总误差 = 偏差² + 方差 + 不可避免的噪声

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression, Ridge
from sklearn.metrics import mean_squared_error
from sklearn.pipeline import make_pipeline

# 1. 生成带噪声的数据集
np.random.seed(42)
X = np.sort(5 * np.random.rand(100, 1), axis=0)
y = np.sin(X).ravel() + np.random.normal(0, 0.2, 100)  # 真实模式 + 噪声

# 2. 划分训练集和测试集
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

# 3. 创建不同复杂度的模型
degrees = [1, 4, 15]  # 低中高复杂度
models = []

plt.figure(figsize=(12, 4))

for i, degree in enumerate(degrees):
    # 创建多项式回归模型
    model = make_pipeline(PolynomialFeatures(degree), LinearRegression())
    model.fit(X_train, y_train)
    models.append(model)
    
    # 预测
    y_train_pred = model.predict(X_train)
    y_test_pred = model.predict(X_test)
    
    # 计算误差
    train_error = mean_squared_error(y_train, y_train_pred)
    test_error = mean_squared_error(y_test, y_test_pred)
    
    # 可视化
    plt.subplot(1, 3, i + 1)
    X_plot = np.linspace(0, 5, 100).reshape(-1, 1)
    y_plot = model.predict(X_plot)
    
    plt.scatter(X_train, y_train, color='blue', s=10, label='训练数据')
    plt.scatter(X_test, y_test, color='green', s=10, label='测试数据')
    plt.plot(X_plot, y_plot, color='red', linewidth=2, label=f'模型 (度={degree})')
    plt.plot(X_plot, np.sin(X_plot), color='black', linestyle='--', alpha=0.5, label='真实函数')
    
    plt.title(f'多项式度={degree}\n训练MSE: {train_error:.3f}, 测试MSE: {test_error:.3f}')
    plt.xlabel('X')
    plt.ylabel('y')
    plt.legend()
    plt.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# 4. 模拟偏差-方差权衡的直观展示
print("=" * 60)
print("模型复杂度分析:")
print("=" * 60)

# 使用多个随机数据集模拟方差
n_datasets = 50
train_errors = {d: [] for d in degrees}
test_errors = {d: [] for d in degrees}

for _ in range(n_datasets):
    # 生成新数据集
    X_new = np.sort(5 * np.random.rand(100, 1), axis=0)
    y_new = np.sin(X_new).ravel() + np.random.normal(0, 0.2, 100)
    X_train_new, X_test_new, y_train_new, y_test_new = train_test_split(
        X_new, y_new, test_size=0.3, random_state=42
    )
    
    for degree in degrees:
        model = make_pipeline(PolynomialFeatures(degree), LinearRegression())
        model.fit(X_train_new, y_train_new)
        
        train_errors[degree].append(mean_squared_error(y_train_new, model.predict(X_train_new)))
        test_errors[degree].append(mean_squared_error(y_test_new, model.predict(X_test_new)))

print("\n不同复杂度模型的平均误差 (50次随机数据集):")
print(f"{'模型复杂度':<15} {'平均训练误差':<15} {'平均测试误差':<15} {'诊断':<10}")
print("-" * 55)

for degree in degrees:
    avg_train = np.mean(train_errors[degree])
    avg_test = np.mean(test_errors[degree])
    
    if avg_train > 0.5 and avg_test > 0.5:
        diagnosis = "高偏差"
    elif avg_train < 0.1 and avg_test > 0.5:
        diagnosis = "高方差"
    else:
        diagnosis = "平衡"
    
    print(f"多项式度={degree:<10} {avg_train:<15.3f} {avg_test:<15.3f} {diagnosis:<10}")