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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>分段数对拟合效果的影响</title>
<style type="text/css">
div.cv {
width:720px;
height:480px;
margin:auto;
}
/*!
*
* Twitter Bootstrap
*
*//*! normalize.css v3.0.2 | MIT License | git.io/normalize */html{font-family:sans-serif;-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%;font-size:10px;-webkit-tap-highlight-color:transparent}article,aside,details,figcaption,figure,footer,header,hgroup,main,menu,nav,section,summary{display:block}audio,canvas,progress,video{display:inline-block;vertical-align:baseline}audio:not([controls]){display:none;height:0}[hidden],template{display:none}a{background-color:transparent}a:active,a:hover{outline:0}abbr[title]{border-bottom:1px dotted}b,optgroup,strong{font-weight:700}dfn{font-style:italic}h1{font-size:2em;margin:.67em 0}mark{background:#ff0;color:#000}small{font-size:80%}sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}sup{top:-.5em}sub{bottom:-.25em}img{border:0;vertical-align:middle}svg:not(:root){overflow:hidden}hr{-moz-box-sizing:content-box;box-sizing:content-box;height:0}pre,textarea{overflow:auto}code,kbd,pre,samp{font-size:1em}button,input,optgroup,select,textarea{color:inherit;font:inherit;margin:0}button{overflow:visible}button,select{text-transform:none}button,html input[type=button],input[type=reset],input[type=submit]{-webkit-appearance:button;cursor:pointer}button[disabled],html input[disabled]{cursor:default}button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}input{line-height:normal}input[type=checkbox],input[type=radio]{box-sizing:border-box;padding:0}input[type=number]::-webkit-inner-spin-button,input[type=number]::-webkit-outer-spin-button{height:auto}input[type=search]::-webkit-search-cancel-button,input[type=search]::-webkit-search-decoration{-webkit-appearance:none}table{border-collapse:collapse;border-spacing:0}td,th{padding:0}/*! Source: https://github.com/h5bp/html5-boilerplate/blob/master/src/css/main.css */@media print{*,:after,:before{background:0 0!important;color:#000!important;box-shadow:none!important;text-shadow:none!important}a,a:visited{text-decoration:underline}a[href]:after{content:" (" attr(href)")"}abbr[title]:after{content:" (" attr(title)")"}a[href^="javascript:"]:after,a[href^="#"]:after{content:""}blockquote,pre{border:1px solid #999;page-break-inside:avoid}thead{display:table-header-group}img,tr{page-break-inside:avoid}img{max-width:100%!important}h2,h3,p{orphans:3;widows:3}h2,h3{page-break-after:avoid}select{background:#fff!important}.navbar{display:none}.btn>.caret,.dropup>.btn>.caret{border-top-color:#000!important}.label{border:1px solid #000}.table{border-collapse:collapse!important}.table td,.table th{background-color:#fff!important}.table-bordered td,.table-bordered th{border:1px solid #ddd!important}}@font-face{font-family:'Glyphicons Halflings';src:url(../components/bootstrap/fonts/glyphicons-halflings-regular.eot);src:url(../components/bootstrap/fonts/glyphicons-halflings-regular.eot?#iefix)format('embedded-opentype'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.woff2)format('woff2'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.woff)format('woff'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.ttf)format('truetype'),url(../components/bootstrap/fonts/glyphicons-halflings-regular.svg#glyphicons_halflingsregular)format('svg')}.glyphicon{position:relative;top:1px;display:inline-block;font-family:'Glyphicons 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<h1 id="说明">说明<a class="anchor-link" href="#说明">¶</a></h1><p><strong> 目的：</strong> 为探讨在同一个样本点集下，cma-es算法对$f(x)$拟合效果与分段数的关系，实验记录每个分段方案下cma-es过程的迭代时间，关系矩阵变化过程，计算结果，cpu计算时间等进行对比。<br>
<strong> 函数： </strong></p>
<p><center>$ f(x)=10sin0.6x+uniform(-1.5,1.5)gauss(0,5),x \in[-7,7)$ </center>
<strong> 分段： </strong> 为保证每段点数的一致性，通过$f(x)$在定义域内均匀分布300个样本点，段数分别为$PN(partition\ number) \in \{5,10,15,20,25,30\}$，对应CMA问题的求解维度从15至90，每段点数不小于10.<br>
<strong>评价函数：</strong></p>
<ol>
<li>$M1$：分段测试函数的mse；</li>
<li>$M2$：在f1基础上增加间断点连续性判断指标：$\sum_{i=1}^{k} (e^{\Delta Y_i-\alpha}-1)$,其中$\Delta Y_i$ 代表拟合分段函数在间断点处的左右间断点差的绝对值,$\alpha $默认代表$Y$值域范围的$1\%$大小；</li>
<li>$M3$：在f2基础上增加间断点一阶导数评价指标：$\sum_{i=1}^{k}(e^{\frac{\Delta \sigma_i-\beta}{10e}}-1)$,其中$\Delta \sigma_i$代表左右间断点处的左右导数的$arctan$差的绝对值，$\beta$如未特殊说明全局默认为10度($\frac {\pi}{18}$)； </li>
</ol>
<h1 id="实验">实验<a class="anchor-link" href="#实验">¶</a></h1><h3 id="迭代次数与分段数关系图表">迭代次数与分段数关系图表<a class="anchor-link" href="#迭代次数与分段数关系图表">¶</a></h3>
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">makeData</span> <span class="kn">as</span> <span class="nn">md</span>
<span class="o">%</span><span class="k">pylab</span> inline
<span class="n">plt</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s">'figure'</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
