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

step1:计算香农熵

from math import logimport 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featuredef 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

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