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Sliding-window filtering an efficient algorithm for incremental mining

Sliding-window filtering an efficient algorithm for incremental mining
Sliding-window filtering an efficient algorithm for incremental mining

Sliding-Window Filtering:An Ef?cient Algorithm for

Incremental Mining

Chang-Hung Lee,Cheng-Ru Lin,and Ming-Syan Chen

Department of Electrical Engineering

National T aiwan University

T aipei,T aiwan,ROC

E-mail:mschen@https://www.doczj.com/doc/f28855980.html,.tw,{chlee,owenlin}@https://www.doczj.com/doc/f28855980.html,.tw

ABSTRACT

We explore in this paper an e?ective sliding-window?ltering (abbreviatedly as SWF)algorithm for incremental mining of association rules.In essence,by partitioning a transaction database into several partitions,algorithm SWF employs a ?ltering threshold in each partition to deal with the can-didate itemset generation.Under SWF,the cumulative in-formation of mining previous partitions is selectively carried over toward the generation of candidate itemsets for the sub-sequent partitions.Algorithm SWF not only signi?cantly reduces I/O and CPU cost by the concepts of cumulative?l-tering and scan reduction techniques but also e?ectively con-trols memory utilization by the technique of sliding-window partition.Algorithm SWF is particularly powerful for e?-cient incremental mining for an ongoing time-variant trans-action database.By utilizing proper scan reduction tech-niques,only one scan of the incremented dataset is needed by algorithm SWF.The I/O cost of SWF is,in orders of mag-nitude,smaller than those required by prior methods,thus resolving the performance bottleneck.Experimental studies are performed to evaluate performance of algorithm SWF.It is noted that the improvement achieved by algorithm SWF is even more prominent as the incremented portion of the dataset increases and also as the size of the database in-creases.

Keywords

Data mining,association rules,time-variant database,in-cremental mining

1.INTRODUCTION

The importance of data mining is growing at rapid pace recently.Analysis of past transaction data can provide very valuable information on customer buying behavior,and busi-ness decisions.In essence,it is necessary to collect and ana-lyze a su?cient amount of sales data before any meaningful Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro?t or commercial advantage and that copies bear this notice and the full citation on the?rst page.To copy otherwise,to republish,to post on servers or to redistribute to lists,requires prior speci?c permission and/or a fee.

Copyright2001ACM X-XXXXX-XX-X/XX/XX

...$5.00.

db i, j

P

i+1

P

j

P

j+

1

db i+1, j+1

P

i

Figure1:Incremental mining for an ongoing time-variant transaction database.

conclusion can be drawn therefrom.Since the amount of these processed data tends to be huge,it is important to devise e?cient algorithms to conduct mining on these data. Mining association rules was?rst introduced in[2],where it was shown that the problem of mining association rules is composed of the following two subproblems:(1)discover-ing the frequent itemsets,i.e.,all sets of itemsets that have transaction support above a pre-determined minimum sup-port s,and(2)using the frequent itemsets to generate the association rules for the database.The overall performance of mining association rules is in fact determined by the?rst subproblem.After the frequent itemsets are identi?ed,the corresponding association rules can be derived in a straight-forward manner[2].Among others,Apriori[2],DHP[16], and partition-based ones[15,19]are proposed to solve the ?rst subproblem e?ciently.In addition,several novel min-ing techniques,including TreeProjection[1],FP-tree[11,12, 18],and constraint-based ones[10,14,17,23,24]also re-ceived a signi?cant amount of research attention.

Recent important applications have called for the need of incremental mining.This is due to the increasing use of the record-based databases whose data are being continu-ously added.Examples of such applications include Web log records,stock market data,grocery sales data,transactions in electronic commerce,and daily weather/tra?c records,to name a few.In many applications,we would like to mine the transaction database for a?xed amount of most recent data (say,data in the last12months).That is,in the incremen-tal mining,one has to not only include new data(i.e.,data in the new month)into,but also remove the old data(i.e., data in the most obsolete month)from the mining process.

