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<meta property="og:description" content="线性回归简介 用于预测一个连续的目标变量（因变量），与一个或多个特征（自变量）之间存在线性关系。 假设函数：$$y = w_1x_1 + w_2x_2 + \cdot\cdot\cdot+w_nx_n$$ $y$ 是目标变量（因变量），即我们希望预测的值。 $x1,x2,…,xn$ 是特征变量（自变量），即输入的值。 损失函数为了找到最佳的线性模型，我们需要通过最小化损失函数">
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<p>用于预测一个连续的目标变量（因变量），与一个或多个特征（自变量）之间存在线性关系。</p>
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</blockquote>
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<p>假设函数：<br>$$y = w_1x_1 + w_2x_2 + \cdot\cdot\cdot+w_nx_n$$</p>
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<ul>
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<li>$y$ 是目标变量（因变量），即我们希望预测的值。</li>
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<li>$x1,x2,…,xn$ 是特征变量（自变量），即输入的值。</li>
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</ul>
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<h3 id="损失函数"><a href="#损失函数" class="headerlink" title="损失函数"></a>损失函数</h3><p>为了找到最佳的线性模型，我们需要通过最小化损失函数来优化模型参数。在线性回归中，常用的损失函数是 <strong>均方误差（MSE）</strong>：<br>$$MSE = \frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{y}_i)^2$$</p>
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<ul>
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<li>m 是样本的数量。</li>
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<li>$y_i$ 是第 i 个样本的真实值。</li>
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<li>$\hat{y}_i$ 是模型预测的第 i 个样本的值。</li>
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</ul>
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<h3 id="线性回归优化"><a href="#线性回归优化" class="headerlink" title="线性回归优化"></a>线性回归优化</h3><ul>
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<li><p>梯度下降法</p>
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<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sklearn.datasets <span class="keyword">import</span> fetch_california_housing</span><br><span class="line"><span class="keyword">from</span> sklearn.model_selection <span class="keyword">import</span> train_test_split</span><br><span class="line"><span class="keyword">from</span> sklearn.preprocessing <span class="keyword">import</span> StandardScaler</span><br><span class="line"><span class="keyword">from</span> sklearn.linear_model <span class="keyword">import</span> SGDRegressor</span><br><span class="line"><span class="keyword">from</span> sklearn.metrics <span class="keyword">import</span> mean_squared_error</span><br><span class="line"></span><br><span class="line"><span class="comment"># 1. 获取数据集</span></span><br><span class="line">housing = fetch_california_housing()</span><br><span class="line"></span><br><span class="line"><span class="comment"># 2. 数据集处理</span></span><br><span class="line"><span class="comment"># 2.1 分割数据集</span></span><br><span class="line">X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=<span class="number">0.25</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 3. 特征工程</span></span><br><span class="line"><span class="comment"># 3.1 标准化</span></span><br><span class="line">transfer = StandardScaler()</span><br><span class="line">X_train = transfer.fit_transform(X_train)</span><br><span class="line">X_test = transfer.transform(X_test) <span class="comment"># 使用 transform() 而不是 fit_transform()</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># 4.机器学习- 梯度下降法</span></span><br><span class="line">estimater = SGDRegressor(max_iter=<span class="number">1000</span>, eta0=<span class="number">0.01</span>)</span><br><span class="line">estimater.fit(X_train, y_train)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"SGD模型的偏置是：<span class="subst">{estimater.intercept_}</span>"</span>)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"SGD模型的系数是：<span class="subst">{estimater.coef_}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 5. 模型评估</span></span><br><span class="line">y_pred = estimater.predict(X_test)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"SGD模型预测值：<span class="subst">{y_pred}</span>"</span>)</span><br><span class="line">mse = mean_squared_error(y_test, y_pred)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"SGD模型均方误差:<span class="subst">{mse}</span>"</span>)</span><br></pre></td></tr></table></figure>
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</li>
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<li><p>正规方程</p>
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<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sklearn.datasets <span class="keyword">import</span> fetch_california_housing</span><br><span class="line"><span class="keyword">from</span> sklearn.model_selection <span class="keyword">import</span> train_test_split</span><br><span class="line"><span class="keyword">from</span> sklearn.preprocessing <span class="keyword">import</span> StandardScaler</span><br><span class="line"><span class="keyword">from</span> sklearn.linear_model <span class="keyword">import</span> LinearRegression</span><br><span class="line"><span class="keyword">from</span> sklearn.metrics <span class="keyword">import</span> mean_squared_error</span><br><span class="line"></span><br><span class="line"><span class="comment"># 1. 获取数据集</span></span><br><span class="line">housing = fetch_california_housing()</span><br><span class="line"></span><br><span class="line"><span class="comment"># 2. 