Apriori算法第二篇----详细分析和代码实现

1 Apriori介绍

Apriori算法使用频繁项集的先验知识，使用一种称作逐层搜索的迭代方法，k项集用于探索(k+1)项集。首先，通过扫描事务（交易）记录，找出所有的频繁1项集，该集合记做L₁，然后利用L₁找频繁2项集的集合L₂，L₂找L₃，如此下去，直到不能再找到任何频繁k项集。最后再在所有的频繁集中找出强规则，即产生用户感兴趣的关联规则。

其中，Apriori算法具有这样一条性质：任一频繁项集的所有非空子集也必须是频繁的。因为假如P(I)< 最小支持度阈值，当有元素A添加到I中时，结果项集（A∩I）不可能比I出现次数更多。因此A∩I也不是频繁的。

2   连接步和剪枝步

在上述的关联规则挖掘过程的两个步骤中，第一步往往是总体性能的瓶颈。Apriori算法采用连接步和剪枝步两种方式来找出所有的频繁项集。

1）  连接步

为找出L_k（所有的频繁k项集的集合），通过将L_k-1（所有的频繁k-1项集的集合）与自身连接产生候选k项集的集合。候选集合记作C_k。设l₁和l₂是L_k-1中的成员。记l_i[j]表示l_i中的第j项。假设Apriori算法对事务或项集中的项按字典次序排序，即对于（k-1）项集l_i，l_i[1]<l_i[2]<……….<l_i[k-1]。将L_k-1与自身连接，如果(l₁[1]=l₂[1])&&( l₁[2]=l₂[2])&&……..&& (l₁[k-2]=l₂[k-2])&&(l₁[k-1]<l₂[k-1])，那认为l₁和l₂是可连接。连接l₁和l₂ 产生的结果是{l₁[1],l₁[2],……,l₁[k-1],l₂[k-1]}。

2）  剪枝步

C_K是L_K的超集，也就是说，C_K的成员可能是也可能不是频繁的。通过扫描所有的事务（交易），确定C_K中每个候选的计数，判断是否小于最小支持度计数，如果不是，则认为该候选是频繁的。为了压缩C_k,可以利用Apriori性质：任一频繁项集的所有非空子集也必须是频繁的，反之，如果某个候选的非空子集不是频繁的，那么该候选肯定不是频繁的，从而可以将其从C_K中删除。

（Tip：为什么要压缩C_K呢？因为实际情况下事务记录往往是保存在外存储上，比如数据库或者其他格式的文件上，在每次计算候选计数时都需要将候选与所有事务进行比对，众所周知，访问外存的效率往往都比较低，因此Apriori加入了所谓的剪枝步，事先对候选集进行过滤，以减少访问外存的次数。）

3   Apriori算法实例

交易ID	商品ID列表
T100	I1，I2，I5
T200	I2，I4
T300	I2，I3
T400	I1，I2，I4
T500	I1，I3
T600	I2，I3
T700	I1，I3
T800	I1，I2，I3，I5
T900	I1，I2，I3

上图为某商场的交易记录，共有9个事务，利用Apriori算法寻找所有的频繁项集的过程如下:

详细介绍下候选3项集的集合C3的产生过程：从连接步，首先C3={{I1,I2,I3}，{I1,I2,I5}，{I1,I3,I5}，{I2,I3,I4}，{I2,I3,I5}，{I2,I4,I5}}（C3是由L2与自身连接产生）。根据Apriori性质，频繁项集的所有子集也必须频繁的，可以确定有4个候选集{I1,I3,I5}，{I2,I3,I4}，{I2,I3,I5}，{I2,I4,I5}}不可能时频繁的，因为它们存在子集不属于频繁集，因此将它们从C3中删除。注意，由于Apriori算法使用逐层搜索技术，给定候选k项集后，只需检查它们的（k-1）个子集是否频繁。

