汽车油耗效率 MPG预测拟牛顿法

### 使用拟牛顿法实现汽车燃油效率MPG预测 为了构建一个基于拟牛顿法的机器学习模型来预测汽车燃油效率(MPG)，可以采用BFGS算法作为优化器。该方法属于二阶最优化技术，能够有效处理复杂的非线性关系并加速收敛过程[^1]。 #### 数据预处理阶段 数据集通常包含多个特征变量以及目标标签——即每加仑行驶里程数（MPG）。在实际操作前需完成如下工作： - 清洗缺失值或异常点； - 对类别型字段执行独热编码转换； - 将数值属性标准化至相同尺度范围； ```python import pandas as pd from sklearn.model_selection import train_test_split from sklearn.preprocessing import StandardScaler, OneHotEncoder from sklearn.compose import ColumnTransformer from sklearn.pipeline import Pipeline # 加载数据集 data = pd.read_csv('auto-mpg.csv') # 处理缺失值 (假设 horsepower 中有缺失值) data['horsepower'] = data['horsepower'].replace('?', None).astype(float) data.dropna(inplace=True) X = data.drop(columns=['mpg']) y = data[['mpg']].values.ravel() numeric_features = ['cylinders', 'displacement', 'horsepower', 'weight', 'acceleration'] categorical_features = ['model year', 'origin'] preprocessor = ColumnTransformer( transformers=[ ('num', StandardScaler(), numeric_features), ('cat', OneHotEncoder(handle_unknown='ignore'), categorical_features)]) pipeline = Pipeline(steps=[('preprocessor', preprocessor)]) X_processed = pipeline.fit_transform(X) ``` #### 构建与训练模型 选用支持自定义优化器的回归库如`scikit-optimize`中的`GP_minimize`接口配合SciPy包下的`minimize`函数实现BFGS求解路径。此处展示如何利用Scipy内置功能创建最小化问题实例并通过调用其solve方法获取最优参数向量w*: ```python from scipy.optimize import minimize from sklearn.metrics import mean_squared_error def objective_function(w, X_train, y_train): predictions = np.dot(X_train, w.T) mse_loss = mean_squared_error(y_train, predictions) return mse_loss initial_guess = np.zeros((X_processed.shape[1],)) result = minimize(objective_function, initial_guess, args=(X_processed, y), method="L-BFGS-B", options={'disp': True}) optimal_weights = result.x print(f'Optimized Weights: {optimal_weights}') ``` 上述代码片段展示了通过L-BFGS-B变体找到使均方误差损失达到极小化的权重组合方案[^2]。 #### 性能评估环节 最后一步是对测试样本集上的表现情况进行评测分析，计算诸如R²得分之类的统计指标衡量泛化能力优劣程度。 ```python from sklearn.metrics import r2_score predictions = np.dot(X_processed, optimal_weights.T) r2 = r2_score(y, predictions) print(f'R^2 Score on Training Data: {r2:.4f}') # 划分训练/验证集合用于更公正评价 X_train, X_val, y_train, y_val = train_test_split(X_processed, y, test_size=0.2, random_state=42) val_predictions = np.dot(X_val, optimal_weights.T) validation_r2 = r2_score(y_val, val_predictions) print(f'R^2 Score on Validation Data: {validation_r2:.4f}') ```