Consider the example transaction database in Figure1. Note that db i,j is the part of the transaction database formed by a continuous region from partition P i to partition P j. Suppose we have conducted the mining for the transaction database db i,j.As time advances,we are given the new data of January of2001,and are interested in conducting an in-cremental mining against the new data.Instead of taking all the past data into consideration,our interest is limited to mining the data in the last12months.As a result, the mining of the transaction database db i+1,j+1is called for.Note that since the underlying transaction database has been changed as time advances,some algorithms,such as Apriori,may have to resort to the regeneration of can-didate itemsets for the determination of new frequent item-sets,which is,however,very costly even if the incremental data subset is small.On the other hand,while FP-tree-based methods[11,18]are shown to be e?cient for small databases,it is expected that their de?ciency of memory overhead due to the need of keeping a portion of database in memory,as indicated in[13],could become more severe in the presence of a large database upon which an incremental mining process is usually performed.

To the best of our knowledge,there is little progress made thus far to explicitly address the problem of incremental mining except noted below.In[8],the FUP algorithm up-dates the association rules in a database when new transac-tions are added to the database.Algorithm FUP is based on the framework of Apriori and is designed to discover the new frequent itemsets iteratively.The idea is to store the counts of all the frequent itemsets found in a previous min-ing https://www.doczj.com/doc/f28855980.html,ing these stored counts and examining the newly added transactions,the overall count of these can-didate itemsets are then obtained by scanning the original database.An extension to the work in[8]was reported in [9]where the authors propose an algorithm FUP2for up-dating the existing association rules when transactions are added to and deleted from the database.In essence,FUP2 is equivalent to FUP for the case of insertion,and is,how-ever,a complementary algorithm of FUP for the case of deletion.It is shown in[9]that FUP2outperforms Apri-ori algorithm which,without any provision for incremental mining,has to re-run the association rule mining algorithm on the whole updated database.Another FUP-based algo-rithm,call FUP2H,was also devised in[9]to utilize the hash technique for performance improvement.Furthermore,the concept of negative borders in[21]and that of UWEP,i.e., update with early pruning,in[4]are utilized to enhance the e?ciency of FUP-based algorithms.

However,as will be shown by our experimental results, the above mentioned FUP-based algorithms tend to su?er from two inherent problems,namely(1)the occurrence of a potentially huge set of candidate itemsets,and(2)the need of multiple scans of database.First,consider the problem of a potentially huge set of candidate itemsets.Note that the FUP-based algorithms deal with the combination of two sets of candidate itemsets which are independently gener-ated,i.e.,from the original data set and the incremental data subset.Since the set of candidate itemsets includes all the possible permutations of the elements,FUP-based algo-rithms may su?er from a very large set of candidate itemsets, especially from candidate2-itemsets.More importantly,in many applications,one may encounter new itemsets in the incremented dataset.While adding some new products in the transaction database,FUP-based algorithms will need to resort to multiple scans of database.Speci?cally,in the presence of a new frequent itemset L k generated in the data subset,k scans of the database are needed by FUP-based algorithms in the worst case.That is,the case of k=8 means that the database has to be scanned8times,which is very costly,especially in terms of I/O cost.The problem of a large set of candidate itemsets will hinder an e?ective use of the scan reduction technique[16]by an FUP-based algorithm.

To remedy these problems,we shall devise in this paper an algorithm based on sliding-window?ltering(abbreviat-edly as SWF)for incremental mining of association rules.In essence,by partitioning a transaction database into several partitions,algorithm SWF employs a?ltering threshold in each partition to deal with the candidate itemset genera-tion.For ease of exposition,the processing of a partition is termed a phase of processing.Under SWF,the cumulative information in the prior phases is selectively carried over to-ward the generation of candidate itemsets in the subsequent phases.After the processing of a phase,algorithm SWF outputs a cumulative?lter,denoted by CF,which consists of a progressive candidate set of itemsets,their occurrence counts and the corresponding partial support required.As will be seen,the cumulative?lter produced in each process-ing phase constitutes the key component to realize the incre-mental mining.An illustrative example for the operations of SWF is presented in Section3.1and a detailed description of algorithm SWF is given in Section3.2.It will be seen that algorithm SW F proposed has several important ad-vantages.First,with employing the prior knowledge in the previous phase,SWF is able to reduce the amount of candi-date itemsets e?ciently which in turn reduces the CPU and memory overhead.The second advantage of SWF is that owing to the small number of candidate sets generated,the scan reduction technique[16]can be applied e?ciently.As a result,only one scan of the ongoing time-variant database is required.As will be validated by our experimental results, this very advantage of SWF enables SWF to signi?cantly outperform FUP-based algorithms.The third advantage of SWF is the capability of SWF to avoid the data skew in na-ture.As mentioned in[15],such instances as severe whether conditions may cause the sales of some items to increase rapidly within a short period of time.The performance of SWF will be less a?ected by the data skew since SWF em-ploys the cumulative information for pruning false candidate itemsets in the early stage.

