Extracting Repitative Patterns from Fuzzy Temporal Data

Transcription

1 IOSR Joural of Computer Egieerig (IOSR-JCE) e-issn: ,p-ISSN: , Volume 8, Issue 6, Ver. I (Nov. - Dec. 26), PP 54-6.iosrjourals.org Extractig Repitative Patters from Fuzzy Temporal Data Fokrul Alom Mazarbhuiya Departmet of Iformatio Techology College of Computer Sciece & IT Albaha Uiversity Albaha, KSA Abstract: Associatio rules miig from temporal dataset is to fid associatios betee items that hold ithi certai time frame but ot throughout the dataset. This problem ivolves first discoverig frequet itemsets hich are frequet at certai time itervals ad the extractig associatio rules from such frequet itemsets. I practice, e may have datasets havig imprecise or fuzzy time attributes, e term such datasets as fuzzy temporal datasets. I such datasets, as the time of trasactio is imprecise, e may have frequet itemsets that are frequet i certai fuzzy time itervals. The algorithm [] fids all such frequet itemsets alog ith a collectio of list of fuzzy time itervals here each frequet itemset is havig a associated list of fuzzy time itervals here it is frequet. The list of fuzzy time itervals may sho some iterestig features e.g. the itemsets may be repetitive i ature. I this paper e propose a method of fidig all repetitive frequet itemsets. The method's efficacy is demostrated ith experimetal results. Keyords: Fuzzy time itervals, Fuzzy temporal datasets, Frequet itemsets, Similarity measure o fuzzy sets, Similar fuzzy itervals, Similar trapezoidal fuzzy umbers, Fuzzy distace betee fuzzy itervals, Aggregatio of fuzzy itervals I. Itroductio Miig associatio rules from datasets has bee proposed iitially [2] by R. Agaral et al for applicatio i large super markets. Large supermarkets have huge collectio of records of customer's daily sales. Aalyzig the buyig patters of the customers may help the maagemet i takig strategic busiess decisios such as hat are the thigs to put o sale, ho to put the goods o the shelves, ho to make strategy for future purchase etc. Associatio rules miig i temporal databases has bee cosidered as a vital data-miig problem. Super market trasactio data are ormally temporal because each trasactio (purchase of customer) ill have a associated time of trasactio (purchase). The problem of super market data miig has bee cosidered as ell-researched problem. Miig from dataset havig fuzzy temporal features have bee cosidered as a excitig data miig problem. A method of fidig frequet itemsets from such datasets have bee discussed i []. The method [] mies all frequet itemsets alog ith a collectio of list of fuzzy time itervals here every frequet itemset is coupled ith a fuzzy time itervals list. The fuzzy time itervals list associated ith ay frequet itemsets ca be exploited to extract some iterestig patters. For example, some fuzzy itervals may be repetitive i ature i.e. it may be repeatig or re-appears i the list after certai gap. We call the associated frequet itemset as repetitive frequet itemsets. I this article, e ited to study the problem of repetitive ature of a fuzzy time itervals list hich is associated ith a frequet itemset. Here e devise a techique to extract all frequet itemsets hich are repetitive i ature. First, e defie a similarity measure o the fuzzy time itervals preset i the list. To cosecutive fuzzy time itervals preset i a list are said to be similar if their similarity measure is less tha or equal to a pre-assig threshold ad similarly to cosecutive fuzzy time gaps (gap betee to cosecutive fuzzy time itervals) are said to be similar if their similarity measure is also less tha or equal to a pre-assig threshold. The article is orgaized as follos. I sectio II, e give a brief discussio o the recet developmets of the orks hich are closely related to our ork. I sectio III, e give importat defiitios, terms ad otatios available i the literature