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<a href="/2024/08/14/shuxuejianmo/" class="post-title-link" itemprop="url">数学建模相关笔记</a>
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<time title="Created: 2024-08-14 03:19:00 / Modified: 03:52:55" itemprop="dateCreated datePublished" datetime="2024-08-14T03:19:00+08:00">2024-08-14</time>
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<h1 id="数学建模学习笔记">数学建模学习笔记</h1>
<h2 id="拟合">1. 拟合</h2>
<h3 id="拟合方法">拟合方法</h3>
<ol type="1">
<li><p>最小二乘法 <span class="math display">\[
\sum_{i=1}^n (f_i-f(x_i))^2 = S
\]</span> 用原数据进行拟合，使得模型变量偏导数为0</p></li>
<li><p>变换后最小二乘</p>
<p>与最小二乘法一样，但使用的数据是经过变换的</p>
<p>例如 <span class="math display">\[
lny与lnx
\]</span></p></li>
<li><p>Chebyshev <span class="math display">\[
\sum_{i=1}^n (f_i-f(x_i)) = S
\]</span></p></li>
</ol>
<h3 id="拟合模型">拟合模型</h3>
<ol type="1">
<li><p>多项式</p>
<p><strong>优点</strong>：可微分</p>
<p><strong>缺点</strong>：端点图形摆动严重、对数据微小变化的敏感度高（误差数据可能导致图形趋势相差很大）</p>
<h4 id="使用技巧">使用技巧</h4>
<ol type="1">
<li><p>使用低项式</p>
<p>如何确认合适的阶数？使用<u>导数</u></p>
<figure>
<img src="../images/shuxuejianmo/image-20240813233244502.png"
alt="image-20240813233244502" />
<figcaption aria-hidden="true">image-20240813233244502</figcaption>
</figure>
<p>如图，当开导开到如图所有值都差不多是一个常数时，可以确认阶数(图中为2阶)</p></li>
<li><p>三阶样条插值</p>
<p>由多个三阶多项式拟合组成的函数</p>
<p>两个点之间用一个三阶多项式拟合，并保证多项式之间连接的端点平滑</p>
<p><img src="../images/shuxuejianmo/image-20240814003659336.png"
alt="image-20240814003659336" /> <span class="math display">\[
\begin{aligned}
& \begin{cases}
y_1=S_1(x_1)\\
y_2=S_1(x_2)\\
y_2=S_2(x_2)\\
y_3=S_3(x_3)
\end{cases} \\
& \begin{cases}
S_1'(x_2)=S_2'(x_2)\\
S_1''(x_2)=S_2''(x_2)\\
\end{cases}\\
& \begin{cases}
S_1'(x_1)=f'(x_1)\\
S_2'(x_3)=f'(x_3)
\end{cases}
\\&当f'(x)=0时为自然样条，当 f'(x)\ne0时为强制样条
\end{aligned}
\]</span></p></li>
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