Extensive experiments are performed to assess the per-formance of SWF.As shown in the experimental results, SWF produces a signi?cantly smaller amount of candidate 2-itemsets than prior algorithms.In fact,the number of the candidate itemsets C k s generated by SW F approaches to its theoretical minimum,i.e.,the number of frequent k-itemsets,as the value of the minimal support increases.It is shown by our experiments that SWF in general signi?cantly outperforms FUP-based algorithms.Explicitly,the execu-tion time of SWF is,in orders of magnitude,smaller than those required by prior algorithms.Sensitivity analysis on various parameters of the database is also conducted to pro-vide many insights into algorithm SWF.The advantage of SWF becomes even more prominent not only as the amount of incremented dataset increases but also as the size of the database increases.

The rest of this paper is organized as follows.Prelimi-naries and related works are given in Section2.Algorithm

SW F is described in Section3.Performance studies on

various schemes are conducted in Section4.This paper

concludes with Section5.

2.PRELIMINARIES

Let I={i1,i2,...,i m}be a set of literals,called items.Let D be a set of transactions,where each transaction T is a set

of items such that T?I.Note that the quantities of items

bought in a transaction are not considered,meaning that

each item is a binary variable representing if an item was

bought.Each transaction is associated with an identi?er,

called TID.Let X be a set of items.A transaction T is said to contain X if and only if X?T.An association rule is an

implication of the form X=?Y,where X?I,Y?I and

X T Y=φ.The rule X=?Y holds in the transaction set D with confidence c if c%of transactions in D that contain

X also contain Y.The rule X=?Y has support s in the

transaction set D if s%of transactions in D contain X S Y. For a given pair of con?dence and support thresholds,the problem of mining association rules is to?nd out all the association rules that have con?dence and support greater than the corresponding thresholds.This problem can be reduced to the problem of?nding all frequent itemsets for the same support threshold[2].

Most of the previous studies,including those is[2,5,8,

16,20,22],belong to Apriori-like approaches.Basically,an

Apriori-like approach is based on an anti-monotone Apriori

heuristic[2],i.e.,if any itemset of length k is not frequent

in the database,its length(k+1)super-itemset will never be frequent.The essential idea is to iteratively generate the set of candidate itemsets of length(k+1)from the set of frequent itemsets of length k(for k≥1),and to check their corresponding occurrence frequencies in the database.As a result,if the largest frequent itemset is a j-itemset,then an Apriori-like algorithm may need to scan the database up to (j+1)times.

In Apriori-like algorithms,C3is generated from L2?L2.In

fact,a C2can be used to generate the candidate3-itemsets.

This technique is referred to as scan reduction in[6].Clearly,

a C03generated from C2?C2,instead of from L2?L2,will have a size greater than|C3|where C3is generated from L2?L2.However,if|C03|is not much larger than|C3|,and both C2and C3can be stored in main memory,we can ?nd L2and L3together when the next scan of the database is performed,thereby saving one round of database scan. It can be seen that using this concept,one can determine all L k s by as few as two scans of the database(i.e.,one initial scan to determine L1and a?nal scan to determine all other frequent itemsets),assuming that C0k for k≥3is generated from C0k?1and all C0k for k>2can be kept in the memory.In[7],the technique of scan-reduction was utilized and shown to result in prominent performance improvement.