ad are used i this paper. I sectio IV, e preset our algorithm for miig repetitive frequet itemsets. Experimetal results are discussed o sectio V. Fially, e coclude the paper ith coclusio ad possible future extesio i sectio VI. II. Recet Works Agraal et al, has formulated the problem of associatio rule miig i 993 [2]. Give a large set I, of items ad a large trasactios sets D of ivolvig the items, the problem of associatio rule miig is to fid relatioships amog the items i.e. the occurrece of various items i the trasactios. A trasactio T supports a item 'a' if 'a' is preset i T. A itemset {a, b} is said to be supported by a trasactio T if T cotais each of 'a' ad 'b' of the itemset. A associatio rule is a expressio of the form A B here A I ad B I. We say, the associatio rule A B holds ith cofidece if % of the trasactio of D supportig A also supports B. The rule has support if % of the trasactios supports DOI:.979/ iosrjourals.org 54 Page

2 Extractig Repitative Patters from Fuzzy Temporal Data A B. A techique of associatio rules discovery is discussed i [3], ko as the A-priori algorithm. The there is a lot of subsequet refiemets, geeralizatios, extesios ad improvemets of orks [3]. Miig from temporal Data is cosidered as a extesio of covetioal data miig. Lately it has bee able to attract more people to ork i the area. By cosiderig ito accout the time features, some more appealig patters that are depedet o time ca be extracted. Usually there are to ide directios of temporal data miig [4]. Oe deals ith the discovery of causal relatioships amog temporally orieted evets. The other deals ith the fidig of similar patters ithi the same time sequece or amog differet time sequeces. The uderlyig problem is to extract frequet sequetial patters i the temporal datasets. The uderlyig problem is termed as sequece miig problem. I [5] the authors have discussed the problem of recogizig frequet episodes i a evet sequece. Icorporatig the time aspects the traditioal associatio rule miig process is exteded hich is termed as temporal associatio rule miig. I temporal associatio rules every rule is associated ith a time iterval i hich the rule holds. I such problem, it is required to fid valid time itervals i hich associatio rules hold. I [6], [7], [8] ad [9], the authors have ot oly discussed the problem of temporal data miig i details but also devised techiques ad algorithms for fidig such patters. I [6], a method for temporal associatio rules discovery is described. I [], to algorithms for fidig cyclic associatio rules from temporal dataset are discussed. The method discussed i [], requires the time iterval to be specified by user to split the data ito disjoit segmets like moths, eeks, days etc. The orks similar to [], are doe i [] ad [2] addig multiple graularities of time itervals (e.g. first orkig day of every moth) from hich both cyclic ad user defied caledar patters ca be foud. I [3], a algorithm for discoverig locally ad periodically frequet itemsets ad periodic associatio rules are discussed. The method [3], is a improvemet of other existig methods i the sese it is dyamic i ature ad it is less user-depedet.. Fuzzy caledric data miig ad fuzzy temporal data miig are discussed i [4] ad [5]. I [4], a method is discussed hich extracts user specified ill-defied fuzzy temporal patters from temporal datasets. The algorithm discussed i [5], discovers user-defied caledric patters from the similar dataset. Miig from fuzzy temporal dataset is discussed iitially [], ad subsequetly [6] ad [7]. I [6], a algorithm for clusterig of frequet itemsets from fuzzy temporal dataset is discussed. I their ork they have used a similarity measure o the fuzzy time itervals hich is defied i terms of variace of fuzzy itervals [8]. I [7], similar orks are doe usig a e similarity measures ad aggregatio of fuzzy umbers [9]. A efficiet hash tree based implemetatio