机器学习 | 决策树 Decision Tree

—— 分而治之，逐个击破

??????????????? 把特征空间划分区域

??????????????? 每个区域拟合简单模型

??????????????? 分级分类决策

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1、核心思想和原理

• 
• 举例：
  • 特征选择、节点分类、阈值确定
  • 

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2、信息嫡

???????

????????熵本身代表不确定性，是不确定性的一种度量。

??????? 熵越大，不确定性越高，信息量越高。

???????

??????? 为什么用log？—— 两种解释，可能性的增长呈指数型；log可以将乘法变为加减法。

????????

??????? 联合熵 的物理意义：观察一个多变量系统获得的信息量。

??????? 条件熵 的物理意义：知道其中一个变量的信息后，另一个变量的信息量。

??????????????? 给定了训练样本 X ，分类标签中包含的信息量是什么。

?????????

????????

??????? 信息增益（互信息）

??????????????? 代表了一个特征能够为一个系统带来多少信息。

????????

????????

??????? 熵的分类

????????

????????

??????? 熵的本质：特殊的衡量分布的混乱程度与分散程度的距离

????????

????????

? ? ? ? 二分类信息熵：

二分类信息熵

import numpy as np
import matplotlib.pyplot as plt

def entropy(p):
    return -(p * np.log2(p) + (1 - p) * np.log2(1 - p))

plot_x = np.linspace(0.001, 0.999, 100)
plt.plot(plot_x, entropy(plot_x))
plt.show()

????????

?

???????? 决策树的本质

????????

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?3、决策树分类代码实现

?

数据集

from sklearn.datasets import load_iris

iris = load_iris()
x = iris.data[:, 1:3]
y = iris.target

plt.scatter(x[:,0], x[:,1], c = y)
plt.show()

?

3.1、sklearn中的决策树

from sklearn.tree import DecisionTreeClassifier

clf = DecisionTreeClassifier(max_depth=2, criterion='entropy')
clf.fit(x, y)

DecisionTreeClassifier

DecisionTreeClassifier(criterion='entropy', max_depth=2)

决策边界绘制的代码：?

def decision_boundary_plot(X, y, clf):
    axis_x1_min, axis_x1_max = X[:,0].min() - 1, X[:,0].max() + 1
    axis_x2_min, axis_x2_max = X[:,1].min() - 1, X[:,1].max() + 1
    
    x1, x2 = np.meshgrid( np.arange(axis_x1_min,axis_x1_max, 0.01) , np.arange(axis_x2_min,axis_x2_max, 0.01))
    z = clf.predict(np.c_[x1.ravel(),x2.ravel()])
    z = z.reshape(x1.shape)
    
    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#F5B9EF','#BBFFBB','#F9F9CB'])

    plt.contourf(x1, x2, z, cmap=custom_cmap)
    plt.scatter(X[:,0], X[:,1], c=y)
    plt.show()

decision_boundary_plot(x, y, clf)

from sklearn.tree import plot_tree
plot_tree(clf)

[Text(0.4, 0.8333333333333334, 'X[1] <= 2.45\nentropy = 1.585\nsamples = 150\nvalue = [50, 50, 50]'),
 Text(0.2, 0.5, 'entropy = 0.0\nsamples = 50\nvalue = [50, 0, 0]'),
 Text(0.6, 0.5, 'X[1] <= 4.75\nentropy = 1.0\nsamples = 100\nvalue = [0, 50, 50]'),
 Text(0.4, 0.16666666666666666, 'entropy = 0.154\nsamples = 45\nvalue = [0, 44, 1]'),
 Text(0.8, 0.16666666666666666, 'entropy = 0.497\nsamples = 55\nvalue = [0, 6, 49]')]

?

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?

3.2、最优划分条件

from collections import Counter
Counter(y)

Counter({0: 50, 1: 50, 2: 50})

def calc_entropy(y):
    counter = Counter(y)
    sum_ent = 0
    for i in counter:
        p = counter[i] / len(y)
        sum_ent += (-p * np.log2(p))
    return sum_ent

calc_entropy(y)

1.584962500721156

def split_dataset(x, y, dim, value):
    index_left = (x[:, dim] <= value)
    index_right = (x[:, dim] > value)
    return x[index_left], y[index_left], x[index_right], y[index_right]

def find_best_split(x, y):
    best_dim = -1
    best_value = -1
    best_entropy = np.inf
    best_entropy_left, best_entropy_right = -1, -1
    for dim in range(x.shape[1]):
        sorted_index = np.argsort(x[:, dim])
        for i in range(x.shape[0] - 1): # x列数
            value_left, value_right = x[sorted_index[i], dim], x[sorted_index[i + 1], dim]
            if value_left != value_right:
                value = (value_left + value_right) / 2
                x_left, y_left, x_right, y_right = split_dataset(x, y, dim, value)
                entropy_left, entropy_right = calc_entropy(y_left), calc_entropy(y_right)
                entropy = (len(x_left) * entropy_left + len(x_right) * entropy_right) / x.shape[0]
                if entropy < best_entropy:
                    best_dim = dim
                    best_value = value
                    best_entropy = entropy
                    best_entropy_left, best_entropy_right = entropy_left, entropy_right
    return best_dim, best_value, best_entropy, best_entropy_left, best_entropy_right
            

find_best_split(x, y)

(1, 2.45, 0.6666666666666666, 0.0, 1.0)

x_left, y_left, x_right, y_right = split_dataset(x, y, 1, 2.45)

find_best_split(x_right, y_right)

(1, 4.75, 0.34262624992678425, 0.15374218032876188, 0.4971677614160753)

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4、基尼系数

????????

