统计学习方法第二章：感知机(perceptron)算法及python实现
统计学习方法第三章：k近邻法(k-NN),kd树及python实现
统计学习方法第四章：朴素贝叶斯法(naive Bayes)，贝叶斯估计及python实现
统计学习方法第五章：决策树(decision tree),CART算法，剪枝及python实现
统计学习方法第五章：决策树(decision tree),ID3算法，C4.5算法及python实现

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决策树(decision tree)是一种基本的分类与回归方法。决策树模型呈树状结构，在分类问题中，表示基于特征对实例进行分类的过程。

它可以认为是if-then规则的集合，也可以认为是定义在特征空间与类空间上的条件概率分布。

其主要优点是模型具有可读性，分类速度快。

学习时，利用训练数据，根据损失函数最小化的原则建立决策树模型。预测时，对新的数据，利用决策树模型进行分类。

决策树学习通常包括3个步骤：特征选择，决策树的生成和决策树的修剪

主要的决策树算法包括ID3,C4.5,CART。

ID3算法的核心是在决策树各个结点上应用信息增益准则选择特征，递归的构建决策树。具体方法是：从根结点开始，对结点计算所有特征的信息增益，选择信息增益最大的特征作为结点的特征，由该特征的不同取值建立子结点；再对子结点递归的调用以上方法，构建决策树直到所有特征的信息增益很小或没有特征可以选择为止。最后得到一个决策树。ID3相当于用极大似然法进行概率模型的选择。

ID3算法

ID3算法

代码如下：

import cv2
import time
import logging
import numpy as np
import pandas as pd
import random

from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score

total_class = 10


def log(func):
    def wrapper(*args, **kwargs):
        start_time = time.time()
        logging.debug('start %s()' % func.__name__)
        ret = func(*args, **kwargs)

        end_time = time.time()
        logging.debug('end %s(), cost %s seconds' % (func.__name__, end_time - start_time))

        return ret

    return wrapper


# 二值化
def binaryzation(img):
    cv_img = img.astype(np.uint8)
    cv2.threshold(cv_img, 50, 1, cv2.THRESH_BINARY, cv_img)
    return cv_img


@log
def binaryzation_features(trainset):
    features = []

    for img in trainset:
        img = np.reshape(img, (10, 10))
        cv_img = img.astype(np.uint8)

        img_b = binaryzation(cv_img)
        # hog_feature = np.transpose(hog_feature)
        features.append(img_b)

    features = np.array(features)
    features = np.reshape(features, (-1, 100))

    return features


class Tree(object):
    def __init__(self, node_type, Class=None, feature=None):
        self.node_type = node_type
        self.dict = {}
        self.Class = Class
        self.feature = feature

    def add_tree(self, val, tree):
        self.dict[val] = tree

    def predict(self, features):
        if self.node_type == 'leaf':
            return self.Class
        if (features[self.feature] in self.dict.keys()):
            tree = self.dict[features[self.feature]]
        else:
            if (self.Class is None):
                return random.randint(0, 1)
            else:
                return self.Class
        return tree.predict(features)


def calc_ent(x):
    """
        calculate empirical entropy of x
    """

    x_value_list = set([x[i] for i in range(x.shape[0])])
    ent = 0.0
    for x_value in x_value_list:
        p = float(x[x == x_value].shape[0]) / x.shape[0]
        logp = np.log2(p)
        ent -= p * logp

    return ent


def calc_condition_ent(train_feature, train_label):
    """
        calculate empirical entropy H(y|x)
    """

    # calc ent(y|x)

    ent = 0
    train_feature_set = set(train_feature)
    # print("train_feature_set", train_feature_set)
    for train_feature_value in train_feature_set:
        Di = train_feature[train_feature == train_feature_value]
        label_i = train_label[train_feature == train_feature_value]
        # print("Di", Di)
        train_label_set = set(train_label)
        temp = 0
        # print("train_label_set", train_label_set)
        for train_label_value in train_label_set:
            Dik = Di[label_i == train_label_value]
            # print(Dik)
            if (len(Dik) != 0):
                p = float(len(Dik)) / len(Di)
                logp = np.log2(p)
                temp -= p * logp
        ent += float(len(Di)) / len(train_feature) * temp
    return ent


def recurse_train(train_set, train_label, features, epsilon):
    global total_class

    LEAF = 'leaf'
    INTERNAL = 'internal'

    # 步骤1——如果train_set中的所有实例都属于同一类Ck
    label_set = set(train_label)
    # print(label_set)
    if len(label_set) == 1:
        return Tree(LEAF, Class=label_set.pop())

    # 步骤2——如果features为空

    class_count0 = 0
    class_count1 = 0

    for i in range(len(train_label)):
        if (train_label[i] == 1):
            class_count1 += 1
        else:
            class_count0 += 1

    if (class_count0 >= class_count1):
        max_class = 0
    else:
        max_class = 0

    if features is None:
        return Tree(LEAF, Class=max_class)

    if len(features) == 0:
        return Tree(LEAF, Class=max_class)

