【数据挖掘】关联分析之Apriori（转载）

【数据挖掘】关联分析之Apriori

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：

1. 生成频繁项集，即满足最小支持度阈值的所有项集；
2. 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('\n')]  

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.