本课程是中国大学慕课《机器学习》的“决策树”章节的课后代码。

课程地址：

https://www.icourse163.org/course/WZU-1464096179

课程完整代码：

https://github.com/fengdu78/WZU-machine-learning-course

代码修改并注释：黄海广，haiguang2000@wzu.edu.cn

机器学习练习7 决策树

代码修改并注释：黄海广，haiguang2000@wzu.edu.cn

1．分类决策树模型是表示基于特征对实例进行分类的树形结构。决策树可以转换成一个if-then规则的集合，也可以看作是定义在特征空间划分上的类的条件概率分布。

2．决策树学习旨在构建一个与训练数据拟合很好，并且复杂度小的决策树。因为从可能的决策树中直接选取最优决策树是NP完全问题。现实中采用启发式方法学习次优的决策树。

决策树学习算法包括3部分：特征选择、树的生成和树的剪枝。常用的算法有ID3、 C4.5和CART。

3．特征选择的目的在于选取对训练数据能够分类的特征。特征选择的关键是其准则。常用的准则如下：

（1）样本集合对特征的信息增益（ID3）

其中，是数据集的熵，是数据集的熵，是数据集对特征的条件熵。是中特征取第个值的样本子集，是中属于第类的样本子集。是特征取 值的个数，是类的个数。

（2）样本集合对特征的信息增益比（C4.5）

其中，是信息增益，是数据集的熵。

（3）样本集合的基尼指数（CART）

特征条件下集合的基尼指数：

4．决策树的生成。通常使用信息增益最大、信息增益比最大或基尼指数最小作为特征选择的准则。决策树的生成往往通过计算信息增益或其他指标，从根结点开始，递归地产生决策树。这相当于用信息增益或其他准则不断地选取局部最优的特征，或将训练集分割为能够基本正确分类的子集。

5．决策树的剪枝。由于生成的决策树存在过拟合问题，需要对它进行剪枝，以简化学到的决策树。决策树的剪枝，往往从已生成的树上剪掉一些叶结点或叶结点以上的子树，并将其父结点或根结点作为新的叶结点，从而简化生成的决策树。

import numpy as np
import pandas as pd
import math
from math import log

创建数据

def create_data():datasets = [['青年', '否', '否', '一般', '否'],['青年', '否', '否', '好', '否'],['青年', '是', '否', '好', '是'],['青年', '是', '是', '一般', '是'],['青年', '否', '否', '一般', '否'],['中年', '否', '否', '一般', '否'],['中年', '否', '否', '好', '否'],['中年', '是', '是', '好', '是'],['中年', '否', '是', '非常好', '是'],['中年', '否', '是', '非常好', '是'],['老年', '否', '是', '非常好', '是'],['老年', '否', '是', '好', '是'],['老年', '是', '否', '好', '是'],['老年', '是', '否', '非常好', '是'],['老年', '否', '否', '一般', '否'],]labels = [u'年龄', u'有工作', u'有自己的房子', u'信贷情况', u'类别']# 返回数据集和每个维度的名称return datasets, labels

datasets, labels = create_data()

train_data = pd.DataFrame(datasets, columns=labels)

train_data

熵

def calc_ent(datasets):data_length = len(datasets)label_count = {}for i in range(data_length):label = datasets[i][-1]if label not in label_count:label_count[label] = 0label_count[label] += 1ent = -sum([(p / data_length) * log(p / data_length, 2)for p in label_count.values()])return ent

条件熵

def cond_ent(datasets, axis=0):data_length = len(datasets)feature_sets = {}for i in range(data_length):feature = datasets[i][axis]if feature not in feature_sets:feature_sets[feature] = []feature_sets[feature].append(datasets[i])cond_ent = sum([(len(p) / data_length) * calc_ent(p)for p in feature_sets.values()])return cond_ent

calc_ent(datasets)

0.9709505944546686

信息增益

def info_gain(ent, cond_ent):return ent - cond_ent

def info_gain_train(datasets):count = len(datasets[0]) - 1ent = calc_ent(datasets)best_feature = []for c in range(count):c_info_gain = info_gain(ent, cond_ent(datasets, axis=c))best_feature.append((c, c_info_gain))print('特征({}) 的信息增益为： {:.3f}'.format(labels[c], c_info_gain))# 比较大小best_ = max(best_feature, key=lambda x: x[-1])return '特征({})的信息增益最大，选择为根节点特征'.format(labels[best_[0]])

info_gain_train(np.array(datasets))

特征(年龄) 的信息增益为：0.083
特征(有工作) 的信息增益为：0.324
特征(有自己的房子) 的信息增益为：0.420
特征(信贷情况) 的信息增益为：0.363'特征(有自己的房子)的信息增益最大，选择为根节点特征'