3． Apriori伪代码

算法：Apriori

输入：D - 事务数据库；min_sup - 最小支持度计数阈值

输出：L - D中的频繁项集

方法：

     L₁=find_frequent_1-itemsets(D); // 找出所有频繁1项集

     For(k=2;L_k-1!=null;k++){

        C_k=apriori_gen(L_k-1); // 产生候选，并剪枝

        For each 事务t in D{ // 扫描D进行候选计数

            C_t =subset(C_k,t); // 得到t的子集

            For each 候选c 属于 C_t

                         c.count++;

        }

        L_k={c属于C_k | c.count>=min_sup}

}

Return L=所有的频繁集；

 

Procedure apriori_gen(L_k-1:frequent(k-1)-itemsets)

      For each项集l₁属于L_k-1

              For each项集 l₂属于L_k-1

                       If((l₁[1]=l₂[1])&&( l₁[2]=l₂[2])&&……..

&& (l₁[k-2]=l₂[k-2])&&(l₁[k-1]<l₂[k-1])) then{

                   c=l₁连接l₂ //连接步：产生候选

                   if has_infrequent_subset(c,L_k-1) then

                       delete c; //剪枝步：删除非频繁候选

                   else add c to C_k;

                  }

          Return C_k;

 

     Procedure has_infrequent_sub(c:candidate k-itemset; L_k-1:frequent(k-1)-itemsets)

        For each(k-1)-subset s of c

            If s不属于L_k-1 then

               Return true;

        Return false;

 

 

 

4． 由频繁项集产生关联规则

Confidence(A->B)=P(B|A)=support_count(AB)/support_count(A)

关联规则产生步骤如下：

1）  对于每个频繁项集l，产生其所有非空真子集；

2）  对于每个非空真子集s,如果support_count(l)/support_count(s)>=min_conf，则输出 s->(l-s)，其中，min_conf是最小置信度阈值。

例如，在上述例子中，针对频繁集{I1，I2，I5}。可以产生哪些关联规则？该频繁集的非空真子集有{I1，I2}，{I1，I5}，{I2，I5}，{I1 }，{I2}和{I5}，对应置信度如下：

I1&&I2->I5            confidence=2/4=50%

I1&&I5->I2            confidence=2/2=100%

I2&&I5->I1            confidence=2/2=100%

I1 ->I2&&I5            confidence=2/6=33%

I2 ->I1&&I5            confidence=2/7=29%

I5 ->I1&&I2            confidence=2/2=100%

如果min_conf=70%,则强规则有I1&&I5->I2，I2&&I5->I1，I5 ->I1&&I2。

5． Apriori Java代码

package com.apriori;

 

import java.util.ArrayList;

import java.util.Collections;

import java.util.HashMap;

import java.util.List;

import java.util.Map;

import java.util.Set;

 

public class Apriori {

 

         private final static int SUPPORT = 2; // 支持度阈值

         private final static double CONFIDENCE = 0.7; // 置信度阈值

 

         private final static String ITEM_SPLIT=";"; // 项之间的分隔符

         private final static String CON="->"; // 项之间的分隔符

 

         private final static List<String> transList=new ArrayList<String>(); //所有交易

 

         static{//初始化交易记录

                   transList.add("1;2;5;");

                   transList.add("2;4;");

                   transList.add("2;3;");

                   transList.add("1;2;4;");

                   transList.add("1;3;");

                   transList.add("2;3;");

                   transList.add("1;3;");

                   transList.add("1;2;3;5;");

                   transList.add("1;2;3;");

         }

 

        

         public Map<String,Integer> getFC(){

        Map<String,Integer> frequentCollectionMap=new HashMap<String,Integer>();//所有的频繁集

 

        frequentCollectionMap.putAll(getItem1FC());

 

        Map<String,Integer> itemkFcMap=new HashMap<String,Integer>();

        itemkFcMap.putAll(getItem1FC());

        while(itemkFcMap!=null&&itemkFcMap.size()!=0){

          Map<String,Integer> candidateCollection=getCandidateCollection(itemkFcMap);

          Set<String> ccKeySet=candidateCollection.keySet();

 

