决策树（decision tree）

决策树的主要思想：计算数据集上的熵，划分数据集以获得最大的信息增益，再在子数据集上划分数据集以获得最大的信息增益，一直到子数据集下的所有实例都具有相同的类型，从而获得了一棵“决策树”。当需要使用决策树时，只需要根据新个体的各个特征，在决策树上从上往下走，最后到达的叶子节点就是新个体的分类。（看完下面的实现再来看这段话应该会更有感觉）
有关信息、熵和信息增益有详细资料可以看这儿。信息&熵&信息增益

优点：计算复杂度不高，输出结果容易理解，可以处理不相关特征数据。
缺点：可能会产生过度匹配的问题。
适用数据类型：数值型和标称型。

这里使用ID3算法划分数据集。

from math import log
import operator


def calShannonEnt(dataSet):
    """
    计算给定数据集的香农熵
    :param dataSet:
    :return:
    """
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet:
        currentLabel = featVec[-1]
        if currentLabel not in labelCounts.keys():
            labelCounts[currentLabel] = 0
        labelCounts[currentLabel] += 1
    shannonEnt = 0.0
    for key in labelCounts:
        prob = float(labelCounts[key]) / numEntries
        shannonEnt -= prob * log(prob, 2)
    return shannonEnt


def createDataset():
    dataSet = [[1, 1, 'yes'],
               [1, 1, 'yes'],
               [1, 0, 'no'],
               [0, 1, 'no'],
               [0, 1, 'no']]
    labels = ['no surfacing', 'flippers']
    return dataSet, labels


def splitDataSet(dataSet, axis, value):
    """
    按照给定特征划分数据集
    :param dataSet:
    :param axis:
    :param value:
    :return:
    """
    retDataSet = []
    for featVec in dataSet:
        if featVec[axis] == value:
            reducedFeatVec = featVec[:axis]
            reducedFeatVec.extend(featVec[axis + 1:])
            retDataSet.append(reducedFeatVec)
    return retDataSet


def chooseBestFeatureToSplit(dataSet):
    """
    选择最好的数据集划分方式
    :param dataSet:
    :return:
    """
    numFeatures = len(dataSet[0]) - 1
    # 信息熵
    baseEntropy = calShannonEnt(dataSet)
    bestInfoGain = 0.0
    bestFeature = -1
    for i in range(numFeatures):
        featList = [example[i] for example in dataSet]
        uniqueVals = set(featList)
        # 条件熵
        newEntropy = 0.0
        for value in uniqueVals:
            subDataSet = splitDataSet(dataSet, i, value)
            prob = len(subDataSet) / float(len(dataSet))
            newEntropy += prob * calShannonEnt(subDataSet)
        # 信息增益 = 信息熵 - 条件熵
        infoGain = baseEntropy - newEntropy
        if infoGain > bestInfoGain:
            bestInfoGain = infoGain
            bestFeature = i
    return bestFeature


def majorityCnt(classList):
    classCount = {}
    for vote in classList:
        if vote not in classCount.keys():
            classCount[vote] = 0
        classCount[vote] += 1
    sortedClassCount = sorted(classCount.items(),
                              key=operator.itemgetter(1),
                              reverse=True)
    return sortedClassCount[0][0]


def createTree(dataSet, labels):
    """
    创建树
    :param dataSet:
    :param labels:
    :return:
    """
    classList = [example[-1] for example in dataSet]
    if classList.count(classList[0]) == len(classList):
        return classList[0]
    if len(dataSet[0]) == 1:
        return majorityCnt(classList)
    bestFeat = chooseBestFeatureToSplit(dataSet)
    bestFeatLabel = labels[bestFeat]
    myTree = {bestFeatLabel: {}}
    del (labels[bestFeat])
    featValues = [example[bestFeat] for example in dataSet]
    uniqueVals = set(featValues)
    for value in uniqueVals:
        subLabels = labels[:]
        myTree[bestFeatLabel][value] = createTree(
            splitDataSet(dataSet, bestFeat, value), subLabels)
    return myTree


def main():
    myData, labels = createDataset()
    myTree = createTree(myData, labels)
    print(myTree)


if __name__ == '__main__':
    main()

参考书籍：
哈林顿李锐. 机器学习实战 : Machine learning in action[M]. 人民邮电出版社, 2013.