利用ID3算法生成决策树

# 定义节点类 二叉树
class Node:def __init__(self, root=True, label=None, feature_name=None, feature=None):self.root = rootself.label = labelself.feature_name = feature_nameself.feature = featureself.tree = {}self.result = {'label:': self.label,'feature': self.feature,'tree': self.tree}def __repr__(self):return '{}'.format(self.result)def add_node(self, val, node):self.tree[val] = nodedef predict(self, features):if self.root is True:return self.labelreturn self.tree[features[self.feature]].predict(features)class DTree:def __init__(self, epsilon=0.1):self.epsilon = epsilonself._tree = {}# 熵@staticmethoddef calc_ent(datasets):data_length = len(datasets)label_count = {}for i in range(data_length):label = datasets[i][-1]if label not in label_count:label_count[label] = 0label_count[label] += 1ent = -sum([(p / data_length) * log(p / data_length, 2)for p in label_count.values()])return ent# 经验条件熵def cond_ent(self, datasets, axis=0):data_length = len(datasets)feature_sets = {}for i in range(data_length):feature = datasets[i][axis]if feature not in feature_sets:feature_sets[feature] = []feature_sets[feature].append(datasets[i])cond_ent = sum([(len(p) / data_length) * self.calc_ent(p)for p in feature_sets.values()])return cond_ent# 信息增益@staticmethoddef info_gain(ent, cond_ent):return ent - cond_entdef info_gain_train(self, datasets):count = len(datasets[0]) - 1ent = self.calc_ent(datasets)best_feature = []for c in range(count):c_info_gain = self.info_gain(ent, self.cond_ent(datasets, axis=c))best_feature.append((c, c_info_gain))# 比较大小best_ = max(best_feature, key=lambda x: x[-1])return best_def train(self, train_data):"""input:数据集D(DataFrame格式)，特征集A，阈值etaoutput:决策树T"""_, y_train, features = train_data.iloc[:, :-1], train_data.iloc[:,-1], train_data.columns[:-1]# 1,若D中实例属于同一类Ck，则T为单节点树，并将类Ck作为结点的类标记，返回Tif len(y_train.value_counts()) == 1:return Node(root=True, label=y_train.iloc[0])# 2, 若A为空，则T为单节点树，将D中实例树最大的类Ck作为该节点的类标记，返回Tif len(features) == 0:return Node(root=True,label=y_train.value_counts().sort_values(ascending=False).index[0])# 3,计算最大信息增益 同5.1,Ag为信息增益最大的特征max_feature, max_info_gain = self.info_gain_train(np.array(train_data))max_feature_name = features[max_feature]# 4,Ag的信息增益小于阈值eta,则置T为单节点树，并将D中是实例数最大的类Ck作为该节点的类标记，返回Tif max_info_gain < self.epsilon:return Node(root=True,label=y_train.value_counts().sort_values(ascending=False).index[0])# 5,构建Ag子集node_tree = Node(root=False, feature_name=max_feature_name, feature=max_feature)feature_list = train_data[max_feature_name].value_counts().indexfor f in feature_list:sub_train_df = train_data.loc[train_data[max_feature_name] ==f].drop([max_feature_name], axis=1)# 6, 递归生成树sub_tree = self.train(sub_train_df)node_tree.add_node(f, sub_tree)# pprint.pprint(node_tree.tree)return node_treedef fit(self, train_data):self._tree = self.train(train_data)return self._treedef predict(self, X_test):return self._tree.predict(X_test)

datasets, labels = create_data()
data_df = pd.DataFrame(datasets, columns=labels)
dt = DTree()
tree = dt.fit(data_df)

tree

{'label:': None, 'feature': 2, 'tree': {'否': {'label:': None, 'feature': 1, 'tree': {'否': {'label:': '否', 'feature': None, 'tree': {}}, '是': {'label:': '是', 'feature': None, 'tree': {}}}}, '是': {'label:': '是', 'feature': None, 'tree': {}}}}

dt.predict(['老年', '否', '否', '一般'])

'否'

Scikit-learn实例

from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from collections import Counter

使用Iris数据集，我们可以构建如下树：

# data
def create_data():iris = load_iris()df = pd.DataFrame(iris.data, columns=iris.feature_names)df['label'] = iris.targetdf.columns = ['sepal length', 'sepal width', 'petal length', 'petal width', 'label']data = np.array(df.iloc[:100, [0, 1, -1]])# print(data)return data[:, :2], data[:, -1],iris.feature_names[0:2]X, y,feature_name= create_data()
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)

决策树分类

from sklearn.tree import DecisionTreeClassifier
from sklearn.tree import export_graphviz
import graphviz
from sklearn import treeclf = DecisionTreeClassifier()
clf.fit(X_train, y_train,)clf.score(X_test, y_test)

0.9666666666666667

一旦经过训练，就可以用 plot_tree函数绘制树：

tree.plot_tree(clf)