          //对候选集项进行累加计数

          for(String trans:transList){

             for(String candidate:ccKeySet){

                      boolean flag=true;// 用来判断交易中是否出现该候选项，如果出现，计数加1

                      String[] candidateItems=candidate.split(ITEM_SPLIT);

                      for(String candidateItem:candidateItems){

                               if(trans.indexOf(candidateItem+ITEM_SPLIT)==-1){

                                         flag=false;

                                         break;

                               }

                      }

                      if(flag){

                               Integer count=candidateCollection.get(candidate);

                               candidateCollection.put(candidate, count+1);

                      }

             }

          }

 

          //从候选集中找到符合支持度的频繁集项

          itemkFcMap.clear();

          for(String candidate:ccKeySet){

             Integer count=candidateCollection.get(candidate);

             if(count>=SUPPORT){

                 itemkFcMap.put(candidate, count);

             }

          }

 

          //合并所有频繁集

          frequentCollectionMap.putAll(itemkFcMap);

 

        }

 

        return frequentCollectionMap;

         }

 

        

         private Map<String,Integer> getCandidateCollection(Map<String,Integer> itemkFcMap){

                   Map<String,Integer> candidateCollection=new HashMap<String,Integer>();

                   Set<String> itemkSet1=itemkFcMap.keySet();

                   Set<String> itemkSet2=itemkFcMap.keySet();

 

                   for(String itemk1:itemkSet1){

                            for(String itemk2:itemkSet2){

                                     //进行连接

                                     String[] tmp1=itemk1.split(ITEM_SPLIT);

                                     String[] tmp2=itemk2.split(ITEM_SPLIT);

 

                                     String c="";

                                     if(tmp1.length==1){

                                               if(tmp1[0].compareTo(tmp2[0])<0){

                                                        c=tmp1[0]+ITEM_SPLIT+tmp2[0]+ITEM_SPLIT;

                                               }

                                     }else{

                                               boolean flag=true;

                    for(int i=0;i<tmp1.length-1;i++){

                           if(!tmp1[i].equals(tmp2[i])){

                                    flag=false;

                                    break;

                           }

                    }

                    if(flag&&(tmp1[tmp1.length-1].compareTo(tmp2[tmp2.length-1])<0)){

                           c=itemk1+tmp2[tmp2.length-1]+ITEM_SPLIT;

                    }

                                     }

 

                                     //进行剪枝

                                     boolean hasInfrequentSubSet = false;

                                     if (!c.equals("")) {

                                               String[] tmpC = c.split(ITEM_SPLIT);

                                               for (int i = 0; i < tmpC.length; i++) {

                                                        String subC = "";

                                                        for (int j = 0; j < tmpC.length; j++) {

                                                                 if (i != j) {

                                                                           subC = subC+tmpC[j]+ITEM_SPLIT;

                                                                 }

                                                        }

                                                        if (itemkFcMap.get(subC) == null) {

                                                                 hasInfrequentSubSet = true;

                                                                 break;

                                                        }

                                               }

                                     }else{

                                               hasInfrequentSubSet=true;

                                     }

 

                                     if(!hasInfrequentSubSet){

                                               candidateCollection.put(c, 0);

                                     }

                            }

                   }

 

                   return candidateCollection;

         }

 

        

         private Map<String,Integer> getItem1FC(){

                   Map<String,Integer> sItem1FcMap=new HashMap<String,Integer>();

                   Map<String,Integer> rItem1FcMap=new HashMap<String,Integer>();//频繁1项集

 

                   for(String trans:transList){

                            String[] items=trans.split(ITEM_SPLIT);

                            for(String item:items){

                                     Integer count=sItem1FcMap.get(item+ITEM_SPLIT);

                                     if(count==null){

                                               sItem1FcMap.put(item+ITEM_SPLIT, 1);

                                     }else{

                                               sItem1FcMap.put(item+ITEM_SPLIT, count+1);

                                     }

                            }

                   }

 

                   Set<String> keySet=sItem1FcMap.keySet();

                   for(String key:keySet){

                            Integer count=sItem1FcMap.get(key);

