決策樹模型
決策樹是一種基本的分類和迴歸方法,本文主要討論用戶分類的決策樹。決策樹模型呈現樹樁結構,在分類問題中,它表示基於特徵對實例進行分類的過程。它可以認爲是if-then的規則的集合也可以認爲是定義在特徵空間與類空間上的條件概率分佈。
決策樹學習通常包括3個步驟:特徵選擇,決策樹的生成和決策樹的修剪。
- 優點:模型具有可讀性,解釋性較強,分類速度快,準確性高,可以處理連續和種類字段,不需要任何領域知識和參數假設,適合高維數據。
- 缺點:對於各類別樣本數量不一致的數據, 信息增益偏向於那些更多數值的特徵,容易過擬合,忽略屬性之間的相關性
決策樹的生成
劃分選擇
決策樹學習的關鍵是如何選擇最優劃分屬性。也就是選擇每一步具體選擇哪個屬性進行切分。我們希望決策樹的分支結點所包含的樣本儘可能屬於同一類別。即結點的“純度”越來越高。
假定當前樣本集合D中的第k類樣本所佔比例爲,
則D的信息熵定義爲:, 其中值越小,則D的純度越高。
而信息增益表示得知特徵A的信息後,使得類X的不確定性減少的程度,一般而言,信息增益越大,則意味着使用屬性a來進行劃分所獲得的“純度提升”越大。因此我們可以用信息增益來進行決策樹的劃分屬性選擇。
假定離散數學a有個可能的取值,若使用a來對樣本集進行切分,則會產生個分支結點,其中第v個分支結點包含了中所有在數學a上取值爲的樣本,記爲,我們可由上式計算得來的信息熵,再考慮到不同分支結點所包含的樣本數不同,給分支結點賦予不同的權重,即樣本越多的分支結點的影響越大,於是可計算出用數學a對樣本集進行劃分所獲得的"信息增益",
信息增益表達式:
Python代碼
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
from collections import Counter
import math
from math import log
import pprint
# 書上題目5.1
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'年齡', u'有工作', u'有自己的房子', u'信貸情況', u'類別']
# 返回數據集和每個維度的名稱
return datasets, labels
datasets, labels = create_data()
train_data = pd.DataFrame(datasets, columns=labels)
計算信息熵
# 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] = 0
label_count[label] += 1
ent = -sum([(p/data_length)*log(p/data_length, 2) for p in label_count.values()])
return ent
# 經驗條件熵
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
# 信息增益
def info_gain(ent, cond_ent):
return ent - cond_ent
def info_gain_train(datasets):
count = len(datasets[0]) - 1
ent = 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('特徵({}) - info_gain - {:.3f}'.format(labels[c], c_info_gain))
# 比較大小
best_ = max(best_feature, key=lambda x: x[-1])
return '特徵({})的信息增益最大,選擇爲根節點特徵'.format(labels[best_[0]])
# 輸入數據集進行測試
info_gain_train(np.array(datasets))
結果
利用ID3算法生成決策樹,例5.3
# 定義節點類 二叉樹
class Node:
def __init__(self, root=True, label=None, feature_name=None, feature=None):
self.root = root
self.label = label
self.feature_name = feature_name
self.feature = feature
self.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] = node
def predict(self, features):
if self.root is True:
return self.label
return self.tree[features[self.feature]].predict(features)
class DTree:
def __init__(self, epsilon=0.1):
self.epsilon = epsilon
self._tree = {}
# 熵
@staticmethod
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] = 0
label_count[label] += 1
ent = -sum([(p/data_length)*log(p/data_length, 2) for p in label_count.values()])
return ent
# 經驗條件熵
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
@staticmethod
def info_gain(ent, cond_ent):
return ent - cond_ent
def info_gain_train(self, datasets):
count = len(datasets[0]) - 1
ent = 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))
# 比較大小
best_ = max(best_feature, key=lambda x: x[-1])
return best_
def train(self, train_data):
"""
input:數據集D(DataFrame格式),特徵集A,閾值eta
output:決策樹T
"""
_, y_train, features = train_data.iloc[:, :-1], train_data.iloc[:, -1], train_data.columns[:-1]
# 1,若D中實例屬於同一類Ck,則T爲單節點樹,並將類Ck作爲結點的類標記,返回T
if len(y_train.value_counts()) == 1:
return Node(root=True,
label=y_train.iloc[0])
# 2, 若A爲空,則T爲單節點樹,將D中實例樹最大的類Ck作爲該節點的類標記,返回T
if len(features) == 0:
return Node(root=True, label=y_train.value_counts().sort_values(ascending=False).index[0])
# 3,計算最大信息增益 同5.1,Ag爲信息增益最大的特徵
max_feature, max_info_gain = self.info_gain_train(np.array(train_data))
max_feature_name = features[max_feature]
# 4,Ag的信息增益小於閾值eta,則置T爲單節點樹,並將D中是實例數最大的類Ck作爲該節點的類標記,返回T
if max_info_gain < self.epsilon:
return Node(root=True, label=y_train.value_counts().sort_values(ascending=False).index[0])
# 5,構建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().index
for f in feature_list:
sub_train_df = train_data.loc[train_data[max_feature_name] == f].drop([max_feature_name], axis=1)
# 6, 遞歸生成樹
sub_tree = self.train(sub_train_df)
node_tree.add_node(f, sub_tree)
# pprint.pprint(node_tree.tree)
return node_tree
def fit(self, train_data):
self._tree = self.train(train_data)
return self._tree
def 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)
顯示樹結構
tree
運行結果
預測
dt.predict(['老年', '否', '否', '一般'])
預測結果:
利用sklearn實現
# data
def create_data():
iris = load_iris()
df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['label'] = iris.target
df.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]
X, y = create_data()
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)
from sklearn.tree import DecisionTreeClassifier
from sklearn.tree import export_graphviz
import graphviz
進行擬合
clf = DecisionTreeClassifier()
clf.fit(X_train, y_train,)
擬合結果:
查看分數
clf.score(X_test, y_test)
顯示結果
輸出圖像結果:
tree_pic = export_graphviz(clf, out_file="mytree.pdf")
with open('mytree.pdf') as f:
dot_graph = f.read()