[Text(197.83636363636364, 195.696, 'X[0] <= 5.45\ngini = 0.5\nsamples = 70\nvalue = [36, 34]'),Text(121.74545454545455, 152.208, 'X[1] <= 2.8\ngini = 0.157\nsamples = 35\nvalue = [32, 3]'),Text(60.872727272727275, 108.72, 'X[0] <= 4.75\ngini = 0.444\nsamples = 3\nvalue = [1, 2]'),Text(30.436363636363637, 65.232, 'gini = 0.0\nsamples = 1\nvalue = [1, 0]'),Text(91.30909090909091, 65.232, 'gini = 0.0\nsamples = 2\nvalue = [0, 2]'),Text(182.61818181818182, 108.72, 'X[0] <= 5.3\ngini = 0.061\nsamples = 32\nvalue = [31, 1]'),Text(152.1818181818182, 65.232, 'gini = 0.0\nsamples = 29\nvalue = [29, 0]'),Text(213.05454545454546, 65.232, 'X[1] <= 3.2\ngini = 0.444\nsamples = 3\nvalue = [2, 1]'),Text(182.61818181818182, 21.744, 'gini = 0.0\nsamples = 1\nvalue = [0, 1]'),Text(243.4909090909091, 21.744, 'gini = 0.0\nsamples = 2\nvalue = [2, 0]'),Text(273.92727272727274, 152.208, 'X[1] <= 3.5\ngini = 0.202\nsamples = 35\nvalue = [4, 31]'),Text(243.4909090909091, 108.72, 'gini = 0.0\nsamples = 31\nvalue = [0, 31]'),Text(304.3636363636364, 108.72, 'gini = 0.0\nsamples = 4\nvalue = [4, 0]')]

也可以导出树

tree_pic = export_graphviz(clf, out_file="mytree.pdf")
with open('mytree.pdf') as f:dot_graph = f.read()

graphviz.Source(dot_graph)

或者，还可以使用函数 export_text以文本格式导出树。此方法不需要安装外部库，而且更紧凑：

from sklearn.tree import export_text

r = export_text(clf,feature_name)

print(r)

|--- sepal width (cm) <= 3.15
|   |--- sepal length (cm) <= 4.95
|   |   |--- sepal width (cm) <= 2.65
|   |   |   |--- class: 1.0
|   |   |--- sepal width (cm) >  2.65
|   |   |   |--- class: 0.0
|   |--- sepal length (cm) >  4.95
|   |   |--- class: 1.0
|--- sepal width (cm) >  3.15
|   |--- sepal length (cm) <= 5.85
|   |   |--- class: 0.0
|   |--- sepal length (cm) >  5.85
|   |   |--- class: 1.0

决策树回归

import numpy as np
from sklearn.tree import DecisionTreeRegressor
import matplotlib.pyplot as plt

# Create a random dataset
rng = np.random.RandomState(1)
X = np.sort(5 * rng.rand(80, 1), axis=0)
y = np.sin(X).ravel()
y[::5] += 3 * (0.5 - rng.rand(16))

# Fit regression model
regr_1 = DecisionTreeRegressor(max_depth=2)
regr_2 = DecisionTreeRegressor(max_depth=5)
regr_1.fit(X, y)
regr_2.fit(X, y)# Predict
X_test = np.arange(0.0, 5.0, 0.01)[:, np.newaxis]
y_1 = regr_1.predict(X_test)
y_2 = regr_2.predict(X_test)# Plot the results
plt.figure()
plt.scatter(X, y, s=20, edgecolor="black", c="darkorange", label="data")
plt.plot(X_test, y_1, color="cornflowerblue", label="max_depth=2", linewidth=2)
plt.plot(X_test, y_2, color="yellowgreen", label="max_depth=5", linewidth=2)
plt.xlabel("data")
plt.ylabel("target")
plt.title("Decision Tree Regression")
plt.legend()
plt.show()

决策树调参

# 导入库
from sklearn.tree import DecisionTreeClassifier
from sklearn import datasets
from sklearn.model_selection import train_test_split
import matplotlib.pyplot as plt
from sklearn.model_selection import GridSearchCV
from sklearn.tree import DecisionTreeRegressor
from sklearn import metrics

# 导入数据集
X = datasets.load_iris()  # 以全部字典形式返回,有data,target,target_names三个键
data = X.data
target = X.target
name = X.target_names
x, y = datasets.load_iris(return_X_y=True)  # 能一次性取前2个
print(x.shape, y.shape)

(150, 4) (150,)

# 数据分为训练集和测试集
x_train, x_test, y_train, y_test = train_test_split(x,y,test_size=0.2,random_state=100)

# 用GridSearchCV寻找最优参数（字典）
param = {'criterion': ['gini'],'max_depth': [30, 50, 60, 100],'min_samples_leaf': [2, 3, 5, 10],'min_impurity_decrease': [0.1, 0.2, 0.5]
}
grid = GridSearchCV(DecisionTreeClassifier(), param_grid=param, cv=6)
grid.fit(x_train, y_train)
print('最优分类器:', grid.best_params_, '最优分数:', grid.best_score_)  # 得到最优的参数和分值

最优分类器: {'criterion': 'gini', 'max_depth': 30, 'min_impurity_decrease': 0.2, 'min_samples_leaf': 3} 最优分数: 0.9416666666666665

参考

• https://github.com/fengdu78/lihang-code

• 《统计学习方法》，清华大学出版社，李航著，2019年出版

• https://scikit-learn.org