                            if(count>=SUPPORT){

                                     rItem1FcMap.put(key, count);

                            }

                   }

                   return rItem1FcMap;

         }

 

   

         public Map<String,Double> getRelationRules(Map<String,Integer> frequentCollectionMap){

                   Map<String,Double> relationRules=new HashMap<String,Double>();

                   Set<String> keySet=frequentCollectionMap.keySet();

                   for (String key : keySet) {

                            double countAll=frequentCollectionMap.get(key);

                            String[] keyItems = key.split(ITEM_SPLIT);

                            if(keyItems.length>1){

                                     List<String> source=new ArrayList<String>();

                                     Collections.addAll(source, keyItems);

                                     List<List<String>> result=new ArrayList<List<String>>();

 

                                     buildSubSet(source,result);//获得source的所有非空子集

 

                                     for(List<String> itemList:result){

                    if(itemList.size()<source.size()){//只处理真子集

                           List<String> otherList=new ArrayList<String>();

                           for(String sourceItem:source){

                                    if(!itemList.contains(sourceItem)){

                                             otherList.add(sourceItem);

                                    }

                           }

                        String reasonStr="";//前置

                        String resultStr="";//结果

                        for(String item:itemList){

                                reasonStr=reasonStr+item+ITEM_SPLIT;

                        }

                        for(String item:otherList){

                                resultStr=resultStr+item+ITEM_SPLIT;

                        }

 

                        double countReason=frequentCollectionMap.get(reasonStr);

                        double itemConfidence=countAll/countReason;//计算置信度

                        if(itemConfidence>=CONFIDENCE){

                                String rule=reasonStr+CON+resultStr;

                                relationRules.put(rule, itemConfidence);

                        }

                    }

                                     }

                            }

                   }

 

                   return relationRules;

         }

 

        

         private  void buildSubSet(List<String> sourceSet, List<List<String>> result) {

                   // 仅有一个元素时，递归终止。此时非空子集仅为其自身，所以直接添加到result中

                   if (sourceSet.size() == 1) {

                            List<String> set = new ArrayList<String>();

                            set.add(sourceSet.get(0));

                            result.add(set);

                   } else if (sourceSet.size() > 1) {

                            // 当有n个元素时，递归求出前n-1个子集，在于result中

                            buildSubSet(sourceSet.subList(0, sourceSet.size() - 1), result);

                            int size = result.size();// 求出此时result的长度，用于后面的追加第n个元素时计数

                            // 把第n个元素加入到集合中

                            List<String> single = new ArrayList<String>();

                            single.add(sourceSet.get(sourceSet.size() - 1));

                            result.add(single);

                            // 在保留前面的n-1子集的情况下，把第n个元素分别加到前n个子集中，并把新的集加入到result中;

                            // 为保留原有n-1的子集，所以需要先对其进行复制

                            List<String> clone;

                            for (int i = 0; i < size; i++) {

                                     clone = new ArrayList<String>();

                                     for (String str : result.get(i)) {

                                               clone.add(str);

                                     }

                                     clone.add(sourceSet.get(sourceSet.size() - 1));

 

                                     result.add(clone);

                            }

                   }

         }

 

         public static void main(String[] args){

                   Apriori apriori=new Apriori();

                   Map<String,Integer> frequentCollectionMap=apriori.getFC();

                   System.out.println("----------------频繁集"+"----------------");

                   Set<String> fcKeySet=frequentCollectionMap.keySet();

                   for(String fcKey:fcKeySet){

                            System.out.println(fcKey+"  :  "+frequentCollectionMap.get(fcKey));

                   }

        Map<String,Double> relationRulesMap=apriori.getRelationRules(frequentCollectionMap);

        System.out.println("----------------关联规则"+"----------------");

        Set<String> rrKeySet=relationRulesMap.keySet();

        for(String rrKey:rrKeySet){

                            System.out.println(rrKey+"  :  "+relationRulesMap.get(rrKey));

                   }

         }

}

转载自：http://blog.csdn.net/zjd950131/article/details/8071414

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