【数据挖掘】关联分析之Apriori（转载）

1.Apriori算法

如果一个事务中有X，则该事务中则很有可能有Y，写成关联规则

{X}→{Y}

将这种找出项目之间联系的方法叫做关联分析。关联分析中最有名的问题是购物蓝问题，在超市购物时，有一个奇特的现象——顾客在买完尿布之后通常会买啤酒，即{尿布}→{啤酒}。原来，妻子嘱咐丈夫回家的时候记得给孩子买尿布，丈夫买完尿布后通常会买自己喜欢的啤酒。

考虑到规则的合理性，引入了两个度量：支持度（support）、置信度（confidence），定义如下

支持度保证项集(X, Y)在数据集出现的频繁程度，置信度确定Y在包含X中出现的频繁程度。

对于包含有d个项的数据集，可能的规则数为

如果用brute-force的方法，计算代价太大了。为此，R. Agrawal与R. Srikant提出了Apriori算法。同大部分的关联分析算法一样，Apriori算法分为两步：

生成频繁项集，即满足最小支持度阈值的所有项集；
生成关联规则，从上一步中找出的频繁项集中找出搞置信度的规则，即满足最小置信度阈值。

A priori在拉丁语中是“from before”（先验）的意思。Apriori算法是用到了一个简单到不能再简单的先验：一个频繁项集的子集也是频繁的。

生成频繁项集、关联规则用到了剪枝，具体参看[2]。

class associationRule:  
    def __init__(self,dataSet):  
        self.sentences=map(set,dataSet)  
        self.minSupport=0.5  
        self.minConf=0.98  
        self.numSents=float(len(self.sentences))  
        self.supportData={}  
        self.L=[]  
        self.ruleList=[]  
  
    def createC1(self):  
        """create candidate itemsets of size 1 C1"""  
  
        C1=[]  
        for sentence in self.sentences:  
            for word in sentence:  
                if not [word] in C1:  
                    C1.append([word])  
        C1.sort()  
        return map(frozenset,C1)  
  
    def scan(self,Ck):  
        """generate frequent itemsets Lk from candidate itemsets Ck"""  
  
        wscnt={}   
        retList=[]  
        #calculate support for every itemset in Ck  
        for words in Ck:  
            for sentence in self.sentences:  
                if words.issubset(sentence):  
                    if not wscnt.has_key(words): wscnt[words]=1  
                    else: wscnt[words]+=1  
  
        for key in wscnt:  
            support=wscnt[key]/self.numSents  
            if support>=self.minSupport:  
                retList.append(key)  
            self.supportData[key]=support  
        self.L.append(retList)  
  
    def aprioriGen(self,Lk,k):  
        """the candidate generation: merge a pair of frequent (k − 1)-itemsets  
        only if their first k − 2 items are identical 
        """  
  
        retList=[]  
        lenLk=len(Lk)  
        for i in range(lenLk):  
            for j in range(i+1,lenLk):  
                L1=list(Lk[i])[:k-2]; L2=list(Lk[j])[:k-2]  
                L1.sort(); L2.sort()  
                if L1==L2:  
                    retList.append(Lk[i]|Lk[j])  
        return retList  
  
    def apriori(self):  
        """generate a list of frequent itemsets"""  
  
        C1=self.createC1()  
        self.scan(C1)  
        k=2  
        while(k<=3):  
            Ck=self.aprioriGen(self.L[k-2],k)  
            self.scan(Ck)  
            k+=1       
  
    def generateRules(self):  
        """generate a list of rules"""  
  
        for i in range(1,len(self.L)):    #get only sets with two or more items  
            for freqSet in self.L[i]:  
                H1=[frozenset([word]) for word in freqSet]  
                if(i>1): self.rulesFromConseq(freqSet,H1)  
                else: self.calcConf(freqSet,H1)  #set with two items  
  
    def calcConf(self,freqSet,H):  
        """calculate confidence, eliminate some rules by confidence-based pruning"""  
  
        prunedH=[]  
        for conseq in H:  
            conf=self.supportData[freqSet]/self.supportData[freqSet-conseq]  
            if conf>=self.minConf:  
                print "%s --> %s, conf=%.3f"%(map(str,freqSet-conseq), map(str,conseq), conf)  
                self.ruleList.append((freqSet-conseq,conseq,conf))  
                prunedH.append(conseq)  
        return prunedH  
  
    def rulesFromConseq(self,freqSet,H):  
        """generate more association rules from freqSet+H"""  
  
        m=len(H[0])  
        if len(freqSet)>m+1:                #try further merging  
            Hmp1=self.aprioriGen(H,m+1)     #create new candidate Hm+1  
            Hmp1=self.calcConf(freqSet,Hmp1)  
            if len(Hmp1)>1:  
                self.rulesFromConseq(freqSet,Hmp1)

读取mushroom.dat数据集

def read_file(raw_file):  
    """read file"""  
  
    return [sorted(list(set(e.split()))) for e in   
            open(raw_file).read().strip().split('
')]  
  
def main():  
    sentences=read_file('test.txt')  
    assrules=associationRule(sentences)  
    assrules.apriori()  
    assrules.generateRules()  
  
if __name__=="__main__":  
    main()

生成的规则

['76'] --> ['34'], conf=1.000
['34'] --> ['85'], conf=1.000
['36'] --> ['85'], conf=1.000
['24'] --> ['85'], conf=1.000
['53'] --> ['90'], conf=1.000
['53'] --> ['34'], conf=1.000
['2'] --> ['85'], conf=1.000
['76'] --> ['85'], conf=1.000
['67'] --> ['86'], conf=1.000
['76'] --> ['86'], conf=1.000
['67'] --> ['34'], conf=1.000
['67'] --> ['85'], conf=1.000
['90'] --> ['85'], conf=1.000
['86'] --> ['85'], conf=1.000
['53'] --> ['85'], conf=1.000
['53'] --> ['86'], conf=1.000
['39'] --> ['85'], conf=1.000
['34'] --> ['86'], conf=0.999
['86'] --> ['34'], conf=0.998
['63'] --> ['85'], conf=1.000
['59'] --> ['85'], conf=1.000
['53'] --> ['86', '85'], conf=1.000
['76'] --> ['34', '85'], conf=1.000
['53'] --> ['90', '34'], conf=1.000
['76'] --> ['86', '85'], conf=1.000
['53'] --> ['34', '85'], conf=1.000
['67'] --> ['34', '85'], conf=1.000
['76'] --> ['86', '34'], conf=1.000
['53'] --> ['86', '34'], conf=1.000
['67'] --> ['86', '34'], conf=1.000
['53'] --> ['90', '85'], conf=1.000
['67'] --> ['86', '85'], conf=1.000
['53'] --> ['90', '86'], conf=1.000
['86'] --> ['85', '34'], conf=0.998
['34'] --> ['86', '85'], conf=0.999

源代码在有些数据集上跑得很慢，还需要做一些优化。这里有一些用作关联分析测试的数据集。

2. Referrence

[1] Peter Harrington, machine learning in action.

[2] Tan, et al., Introduction to data minging.