And Ph.D. Candidate of Computer Science, University of Putra Malaysia 2 Faculty of Computer Science and Information Technology,

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1 (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, Efficient Candidacy Reduction Fo Fequent Patten Mining M.H Nadimi-Shahaki 1, Nowati Mustapha 2, Md Nasi B Sulaiman 2, Ali B Mamat 2 1 Faculty of Compute Engineeing, Islamic Azad Univesity, Najafabad banch, Ian, And Ph.D. Candidate of Compute Science, Univesity of Puta Malaysia 2 Faculty of Compute Science and Infomation Technology, Univesity of Puta Malaysia (UPM), Selango, Malaysia. Abstact Cetainly, nowadays knowledge discovey o extacting knowledge fom lage amount of data is a desiable task in competitive businesses. Data mining is a main step in knowledge discovey pocess. Meanwhile fequent pattens play cental ole in data mining tasks such as clusteing, classification, and association analysis. Identifying all fequent pattens is the most time consuming pocess due to a massive numbe of candidate pattens. Fo the past decade thee have been an inceasing numbe of efficient algoithms to mine the fequent pattens. Howeve educing the numbe of candidate pattens and compaisons fo suppot counting ae still two poblems in this field which have made the fequent patten mining one of the active eseach themes in data mining. A easonable solution is identifying a small candidate patten set fom which can geneate all fequent pattens. In this pape, a method is poposed based on a new candidate set called candidate head set o H which foms a small set of candidate pattens. The expeimental esults veify the accuacy of the poposed method and eduction of the numbe of candidate pattens and compaisons. Keywods- Data mining; Fequent pattens; Maximal fequent pattens; Candidate patten I. INTRODUCTION The explosive gowth of data in all business, govenment and scientific applications ceates enomous hidden knowledge in thei databases. Cetainly, in this decade knowledge discovey o extacting knowledge fom lage amount of data is a desiable task in competitive businesses. Fo example, daily a lage amount of puchase data called maket basket tansactions ae collected in the cashie countes of huge makets. The maket management systems ae inteested in analyzing the puchase data to undestand moe about the behavio of thei customes. The association analysis can epesent inteesting elationships hidden in lage data set in the fom of association ules. Fo example, 75% of customes who buy diapes also buy oange juice. These ules can be used to identify new oppotunities fo coss-selling makets poducts to thei customes. The association analysis is also useful in othe applications such as web mining scientific applications. Data mining theefoe appeas to addess the need of sifting useful infomation such as inteesting elationships hidden in lage databases. As shown in Figue 1, data mining is an essential step in the pocess of knowledge discovey fom data (KDD) to extact data pattens. It is a composite pocess of multiple disciplines including statistics, database systems, machine leaning, intelligent computing and infomation technology. Figue 1. Data mining as a main step in KDD pocess Since the fist intoducing [2], fequent pattens mining plays an impotant ole in data mining tasks such as clusteing, classification, pediction and especially association analysis. Fequent pattens ae itemsets o substuctues that exist in a data set with fequency no less than a use specified theshold. Identifying all fequent pattens is the most time consuming pocess due to a massive numbe of candidate pattens. In geneal, 2 i -1 candidate pattens can be geneated fom a data set contains i items. Theefoe the computational equiements fo fequent pattens mining ae vey expensive. Fo the past decade thee have been an inceasing numbe of efficient algoithms to mine the fequent pattens by satisfying the minimum suppot theshold. They ae almost based on thee fundamental fequent pattens mining methodologies: Apioi, FP-tee and Eclat [9]. The Apioibased algoithms significantly educe the size of candidate sets using the Apioi pinciple that says all subsets of an infequent itemset must be infequent. But they still suffe fom the geneate-and-test stategy. They mine fequent pattens by geneating candidates and checking thei fequency against the tansaction database. The FP-tee keeps only fequent items espect to the minimum suppot theshold by two database scan. Recently some FP-tee-based algoithms have been developed to captue the content of the tansaction database only by one database scan which can be vey useful fo incemental updating of fequent pattens [12]. They usually tavese the tee to mine fequent pattens without candidate geneation in same fashion. Only a few of them can fit the 1Coesponding Autho 230

