304px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p> </p></div></div>
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