本文链接： https://blog.csdn.net/juwikuang/article/details/100652089

Prerequisite 必备的知识

要理解GBDT，首先要理解DT和线性回归。可以参考一下文章：

《python机器学习手写算法系列——线性回归》

《python机器学习手写算法系列——决策树》

《python机器学习手写算法系列——梯度提升GBDT回归》

概述

GBDT全称Gradent Boosting Decision Tree, 也叫Gradient Boosting Tree。

GBDT是一种ensemble方法。所以如果你使用sklearn，可以在sklearn.ensemble下面找到它。我们知道一般ensemble方法的表现会比较好一点。

代码

和GBDT回归相比，分类要复杂一些。在回归的时候，我们把残差叠加到前一次的预测值上面。而在分类问题中，我们把gamma值叠加到log of odds上面。（odds是分类1的数量除以分类2的数量。）为了使得新的损失函数取值最小，经过微分和泰勒多项式(2nd Order Taylor Polynomial)。得到gamma的值如下：

在这里插入图片描述

得到gamma值以后，把gamma值和原来的log of odds相加，得到新的log of odds。


#now let's get started
n_times = 20
learning_rate=0.1
log_of_odds=np.zeros([n_times+1,n_samples])

residuals = np.zeros([n_times+1,n_samples])
loss = [0] * (n_times+1)
prediction = np.zeros([n_times+1,n_samples])
score = np.zeros([n_times+1])
#calculation
log_of_odds[0] = [log_of_odds0] * n_samples
prediction[0] = [probability0] * n_samples
score[0]=np.sum((prediction[0]>0.5)*1.0==y) / n_samples
residuals[0] = y - prediction[0]
loss[0] = loss0.sum()
trees = []
gamma_value = np.zeros([n_times+1,8])
gamma = np.zeros([n_times+1,n_samples])

for i in range(n_times):
    dt = DecisionTreeRegressor(max_depth=2)
    dt=dt.fit(X, residuals[i])
    tree = dt.tree_
    trees.append(tree)
    leaf_indeces=dt.apply(X)
    unique_leafs=np.unique(leaf_indeces)
    n_leaf=len(unique_leafs)
    #for leaf 1
    for ileaf in range(n_leaf):
        leaf_index=unique_leafs[ileaf]
        n_leaf=len(leaf_indeces[leaf_indeces==leaf_index])
        previous_probability = prediction[i][leaf_indeces==leaf_index]
        denominator = np.sum(previous_probability * (1-previous_probability))
        igamma = tree.value[ileaf+1][0][0] * n_leaf / denominator
        gamma_value[i][ileaf]=igamma

    gamma[i] = [gamma_value[i][np.where(unique_leafs==index)] for index in leaf_indeces]
    #next 
    log_of_odds[i+1] = log_of_odds[i] + learning_rate * gamma[i]

    prediction[i+1] = np.array([np.exp(odds)/(np.exp(odds)+1) for odds in log_of_odds[i+1]])
    score[i+1]=np.sum((prediction[i+1]>0.5)*1.0==y) / n_samples
    residuals[i+1] = y - prediction[i+1]
    loss[i+1]=np.sum(-y * log_of_odds[i+1] + np.log(1+np.exp(log_of_odds[i+1])))