2 content of the tansaction database in memoy to eliminate the database escanning and the mining model estuctuing. Meanwhile educing the numbe of candidate pattens and the compaisons fo suppot counting ae still two poblems in this field which have made the fequent patten mining one of the active eseach themes in data mining. A easonable solution is identifying a small candidate patten set fom which can geneate all fequent pattens. In this pape, a method is poposed based on a new candidate set called candidate head set o H which fom a small set of candidate pattens. It is an impovement of ou pevious method pesented fo maximal fequent patten mining [13]. The poposed method is based on pime numbe chaacteistics fo fequent patten mining including a data tansfomation technique, an efficient tee stuctue called Pime-based encoded and Compessed Tee o PC_Tee and mining algoithm PC_Mine. The salient diffeence is that mining pocess makes use of the candidate head set and its popeties to educe the numbe of the candidate sets and compaisons. The PC_Mine algoithm stives to find long pomising pattens duing of the initial steps based on the candidate head set and its popeties. Consequently, it punes the seach space efficiently. The est of this pape is oganized as follows. Section 2 intoduces the poblem and eviews some efficient elated woks. The poposed method is descibed in section 3. The expeimental esults and evaluation show in section 4. Finally, section 5 contains the conclusions and futue woks. (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, contains a numbe of shote fequent sub-pattens. Given the itemset lattice shown in Figue 2, which pesents the list of all possible itemsets fo L= {A, B, C, D, E}. A bute-foce appoach fo mining fequent itemsets is to count the suppot evey candidate itemsets in the lattice stuctue. It needs to compae each itemsets against evey tansaction. Obviously, this appoach can be vey expensive and it needs O (NML) compaisons, whee N is the numbe of tansactions, M is equal 2 i -1 candidate itemsets fo i items and L is maximum length of tansactions. Theefoe thee ae two main ideas to educe the computational complexity of fequent itemsets mining. Fistly, educing the numbe of candidate itemsets and secondly educing the numbe of tansactions those must be compaed to count the suppot of the candidate itemsets. II. PROBLEM DEFINITION AND RELATED WORK Simply, fequent pattens ae itemsets o substuctues that exist in a dataset with fequency no less than a use specified theshold. The fist definition of fequent itemset was intoduced fo mining tansaction databases (Agawal et al. 1993). A. Poblem Definition Let L= {i 1, i 2 i n } be a set of items and D be a tansaction database whee each tansaction T is a set of items and D be the numbe of tansactions in D. Given X= {i j i k } be a subset of L (j k and 1 j, k n) is called a patten. The suppot of the patten X o Sup (X) in D is the numbe of tansactions in D that contains X. The patten X will be called fequent if its suppot is no less than a use specified suppot theshold min_sup σ (0 σ D ). The poblem of fequent patten mining is finding all fequent pattens fom dataset D with espect to specified min_sup σ. Vaious kinds of fequent pattens can be mined fom diffeent kinds of data sets. In this eseach, we use itemsets (sets of items) as a data set and the poposed method is fo fequent itemset mining, that is, the mining of fequent itemsets fom tansactional data sets. Howeve, it can be extended fo othe kinds of fequent pattens. The complexity of fequent pattens mining fom a lage amount of data is geneating a huge numbe of pattens satisfying the minimum suppot theshold, especially when min_sup σ is specified low. This is because, all sub-patten of a fequent patten ae fequent as well. Theefoe a long patten Figue 2. The itemset lattice fo item set L= {A, B, C, D, E} B. Related