of the algorithm for fidig patters from fuzzy temporal data is discussed i [2]. A trie-based implemetatio of the algorithm for fidig patters from fuzzy temporal dataset is discussed i [2] III. Problem Defiitio A. Fuzzy sets A fuzzy set A i E (uiverse of discourse) is characterized by a membership fuctio A (x) [,]. A (x) for x E represets the membership grade of x i A. Therefore, a fuzzy set A is defied as A={(x, A (x)), x E } A fuzzy set A is termed as ormal if A (x) = for at least oe x i E I geeral, a geeralized fuzzy umber A is expressed as ay fuzzy subset of the real lie R, hose membership fuctio A (x) satisfyig the folloig coditios. () A (x) is cotiuous mappig from R to the closed iterval [, ] (2) A (x)=, - x c (3) A (x)= L(x) is strictly icreasig o [c, a] (4) A (x)=, a x b (5) A (x)= R(x) is strictly decreasig o [b, d] (6) A (x)=, d x, here, a, b, c ad d are all real umbers. The above type of geeralized fuzzy umber is deoted by A=(c, a, b, d; ) LR. For =, it ill be a fuzzy iterval deoted by A=(c, a, b, d) LR. Whe the fuctios L(x) ad R(x) are straight lie, the A ill be a trapezoidal fuzzy umber deoted by (c, a, b, d). If a=b, the the above trapezoidal umber ill become a triagular fuzzy umber deoted by (c, a, d). The -cut of the fuzzy umber [s -a, s, s +a] is a closed iterval of the form [s +( -).a, s +(- ).a]. Similarly the -cut of the fuzzy iterval [s -a, s, s 2, s 2 +a] is also a closed iterval of the form [s +( -).a, s 2 +(- ).a]. DOI:.979/ iosrjourals.org 55 Page

3 Extractig Repitative Patters from Fuzzy Temporal Data B. Fuzzy distace The cocept fuzzy distace betee ay to trapezoidal fuzzy umbers is proposed i [22] as follos: Suppose A=(a, a 2, a 3, a 4 ), B=(b, b 2, b 3, b 4 ) be to trapezoidal fuzzy umbers (fuzzy itervals), havig graded mea itegratio represetatio are P(A), P(B) respectively. Also suppose d i =(a i -P(A)+b i -P(B))/2, i=, 2, 3, 4; c i = P(A)-P(B) + d i, i=, 2, 3, 4;... () the the fuzzy distace of A, B is C=(c, c 2, c 3, c 4 ) ad is expressed i equatio (). Obviously the fuzzy distace betee to trapezoidal umbers (fuzzy itervals) is also a trapezoidal umber (fuzzy itervals). Where the graded mea itegratio P(A) ad P(B) for the fuzzy itervals are give belo: L ( ) R ( ) P(A)= ( ) d / d 2... (2) L ( ) R ( ) ad P(B)= ( ) d / d 2 [see e.g. [23], [24], [25]]... (3), are betee ad, < C. Similarity measure betee to fuzzy itervals. Let F be a frequet itemset hich is frequet i to fuzzy itervals T=[a, a 2, a 3, a 4 ] ad S=[b, b 2, b 3, b 4 ], the the degree of similarity betee T ad S is give by 4 a b i sim(t, S)= i [see e.g. [23]]... (4) 4 If the value of sim(t, S) is less tha or equal to some pre-assiged threshold say, the S ad T are said to be almost similar. D. Aggregatio operator Suppose that [a, b, c, d ] ad [a 2, b 2, c 2, d 2 ],... [a, b, c, d ] be trapezoidal fuzzy umbers (fuzzy itervals), the arithmetic mea aggregatio operator [26] ill produces a trapezoidal fuzzy umber (fuzzy iterval) say [a, b, c, d] here a a i i, b b i i, c c i i ad d i i DOI:.979/ iosrjourals.org 56 Page d... (5) For a particular frequet itemset say F, if it's all fuzzy time itervals (hich are actually trapezoidal fuzzy umbers) i hich F is frequet are foud to be almost similar ad if ay to cosecutive fuzzy time itervals of F are foud to be equi-distat i terms similarity measure, the F is called as repetitive frequet itemset ith a period average of all its fuzzy time itervals. The averagig or aggregatio is doe usig the method defied i equatio (5). IV. Algorithm Proposed For fidig repetitive frequet itemsets, e apply to measures o the list of time itervals associated ith a frequet itemset amely the fuzzy distace ad similarity measure o ay to cosecutive fuzzy time itervals associated ith that frequet itemset (the defiitios of fuzzy distace ad similarity measure are give by equatio () ad equatio (4) respectively). The frequet itemsets