数据集处理</span></span><br><span class="line"><span class="comment"># 2.1 分割数据集</span></span><br><span class="line">X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=<span class="number">0.25</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 3. 特征工程</span></span><br><span class="line"><span class="comment"># 3.1 标准化</span></span><br><span class="line">transfer = StandardScaler()</span><br><span class="line">X_train = transfer.fit_transform(X_train)</span><br><span class="line">X_test = transfer.fit_transform(X_test)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 4.机器学习- 线性回归</span></span><br><span class="line">estimater = LinearRegression()</span><br><span class="line">estimater.fit(X_train, y_train)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"模型的偏置是：<span class="subst">{estimater.intercept_}</span>"</span>)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"模型的系数是：<span class="subst">{estimater.coef_}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 5. 模型评估</span></span><br><span class="line">y_pred = estimater.predict(X_test)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"模型预测值：<span class="subst">{y_pred}</span>"</span>)</span><br><span class="line">mse = mean_squared_error(y_test, y_pred)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"模型均方误差:<span class="subst">{mse}</span>"</span>)</span><br></pre></td></tr></table></figure>
|
||
</li>
|
||
<li><p>岭回归</p>
|
||
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sklearn.datasets <span class="keyword">import</span> fetch_california_housing</span><br><span class="line"><span class="keyword">from</span> sklearn.model_selection <span class="keyword">import</span> train_test_split</span><br><span class="line"><span class="keyword">from</span> sklearn.preprocessing <span class="keyword">import</span> StandardScaler</span><br><span class="line"><span class="keyword">from</span> sklearn.linear_model <span class="keyword">import</span> Ridge, RidgeCV</span><br><span class="line"><span class="keyword">from</span> sklearn.metrics <span class="keyword">import</span> mean_squared_error</span><br><span class="line"></span><br><span class="line"><span class="comment"># 1. 获取数据集</span></span><br><span class="line">housing = fetch_california_housing()</span><br><span class="line"></span><br><span class="line"><span class="comment"># 2. 数据集处理</span></span><br><span class="line"><span class="comment"># 2.1 分割数据集</span></span><br><span class="line">X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=<span class="number">0.25</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 3. 特征工程</span></span><br><span class="line"><span class="comment"># 3.1 标准化</span></span><br><span class="line">transfer = StandardScaler()</span><br><span class="line">X_train = transfer.fit_transform(X_train)</span><br><span class="line">X_test = transfer.transform(X_test) <span class="comment"># 使用 transform() 而不是 fit_transform()</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># 4.机器学习- 岭回归 使用了Ridge的alpha的搜索</span></span><br><span class="line"><span class="comment"># estimater = Ridge(alpha=1.0)</span></span><br><span class="line">estimater = RidgeCV(alphas=[<span class="number">0.001</span>, <span class="number">0.01</span>, <span class="number">0.1</span>, <span class="number">1</span>, <span class="number">10</span>, <span class="number">100</span>])</span><br><span class="line">estimater.fit(X_train, y_train)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"Ridge模型的偏置是：<span class="subst">{estimater.intercept_}</span>"</span>)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"Ridge模型的系数是：<span class="subst">{estimater.coef_}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 查看最佳 alpha</span></span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"最佳 alpha 值是：<span class="subst">{estimater.alpha_}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 5. 模型评估</span></span><br><span class="line">y_pred = estimater.predict(X_test)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"Ridge模型预测值：<span class="subst">{y_pred}</span>"</span>)</span><br><span class="line">mse = mean_squared_error(y_test, y_pred)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">f"Ridge模型均方误差:<span class="subst">{mse}</span>"</span>)</span><br></pre></td></tr></table></figure></li>
|
||
</ul>
|
||
<p>这样每个代码块的缩进保持一致，便于阅读和理解。如果有其他优化需求，随时告诉我！</p>
|
||
<p><img src="/img/machinelearning/linear.png"></p>
|
||
<p><img src="/img/machinelearning/fitting.png"></p>
|
||
<h3 id="模型保存和加载"><a href="#模型保存和加载" class="headerlink" title="模型保存和加载"></a>模型保存和加载</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">from</span> sklearn.datasets <span class="keyword">import</span> fetch_california_housing</span><br><span class="line"><span class="keyword">from</span> sklearn.model_selection <span class="keyword">import</span> train_test_split</span><br><span class="line"><span class="keyword">from</span> sklearn.preprocessing <span class="keyword">import</span> StandardScaler</span><br><span class="line"><span class="keyword">from</span> sklearn.linear_model <span class="keyword">import</span> Ridge, RidgeCV</span><br><span class="line"><span class="keyword">from</span> sklearn.metrics <span class="keyword">import</span> mean_squared_error</span><br><span class="line"><span class="keyword">import</span> joblib</span><br><span class="line"></span><br><span class="line"><span class="keyword">def</span> <span class="title function_">save_model</span>():</span><br><span class="line"> <span class="comment"># 1. 