wok the Agawal and Sikant [3] intoduced an inteesting popety called Apioi pinciple, to educe the numbe of candidate sets among fequent k-itemsets: if a k-itemset is fequent then all of its sub-itemsets must also be fequent. The Apioi algoithm find the fequent 1-itemsets by fist scanning the database, then geneating the candidate fequent 2-itemsets by using the fequent 1-itemsets, and check them against the database to find the fequent 2-itemsets. This pocess is iteates until it can not geneate any fequent k-itemsets fo some k. The Apioi algoithm has been extended by seveal extension fo impoving efficiency and scalability. The most impotant impovement of the Apioi which intoduced new technique o method such as patitioning technique [16], hashing technique [14], sampling appoach [17], dynamic itemset counting [5], use the Apioi pinciple appoach as well. In many cases, the Apioi appoach showed a good pefomance to educe the size of candidate sets. Howeve, in condition with a lage numbe of fequent pattens o low minimum suppot thesholds, it almost suffes inheently fom two poblems; multiple database scans that ae costly and geneating lots of candidates [9]

3 Han et al. [8] poposed fequent patten tee o FP-Tee as a pefix-based tee stuctue, and an algoithm called FP-gowth. The FP-Tee stoes only the fequent items in a fequencydescending ode. The highly compact natue of FP-tee enhances the pefomance of the FP-gowth. The FP-Tee constuction equies two data scans. The FP-gowth unlike the Apioi algoithm mines the complete set of fequent pattens without candidate geneation. Thee have been intoduced many extensions based on the FP-Tee appoach such as depthfist geneation by Agawal et al. [1], H-Mine by Pei et al. [15], aay-based implementation of pefix-tee-stuctue by Gahne and Zhu [7] and CanTee by leung et al [10]. The expeimental esults showed that FP-Tee and almost all its extensions have a high compactness ate fo dense data set. Howeve, they need a lage amount of memoy fo spase data set whee pobability fo shaing common paths is low [10, 12]. The pesentation of data which will be mined is an essential consideation in almost all algoithms. The mining algoithms can be classified accoding to two hoizontal and vetical database layouts. Both the Apioi and FP-gowth methods use hoizontal data fomat (i.e., {TID: itemset}) to mine fequent pattens. Zaki [18] poposed Eclat algoithm o Equivalence CLASS Tansfomation by using the vetical data fomat (i.e., {item: TID_set}). The Eclat uses the lattice theoy to epesent the database items. The esults showed that Eclat outpefoms Apioi significantly. Howeve, it needs an additional convesion step. This is because most databases use a hoizontal fomat. Moeove, it uses a Boolean powe set lattice that needs to much space to stoe the labels and tid-lists. Consequently, thee have been intoduced some efficient algoithms based on vetical layout. The Flex [11] is a lexicogaphic tee designed in vetical layout to stoe patten X and list of tansaction identifie whee patten X appeas. Its stuctue is esticted test-and-geneation instead of Apioi-like is esticted geneation-and-test. Thus nodes geneated ae cetainly fequent. The Flex tee is constucted in depth-fist fashion. The expeimental esults showed the Flex is an efficient algoithm to find long and maximal fequent pattens. Howeve, it needs a lage amount of memoy especially to stoe the list of tansaction identifie. (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, Vol. 6, No.3, 2009 technique, an efficient tee stuctue called Pime-based encoded and Compessed Tee o PC_Tee and mining algoithm PC_Mine. The salient diffeence is that mining pocess makes use of the candidate head set and its popeties to educe the numbe of candidate sets and compaisons. In fact the PC_Mine algoithm pune the seach space by using the candidate head set and it finds the most pomising candidate set efficiently. This section is followed by eviewing of the data tansfomation technique and the PC_Tee. Then the candidate head set and its popeties ae explained to show how they can use in the PC_Mine algoithm to educe the numbe of candidate sets. A. Data Tansfomation Technique As shown in Fig 1 the data tansfomation is an essential pocess in data pepocessing step which can educe the size of database. Obviously, educing of the size of database can