    # 步骤3——计算信息增益
    max_feature = 0
    max_gda = 0

    D = train_label
    HD = calc_ent(D)
    for feature in features:
        A = np.array(train_set[:, feature].flat)
        gda = HD - calc_condition_ent(A, D)

        if gda > max_gda:
            max_gda, max_feature = gda, feature

    # 步骤4——小于阈值
    if max_gda < epsilon:
        return Tree(LEAF, Class=max_class)

    # 步骤5——构建非空子集
    sub_features = features.remove(max_feature)
    tree = Tree(INTERNAL, feature=max_feature)

    feature_col = np.array(train_set[:, max_feature].flat)
    feature_value_list = set([feature_col[i] for i in range(feature_col.shape[0])])
    for feature_value in feature_value_list:

        index = []
        for i in range(len(train_label)):
            if train_set[i][max_feature] == feature_value:
                index.append(i)

        sub_train_set = train_set[index]
        sub_train_label = train_label[index]

        sub_tree = recurse_train(sub_train_set, sub_train_label, sub_features, epsilon)
        tree.add_tree(feature_value, sub_tree)

    return tree


@log
def train(train_set, train_label, features, epsilon):
    # print(features)
    return recurse_train(train_set, train_label, features, epsilon)


@log
def predict(test_set, tree):
    result = []
    for features in test_set:
        tmp_predict = tree.predict(features)
        result.append(tmp_predict)
    return np.array(result)


if __name__ == '__main__':
    logger = logging.getLogger()
    logger.setLevel(logging.DEBUG)

    raw_data = pd.read_csv('../data/train_binary2.csv', header=0)
    data = raw_data.values

    images = data[0:, 1:]
    labels = data[:, 0]

    # 图片二值化
    features = binaryzation_features(images)

    # 选取 2/3 数据作为训练集， 1/3 数据作为测试集
    train_features, test_features, train_labels, test_labels = train_test_split(features, labels, test_size=0.33,randomstate=1)

    # print(train_features.shape)
    tree = train(train_features, train_labels, [i for i in range(99)], 0.1)
    test_predict = predict(test_features, tree)
    # print(test_predict)
    score = accuracy_score(test_labels, test_predict)

    print("The accuracy score is ", score)

实现中最主要的还是对信息增益的计算，特征选择后结点的分类以及递归调用

但我这个正确率其实挺低的，自己也百思不得其解，我觉得关键算法应该是没有问题，如果有大神看出问题了希望指正！

C4.5算法

C4.5算法其实与ID3类似，主要的不同就是C4.5使用信息增益比来选择特征

C4.5算法

代码如下：

def recurse_train(train_set, train_label, features, epsilon):

    LEAF = 'leaf'
    INTERNAL = 'internal'

    # 步骤1——如果train_set中的所有实例都属于同一类Ck
    label_set = set(train_label)
    # print(label_set)
    if len(label_set) == 1:
        return Tree(LEAF, Class=label_set.pop())

    # 步骤2——如果features为空

    class_count0 = 0
    class_count1 = 0

    for i in range(len(train_label)):
        if (train_label[i] == 1):
            class_count1 += 1
        else:
            class_count0 += 1

    if (class_count0 >= class_count1):
        max_class = 0
    else:
        max_class = 0

    if features is None:
        return Tree(LEAF, Class=max_class)

    if len(features) == 0:
        return Tree(LEAF, Class=max_class)

    # 步骤3——计算信息增益
    max_feature = 0
    max_grda = 0

    D = train_label
    HD = calc_ent(D)
    for feature in features:
        A = np.array(train_set[:, feature].flat)
        gda = HD - calc_condition_ent(A, D)
        had = calc_ent(A)
        grda = gda / had

        if grda > max_grda:
            max_grda, max_feature = grda, feature

    # 步骤4——小于阈值
    if max_grda < epsilon:
        return Tree(LEAF, Class=max_class)

    # 步骤5——构建非空子集
    sub_features = features.remove(max_feature)
    tree = Tree(INTERNAL, feature=max_feature)

    feature_col = np.array(train_set[:, max_feature].flat)
    feature_value_list = set([feature_col[i] for i in range(feature_col.shape[0])])
    for feature_value in feature_value_list:

        index = []
        for i in range(len(train_label)):
            if train_set[i][max_feature] == feature_value:
                index.append(i)

        sub_train_set = train_set[index]
        sub_train_label = train_label[index]

        sub_tree = recurse_train(sub_train_set, sub_train_label, sub_features, epsilon)
        tree.add_tree(feature_value, sub_tree)

    return tree

与ID3代码的不同只有在特征选择时的标准不同

在写代码的过程中参考了别人的实现，水平有限，如有错误，希望指出