alog ith their list time itervals are extracted by the method discussed i []. It is to be metioed here the fuzzy distace betee ay to fuzzy itervals is also a fuzzy iterval. If the fuzzy distace (fuzzy time gap) betee ay to cosecutive fuzzy time itervals associated ay frequet itemset are foud to be almost similar ad also the same fuzzy time itervals are almost similar (the defiitio of almost similar fuzzy time iterval is give i sectio III) the e call such frequet itemsets as repetitive frequet itemsets. No, to fid out such type of repetitive patters from frequet itemsets extracted by the method discussed i [], e proceed as follos. If the first fuzzy time iterval is almost similar to the secod fuzzy time iterval the e see hether the fuzzy distace (fuzzy time gap) betee the first ad the secod fuzzy time iterval is almost similar to the fuzzy distace (fuzzy time gap) betee the secod ad third fuzzy time itervals. If both fuzzy time itervals ad their fuzzy time gaps are almost similar, the e take the aggregate of the first fuzzy time gap ( fuzzy distace betee first to fuzzy time itervals) ad secod fuzzy time gap (fuzzy distace betee secod ad third fuzzy time itervals) ad see hether it is almost similar to the fuzzy distace (fuzzy time gap) betee the third ad the fourth fuzzy time itervals associated ith frequet itemset. Also e take aggregate of first ad secod fuzzy time itervals to check hether it is almost similar to third fuzzy time itervals. If the aggregate of first ad secod fuzzy time

4 Extractig Repitative Patters from Fuzzy Temporal Data iterval is almost similar to the third fuzzy time iterval ad if the aggregate of the fuzzy distace of the first to fuzzy time itervals is almost similar to the fuzzy distace of the third ad fourth fuzzy time iterval e proceed further. Otherise the process ill be stopped ith message that the give frequet itemset is ot repetitive. I geeral if the aggregate of the first (-) fuzzy time itervals associated ith a frequet itemset is almost similar to the -th fuzzy time iterval ad that of first (-2) fuzzy distaces (fuzzy time gaps) are almost similar to the (-)-th fuzzy time gap, the the aggregate of fuzzy time itervals is compared ith (+)-th fuzzy time iterval ad that of the first - fuzzy time gaps is compared ith the -th fuzzy time gap to check the repetitiveess of the frequet itemsets. Usig this procedure, e ca extract repetitive frequet itemsets if such frequet itemsets exist i the fuzzy temporal datasets. We describe belo the algorithm's pseudo code for extractig such repetitive frequet itemsets. Algorithm for fidig repetitive frequet itemsets from fuzzy temporal datasets for each frequet itemset i havig a list fuzzy time itervals i hich i is frequet do { I first fuzzy time iterval for i I 2 secod fuzzy time iterval for i compute sim(i, I 2 ) if (sim(i, I 2 )! ) { the I,I 2 are ot almost similar report that i is ot repetitive i ature cotiue /* proceed to the ext frequet itemset */ } = fig fuzzydist(i,i 2 ) avgit aggregate(i,i 2 ) flag = hile ( ot ed of fuzzy iterval list for i) do { Cit curret fuzzy time iterval fcit fuzzydist(cit, I 2 ) Compute sim(cit, I 2 ), if (sim(,fig, fcit) ) the avv(gftg) aggregatr(* fig, fcit) else { flag = ; break;} avg=aggregate(cit, I 2 ) if ( upto th fuzzy time itervals are almost similar) the avg aggregate((+)* avgit, avg) else {flag = ; break;} I 2 tit; +; } if (flag == ) report that i is ot repetitive i ature else report that i is repetitive i ature } Here the fuctio, fuzzydist() returs fuzzy distace betee ay to cosecutive fuzzy time itervals associated ith a frequet itemset i.e. fuzzy time gap, sim() returs the similarity value of ay to fuzzy time itervals aggregate() returs the aggregate of the fuzzy time itervals, is the threshold value upto hich the variatio of similarity measure ca be acceptable. V. Experimetal Results Ad Discussios For experimets, e