enhance the pefomance of mining algoithms. Ou method uses a pime-based data tansfomation technique to educe the size of tansaction database. It tansfoms each tansaction into a positive intege called Tansaction Value (TV) duing of the PC_Tee constuction as follows: Given tansaction T = (tid, X) whee tid is the tansaction-id and X = {i j i k } is the tansaction-items o patten X. While the PC_Tee algoithm scans tansaction T, the tansfome pocedue consides a pime numbe p fo each item i in patten X, and then TV tid is computed by Equation 1 whee T= (tid, X), X = {i j i k } and i is pesented by p. = k tid p j TV (1) The data tansfomation technique utilizes Equation 1 based on simple following definitions: A positive intege N can be expessed by unique poduct N = pm1 pm2 K pm whee 1 2 p i is pime numbe, p 1 p p 2 pl p and mi is a positive intege, called the multiplicity of p [6]. i III. PROPOSED METHOD Based on elated woks esults, educing the numbe of candidate sets and compaisons (to count the suppot) ae two effective way to enhance the pefomance of mining pocess. They showed using the Apioi pinciple can educe the numbe of candidate sets and well-oganized tee stuctue such as FP-Tee which captues the content of the tansaction database educes the numbe of compaisons of the suppot counting. Theefoe, in this eseach, we aim to use both the Apioi pinciple and well-oganized tee stuctue in ou poposed method. We poposed a method by using a simple and effective tee stuctue fo maximal fequent patten mining which can captue all content of the tansaction database [13]. This eseach poposes an impovement of pevious vesion using a new efficient candidate set called candidate head set based on the Apioi pinciple to educe the numbe of candidate sets. The poposed method is also based on pime numbe chaacteistics including a data tansfomation Fo example, N = 1800=2 3 * 32 * 52. Fundamentally, thee is no duplicated item in tansaction T. Hence we estict the multiplicity only to m i = 1 without losing any significant infomation. Theefoe N can be poduced by P P K 1 2 P. To facilitate the tansfomation pocess used in ou method, let s examine it though an example. Let item set L= {A, B, C, D, E, F} and the tansaction database, DB, be the fist two columns of Table 1 with eight tansactions. The fouth column of Table 1 shows TV tid tansfomed fo all tansactions. Geneally the aveage length of items used in the benchmak datasets is smalle than those in the eal applications. Fo example custome puchase tansaction T c = (3, { , , }) fom a maket can be pesented by thid tansaction T= (3, {A, B, E}) in Table 1. Although, the length of items in tansaction T c is bigge than T but both T and T c can be tansfomed into the same TV 66 by using ou data tansfomation technique. Hence it is an item-length 232

4 (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, independent tansfomation technique. The expeiments TV ( n ) R = ( oot, n, n,..., n ) whee 0 j i. It means j j 0 1 j showed that by applying this data tansfomation technique, the size of eal tansaction databases can be educed moe than any TV stoed in nodes of the sub half [13]. descendant R = ( oot, n, n,..., n ) can be divided by TV TABLE I. TRANSACTION DATABASE DB AND ITS TRANSACTION VALUES TID Items Tansfomed TV 1 A, B, C, D, E 2, 3, 5, 7, A, B, C, D, F 2, 3, 5, 7, A, B, E 2, 3, A, C, D, E 2, 5, 7, C, D, F 5, 7, A, C, D, F 2, 5, 7, A, C, D 2, 5, C, D, F 5, 7, B. PC_Tee constuction Thee have been intoduced seveal methods to educe the complexity of fequent patten mining pocess using welloganized tee stuctue. Thus the tee stuctues have been consideed as a basic stuctue in pevious data mining eseach [8, 10-13]. Recently, we intoduced a novel tee stuctue called Pime-based encoded and Compessed Tee o PC_Tee [13].It is vey simple but still poweful to captue the content of tansaction database efficiently. Unlike the pevious methods, the PC_Tee is based on pime numbe chaacteistics. Moeove PC_Tee has some nice popeties which used to pune the seach space duing of mining pocess. Let s eview the PC_Tee. A PC_Tee consists of one oot labeled as null and some nodes that fom sub tees as childen of the oot. The node stuctue consists of seveal fields: value, local-count, globalcount, status and link. The value field stoes the TV made by the data tansfomation technique duing of insetion