本文整理自《Python机器学习》
决策树可视为数据从顶向下划分的一种方法,通常为二叉树。
通过决策树算法,从树根开始,基于可获得的最大信息增益(Information Gain, IG)的特征对数据进行划分。
目标函数能够在每次划分时实现对信息增益的最大化,其定义如下:
IG ( D p , f ) = I ( D p ) − ∑ j = 1 m N j N p I ( D j ) \text{IG}(D_p,f)=I(D_p)-\sum_{j=1}^m\frac{N_j}{N_p}I(D_j) IG(Dp,f)=I(Dp)−j=1∑mNpNjI(Dj)
其中 f f f为将要进行划分的特征, D p D_p Dp与 D j D_j Dj分别为父节点和第 j j j个子节点, I I I为不纯度衡量标准, N p N_p Np为父节点中样本数量, N j N_j Nj为第 j j j个子节点中样本的数量。上式即表示,信息增益是父节点的不纯度与所有子节点不纯度总和之差,子节点的不纯度越低,信息增益越大。
对于二叉树(scikit-learn中的实现方式)有:
IG ( D p , a ) = I ( D p ) − N l e f t N p I ( D l e f t ) − N r i g h t N p I ( D r i g h t ) \text{IG}(D_p,a)=I(D_p)-\frac{N_{left}}{N_p}I(D_{left})-\frac{N_{right}}{N_p}I(D_{right}) IG(Dp,a)=I(Dp)−NpNleftI(Dleft)−NpNrightI(Dright)
二叉决策树主要有三类不纯度衡量标准。
熵(entropy):
I H ( t ) = − ∑ i = 1 c p ( i ∣ t ) log 2 p ( i ∣ t ) I_H(t)=-\sum_{i=1}^cp(i|t)\log_2p(i|t) IH(t)=−i=1∑cp(i∣t)log2p(i∣t)
基尼系数(Gini index):
I G ( t ) = 1 − ∑ i = 1 c p ( i ∣ t ) 2 I_G(t)=1-\sum_{i=1}^cp(i|t)^2 IG(t)=1−i=1∑cp(i∣t)2
误分类率(classification error)
I E = 1 − max { p ( i ∣ t ) } I_E=1-\max\{p(i|t)\} IE=1−max{p(i∣t)}
p ( i ∣ t ) p(i|t) p(i∣t)为特定节点 t t t中,属于类别 i i i的样本占特定节点 t t t中样本总数的比例。
实践中,基尼系数和熵会产生非常相似的效果,不会花大量时间用不纯度评判决策树的好坏,而尝试使用不同的剪枝算法,误分类率是对于剪枝方法的一个很好的准则但不建议用于决策树的构建。
样本属于类别1,概率介于[0,1]情况下三种不纯度的图像可由如下代码构建:
import matplotlib.pyplot as plt
import numpy as np
def gini(p):
return (p)*(1-(p)) + (1-p)*(1-(1-p))
def entropy(p):
return -p*np.log2(p)-(1-p)*np.log2((1-p))
def error(p):
return 1-np.max([p, 1-p])
x = np.arange(0, 1, 0.01)
giniVal=gini(x)
ent = [entropy(p) if p !=0 else None for p in x]
sc_ent = [e*0.5 if e else None for e in ent] # 按0.5比例缩放
err = [error(i) for i in x]
fig = plt.figure()
ax = plt.subplot(111)
for i, lab, ls, c in zip([ent, sc_ent, gini(x), err], ['Entropy', 'Entropy (scaled)', 'Gini Impurity', 'Missclassification Error'], ['-', '-', '--','-.'],['black','lightgray', 'red', 'green', 'cyan']):
line = ax.plot(x, i, label=lab, linestyle=ls, lw=2, color=c)
ax.legend(loc='upper center', bbox_to_anchor=(0.5,1.15), ncol=3, fancybox=True, shadow=False)
ax.axhline(y=0.5, linewidth=1, color='k', linestyle='--') # horizon line
ax.axhline(y=1.0, linewidth=1, color='k', linestyle='--')
plt.ylim([0, 1.1])
plt.xlabel('p(i=1)')
plt.ylabel('Impurity Index')
plt.show()
from sklearn import datasets
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from matplotlib.colors import ListedColormap
import matplotlib.pyplot as plt
import numpy as np
from sklearn.tree import DecisionTreeClassifier
from sklearn.tree import export_graphviz
iris = datasets.load_iris()
X = iris.data[:, [2, 3]]
y = iris.target
X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.3,random_state=0)
sc = StandardScaler()
sc.fit(X_train)
X_train_std = sc.transform(X_train)
X_test_std: object = sc.transform(X_test)
def plot_decision_regions(X, y, classifier, test_idx=None, resolution=0.02):
markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
cmap = ListedColormap(colors[:len(np.unique(y))])
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap)
plt.xlim(xx1.min(), xx1.max())
plt.ylim = (xx2.min(), xx2.max())
X_test, y_test = X[test_idx, :], y[test_idx]
for idx, cl in enumerate(np.unique(y)):
plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1], alpha=0.8, c=cmap(idx), marker=markers[idx], label=cl)
if test_idx:
X_test, y_test = X[test_idx, :], y[test_idx]
plt.scatter(X_test[:, 0], X_test[:, 1], c='black', alpha=0.8, linewidths=1, marker='o', s=10, label='test set')
tree = DecisionTreeClassifier(criterion='entropy', max_depth=3, random_state=0)
tree.fit(X_train, y_train)
X_combined=np.vstack((X_train, X_test))
y_combined=np.hstack((y_train, y_test))
plot_decision_regions(X_combined, y_combined,classifier=tree, test_idx=range(105, 150))
plt.xlabel('petal length [cm]')
plt.ylabel('petal width [cm]')
plt.legend(loc='upper left')
plt.show()
export_graphviz(tree, out_file='tree.dot',feature_names=['petal length', 'petal width']) # 导出为dot文件
分类结果如下:
对于输出的tree.dot文件,我们可以通过GraphViz在命令行中输入指令
dot -Tpng tree.dot -o tree.png