回归决策树和线性回归对比

import numpy as np
import matplotlib.pyplot as plt
from sklearn.tree import DecisionTreeRegressor
from sklearn.linear_model import LinearRegression

# 1. 生成数据
np.random.seed(42)
X = np.sort(5 * np.random.rand(100, 1), axis=0)
y = np.sin(X).ravel() + np.random.normal(0, 0.1, X.shape[0])

print("=== 数据信息 ===")
print(f"X形状: {X.shape}")
print(f"y形状: {y.shape}")
print(f"X范围: [{X.min():.2f}, {X.max():.2f}]")
print(f"y范围: [{y.min():.2f}, {y.max():.2f}]")

# 2. 线性回归
print("\n=== 线性回归 ===")
lr = LinearRegression()
lr.fit(X, y)
y_pred_lr = lr.predict(X)
print(f"系数w: {lr.coef_[0]:.4f}")
print(f"截距b: {lr.intercept_:.4f}")
print(f"R²: {lr.score(X, y):.4f}")

# 3. 回归决策树
print("\n=== 回归决策树 ===")
dt = DecisionTreeRegressor(max_depth=4, random_state=42)
dt.fit(X, y)
y_pred_dt = dt.predict(X)
print(f"R²: {dt.score(X, y):.4f}")

# 4. 对比平方损失
print("\n=== 平方损失对比 ===")
loss_lr = np.sum((y - y_pred_lr) ** 2)
loss_dt = np.sum((y - y_pred_dt) ** 2)
print(f"线性回归平方损失: {loss_lr:.4f}")
print(f"回归决策树平方损失: {loss_dt:.4f}")

# 5. 可视化对比
plt.figure(figsize=(15, 5))

# 训练数据
plt.subplot(1, 3, 1)
plt.scatter(X, y, color='blue', s=20, label='训练数据')
plt.xlabel('X')
plt.ylabel('y')
plt.title('原始数据')
plt.legend()
plt.grid(True)

# 线性回归
plt.subplot(1, 3, 2)
plt.scatter(X, y, color='blue', s=20, label='训练数据')
plt.plot(X, y_pred_lr, color='red', linewidth=2, label='线性回归')
plt.xlabel('X')
plt.ylabel('y')
plt.title(f'线性回归 (R²={lr.score(X, y):.4f})')
plt.legend()
plt.grid(True)

# 回归决策树
plt.subplot(1, 3, 3)
plt.scatter(X, y, color='blue', s=20, label='训练数据')
plt.plot(X, y_pred_dt, color='green', linewidth=2, label='回归决策树')
plt.xlabel('X')
plt.ylabel('y')
plt.title(f'回归决策树 (R²={dt.score(X, y):.4f})')
plt.legend()
plt.grid(True)

plt.tight_layout()
plt.show()

# 6. 预测新数据
print("\n=== 预测新数据 ===")
X_test = np.linspace(0, 5, 200).reshape(-1, 1)
y_pred_lr_new = lr.predict(X_test)
y_pred_dt_new = dt.predict(X_test)

# 7. 不同深度的回归决策树
print("\n=== 不同深度的回归决策树 ===")
depths = [1, 2, 4, 10]
plt.figure(figsize=(15, 10))

for i, depth in enumerate(depths, 1):
    dt_depth = DecisionTreeRegressor(max_depth=depth, random_state=42)
    dt_depth.fit(X, y)
    y_pred = dt_depth.predict(X_test)
    
    plt.subplot(2, 2, i)
    plt.scatter(X, y, color='blue', s=20, label='训练数据')
    plt.plot(X_test, y_pred, color='red', linewidth=2, label=f'深度={depth}')
    plt.xlabel('X')
    plt.ylabel('y')
    plt.title(f'回归决策树 (深度={depth}, R²={dt_depth.score(X, y):.4f})')
    plt.legend()
    plt.grid(True)

plt.tight_layout()
plt.show()

# 8. 手动计算平方损失示例
print("\n=== 手动计算平方损失示例 ===")
# 简单数据
X_simple = np.array([1, 2, 3, 4, 5]).reshape(-1, 1)
y_simple = np.array([5.56, 5.70, 5.91, 6.40, 6.80])

print(f"X: {X_simple.ravel()}")
print(f"y: {y_simple}")

# 在2.5切分
split = 2.5
left_mask = X_simple.ravel() <= split
right_mask = X_simple.ravel() > split

left_y = y_simple[left_mask]
right_y = y_simple[right_mask]

left_pred = left_y.mean()
right_pred = right_y.mean()

left_loss = np.sum((left_y - left_pred) ** 2)
right_loss = np.sum((right_y - right_pred) ** 2)
total_loss = left_loss + right_loss

print(f"\n在{split}切分:")
print(f"左侧: {left_y}, 均值={left_pred:.4f}, 损失={left_loss:.4f}")
print(f"右侧: {right_y}, 均值={right_pred:.4f}, 损失={right_loss:.4f}")
print(f"总损失: {total_loss:.4f}")