3.SWF:INCREMENTAL MINING WITH

SLIDING-WINDOW FILTERING

In essence,by partitioning a transaction database into sev-eral partitions,algorithm SWF employs a?ltering threshold in each partition to deal with the candidate itemset gen-eration.As described earlier,under SWF,the cumulative information in the prior phases is selectively carried over to-

Figure2:An illustrative transaction database ward the generation of candidate itemsets in the subsequent phases.In the processing of a partition,a progressive candi-date set of itemsets is generated by SWF.Explicitly,a pro-gressive candidate set of itemsets is composed of the follow-ing two types of candidate itemsets,i.e.,(1)the candidate itemsets that were carried over from the previous progressive candidate set in the previous phase and remain as candidate itemsets after the current partition is taken into consider-ation(Such candidate itemsets are called typeαcandidate itemsets);and(2)the candidate itemsets that were not in the progressive candidate set in the previous phase but are newly selected after only taking the current data partition into account(Such candidate itemsets are called typeβcan-didate itemsets).As such,after the processing of a phase, algorithm SWF outputs a cumulative?lter,denoted by CF, which consists of a progressive candidate set of itemsets, their occurrence counts and the corresponding partial sup-port required.With these design considerations,algorithm SWF is shown to have very good performance for incremen-tal mining.In Section3.1,an illustrative example of SWF is presented.A detailed description of algorithm SWF is given in Section3.2.

3.1An example of incremental mining by SWF Algorithm SW F proposed can be best understood by the illustrative transaction database in Figure2and Figure3 where a scenario of generating frequent itemsets from a transaction database for the incremental mining is given. The minimum transaction support is assumed to be s= 40%.Without loss of generality,the incremental mining problem can be decomposed into two procedures:

1.Preprocessing procedure:This procedure deals with mining on the original transaction database.

2.Incremental procedure:The procedure deals with the update of the frequent itemsets for an ongoing time-variant transaction database.

The preprocessing procedure is only utilized for the ini-tial mining of association rules in the original database,e.g., db1,n.For the generation of mining association rules in db2,n+1,db3,n+2,db i,j,and so on,the incremental proce-dure is employed.Consider the database in Figure2.As-sume that the original transaction database db1,3is seg-mented into three partitions,i.e.,{P1,P2,P3},in the pre-

{A }, {B }, {C }, {D }, {E }, {F }, {B D }, {B E }, {D E }

C a n d id a te s in d b 1,3

:

{A }, {B }, {C }, {D }, {E }, {F }, {B D }, {B E }, {D E }, {D F }, {E F }, {B D E }, {D E F }

L a rg e Item s ets in d b 2,4

:

{A }, {B }, {C }, {D }, {E }, {F }, {A B }, {A C }, {B C }, {B D }, {B E }, {A B C }

L a rg e Item s ets in d b 1,3

:

{A }, {B }, {C }, {D }, {E }, {F }, {A B }, {A C }, {B C }, {B E }C a n d id a te s in d b 1,3:

Figure 3:Large itemsets generation for the incre-mental mining with SWF

processing procedure.Each partition is scanned sequentially for the generation of candidate 2-itemsets in the ?rst scan of the database db 1,3.After scanning the ?rst segment of 3transactions,i.e.,partition P 1,2-itemsets {AB,AC,AE,AF,BC,BE,CE}are generated as shown in Figure 3.In addition,each potential candidate itemset c ∈C 2has two attributes:(1)c.start which contains the identity of the starting partition when c was added to C 2,and (2)c.count which contains the number of occurrences of c since c was added to C 2.Since there are three transactions in P 1,the partial minimal support is d 3?0.4e =2.Such a partial minimal support is called the ?ltering threshold in this pa-per.Itemsets whose occurrence counts are below the ?lter-ing threshold are removed.Then,as shown in Figure 3,only {AB,AC,BC},marked by “°”,remain as candidate item-sets (of type βin this phase since they are newly generated)whose information is then carried over to the next phase of processing.

Similarly,after scanning partition P 2,the occurrence counts of potential candidate 2-itemsets are recorded (of type αand type β).From Figure 3,it is noted that since there are also 3transactions in P 2,the ?ltering threshold of those item-sets carried out from the previous phase (that become type αcandidate itemsets in this phase)is d (3+3)?0.4e =3and that of newly identi ?ed candidate itemsets (i.e.,type βcandidate itemsets)is d 3?0.4e =2.It can be seen from Figure 3that we have 5candidate itemsets in C 2after the processing of partition P 2,and 3of them are type αand 2of them are type β.

