波士顿房价1.使用第一个特征实现线性回归模型，输出loss曲线，并使用至少三个指标评价模型性能，输出性能表格。2.分别使用2、4、6、10、14个特征复现线性回归模型，列出性能表格。3.选择3个比较重要的特征，复现线性回归模型，列出性能表格。详细代码展示

波士顿房价数据集是一个经典的机器学习示例，通常用于说明如何使用线性回归。这里我们将使用Python的`sklearn`库来完成这个任务。首先，确保已安装所需的库：

pip install numpy pandas scikit-learn matplotlib seaborn

然后，我们开始编写代码：

1. **使用单个特征实现线性回归**

import numpy as np
import pandas as pd
from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error

# 加载数据
boston = load_boston()
X = boston.data
y = boston.target

# 划分训练集和测试集
X_train, X_test, y_train, y_test = train_test_split(X[:, [0]], # 使用第一个特征
                                                    y,
                                                    test_size=0.2,
                                                    random_state=42)

# 创建并拟合模型
model = LinearRegression()
model.fit(X_train, y_train)

# 预测并计算损失
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
mae = mean_absolute_error(y_test, y_pred)

# 输出性能指标
print(f"Loss Curve (MSE): {mse}")
print("\nPerformance Metrics:")
print(pd.DataFrame({"Metric": ["RMSE", "R^2", "MAE"],
                    "Value": [np.sqrt(mse), r2, mae]}))

2. **使用多个特征复现线性回归**

features_to_use = [2, 4, 6, 10, 14]  # 特征编号
performance_tables = []

for feature in features_to_use:
    X_train, X_test, y_train, y_test = train_test_split(X[:, :feature], y, ...  # 使用对应特征
                                                        ...)

    model = LinearRegression()
    model.fit(X_train, y_train)
    
    y_pred = model.predict(X_test)
    mse = mean_squared_error(y_test, y_pred)
    r2 = r2_score(y_test, y_pred)
    mae = mean_absolute_error(y_test, y_pred)

    performance_table = {"Feature": feature,
                        "MSE": mse,
                        "R^2": r2,
                        "MAE": mae}
    performance_tables.append(performance_table)

# 将所有结果合并到一个DataFrame中
results_df = pd.DataFrame(performance_tables).set_index("Feature")
print(results_df)

3. **选择重要特征复现线性回归**

# 计算特征重要性（假设Boston Housing数据集没有直接提供，需要额外处理）
# 例如使用Lasso Regression for feature selection
lasso = Lasso(alpha=0.1)  # 或其他适合的选择
lasso.fit(X, y)
important_features = lasso.coef_ > 0  # 看哪些特征系数非零

important_features_indices = np.where(important_features)[0]
X_train_important, _, y_train_, _ = train_test_split(X[:, important_features_indices],
                                                    y,
                                                    ...)

model = LinearRegression()
model.fit(X_train_important, y_train_)
y_pred = model.predict(X_test[:, important_features_indices])

mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
mae = mean_absolute_error(y_test, y_pred)

performance_table = {"Important Features": list(important_features_indices),
                    "MSE": mse,
                    "R^2": r2,
                    "MAE": mae}
print(performance_table)