AdaBoost算法之推导过程

import numpy as np

# 1. 示例数据
print("=== AdaBoost推导示例 ===")
# 10个样本
X = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
y = np.array([1, 1, 1, 1, 1, -1, -1, -1, -1, -1])  # 前5正，后5负

print(f"X: {X}")
print(f"y: {y}")

# 2. 初始化样本权重（均匀分布）
n = len(y)
w = np.ones(n) / n
print(f"\n初始权重: {w}")

# 3. 计算各切分点的错误率
print("\n=== 各切分点错误率 ===")
split_points = [0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5]

best_error = float('inf')
best_split = None

for sp in split_points:
    # 左边预测为正（1），右边预测为负（-1）
    y_pred = np.where(X <= sp, 1, -1)
    
    # 计算加权错误率
    error = np.sum(w * (y_pred != y))
    
    print(f"切分点{sp}: 错误率={error:.4f}")
    
    if error < best_error:
        best_error = error
        best_split = sp

print(f"\n最优切分点: {best_split}, 错误率: {best_error:.4f}")

# 4. 计算模型权重
alpha = 0.5 * np.log((1 - best_error) / best_error)
print(f"\n模型权重 alpha: {alpha:.4f}")

# 5. 更新样本权重
y_pred = np.where(X <= best_split, 1, -1)
correct = (y_pred == y)

# 预测正确：权重 * exp(-alpha)
# 预测错误：权重 * exp(alpha)
w_new = w * np.exp(-alpha * y * y_pred)

# 归一化
Z = np.sum(w_new)
w_normalized = w_new / Z

print(f"\n更新后权重:")
for i in range(n):
    status = "正确" if correct[i] else "错误"
    print(f"  样本{i}: 原权重={w[i]:.4f}, {status}, 新权重={w_normalized[i]:.4f}")

# 6. 完整AdaBoost过程
print("\n=== 完整AdaBoost过程 ===")
def adaboost_train(X, y, T=5):
    """AdaBoost训练过程"""
    n = len(y)
    w = np.ones(n) / n  # 初始化权重
    
    alphas = []
    classifiers = []
    
    for t in range(T):
        # 找最优切分点
        best_error = float('inf')
        best_split = None
        
        for sp in np.arange(0.5, len(X)-0.5, 1):
            y_pred = np.where(X <= sp, 1, -1)
            error = np.sum(w * (y_pred != y))
            
            if error < best_error:
                best_error = error
                best_split = sp
        
        # 计算模型权重
        alpha = 0.5 * np.log((1 - best_error) / (best_error + 1e-10))
        
        # 预测
        y_pred = np.where(X <= best_split, 1, -1)
        
        # 更新样本权重
        w = w * np.exp(-alpha * y * y_pred)
        w = w / np.sum(w)  # 归一化
        
        alphas.append(alpha)
        classifiers.append(best_split)
        
        print(f"第{t+1}轮: 切分点={best_split}, 错误率={best_error:.4f}, alpha={alpha:.4f}")
    
    return alphas, classifiers

# 运行
alphas, classifiers = adaboost_train(X, y, T=5)

# 7. 最终分类器
print("\n=== 最终分类器 ===")
print(f"模型权重: {alphas}")
print(f"切分点: {classifiers}")

# 预测
def adaboost_predict(x, alphas, classifiers):
    """AdaBoost预测"""
    result = 0
    for alpha, sp in zip(alphas, classifiers):
        h = 1 if x <= sp else -1
        result += alpha * h
    return 1 if result > 0 else -1

# 测试
print("\n=== 预测测试 ===")
test_points = [0, 2, 4, 6, 8]
for x in test_points:
    pred = adaboost_predict(x, alphas, classifiers)
    print(f"X={x}: 预测={'正样本' if pred == 1 else '负样本'}")