Finally,partition P 3is processed by algorithm SW F .The resulting candidate 2-itemsets are C 2={AB,AC,BC,BD,BE }as shown in Figure 3.Note that though appearing in the previous phase P 2,itemset {AD }is removed from C 2once P 3is taken into account since its occurrence count does

not meet the ?ltering threshold then,i.e.,2<3.However,we do have

one new itemset,i.e.,BE ,which joins the C 2as a type βcandidate itemset.Consequently,we have 5candidate 2-itemsets generated by SWF,and 4of them are of type αand one of them is of type β.Note that instead of 15candidate itemsets that would be generated if Apriori were used 1,only 5candidate 2-itemsets are generated by SW F .

After generating C 2from the ?rst scan of database db 1,3,we employ the scan reduction technique and use C 2to gen-erate C k (k =2,3,...,n ),where C n is the candidate last -itemsets.It can be veri ?ed that a C 2generated by SWF can be used to generate the candidate 3-itemsets and its sequen-tial C 0k ?1can be utilized to generate C 0k .Clearly,a C 0

3gen-erated from C 2?C 2,instead of from L 2?L 2,will have a size greater than |C 3|where C 3is generated from L 2?L 2.How-ever,since the |C 2|generated by SWF is very close to the

theoretical minimum,i.e.,|L 2|,the |C 0

3|is not much larger

than |C 3|.Similarly,the |C 0k |is close to |C k |.All C 0

k

can be stored in main memory,and we can ?nd L k (k =1,2,...,n )together when the second scan of the database db 1,3is per-formed.Thus,only two scans of the original database db 1,3are required in the preprocessing step.In addition,instead of recording all L k s in main memory,we only have to keep C 2in main memory for the subsequent incremental mining of an ongoing time variant transaction database.Table 1:Meanings of symbols used

The merit of SWF mainly lies in its incremental proce-dure.As depicted in Figure 3,the mining database will be moved from db 1,3to db 2,4.Thus,some transactions,i.e.,t 1,t 2,and t 3,are deleted from the mining database and other transactions,i.e.,t 10,t 11,and t 12,are added.For ease of ex-position,this incremental step can also be divided into three sub-steps:(1)generating C 2in D ?=db 1,3???,(2)gener-ating C 2in db 2,4=D ?+?+and (3)scanning the database db 2,4only once for the generation of all frequent itemsets L k .In the ?rst sub-step,db 1,3???=D ?,we check out the pruned partition P 1,and reduce the value of c.count and set c.start =2for those candidate itemsets c where c.start =1.It can be seen that itemsets {AB,AC,BC}were removed.Next,in the second sub-step,we scan the incremental trans-actions in P 4.The process in D ?+?+=db 2,4is similar to the operation of scanning partitions,e.g.,P 2,in the pre-processing step.Three new itemsets,i.e.,DE,DF,EF,join the C 2after the scan of P 4as type βcandidate itemsets.

Finally,in the third sub-step,we use C 2to generate C 0

k

as mentioned above.With scanning db 2,4

only once,SWF ob-1

The details of the execution procedure by Apriori are omit-ted here.Interested readers are referred to [3].

tains frequent itemsets {A,B,C,D,E,F,BD,BE,DE}in db 2,4.

3.2Algorithm of SWF

For ease exposition,the meanings of various symbols used are given in Table 1.The preprocessing procedure and the incremental procedure of algorithm SWF are described in Section 3.2.1and Section 3.2.2,respectively.

3.2.1Preprocessing procedure of SWF

The preprocessing procedure of Algorithm SWF is out-lined below.Initially,the database db 1,n is partitioned into n partitions by executing the preprocessing procedure (in Step 2),and CF,i.e.,cumulative ?lter,is empty (in Step

3).Let C i,j

2be the set of progressive candidate 2-itemsets generated by database db i,j .It is noted that instead of keep-ing L k s in the main memory,algorithm SWF only records C 1,n

2

which is generated by the preprocessing procedure to be used by the incremental procedure.