获取数据集</span></span><br><span class="line"> housing = fetch_california_housing()</span><br><span class="line"> <span class="comment"># 2. 数据集处理</span></span><br><span class="line"> <span class="comment"># 2.1 分割数据集</span></span><br><span class="line"> X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=<span class="number">0.25</span>)</span><br><span class="line"> <span class="comment"># 3. 特征工程</span></span><br><span class="line"> <span class="comment"># 3.1 标准化</span></span><br><span class="line"> transfer = StandardScaler()</span><br><span class="line"> X_train = transfer.fit_transform(X_train)</span><br><span class="line"> X_test = transfer.transform(X_test) <span class="comment"># 使用 transform() 而不是 fit_transform()</span></span><br><span class="line"> <span class="comment"># 4. 机器学习 - 岭回归 使用了Ridge的alpha的搜索</span></span><br><span class="line"> estimater = RidgeCV(alphas=[<span class="number">0.001</span>, <span class="number">0.01</span>, <span class="number">0.1</span>, <span class="number">1</span>, <span class="number">10</span>, <span class="number">100</span>])</span><br><span class="line"> estimater.fit(X_train, y_train)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型的偏置是：<span class="subst">{estimater.intercept_}</span>"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型的系数是：<span class="subst">{estimater.coef_}</span>"</span>)</span><br><span class="line"> <span class="comment"># 保存模型</span></span><br><span class="line"> joblib.dump(estimater, <span class="string">'ridge_model.pkl'</span>)</span><br><span class="line"> <span class="comment"># 查看最佳 alpha</span></span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"最佳 alpha 值是：<span class="subst">{estimater.alpha_}</span>"</span>)</span><br><span class="line"> <span class="comment"># 5. 模型评估</span></span><br><span class="line"> y_pred = estimater.predict(X_test)</span><br><span class="line"> mse = mean_squared_error(y_test, y_pred)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型均方误差:<span class="subst">{mse}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="keyword">def</span> <span class="title function_">load_model</span>():</span><br><span class="line"> <span class="comment"># 1. 获取数据集</span></span><br><span class="line"> housing = fetch_california_housing()</span><br><span class="line"> <span class="comment"># 2. 数据集处理</span></span><br><span class="line"> <span class="comment"># 2.1 分割数据集</span></span><br><span class="line"> X_train, X_test, y_train, y_test = train_test_split(housing.data, housing.target, test_size=<span class="number">0.25</span>)</span><br><span class="line"> <span class="comment"># 3. 特征工程</span></span><br><span class="line"> <span class="comment"># 3.1 标准化</span></span><br><span class="line"> transfer = StandardScaler()</span><br><span class="line"> X_train = transfer.fit_transform(X_train)</span><br><span class="line"> X_test = transfer.transform(X_test) <span class="comment"># 使用 transform() 而不是 fit_transform()</span></span><br><span class="line"> <span class="comment"># 加载模型</span></span><br><span class="line"> estimater = joblib.load(<span class="string">'ridge_model.pkl'</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型的偏置是：<span class="subst">{estimater.intercept_}</span>"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型的系数是：<span class="subst">{estimater.coef_}</span>"</span>)</span><br><span class="line"> <span class="comment"># 查看最佳 alpha</span></span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"最佳 alpha 值是：<span class="subst">{estimater.alpha_}</span>"</span>)</span><br><span class="line"> <span class="comment"># 5. 模型评估</span></span><br><span class="line"> y_pred = estimater.predict(X_test)</span><br><span class="line"> mse = mean_squared_error(y_test, y_pred)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型预测值：<span class="subst">{y_pred}</span>"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"Ridge模型均方误差:<span class="subst">{mse}</span>"</span>)</span><br><span class="line"></span><br><span class="line"><span class="built_in">print</span>(<span class="string">"训练并保存模型："</span>)</span><br><span class="line">save_model()</span><br><span class="line"><span class="built_in">print</span>(<span class="string">"加载模型"</span>)</span><br><span class="line">load_model()</span><br></pre></td></tr></table></figure></article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta"><i class="fas fa-circle-user fa-fw"></i>文章作者: </span><span class="post-copyright-info"><a href="https://rq.shenjianl.cn">shenjianZ</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta"><i class="fas fa-square-arrow-up-right fa-fw"></i>文章链接: </span><span class="post-copyright-info"><a href="https://rq.shenjianl.cn/posts/52662.html">https://rq.shenjianl.cn/posts/52662.html</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta"><i class="fas fa-circle-exclamation fa-fw"></i>版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://qr.shenjianl.cn/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="https://rq.shenjianl.cn" target="_blank">QuickReference</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/linear-regression/">linear-regression</a></div><div class="post_share"><div 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