8.李航机器学习-AdaBoost梯度提升算法python实现

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AdaBoost梯度提升算法

项目链接:https://github.com/Wchenguang/gglearn/blob/master/AdaBoost/李航机器学习讲解/AdaBoost.ipynb

算法步骤与原理

  1. 训练 m m m 个弱学习分类器,分类器有相同的接口
    G m ( x ) : X → { x 1 , x 2 …   } G_{m}(x) : \mathcal{X} \rightarrow\{x_{1},x_{2} \dots\} G m ( x ) : X { x 1 , x 2 }

  2. 假设数据有均匀的权值分布,即每个样本在分类器中作用相同,$ n $个实例的权重为
    D 1 = ( w 11 , ⋯   , w 1 i , ⋯   , w 1 N ) , w 1 i = 1 N , i = 1 , 2 , ⋯   , N D_{1}=\left(w_{11}, \cdots, w_{1 i}, \cdots, w_{1 N}\right), \quad w_{1 i}=\frac{1}{N}, \quad i=1,2, \cdots, N D 1 = ( w 1 1 , , w 1 i , , w 1 N ) , w 1 i = N 1 , i = 1 , 2 , , N
    对于 m m m 个分类器而言,有$ m \times n $个权重

  3. 进入迭代循环,在每一次循环中进行如下操作

    3.1 计算 m m m 个分类器在加权数据集上的分类错误率
    e m = P ( G m ( x i ) ≠ y i ) = ∑ C n ( x i ) ≠ y i w m i e_{m}=P\left(G_{m}\left(x_{i}\right) \neq y_{i}\right)=\sum_{C_{n}\left(x_{i}\right) \neq y_{i}} w_{m i} e m = P ( G m ( x i ) ̸ = y i ) = C n ( x i ) ̸ = y i w m i
    3.2 计算每个分类器的权重 a l p h a m alpha_{m} a l p h a m ,该权重表明,每个单独分类器在最终分类器中的重要程度
    α m = 1 2 log ⁡ 1 − e m e m \alpha_{m}=\frac{1}{2} \log \frac{1-e_{m}}{e_{m}} α m = 2 1 lo g e m 1 e m

    • 由上式可知,随着分类器的误差率的减小,其权重值越大

    3.3 更新数据集的权重分布

    D m + 1 = ( w m + 1 , 1 , ⋯   , w m + 1 , i , ⋯   , w m + 1 , N ) w m + 1 , i = w m i Z m exp ⁡ ( − α m ( y i = = G m ( x i ) ) ) , i = 1 , 2 , ⋯   , N \begin{array}{c}{D_{m+1}=\left(w_{m+1,1}, \cdots, w_{m+1, i}, \cdots, w_{m+1, N}\right)} \\ {w_{m+1, i}=\frac{w_{m i}}{Z_{m}} \exp \left(-\alpha_{m} (y_{i}== G_{m}\left(x_{i})\right)\right), \quad i=1,2, \cdots, N}\end{array} D m + 1 = ( w m + 1 , 1 , , w m + 1 , i , , w m + 1 , N ) w m + 1 , i = Z m w m i exp ( α m ( y i = = G m ( x i ) ) ) , i = 1 , 2 , , N
    Z m = ∑ i = 1 N w m i exp ⁡ ( − α m y i G m ( x i ) ) Z_{m}=\sum_{i=1}^{N} w_{m i} \exp \left(-\alpha_{m} y_{i} G_{m}\left(x_{i}\right)\right) Z m = i = 1 N w m i exp ( α m y i G m ( x i ) )