Preprocessing procedure of Algorithm SWF 1.n =Number of partitions;2.|db 1,n

|=P k =1,n |P k |;3.CF =?;

4.begin for k =1to n //1st scan of db 1,n

5.begin for each 2-itemset I ∈P k

6.if (I /∈CF )

7.I.count =N p k (I );

8.I.start =k ;

9.if (I.count ≥s ?|P k |)10.CF =CF ∪I ;11.if (I ∈CF )12.I.count =I.count +N p k (I );13.if (I.count

17.select C 1,n

2from I where I ∈CF ;

18.keep C 1,n

2in main memory;19.h =2;//C 1is given

20.begin while (C 1,n

h =?)//Database scan reduction

21.C 1,n h +1=C 1,n h ?C 1,n

h ;22.h =h +1;23.end

24.refresh I.count =0where I ∈C 1,n

h ;25.begin for k =1to n //2nd scan of db 1,n

26.for each itemset I ∈C 1,n

h 27.I.count =I.count +N p k (I );28.end

29.for each itemset I ∈C 1,n

h 30.if (I.count ≥d s ?|db 1,n |e )31.L h =L h ∪I ;32.end

33.return L h ;

From Step 4to Step 16,the algorithm processes one par-tition at a time for all partitions.When partition P i is pro-cessed,each potential candidate 2-itemset is read and saved to CF.The number of occurrences of an itemset I and its starting partition are recorded in I.count and I.start ,re-spectively.An itemset,whose I.count ≥d s ?P m =I.start,k |P m |e ,

will be kept in CF.Next,we select C 1,n

2from I where I ∈CF

and keep C 1,n

2in main memory for the subsequent incremen-tal procedure.With employing the scan reduction technique

from Step 19to Step 23,C 1,n

h s (h ≥3)are generated in main

memory.After refreshing I.count =0where I ∈C 1,n

h

,we begin the last scan of database for the preprocessing proce-dure from Step 25to Step 28.Finally,those itemsets whose I.count ≥d s ?|db 1,n |e are the frequent itemsets.

3.2.2Incremental procedure of SWF

As shown in Table 1,D ?indicates the unchanged portion of an ongoing transaction database.The deleted and added portions of an ongoing transaction database are denoted by 4?and 4+,respectively.It is worth mentioning that the sizes of 4+and 4?,i.e.,|4+|and |4?|respectively,are not required to be the same.The incremental procedure of SWF is devised to maintain frequent itemsets e ?ciently and e ?ectively.This procedure is outlined below.

Incremental procedure of Algorithm SWF 1.Original database =db m,n ;2.New database =db i,j ;3.Database removed 4?=P

k =m,i ?1P k ;

4.Database database 4+=P

k =n +1,j P k ;

5.D ?=P

k =i,n P k ;

6.db i,j =db m,n ?4?+4+;

7.loading C m,n 2

of db m,n into CF where I ∈C m,n

2;8.begin for k =m to i ?1//one scan of 4?

9.begin for each 2-itemset I ∈P k 10.if (I ∈CF and I.start ≤k )11.I.count =I.count ?N p k (I );12.I.start =k +1;13.if (I.count

17.begin for k =n +1to j //one scan of 4+18.begin for each 2-itemset I ∈P k 19.if (I /∈CF )20.I.count =N p k (I );21.I.start =k ;22.if (I.count ≥s ?|P k |)23.CF =CF ∪I ;24.if (I ∈CF )25.I.count =I.count +N p k (I );26.if (I.count

30.select C i,j

2from I where I ∈CF ;

31.keep C i,j

2in main memory 32.h =2//C 1is well known.

33.begin while (C i,j

h =?)//Database scan reduction

34.C i,j h +1=C i,j h ?C i,j

h ;35.h =h +1;36.end

37.refresh I.count =0where I ∈C i,j

h ;38.begin for k =i to j //only one scan of db i,j

39.for each itemset I ∈C i,j

h 40.I.count =I.count +N p k (I );41.end

42.for each itemset I ∈C i,j

h 43.if (I.count ≥d s ?|db i,j |e )44.L h =L h ∪I ;45.end

46.return L h ;

