实验目标

本实验演示多种回归模型（OLS、Ridge、Lasso、ElasticNet、多项式回归）和分类模型（逻辑回归）在房价数据集中的应用，通过对比模型性能指标评估不同算法的拟合效果与预测能力。

实验环境

• Python 3.8
• 核心库：pandas、numpy、matplotlib/seaborn、scikit-learn
• 开发工具：Jupyter Notebook

数据加载与预处理

加载房价数据集（train.csv），包含1460条样本和81个特征。选取4个关键数值特征用于建模，目标变量为房屋售价（SalePrice）。数据分割为训练集和测试集（80:20比例），并进行标准化处理。

import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler

data = pd.read_csv("train.csv")
features = ['OverallQual', 'GrLivArea', 'GarageCars', 'TotalBsmtSF']
X = data[features]
y = data['SalePrice']

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

模型构建与评估

普通最小二乘法（OLS）

from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score

ols = LinearRegression()
ols.fit(X_train_scaled, y_train)
y_pred_ols = ols.predict(X_test_scaled)

print("MSE:", mean_squared_error(y_test, y_pred_ols))
print("R2:", r2_score(y_test, y_pred_ols))

岭回归（Ridge）

from sklearn.linear_model import Ridge

ridge = Ridge(alpha=50)
ridge.fit(X_train_scaled, y_train)
y_pred_ridge = ridge.predict(X_test_scaled)

print("MSE:", mean_squared_error(y_test, y_pred_ridge))
print("R2:", r2_score(y_test, y_pred_ridge))

拉索回归（Lasso）

from sklearn.linear_model import Lasso

lasso = Lasso(alpha=100)
lasso.fit(X_train_scaled, y_train)
y_pred_lasso = lasso.predict(X_test_scaled)

print("MSE:", mean_squared_error(y_test, y_pred_lasso))
print("R2:", r2_score(y_test, y_pred_lasso))
print("Number of zero coefficients:", np.sum(lasso.coef_ == 0))

弹性网络（ElasticNet）

from sklearn.linear_model import ElasticNet

enet = ElasticNet(alpha=50, l1_ratio=0.5)
enet.fit(X_train_scaled, y_train)
y_pred_enet = enet.predict(X_test_scaled)

print("MSE:", mean_squared_error(y_test, y_pred_enet))
print("R2:", r2_score(y_test, y_pred_enet))

多项式回归

from sklearn.preprocessing import PolynomialFeatures

poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly_train = poly.fit_transform(X_train_scaled)
X_poly_test = poly.transform(X_test_scaled)

ols_poly = LinearRegression()
ols_poly.fit(X_poly_train, y_train)
y_pred_poly = ols_poly.predict(X_poly_test)

print("MSE:", mean_squared_error(y_test, y_pred_poly))
print("R2:", r2_score(y_test, y_pred_poly))

逻辑回归（分类任务）

from sklearn.linear_model import LogisticRegression
from sklearn.metrics import accuracy_score, roc_auc_score, roc_curve
import matplotlib.pyplot as plt

median_price = data['SalePrice'].median()
y_class = (y > median_price).astype(int)

X_train_c, X_test_c, y_train_c, y_test_c = train_test_split(X, y_class, test_size=0.2, random_state=42)
X_train_c_scaled = scaler.fit_transform(X_train_c)
X_test_c_scaled = scaler.transform(X_test_c)

log_reg = LogisticRegression()
log_reg.fit(X_train_c_scaled, y_train_c)
y_pred_c = log_reg.predict(X_test_c_scaled)
y_pred_prob = log_reg.predict_proba(X_test_c_scaled)[:, 1]

print("Accuracy:", accuracy_score(y_test_c, y_pred_c))
print("AUC:", roc_auc_score(y_test_c, y_pred_prob))

fpr, tpr, _ = roc_curve(y_test_c, y_pred_prob)
plt.plot(fpr, tpr)
plt.plot([0, 1], [0, 1], '--', color='gray')
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.show()

实验结果分析

• 回归模型中，多项式回归表现最佳（R2=0.86），其次是OLS（R2=0.79）
• 正则化模型中，Lasso表现优于Ridge和ElasticNet
• 逻辑回归在分类任务中取得良好效果（AUC>0.85）
• 可视化分析可通过绘制预测值与真实值散点图、残差图、ROC曲线等进一步验证模型性能