本課程是中國大學(xué)慕課《機(jī)器學(xué)習(xí)》的“決策樹”章節(jié)的課后代碼。

課程地址：

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

課程完整代碼：

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

代碼修改并注釋：黃海廣，[email protected]

機(jī)器學(xué)習(xí)練習(xí)7 決策樹

代碼修改并注釋：黃海廣，[email protected]

1．分類決策樹模型是表示基于特征對實例進(jìn)行分類的樹形結(jié)構(gòu)。決策樹可以轉(zhuǎn)換成一個if-then規(guī)則的集合，也可以看作是定義在特征空間劃分上的類的條件概率分布。

2．決策樹學(xué)習(xí)旨在構(gòu)建一個與訓(xùn)練數(shù)據(jù)擬合很好，并且復(fù)雜度小的決策樹。因為從可能的決策樹中直接選取最優(yōu)決策樹是NP完全問題?，F(xiàn)實中采用啟發(fā)式方法學(xué)習(xí)次優(yōu)的決策樹。

決策樹學(xué)習(xí)算法包括3部分：特征選擇、樹的生成和樹的剪枝。常用的算法有ID3、 C4.5和CART。

3．特征選擇的目的在于選取對訓(xùn)練數(shù)據(jù)能夠分類的特征。特征選擇的關(guān)鍵是其準(zhǔn)則。常用的準(zhǔn)則如下：

（1）樣本集合對特征的信息增益（ID3）

其中，是數(shù)據(jù)集的熵，是數(shù)據(jù)集的熵，是數(shù)據(jù)集對特征的條件熵。是中特征取第個值的樣本子集，是中屬于第類的樣本子集。是特征取 值的個數(shù)，是類的個數(shù)。

（2）樣本集合對特征的信息增益比（C4.5）

其中，是信息增益，是數(shù)據(jù)集的熵。

（3）樣本集合的基尼指數(shù)（CART）

特征條件下集合的基尼指數(shù)：

4．決策樹的生成。通常使用信息增益最大、信息增益比最大或基尼指數(shù)最小作為特征選擇的準(zhǔn)則。決策樹的生成往往通過計算信息增益或其他指標(biāo)，從根結(jié)點開始，遞歸地產(chǎn)生決策樹。這相當(dāng)于用信息增益或其他準(zhǔn)則不斷地選取局部最優(yōu)的特征，或?qū)⒂?xùn)練集分割為能夠基本正確分類的子集。

5．決策樹的剪枝。由于生成的決策樹存在過擬合問題，需要對它進(jìn)行剪枝，以簡化學(xué)到的決策樹。決策樹的剪枝，往往從已生成的樹上剪掉一些葉結(jié)點或葉結(jié)點以上的子樹，并將其父結(jié)點或根結(jié)點作為新的葉結(jié)點，從而簡化生成的決策樹。

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

創(chuàng)建數(shù)據(jù)

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(shù)據(jù)集和每個維度的名稱
????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]?=?0
????????label_count[label]?+=?1
????ent?=?-sum([(p?/?data_length)?*?log(p?/?data_length,?2)
????????????????for?p?in?label_count.values()])
????return?ent

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

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])?-?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('特征({})?的信息增益為：?{:.3f}'.format(labels[c],?c_info_gain))
????#?比較大小
????best_?=?max(best_feature,?key=lambda?x:?x[-1])
????return?'特征({})的信息增益最大，選擇為根節(jié)點特征'.format(labels[best_[0]])

info_gain_train(np.array(datasets))

特征(年齡) 的信息增益為：0.083
特征(有工作) 的信息增益為：0.324
特征(有自己的房子) 的信息增益為：0.420
特征(信貸情況) 的信息增益為：0.363


'特征(有自己的房子)的信息增益最大，選擇為根節(jié)點特征'

利用ID3算法生成決策樹

#?定義節(jié)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(jīng)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(shù)據(jù)集D(DataFrame格式)，特征集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中實例屬于同一類Ck，則T為單節(jié)點樹，并將類Ck作為結(jié)點的類標(biāo)記，返回T
????????if?len(y_train.value_counts())?==?1:
????????????return?Node(root=True,?label=y_train.iloc[0])

????????#?2,?若A為空，則T為單節(jié)點樹，將D中實例樹最大的類Ck作為該節(jié)點的類標(biāo)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為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的信息增益小于閾值eta,則置T為單節(jié)點樹，并將D中是實例數(shù)最大的類Ck作為該節(jié)點的類標(biāo)記，返回T
????????if?max_info_gain?????????????return?Node(
????????????????root=True,
????????????????label=y_train.value_counts().sort_values(
????????????????????ascending=False).index[0])

????????#?5,構(gòu)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)

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實例

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

使用Iris數(shù)據(jù)集，我們可以構(gòu)建如下樹：

#?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],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)

決策樹分類

from?sklearn.tree?import?DecisionTreeClassifier
from?sklearn.tree?import?export_graphviz
import?graphviz
from?sklearn?import?tree

clf?=?DecisionTreeClassifier()
clf.fit(X_train,?y_train,)

clf.score(X_test,?y_test)

0.9666666666666667

一旦經(jīng)過訓(xùn)練，就可以用 plot_tree函數(shù)繪制樹：

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]')]

也可以導(dǎo)出樹

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，還可以使用函數(shù) export_text以文本格式導(dǎo)出樹。此方法不需要安裝外部庫，而且更緊湊：

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

決策樹回歸

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()

決策樹調(diào)參

#?導(dǎo)入庫
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

#?導(dǎo)入數(shù)據(jù)集
X?=?datasets.load_iris()??#?以全部字典形式返回,有data,target,target_names三個鍵
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個
print(x.shape,?y.shape)

(150, 4) (150,)

#?數(shù)據(jù)分為訓(xùn)練集和測試集
x_train,?x_test,?y_train,?y_test?=?train_test_split(x,
????????????????????????????????????????????????????y,
????????????????????????????????????????????????????test_size=0.2,
????????????????????????????????????????????????????random_state=100)

#?用GridSearchCV尋找最優(yōu)參數(shù)（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('最優(yōu)分類器:',?grid.best_params_,?'最優(yōu)分?jǐn)?shù):',?grid.best_score_)??#?得到最優(yōu)的參數(shù)和分值

最優(yōu)分類器: {'criterion': 'gini', 'max_depth': 30, 'min_impurity_decrease': 0.2, 'min_samples_leaf': 3} 最優(yōu)分?jǐn)?shù): 0.9416666666666665

參考

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

• 《統(tǒng)計學(xué)習(xí)方法》，清華大學(xué)出版社，李航著，2019年出版

• https://scikit-learn.org