As mentioned before,this incremental step can also be divided into three sub-steps:(1)generating C 2in D ?=db 1,3???,(2)generating C 2in db 2,4=D ?+?+and (3)scanning the database db 2,4only once for the genera-tion of all frequent itemsets L k .Initially,after some up-date activities,old transactions 4?are removed from the database db m,n and new transactions 4+are added (in Step 6).Note that 4??db m,n .Denote the updated database as db i,j .Note that db i,j =db m,n ?4?+4+.We de-note the unchanged transactions by D ?=db m,n ???=

db i,j ??+.After loading C m,n

2of db m,n into CF where

I ∈C m,n

2

,we start the ?rst sub-step,i.e.,generating C 2in D ?=db m,n ???.This sub-step tries to reverse the cu-mulative processing which is described in the preprocessing procedure.From Step 8to Step 16,we prune the occur-rences of an itemset I ,which appeared before partition P i ,by deleting the value I.count where I ∈CF and I.start

new potential C i,j

2in db i,j =D ?+4+and employs the

scan reduction technique to generate C i,j h

s from C i,j 2.Fi-nally,to generate new L k s in the updated database,we scan db i,j for only once in the incremental procedure to maintain

frequent itemsets.Note that C i,j

2is kept in main memory for the next generation of incremental mining.

Note that SWF is able to ?lter out false candidate itemsets in P i with a hash table.Same as in [16],using a hash table to prune candidate 2-itemsets,i.e.,C 2,in each accumulative ongoing partition set P i of transaction database,the CPU and memory overhead of SWF can be further reduced.

Table 2:Meanings of various parameters

4.EXPERIMENTAL STUDIES

To assess the performance of algorithm SWF,we per-formed several experiments on a computer with a CPU clock rate of 450MHz and 512MB of main memory.The trans-action data resides in the NTFS ?le system and is stored on a 30GB IDE 3.5”drive with a measured sequential through-put of 10MB/second.The simulation program was coded in C++.The methods used to generate synthetic data are described in Section 4.1.The performance comparison of SWF,FUP 2and Apriori is presented in Section 4.2.Section 4.3shows the I/O and CPU overhead among SWF,FUP 2H and Apriori.Results on scaleup experiments are presented in Section 4.4.

4.1Generation of synthetic workload

For obtaining reliable experimental results,the method to generate synthetic transactions we employed in this study is similar to the ones used in prior works [4,8,16,21].Ex-plicitly,we generated several di ?erent transaction databases from a set of potentially frequent itemsets to evaluate

the

Figure 4:Relative performance

performance of SWF.These transactions mimic the transac-tions in the retailing environment.Note that the e ?ciency of algorithm SWF has been evaluated by some real databases,such as Web log records and grocery sales data.However,we show the experimental results from synthetic transaction data so that the work relevant to data cleaning,which is in fact application-dependent and also orthogonal to the in-cremental technique proposed,is hence omitted for clarity.Further,more sensitivity analysis can then be conducted by using the synthetic transaction data.Each database con-sists of |D |transactions,and on the average,each trans-action has |T |items.Table 2summarizes the meanings of various parameters used in the experiments.The mean of the correlation level is set to 0.25for our experiments.

Recall that the sizes of |?+|and |??|are not required to be the same for the execution of SWF.Without loss of generality,we set |d |=|?+|=|??|for simplicity.Thus,by denoting the original database as db 1,n and the new mining database as db i,j ,we have |db i,j |=|db 1,n ???+?+|=|D|,where ??=db 1,i ?1and ?+=db n +1,j .In the following,we use the notation T x ?Iy ?Dm ?dn to represent a database in which D =m thousands,d =n thousands,|T |=x ,and |I |=y .We compare relative performance of three methods,i.e.,Apriori,FUP-based algorithms and SWF.

As mentioned before,without any provision for incre-mental mining,Apriori algorithm has to re-run the associa-tion rule mining algorithm on the whole updated database.As reported in [8,9],with reducing the candidate item-sets,FUP-based algorithms outperform Apriori.As will be shown by our experimental results,with the sliding win-dow technique that carries cumulative information selec-tively,the execution time of SWF is,in orders of magnitude,smaller than those required by prior algorithms.In order to

Figure5:I/O cost performance

Figure6:Reduction on candidate itemsets when the dataset T10-I4-D100-d10was used

conduct our experiments on a database of size db i,j with an increment of?+and a removal of??,a database of db1,j is?rst generated and then db1,i?1,db1,n,db n+1,j,and db i,j are produced separately.