pocedue in the PC_Tee constuction algoithm. In fact, the value field egistes which tansaction this node epesents. The localcount field egistes the numbe of individual tansaction epesented by its node in the whole of tansaction database. It is set to 1 duing inseting cuent TV in a new node and o if thee is a node with same TV then the value of its local-count field is inceased by 1. Hence thee is no duplicated TV in the PC_Tee. The global-count field egistes fequency of its TV in its sub tee (descendant) to use in the suppot computing function of its TV. The status field is to keep tacking of tavesing which is changed fom 0 to 1 when a node visited in the tavesing pocedue. The link field is to fom sub tees. The PC_Tee constuction algoithm foms a PC_Tee by inseting TV(s) based on definitions below: Definition 1: TV of the oot is assumed null and can be divided by all TVs. Definition 2: Sub tee R = ( oot, n, n,..., n ) is a sub i 0 1 i descendant if only if stoed in node n j. j 0 1 j Definition 3: Sub descendant R = ( oot, n, n,..., n ) is a descendant if n i is a leaf. i 0 1 i Definition 4: In a PC_Tee, If TV (n ) = TV (n s ) then = s. The insetion pocedue inceases local - count field of node n by 1 if the cuent TV is equal with TV of n. Based on definition 1-4, the PC_Tee can be constucted by algoithm 1 as follow: Algoithm 1 (PC_Tee constuction). Input: A tansaction database DB. Output: The PC_Tee of DB. Method: The PC_Tee is constucted as follows. 1. Ceate the oot of the PC_Tee and label it as null. 2. Fo each tansaction T in DB, inset T into the PC_Tee as follow: 2.1. Scan T fom input file, tansfom T into its TV and update the item fequency table Add TV in the tee as follows If TV can be an element of existent descendant R then inset TV as follows. If thee is a node in R with same TV then incease its local-count and global-count fields and its childen s global-count field by 1; else ceate a new node, with its local-count field initialized to 1. Link the new node to its paent and childen. Incease its childen s global-count field by 1 and set its global-count field by summation value of its local-count field and values of its paent s global-count field Else; the TV cannot be an element of existent descendants. Ceate a new node, with its local-count and global-count initialized to 1. Link the new node to the oot as its paent and to its childen. Incease its childen s global-count field by 1. Figue 3 shows the PC_Tee constucted fo tansaction database shown in Table 1. Each node pesents a TV o patten followed by two numbes afte : to indicate the local-count and global-count espectively. Thee ae seveal impotant popeties of PC_Tee that can be deived fom the PC_Tee constuction pocedue. Popety 1: In the PC_Tee, all nodes ae aanged accoding to TV-descending ode. Popety 2: Impotant pocedues used in the PC_Tee algoithm ae almost done only by two simple mathematic opeations poduct and division. Obviously using mathematic opeations will enhance the pefomance instead of sting opeations

5 Figue 3. Step by step PC_Tee constuction C. PC_Mine algoithm using candidate head set In ou pevious wok [13], the PC_Tee was mined by the PC_Mine algoithm in a top-down tavesing fashion. It finds the maximal fequent pattens as the smallest epesentative set fom which all fequent pattens can be deived. The PC_Mine makes use fom supeset and subset puning to enhance the pefomance of maximal fequent patten mining. The weakness of PC_Mine is that it consides all sub tees o descendants with same possibility duing of tavesing. To solve this weakness, the PC_Mine is impoved by an efficient heuistic called candidate head set denote H to educe the effective banch facto of the PC_Tee as follows. Without consideing the oot, let R = ( n, n,..., n ) be a 0 1 i descendant in the PC_Tee, the node n 0 as the fist node of the descendant R that egistes the biggest TV in R is called head of the descendant R. In othe wod the immediate childen of the oot make the head set. Definition 6 candidate head: the positive intege h = p 1 K p which made by poduct pime j numbes p 1 K p is the candidate head of the descendant R j if only if h is the lagest subset of n 0 and sup( p ) min_ sup, 1 k k j. Accoding to the data tansfomation technique used in the PC_Tee constuction, TV (n 0 ) is a poduct of pime numbes p 1 K p which thei fequency