【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.