4.2Experiment one:Relative performance We?rst conducted several experiments to evaluate the relative performance of Apriori,FUP2and SWF.For inter-est of space,we only report the results on|L|=2000and N=10000in the following experiments.Figure4shows the relative execution times for the three algorithms as the minimum support threshold is decreased from1%support to 0.1%support.When the support threshold is high,there are only a limited number of frequent itemsets produced.How-ever,as the support threshold decreases,the performance di?erence becomes prominent in that SWF signi?cantly out-performs both FUP2and Apriori.As shown in Figure4, SWF leads to prominent performance improvement for var-ious sizes of|T|,|I|and|d|.Explicitly,SWF is in orders of magnitude faster than FUP2,and the margin grows as the minimum support threshold decreases.Note that from our experimental results,the di?erence between FUP2and Apriori is consistent with that observed in[9].In fact,SWF outperforms FUP2and Apriori in both CPU and I/O costs, which are evaluated next.

4.3Experiment two:Evaluation of I/O and

CPU overhead

To evaluate the corresponding of I/O cost,same as in [18],we assume that each sequential read of a byte of data consumes one unit of I/O cost and each random read

of Figure7:Scaleup performance with the execution

time ratio between SWF and FUP

a byte of data consumes two units of I/O cost.Figure5

shows the number of database scans and the I/O costs of

Apriori,FUP2H,i.e.,hash-type FUP in[9],and SWF over data sets T10?I4?D100?d10.As shown in Figure5,

SWF outperforms Apriori and FUP2H where without loss of generality a hash table of250thousand entries is employed for those methods.Note that the large amount of database scans is the performance bottleneck when the database size does not?t into main memory.In view of that,SWF is advantageous since only one scan of the updated database is required,which is independent of the variance in minimum supports.

As explained before,SWF substantially reduces the num-

ber of candidate itemsets generated.The e?ect is particu-larly important for the candidate2-itemsets.The experi-mental results in Figure6show the candidate itemsets gen-erated by Apriori,FUP2H,and SWF across the whole pro-cessing on the datasets T10?I4?D100?d10with minimum support threshold s=0.1%.As shown in Figure6,SWF leads to a99%candidate reduction rate in C2when being compared to Apriori,and leads to a93%candidate reduc-tion rate in C2when being compared to FUP2H.Similar phenomena were observed when other datasets were used. This feature of SWF enables it to e?ciently reduce the CPU and memory overhead.Note that the number of candidate 2-itemsets produced by SWF approaches to its theoretical minimum,i.e.,the number of frequent2-itemsets.Recall that the C3in either Apriori or FUP2H has to be obtained by L2due to the large size of their C2.As shown in Fig-ure6,the value of|C k|(k≥3)is only slightly larger than that of Apriori or FUP2H,even though SWF only employs C2to generate C k s,thus fully exploiting the bene?t of scan reduction.

4.4Experiment three:Scaleup on the incre-

mental portion

To further understand the impact of|d|to the relative per-

formance of algorithms SWF and FUP-based algorithms,we conduct the scaleup experiments for both SWF and FUP2 with two minimum support thresholds0.2%and0.4%.The results are shown in Figure7where the value in y-axis corre-sponds to the ratio of the execution time of SWF to that of FUP2.Figure7shows the execution-time-ratio for di?erent values of|d|.It can be seen that since the size of|d|has less in?uence on the performance of SWF,the execution-time-

ratio becomes smaller with the growth of the incremental transaction number|d|.This also implies that the advan-tage of SWF over FUP2becomes even more prominent as the amount of incremental portion increases.

5.CONCLUSION

We explored in this paper an e?cient sliding-window?l-tering algorithm for incremental mining of association rules. Algorithm SWF not only signi?cantly reduces I/O and CPU cost by the concepts of cumulative?ltering and scan reduc-tion techniques but also e?ectively controls memory utiliza-tion by the technique of sliding-window partition.Extensive simulations have been performed to evaluate performance of algorithm SWF.With proper sampling and partitioning methods,the technique devised in this paper can be applied to progressive mining.

Acknowledgment

The authors are supported in part by the Ministry of Educa-tion Project No.89-E-FA06-2-4-7and the National Science Council,Project No.NSC89-2219-E-002-028and NSC89-2218-E-002-028,Taiwan,Republic of China.

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福建省福州格致中学2014-2015学年高一上学期期中考试英语试题 Word版无答案

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