egiste in the item k fequency table(see Table 2) by step 2.1 of the algoithm1. Using this table and the minimum suppot theshold, infequent pime numbes can be emoved fom TV (n 0 ) as head of descendant R to make its candidate head. Definition 7 candidate head set: Let R = R, K, R ) be the ( 0 j descendant set of the PC_Tee, (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, H = { h h is candidate head of R, 0 j such that if h, h H and h h h = h } is candidate head set of the s PC_Tee. s s Based on above definitions, the PC_Tee has some inteesting popeties which will facilitate fequent-patten mining. Given σ as min_sup and patten P and Q have been epesented by TV (P) and TV (Q) in descendant R espectively. Popety 3: P, Q R, sup( P) p sup( Q) if and only if TV ( Q) TV ( P) (i.e. TV (P) can be divided by TV (Q)). σ Popety 4: sup (P) and sup (P s paents in its descendants) p σ if and only if P is a maximal fequent patten. Now, this is possible to use the candidate head set to find a smalle candidate set fom which the maximal fequent pattens can be deived. Consequently all fequent pattens can be geneated by the maximal fequent patten set. Theefoe, based on the candidate head set definition and above popeties, we have the following algoithm fo fequent patten mining using PC_Tee. Algoithm 2 (PC_Mine: Mining fequent pattens fom the PC_Tee by candidate head set) Input: A tansaction database DB, epesented by PC_Tee and minimum suppot theshold σ. Output: The complete set of fequent Pattens of DB. Method: The PC_Mine mines fequent pattens as follows. 1: Make the candidate head set H by using the item fequency table. 2: Let k max denote the maximum size of itemsets in H. 3: F = F = { f sup( f ) min_ sup, f = 1}. 4: k = k 1 max 5: fo k downto 2 do 6: H k = { f f H, f = k}. 7: fo each f H and f F do k 8: if Sup ( f ) min_sup // popety 4 and coollay 1 // 9: F=F U all subsets of f. 10: else if k>2 then 11: add all (k-1)-subsets of f to H. 12: end if 13: end fo 14: end fo The coectness and completeness of the pocess in the PC_Mine algoithm should be justified. This is accomplished by fist intoducing impotant lemma and coollay as follow: 234

6 Lemma 1: Let U k i=1 H = H consist of all k-candidate head i sets, then the complete set of the fequent itemsets can be geneated by H. Rationale: Let F be the fequent items in DB, n be head of the descendant R and h (n) is the set of fequent items in n, i.e., h (n) = n F. Accoding to candidate head set definition, h (n) is the candidate head of the descendant R and belongs to H. In othe wod the candidate head set H used in the PC_Mine algoithm includes all fequent items. The elationship among the candidate head set H, fequent and maximal fequent itemsets ae shown in Figue 4. Figue 4. Relationship among candidate head set, fequent and maximal fequent itemsets Coollay 1: The complete set of the fequent itemsets F can be geneated by k U H ) i U 1 ( F i= 2 whee F 1 is set of 1-length fequent itemsets which can be deived fom item fequency table made in the PC_Tee algoithm. The numbe of the candidate head sets is vey smalle than the fequent and even maximal fequent itemsets. Howeve, Figue 4 shows that the candidate head set is a supeset of fequent and maximal itemsets. To illustate let s examine the mining pocess. In the PC_Mine algoithm, the line 3 makes 1-fequent itemsets by using the item fequency table which made duing the tee constuction pocess. In oute fo-loop between lines 5-14, all k-candidate heads (k>2) ae made and investigated espectively. Then duing of line 7-13 all fequent itemsets ae geneated by using maximal fequent itemsets found in line 8 based on popety 4. Accoding to the elationship shown in Figue 4 and steps of the PC_Mine algoithm which investigate all candidate heads; theefoe thee is no missing fequent itemsets. The PC_Mine algoithm make use fom supeset infequent puning based on the candidate head set intoduced in this eseach which is an effective way to eliminate some of the candidate itemsets. In fact the candidate head set definition is based on the Apioi pinciple which all subsets of a fequent itemset must also be fequent. Fo instance thee ae = 63 possible candidate itemsets in the tansaction database DB (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, pesented in Table1. The PC_Mine uses the fequency of items computed