- **The probabilistic approach** - treat all unknown quantities as random variables - it is the optimal approach to decision making under uncertainty <br> >Almost all of machine learning can be viewed in probabilistic terms, making probabilistic thinking fundamental. It is, of course, not the only view. But it is through this view that <font color="red">we can connect what we do in machine learning to every other computational science, whether that be in stochastic optimisation, control theory, operations research, econometrics, information theory, statistical physics or bio-statistics.</font> For this reason alone, mastery of probabilistic thinking is essential. <p align="right">---Shakir Mohamed, DeepMind<p>
<img src="https://scikit-learn.org/stable/_images/grid_search_workflow.png" width=150%>
--- <div align="center"> <table rules="none"> <tr> <td> <div style="width: 300"> ### 聚类（Clustering） - 在数据中查找**聚类**：将输入划分为包含“相似”点的区域。 </div> </td> <td> <div style="width: 500pt"> <img align="center" style="padding-right:10px;" width=100% src="../myfig/L01/L01-clustering.png"> </div> </td> </tr> </table> </div>
--- ## 7. Python实现案例：基础实现 ```python # PCR基础实现 from sklearn.decomposition import PCA from sklearn.linear_model import LinearRegression from sklearn.pipeline import make_pipeline from sklearn.preprocessing import StandardScaler import numpy as np import matplotlib.pyplot as plt # 创建管道 pcr_model = make_pipeline( StandardScaler(), PCA(n_components=5), # 选择主成分数量 LinearRegression() ) # 拟合模型 pcr_model.fit(X_train, y_train) # 预测 predictions = pcr_model.predict(X_test) ``` --- ## 7. Python实现案例：组件选择 ```python # 使用交叉验证选择最佳主成分数量 from sklearn.model_selection import cross_val_score from sklearn.metrics import mean_squared_error # 测试不同的主成分数量 n_components = range(1, X.shape[1] + 1) cv_scores = [] for n in n_components: pcr_model = make_pipeline( StandardScaler(), PCA(n_components=n), LinearRegression() ) # 使用交叉验证评估性能 scores = -cross_val_score(pcr_model, X, y, cv=5, scoring='neg_mean_squared_error') cv_scores.append(scores.mean()) # 绘制结果 plt.figure(figsize=(10, 6)) plt.plot(n_components, cv_scores, marker='o') plt.xlabel('主成分数量') plt.ylabel('MSE') plt.title('PCR: 主成分数量与MSE关系') ``` --- ## 7. Python实现案例：完整示例 ```python # 完整PCR实现示例 import numpy as np import pandas as pd import matplotlib.pyplot as plt from sklearn.preprocessing import StandardScaler from sklearn.decomposition import PCA from sklearn.linear_model import LinearRegression from sklearn.model_selection import train_test_split, cross_val_score from sklearn.metrics import mean_squared_error, r2_score # 1. 加载数据并划分 X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42) # 2. 标准化 scaler = StandardScaler() X_train_scaled = scaler.fit_transform(X_train) X_test_scaled = scaler.transform(X_test) # 3. PCA pca = PCA() X_train_pca = pca.fit_transform(X_train_scaled) # 4. 检查累积方差 explained_variance = pca.explained_variance_ratio_ cumulative_variance = np.cumsum(explained_variance) optimal_n = np.argmax(cumulative_variance >= 0.95) + 1 # 解释95%方差所需的成分数 # 5. 使用选择的组件数构建模型 pca = PCA(n_components=optimal_n) X_train_pca = pca.fit_transform(X_train_scaled) X_test_pca = pca.transform(X_test_scaled) # 6. 拟合回归模型 regressor = LinearRegression() regressor.fit(X_train_pca, y_train) # 7. 预测 y_train_pred = regressor.predict(X_train_pca) y_test_pred = regressor.predict(X_test_pca) # 8. 评估模型 train_mse = mean_squared_error(y_train, y_train_pred) test_mse = mean_squared_error(y_test, y_test_pred) test_r2 = r2_score(y_test, y_test_pred) print(f"使用的主成分数: {optimal_n}") print(f"训练集MSE: {train_mse:.4f}") print(f"测试集MSE: {test_mse:.4f}") print(f"测试集R²: {test_r2:.4f}") ```