????????基尼系数运算稍快；

????????物理意义略有不同，信息熵表示的是随机变量的不确定度；

??????????????? 基尼系数表示在样本集合中一个随机选中的样本被分错的概率，也就是纯度。

??????????????? 基尼系数越小，纯度越高。

????????模型效果上差异不大。

????????

二分类信息熵和基尼系数代码实现：

import numpy as np
import matplotlib.pyplot as plt

def entropy(p):
    return -(p * np.log2(p) + (1 - p) * np.log2(1 - p))

def gini(p):
    return 1 - p ** 2 - (1 - p) ** 2

plot_x = np.linspace(0.001, 0.999, 100)
plt.plot(plot_x, entropy(plot_x), color = 'blue')
plt.plot(plot_x, gini(plot_x), color = 'red')
plt.show()

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5、决策树剪枝

Chapter-07/7-6 决策树剪枝.ipynb · 梗直哥/Machine-Learning - Gitee.com

为什么要剪枝？

??????????????? 复杂度过高。

??????????????????????? 预测复杂度：O(logm)

??????????????????????? 训练复杂度：O(n x m x logm)

??????????????????????? logm为数的深度，n为数据的维度。

??????????????? 容易过拟合

??????????????????????? 为非参数学习方法。

?目标：

????????????????降低复杂度

????????????????解决过拟合

?手段：

????????????????限制深度（结点层数)

????????????????限制广度(叶子结点个数)

?? —— 设置超参数

????????????????????????

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6、决策树回归

??????? 基于一种思想：相似输入必会产生相似输出。

??????? 取节点平均值。

????????

6.1、决策树回归代码实现

import matplotlib.pyplot as plt
import numpy as np

from sklearn import datasets
from sklearn.model_selection import train_test_split
import warnings
warnings.filterwarnings('ignore')

boston = datasets.load_boston()
x = boston.data
y = boston.target
x_train, x_test, y_train, y_test = train_test_split(x, y, random_state=233)

from sklearn.tree import DecisionTreeRegressor

reg = DecisionTreeRegressor()
reg.fit(x_train,y_train)

DecisionTreeRegressor

DecisionTreeRegressor()

reg.score(x_test,y_test)

0.7410680140563546

reg.score(x_train,y_train)

1.0

6.2、绘制学习曲线

from sklearn.metrics import r2_score

plt.rcParams["figure.figsize"] = (12, 8)
max_depth = [2, 5, 10, 20]
    
for i, depth in enumerate(max_depth):
    
    reg = DecisionTreeRegressor(max_depth=depth)
    train_error, test_error = [], []
    for k in range(len(x_train)):
        reg.fit(x_train[:k+1], y_train[:k+1])
        
        y_train_pred = reg.predict(x_train[:k + 1])
        train_error.append(r2_score(y_train[:k + 1], y_train_pred))
        
        y_test_pred = reg.predict(x_test)
        test_error.append(r2_score(y_test, y_test_pred))
    
    plt.subplot(2, 2, i + 1)
    plt.ylim(0, 1.1)
    plt.title("Depth: {0}".format(depth))
    plt.plot([k + 1 for k in range(len(x_train))], train_error, color = "red", label = 'train')
    plt.plot([k + 1 for k in range(len(x_train))], test_error, color = "blue", label = 'test')
    plt.legend()

plt.show()

6.3、网格搜索

from sklearn.model_selection import GridSearchCV

params = {
    'max_depth': [n for n in range(2, 15)],
    'min_samples_leaf': [sn for sn in range(3, 20)],
}

grid = GridSearchCV(
    estimator = DecisionTreeRegressor(), 
    param_grid = params, 
    n_jobs = -1
)

grid.fit(x_train,y_train)

GridSearchCV

GridSearchCV(estimator=DecisionTreeRegressor(), n_jobs=-1,
             param_grid={'max_depth': [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
                                       14],
                         'min_samples_leaf': [3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
                                              13, 14, 15, 16, 17, 18, 19]})

estimator: DecisionTreeRegressor

DecisionTreeRegressor()

DecisionTreeRegressor

DecisionTreeRegressor()

grid.best_params_

{'max_depth': 5, 'min_samples_leaf': 3}

grid.best_score_

0.7327442904059717

reg = grid.best_estimator_

reg.score(x_test, y_test)

0.781690085676063

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7、优缺点和适用条件

优点：

????????符合人类直观思维

????????可解释性强

????????能够处理数值型数据和分类型数据

????????能够处理多输出问题

缺点：

????????容易产生过拟合

????????决策边界只能是水平或竖直方向

????????????????

????????不稳定，数据的微小变化可能生成完全不同的树

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参考于

Chapter-07/7-4 决策树分类.ipynb · 梗直哥/Machine-Learning - Gitee.com