朴素贝叶斯

基础

贝叶斯概率是以18世纪的一位神学家托马斯.贝叶斯(Thomas Bayes)的名字命名。贝叶斯理论引入先验知识和逻辑推理来处理不确定命题。条件概率是贝叶斯理论的理论的基础

贝叶斯公式：

p(ci|x)=p(x|ci)p(ci)p(x)

在贝叶斯分类器中ci为类别i，x为特征。在朴素贝叶斯分类器你中，朴素：条件相互独立，将x展开为独立的特征，可以将p(x|c)记为p(x1,x2,x3,...xn|ci)（也称为后验概率），对应n个特征，在独立的前提下p(x1,x2,x3,...xn|ci)=p(x1|ci)p(x2|ci)...p(xn|ci)， p(x)为总体分布，没有变化，只需要比较分子的大小即可。

• 朴素贝叶斯分类器一般流程
  
  1. 采集各类型样本
  2. 分类别在各特征上进行概率统计，当取值为连续时默认为正太分布
  3. 输入待识别数据
  4. 比较后验概率大小，后验概率最大者即为对应的类别

注意事项

• 在计算p(x1|ci),p(x2|ci),...p(xn|ci)时若其中一项为0，最后乘积也为0。为了避免这种影响，在频数统计中，可以将每一项初始化为1，分母初始化为2
• 同样在计算p(x1|ci),p(x2|ci),...p(xn|ci)时，大部分分子非常小，会出现下溢出的情况，此时的解决方法是取对数。ln(a∗b)=ln(a)+ln(b), 该处理在代数意义上也不会有损失

示例代码

# -*- coding:utf-8 -*-from numpy import *def loadDataSet():    # 词条切分集合    postingList = [['my', 'dog', 'has', 'flea','problem', 'help', 'please'],                   ['maybe', 'not', 'take', 'him','to', 'dog', 'park', 'stupid'],                   ['my', 'dalmation', 'is', 'so', 'cute', 'I', 'love', 'him'],                   ['stop', 'posting', 'stupid', 'worthless', 'garbage'],                   ['mr', 'licks', 'ate', 'my', 'steak', 'how','to', 'stop', 'him'],                   ['quit', 'buying', 'worthless', 'dog', 'food', 'stupid']]    classVec = [0, 1, 0, 1, 0, 1]    #1 代表侮辱性文字, 0 代表正常言论    return postingList, classVec# 创建词汇表def createVocabList(dataSet):    vocabSet = set([])    for document in dataSet:        vocabSet = vocabSet | set(document) # 创建两个合集的并集    return list(vocabSet)def setOfWords2Vec(vocabList, inputSet):    returnVec = [0]*len(vocabList)    for word in inputSet:        if word in vocabList:            returnVec[vocabList.index(word)] = 1        else:            print('the word: %s is not in my Vocabulary!' %word)    return returnVecdef trainNB0(trainMatrix, trainCategory):    numTrainDocs = len(trainMatrix)    numWords = len(trainMatrix[0])    pAbusive = sum(trainCategory)/float(numTrainDocs)    p0Num = ones(numWords); p1Num = ones(numWords) # 为了避免出现其中一个概率值为0，最后的乘积为0    p0Denom = 2.0; p1Denom = 2.0    for i in range(numTrainDocs):        if trainCategory[i] == 1:            p1Num += trainMatrix[i]            p1Denom += sum(trainMatrix[i])        else:            p0Num += trainMatrix[i]            p0Denom += sum(trainMatrix[i])    # 避免下溢出，分子多小的情况    p1Vect = log(p1Num/p1Denom)    p0Vect = log(p0Num/p0Denom)    return p0Vect, p1Vect, pAbusivedef classifyNB(vec2Classify, p0Vec, p1Vec, pClass1):    p1 = sum(vec2Classify*p1Vec) + log(pClass1)    p0 = sum(vec2Classify*p0Vec) + log(1.0 - pClass1)    if p1 > p0:        return 1    else:        return 0def testingNB():    listOPosts,listClasses = loadDataSet()    myVocabList = createVocabList(listOPosts)    trainMat = []    for postinDoc in listOPosts:        trainMat.append(setOfWords2Vec(myVocabList, postinDoc))    p0V, p1V, pAb = trainNB0(array(trainMat), array(listClasses))    testEntry = ['love', 'my', 'dalmation']    thisDoc = array(setOfWords2Vec(myVocabList, testEntry))    print(testEntry, 'classified as: ', classifyNB(thisDoc, p0V, p1V, pAb))    testEntry = ['stupid', 'garbage']    thisDoc = array(setOfWords2Vec(myVocabList, testEntry))    print(testEntry, 'classified as:', classifyNB(thisDoc, p0V, p1V, pAb))# 朴素贝叶斯词袋模型def bagOfWords2VecMN(vocabList, inputSet):    returnVec = [0]*len(vocabList)    for word in inputSet:        if word in vocabList:            returnVec[vocabList.index(word)] += 1    return returnVecdef textParse(bigString):    import re    relistOfTokens = re.split(r'\w*', bigString)    return [tok.lower() for tok in relistOfTokens if len(tok) > 2]def spamTest():    docList = []; classList = []; fullText = []    for i in range(1,26):        print (i)        wordList = textParse(open('email/spam/%d.txt' % i).read())        docList.append(wordList)        fullText.extend(wordList)        classList.append(1)        wordList = textParse(open('email/ham/%d.txt' % i).read())        docList.append(wordList)        fullText.extend(wordList)        classList.append(0)    vocabList = createVocabList(docList)    # 随机构建训练集    trainingSet = list(range(50)); testSet = []    for i in range(10):        randIndex = int(random.uniform(0, len(trainingSet)))        testSet.append(trainingSet[randIndex])        del(trainingSet[randIndex])    trainMat = []; trainClasses = []    for docIndex in trainingSet:        trainMat.append(setOfWords2Vec(vocabList, docList[docIndex]))        trainClasses.append(classList[docIndex])    p0V, p1V, pSpam = trainNB0(array(trainMat), array(trainClasses))    errorCount = 0    # 对测试集分类    for docIndex in testSet:        wordVector = setOfWords2Vec(vocabList, docList[docIndex])        if classifyNB(array(wordVector), p0V, p1V, pSpam) != classList[docIndex]:            errorCount += 1    print('the error rate is:', float(errorCount)/len(testSet))

算法特点

优点： 在数据较少的情况下仍然有效，对数据缺失不敏感
缺点： 对于输入数据的准备方式较为敏感
适用数据：标称型数据