<span class="n">X</span><span class="o">=</span><span class="n">md</span><span class="o">.</span><span class="n">loadData</span><span class="p">(</span><span class="s">'result.tl'</span><span class="p">)</span>
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<pre>Populating the interactive namespace from numpy and matplotlib
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<div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="kn">as</span> <span class="nn">pd</span>
<span class="n">b</span><span class="o">=</span><span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">35</span><span class="p">,</span><span class="mi">5</span><span class="p">):</span>
<span class="n">temp</span><span class="o">=</span><span class="p">[]</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
<span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">][</span><span class="s">'iter'</span><span class="p">])</span>
<span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
<span class="n">bs</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">ind</span><span class="o">=</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">35</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="n">d</span><span class="o">=</span><span class="p">{</span><span class="s">'M1'</span><span class="p">:</span><span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">bs</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">index</span><span class="o">=</span><span class="n">ind</span><span class="p">),</span>
<span class="s">'M2'</span><span class="p">:</span><span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">bs</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">index</span><span class="o">=</span><span class="n">ind</span><span class="p">),</span>
<span class="s">'M3'</span><span class="p">:</span><span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">(</span><span class="n">bs</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span><span class="n">index</span><span class="o">=</span><span class="n">ind</span><span class="p">)}</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">columns</span><span class="o">.</span><span class="n">name</span><span class="o">=</span><span class="s">'function'</span>
<span class="n">df</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">name</span><span class="o">=</span><span class="s">'partition'</span>
<span class="n">df</span>
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<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th>function</th>
<th>M1</th>
<th>M2</th>
<th>M3</th>
</tr>
<tr>
<th>partition</th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
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<th>5</th>
<td>624</td>
<td>1379</td>
<td>1466</td>
</tr>
<tr>
<th>10</th>
<td>2548</td>
<td>7006</td>
<td>5941</td>
</tr>
<tr>
<th>15</th>
<td>6333</td>
<td>19248</td>
<td>19219</td>
</tr>
<tr>
<th>20</th>
<td>11703</td>
<td>38264</td>
<td>41027</td>
</tr>
<tr>
<th>25</th>
<td>18652</td>
<td>6692</td>
<td>73007</td>
</tr>
<tr>
<th>30</th>
<td>28149</td>
<td>15486</td>
<td>66815</td>
</tr>
</tbody>
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<div class=" highlight hl-ipython2"><pre><span class="n">df</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">kind</span><span class="o">=</span><span class="s">'bar'</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="n">leg</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">get_legend</span><span class="p">()</span>
<span class="n">ltext</span> <span class="o">=</span> <span class="n">leg</span><span class="o">.</span><span class="n">get_texts</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">setp</span><span class="p">(</span><span class="n">ltext</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s">'20'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">"iteration counts"</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
<span class="n">leg</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">gca</span><span class="p">()</span><span class="o">.</span><span class="n">get_legend</span><span class="p">()</span>
<span class="n">ltext</span> <span class="o">=</span> <span class="n">leg</span><span class="o">.</span><span class="n">get_texts</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">setp</span><span class="p">(</span><span class="n">ltext</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="s">'20'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">"iteration counts"</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
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<pre><matplotlib.text.Text at 0x109768750></pre>
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