在周志华的西瓜书和李航的统计机器学习中对决策树ID3算法都有很详细的解释，如何实现呢？核心点有如下几个步骤，（参考自机器学习实战）

step1:计算香农熵

from math import log
import operator


# 计算香农熵
def calculate_entropy(data):
    label_counts = {}
    for feature_data in data:
        laber = feature_data[-1]  # 最后一行是laber
        if laber not in label_counts.keys():
            label_counts[laber] = 0
        label_counts[laber] += 1

    count = len(data)
    entropy = 0.0

    for key in label_counts:
        prob = float(label_counts[key]) / count
        entropy -= prob * log(prob, 2)
    return entropy

step2.计算某个feature的信息增益的方法

# 计算某个feature的信息增益
# index:要计算信息增益的feature 对应的在data 的第几列
# data 的香农熵
def calculate_relative_entropy(data, index, entropy):
    feat_list = [number[index] for number in data]  # 得到某个特征下所有值（某列）
    uniqual_vals = set(feat_list)
    new_entropy = 0
    for value in uniqual_vals:
        sub_data = split_data(data, index, value)
        prob = len(sub_data) / float(len(data))  
        new_entropy += prob * calculate_entropy(sub_data)  # 对各子集香农熵求和
    relative_entropy = entropy - new_entropy  # 计算信息增益
    return relative_entropy

step3.选择最大信息增益的feature

# 选择最大信息增益的feature
def choose_max_relative_entropy(data):
    num_feature = len(data[0]) - 1
    base_entropy = calculate_entropy(data)#香农熵
    best_infor_gain = 0
    best_feature = -1
    for i in range(num_feature):
        info_gain=calculate_relative_entropy(data, i, base_entropy)
        #最大信息增益
        if (info_gain > best_infor_gain):
            best_infor_gain = info_gain
            best_feature = i

    return best_feature

step4.构建决策树

def create_decision_tree(data, labels):
    class_list=[example[-1] for example in data]
    # 类别相同，停止划分
    if class_list.count(class_list[-1]) == len(class_list):
        return class_list[-1]
    # 判断是否遍历完所有的特征时返回个数最多的类别
    if len(data[0]) == 1:
        return most_class(class_list)
    # 按照信息增益最高选取分类特征属性
    best_feat = choose_max_relative_entropy(data)
    best_feat_lable = labels[best_feat] # 该特征的label
    decision_tree = {best_feat_lable: {}} # 构建树的字典
    del(labels[best_feat]) # 从labels的list中删除该label
    feat_values = [example[best_feat] for example in data]
    unique_values = set(feat_values)
    for value in unique_values:
        sub_lables=labels[:]
        # 构建数据的子集合，并进行递归
        decision_tree[best_feat_lable][value] = create_decision_tree(split_data(data, best_feat, value), sub_lables)
    return decision_tree

在构建决策树的过程中会用到两个工具方法：

# 当遍历完所有的特征时返回个数最多的类别
def most_class(classList):
    class_count={}
    for vote in classList:
        if vote not in class_count.keys():class_count[vote]=0
        class_count[vote]+=1
    sorted_class_count=sorted(class_count.items,key=operator.itemgetter(1),reversed=True)
    return sorted_class_count[0][0]
    
# 工具函数输入三个变量（待划分的数据集，特征，分类值）返回不含划分特征的子集
def split_data(data, axis, value):
    ret_data=[]
    for feat_vec in data:
        if feat_vec[axis]==value :
            reduce_feat_vec=feat_vec[:axis]
            reduce_feat_vec.extend(feat_vec[axis+1:])
            ret_data.append(reduce_feat_vec)
    return ret_data