    • 由上式可知
      w m + 1 , i = { w m i Z m e − α m , G m ( x i ) = y i w m i Z m e α m , G m ( x i ) ≠ y i w_{m+1, i}=\left\{\begin{array}{ll}{\frac{w_{m i}}{Z_{m}} \mathrm{e}^{-\alpha_{m}},} & {G_{m}\left(x_{i}\right)=y_{i}} \\ {\frac{w_{m i}}{Z_{m}} \mathrm{e}^{\alpha_{m}},} & {G_{m}\left(x_{i}\right) \neq y_{i}}\end{array}\right. w m + 1 , i = { Z m w m i e α m , Z m w m i e α m , G m ( x i ) = y i G m ( x i ) ̸ = y i
      预测错误的实例,权重提升。预测正确的实例,权重下降。
            
              import numpy as np

class testClf:
    def __init__(self, thresold):
        self.thresold = thresold
        self.x = None
        self.y = None
    def fit(self, x, y):
        self.x = x
        self.y = y
        return self
    def predict(self, x):
        y = x.copy()
        less_index = np.where(y[:, 0] < self.thresold)
        greater_index = np.where(y[:, 0] > self.thresold)
        y[less_index] = 1
        y[greater_index] = -1
        return y
    def fit_predict(self, x, y):
        return self.fit(x, y).predict(x)
        
'''
test_x = np.arange(10).reshape(-1, 1)
test_y = np.array([1,1,1,-1,-1,-1,1,1,1,-1]).reshape(-1, 1)
tc = testClf(2.5)
print(tc.fit_predict(test_x, test_y))
'''
import numpy as np
import matplotlib.pyplot as plt

class AdaBoost:
    def __init__(self, clf_list, iteration_times):
        '''
        分类器需要有相同的fit,predict接口用于训练及预测
        '''
        self.clf_list = clf_list
        self.iteration_times = iteration_times
        self.x_weight_matrix = None
        self.clf_weight = None
    def _em(self, y_predict, y, x_weight):
        y_predict_flag = (y_predict != y).astype(int)
        return np.multiply(y_predict_flag, x_weight).sum()
    def _am(self, em):
        return np.log((1- em) / em) * 0.5
    def _update_x_weight(self, y_predict, y, am, x_weight):
        y_predict_flag = (y_predict == y).astype(int)
        y_predict_flag[np.where(y_predict_flag[:, 0] == 0)] = -1
        zm_array = np.multiply(np.exp(y_predict_flag * am * -1),
                                    x_weight)
        zm_array = zm_array / zm_array.sum()
        return zm_array
    def _fit_once(self, x, y, x_weight, clf_weight):
        for index in range(len(self.clf_list)):
            clf = self.clf_list[index]
            y_predict = clf.fit_predict(x, y)
            em = self._em(y_predict, y, x_weight)
            am = self._am(em)
            x_weight = self._update_x_weight(y_predict, y, am, x_weight)
            clf_weight[index] = am
            print('em', em, 'am', am)
            print('更新后权重')
            print(x_weight)
    def fit(self, x, y):
        m = len(self.clf_list)
        n = x.shape[0]
        if(0 == n or 0 == m):
            return
        self.x_weight = np.full((n, 1), 1/n)
        self.clf_weight = np.full((m, 1), 1/m)
        for i in range(self.iteration_times):
            self._fit_once(x, y, self.x_weight, self.clf_weight)
    def transform(self, x):
        if(self.clf_list == None or 0 == len(self.clf_list)):
            return None
        res = self.clf_weight[0] * self.clf_list[0].predict(x)
        for index in range(1, len(self.clf_list)):
            res += (self.clf_weight[index] * 
                            self.clf_list[index].predict(x))
        return res
test_x = np.arange(10).reshape(-1, 1)
test_y = np.array([1,1,1,-1,-1,-1,1,1,1,-1]).reshape(-1, 1)
adaboost = AdaBoost([testClf(2.5), testClf(8.5), testClf(5.5), ], 1)
adaboost.fit(test_x, test_y)
predict = adaboost.transform(test_x)
predict[np.where(predict[:, 0] < 0)] = -1
predict[np.where(predict[:, 0] >= 0)] = 1
print('predict')
print(predict)
print('truth')
print(test_y)

            
          
  • 与书中P140-P41结果相符
  • 书中分类器G3计算应为错误

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