have used a dataset hich is actually a super market dataset. The dataset is a retail dataset from a aoymous Belgia store ad is doated by Tom Brijs [27]. The dataset cotais 7 items ad 8862 trasactios besides other details like miimum umber of items i a trasactio, maximum umber of items i a trasactio ad average umber of items. The dataset does ot cotai fuzzy time stamp (fuzzy temporal features). So, e icorporate the fuzzy time attribute i it hich is are fuzzy umbers to make it compatible for the executio of our algorithm. With these details our experimetal evaluatio ill be as follos. DOI:.979/ iosrjourals.org 57 Page

5 Extractig Repitative Patters from Fuzzy Temporal Data We break the dataset ito differet sizes based o umber of trasactio. For example,, trasactios, 2, trasactios, 3, trasactios, 4, trasactios, 5, trasactios, 6, trasactios, 7, trasactios ad hole dataset, the execute the algorithm discussed i [] to fid frequet itemsets ith their list of fuzzy time itervals i hich they are frequet. No, e apply our algorithm o the list of fuzzy time itervals associated ith each frequet itemsets to fid the repetitive ature of the frequet itemsets. A partial vie of experimetal results are give i table-, figure- ad figure-2. Trasactio sizes No. of repetitive frequet itemsets, 2, 3, 4, 2 5, 3 6, 3 7, Table-: Repetitive frequet itemsets for differet trasactio sizes Figure-: Trasactio vs. Repetitive frequet itemsets Figure-: Trasactio vs. Repetitive frequet itemsets DOI:.979/ iosrjourals.org 58 Page

6 Extractig Repitative Patters from Fuzzy Temporal Data VI. Coclusios Ad Lies For Future Works A algorithm for fidig repetitive frequet itemsets from fuzzy temporal data is proposed i this paper. The algorithm takes iput from the results obtaied by the method discussed i []. The method discussed i [] extracts all frequet itemsets here each frequet itemset is associated ith a sequece of fuzzy time itervals here it is frequet. Our algorithm orks o the list of fuzzy time itervals associated ith every frequet itemsets to fid the repetitive ature of the frequet itemsets. For this purpose, e take to parameters, amely similarity measure of fuzzy time itervals ad fuzzy distace betee ay to cosecutive fuzzy time itervals associated ith a frequet itemset. The algorithm orks o the list of fuzzy time itervals i the folloig maer. For a frequet itemset say A, havig a list of fuzzy time itervals say (T, T 2, T 3,...T.), if T is almost similar to T 2 also fuzzy distace (T, T 2 ) is almost similar to fuzzy distace (T 2, T 3 ), the similarity of aggregate (T, T 2 ) ad T 3 is checked. Similarly the aggregate of first to fuzzy distaces is checked ith the ext. The process cotiues till ed of the fuzzy time iterval list or there is a dissimilarity at certai stage i.e. to cosecutive fuzzy time itervals are ot almost similar or to cosecutive fuzzy time gaps are ot almost similar. The output to the algorithm are those frequet itemsets for hich ay to cosecutive fuzzy time itervals are almost similar ad ay to cosecutive fuzzy time gaps are almost similar. Such frequet itemsets are called as repetitive frequet itemsets It is to be metioed here that the fuzzy distace of ay to cosecutive fuzzy itervals is same as the fuzzy time gap of the itervals hich is a fuzzy umber or fuzzy itervals. I practice, e may have certai frequet itemsets, hich are ot repetitive i ature. For example e may have some itemsets i hich fuzzy time gaps are almost similar, but the fuzzy time itervals i hich the itemsets are frequet may ot be almost similar. We may have some frequet itemsets here the fuzzy time itervals are almost similar, but the correspodig fuzzy time gaps may ot be almost similar. Also e may have frequet items for hich both fuzzy time itervals ad fuzzy time gaps may ot be almost similar. Our algorithm ca be suitably modified to extract all such frequet itemsets if they