in step 2.1 of the PC_Tee algoithm to find the candidate head set. Table 2 shows the fequency of items of DB. Consequently, items B and E ae infequent when min_sup=4. Theefoe, the PC_Mine stats mining with candidate head set H includes only 4-candidate itemset ACDF and the PC_Mine only examined 6 candidate itemsets {ACDF, ACD, ACF, ADF, CDF, AF}. Above definitions and PC_Mine algoithm show that using the candidate head set can educe the numbe of candidate itemsets. Moeove the expeimental esults suppot the accuacy and efficiency of ou method. IV. EXPERIMENTAL RESULTS In this section, we evaluate the accuacy and pefomance of ou method by seveal expeiments. All expeiments wee pefomed in a time-shaing envionment in a 2.4 GHz PC with 2 GB memoy. We used seveal spase and dense datasets which used in pevious woks as benchmak datasets. The synthetic spase datasets ae geneated by the pogam developed at IBM Almaden Reseach Cente [3] and eal dense datasets ae download fom UC Ivine Machine Leaning Repositoy [4]. The esults epoted in figues wee computed by the aveage of multiple uns. Accoding to the space limitation and the poblem specifications, only the esults of expeiments by using synthetic spase dataset T10I5D100K and eal dense dataset mushoom which ae the most popula benchmak datasets in this field ae pesented in this pape. The numbe of tansactions, the aveage tansaction length, the numbe of items and the aveage fequent patten length of T10I5D100k ae set to 100k, 10, 1000 and 4 espectively. The mushoom dataset consists of the chaacteistics of vaious mushoom species. The numbe of ecods, the numbe of items and the aveage ecod length ae set to 8124, 119 and 23 espectively. The fist expeiment is to poof the accuacy of the poposed method. Howeve the coectness and completeness of the pocess in the PC_Mine algoithm was justified in pevious section. The numbe of fequent pattens mined by the PC_Mine algoithm vesus suppot using T10I5D100k and mushoom datasets is shown in Figues 5 and 6 espectively. We compaed all fequent pattens mined by the poposed method using the candidate head set with those mined by the Apioi algoithm in seveal datasets. They wee exactly equal. The second expeiment evaluates the numbe of the candidate sets geneated by the PC_Mine and Apioi which is one of the most efficient algoithms in tem of candidacy eduction. Figue 7 shows the numbe of candidate sets geneated by PC_Mine and Apioi algoithms vesus diffeent minimum suppot thesholds ove dataset T10I5D100k. The efficiency of using the candidate head set in the PC_Mine algoithm is veified by thid expeiment which compaes the un time vesus the suppot. Figue 8 and 9 pesent the efficiency of the PC_Mine algoithm using the candidate head set vesus the Apioi algoithm in T10I5D100k and mushoom datasets ove diffeent minimum suppot thesholds espectively

7 (IJCSIS) Intenational Jounal of Compute Science and Infomation Secuity, T10I5D100K T10I5D100k #Fequent Pattens Run time (sec) PC_Mine Apioi Suppot (%) Suppot(%) Figue 5. #Fequent Pattens Vs. Suppot Figue 8. Run time Vs. Suppot #Fequent Pattens Mushoom Suppot (%) Run time (sec) PC_Mine Apioi Mushoom Suppot(%) Figue 6. #Fequent Pattens Vs. Suppot Figue 9. Run time Vs. Suppot Candidate sets T10I4D100k geneated by PC_Mine Candidate sets geneated by Apioi Suppot(%) Figue 7. The numbe of candidate sets Vs. Suppot V. CONCLUSION AND FUTURE WORKS In this pape we intoduced the candidate head set to educe the numbe of candidate sets in mining pocess. Ou pevious method [13] was impoved by using the candidate head sets to popose an efficient method fo fequent patten mining. The expeimental esults veified the accuacy and efficiency compaing with the Apioi algoithm which is one of the most efficient algoithms in tem of candidacy eduction. Paticulaly, we intoduced a new method based on pime numbe chaacteistics using candidate head sets to find completed fequent pattens by using maximal fequent pattens. The poposed method can be impoved fo incemental mining of fequent pattens whee database tansactions can be inseted, deleted, and/o modified incementally. Moeove it can be impoved fo inteactive mining of fequent pattens whee minimum suppot theshold can be changed to find new coelation between pattens without eunning the mining pocess fom scatch

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