exist i the datasets. I future, e may ork i the folloig lies () We may extract associatio rules from such frequet itemsets. (2) We may optimize our algorithm to capture the patters more efficietly (3) We may do clusterig of frequet itemsets based some parameters. (4) We may classify the frequet sets. (5) We may also do outlier aalysis o the set of such frequet itemsets. Refereces []. F. A. Mazarbhuiya, ad Ekramul Hamid; Fidig Fuzzy Locally Frequet itemsets, Iteratioal Joural of Computer Sciece ad Iformatio Security, ISSN: , LJS Publisher ad IJCSIS Press, USA, 2. [2]. R. Agraal, T. Imieliski ad A. Sami; Miig associatio rules betee sets of items i large databases; Proceedigs of the ACM SIGMOD 93, Washigto, USA, May 993. [3]. R. Agraal ad R. Srikat; Fast algorithms for miig associatio rules, Proceedigs of the 2 th Iteratioal Coferece o Very Large Databases (VLDB 94), Satiago, Chile, Jue 994. [4]. J. F. Roddick, M. Spillopoulou; A Biblography of Temporal, Spatial ad Spatio-Temporal Data Miig Research; ACM SIGKDD, Jue 999. [5]. H. Mailla, H. Toivoe ad I. Verkamo; Discoverig frequet episodes i sequeces; KDD 95; AAAI, 2-25, August 995. [6]. J. M. Ale ad G.H. Rossi; A approach to discoverig temporal associatio rules; Proceedigs of the 2 ACM symposium o Applied Computig, March 2. [7]. X. Che ad I. 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7 Extractig Repitative Patters from Fuzzy Temporal Data [7]. F. A. Mazarbhuiya; Clusterig of Locally Frequet Paters over Fuzzy Temporal Datasets u s i g a N e S i m i l a r i t y M e a s u r e, Iteratioal Joural of Advace Research i Computer Sciece ad Maagemet Studies (IJARCSMS), Vol.. Issue, Jauary, 26, pp [8]. C. Carlsso ad R. Fuller, O possibilistic mea value ad variace of fuzzy umbers, Fuzzy Sets ad Systems, vol.22, pp , 2. [9]. George J. Klir ad Tia A. Folger, Fuzzy Sets, Ucertaity, ad Iformatio, Pretice-Hall of Idia Pvt. Ltd., Ne Delhi, 988. [2]. F. A. Mazarbhuiya; Miig local patters from fuzzy temporal data, Iteratioal Joural of Egieerig ad Applied Scieces (IJEAS), ISSN: , Volume-, Issue-, Jauary 26, INDI, pp [2]. F. A. Mazarbhuiya; A Efficiet Algorithm for Miig Fuzzy Temporal Data, Iteratioal Joural of Mathematics, Tred ad Techology (IJMTT), August, 26, Idia (accepted). [22]. S. H. Che ad C. H. Hsieh; Represetatio, Rakig, Distace, Similarity of L-R type fuzzy umber ad Applicatio, Australia Joural of Itelliget Iformatio Processig Systems, Vol. 6, No. 4, pp , Australia. [23]. S. M. Che; Ne methods for subjective metal orkload assessmet ad fuzzy risk aalysis. Cyberet. Syst. 27, S. [24]. H. Che ad C. H. Hsieh; Graded mea itegratio represetatio of geeralised fuzzy umber, Proc. of Coferece of Taia Fuzzy System Associatio, Taia, 998. [25]. S. H. Che ad C. H. Hsieh; Graded mea itegratio represetatio of geeralised fuzzy umber, Joural of the Chiese Fuzzy Sytem Associatio, Vol. 5, No. 2, pp. -7, Taia, 999. [26]. George J. Klir ad Tia A. Folger, Fuzzy Sets, Ucertaity, ad Iformatio, Pretice-Hall of Idia Pvt. Ltd., Ne Delhi, 988. [27]. Tom Brijs, G. Sie, K. Vahoof, ad G. Wets; Usig associatio rules for product assortmet decisios: A case study. I Koledge Discovery ad Data Miig (999), pp AUTHOR S PROFILE Fokrul Alom Mazarbhuiya received B.Sc. degree i Mathematics from Assam Uiversity, Idia ad M.Sc. degree i Mathematics from Aligarh Muslim Uiversity, Idia. After this he obtaied his Ph.D. degree i Computer Sciece from Gauhati Uiversity, Idia. He had bee servig as a Assistat Professor i College of Computer Sciece ad Iformatio Systems, Kig Khalid Uiversity, Abha, kigdom of Saudi Arabia from 28 to 2. Curretly, he is orkig as a Assistat Professor, Iformatio Techology, College of Computer Sciece aid Iformatio Techology, Al Baha Uiversity, Al Baha, Kigdom of Saudi Arabia. His research iterest icludes Data Miig, Iformatio security, Fuzzy Mathematics ad Fuzzy logic. DOI:.979/ iosrjourals.org 6 Page