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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 Mining Maximal Dense Intervals from Temporal Interval Data F. A. Mazarbhuiya1 M.A.Khaleel1 A. K. Mahanta2 H. K. Baruah2 1 2 Dept. of Computer Science Department of Computer Science 1 2 College of Computer Science Gauhati University, India 1 2 King Khalid University, Abha Saudi Arabia Email: anjanagu@yahoo.co.in, hemanta_bh@yahoo.com 1 Email:{fokrul_2005, khaleel_dm}@yahoo.com Abstract- Some real life data are associated with duration of algorithm to mine maximal dense fuzzy intervals. In such cases, events instead of point events. The most common example of such we define the amount of contribution (also called vote) of a data is data of cellular industry where each transaction is transaction t associated with time interval [t1, t2] for a given associated with a time interval. Mining maximal fuzzy intervals fuzzy interval A as the ratio of the area bounded by the from such data allows the user to group the transactions with membership function A(x) (associated with the fuzzy interval) similar behavior together. Earlier works were devoted to mining frequent as well as maximal frequent non-fuzzy intervals. We and the real line included within the interval [t1, t2] to the total propose here a method of mining maximal dense fuzzy intervals area covered by A(x) and the real line. If the total average of the where density of an interval quite similar to the frequency of an votes of all the transactions in a fuzzy interval A exceeds a pre- interval. defined threshold, then the fuzzy interval is called a dense fuzzy interval. Similarly a dense fuzzy interval will be maximal if no Keywords- Frequent intervals, Maximal frequent intervals, Density dense fuzzy interval contains it. The well-known A-priori of a fuzzy interval, Minimum density, Contribution (vote) of a algorithm cannot be used here directly as the downward and transaction on a fuzzy interval, join of two fuzzy intervals. upward closure property of frequent sets does not hold in this case (it is proved with an example). We propose a variation of the A-priori algorithm that works in this situation and gives us I INTRODUCTION the maximal dense fuzzy intervals. Among the various types of data mining applications, analysis of transactional data has been considered important. One important extension of this mining problem is to include a II. RELATED WORKS temporal dimension. Most of the earlier works done in this area do not take into account the time factor. By taking into account One of the very useful extensions of conventional data mining the time aspect, more interesting patterns that are time dependent is temporal data mining. In recent times it has been able to attract can be extracted. Recently data mining in temporal data sets has a lot of researcher to work in this area. Considering the time arisen as an important data mining problem [[2], [10]]. dimension in the conventional data mining problem, more interesting patterns can be extracted that are time dependent. Many real life problems are associated with duration events There are mainly two broad directions of temporal data mining instead of point events. In this paper we are considering such [7]. One concerns the discovery of causal relationships among datasets i.e. dataset having time intervals. Such datasets are temporally oriented events. Ordered events from sequences and called as temporal interval datasets. A record in such data the cause of an event always occur before it. The other concerns typically consists of the starting time and ending time (or the the discovery of similar patterns within the same time sequence length of the transaction) in addition to other fields. In [5] an or among different time sequences. The underlying problem is to algorithm for mining maximal frequent intervals from such data find frequent sequential pattern in the temporal databases. sets has been given Wong et al [9] introduced the fuzzy concept into the In practice however most of the time people make statements association rule mining to deal with quantitative attributes. using vague terms like the early morning, late evening etc Quantitative attributes are normally handled by partitioning the instead of mentioning strict time intervals. There is no strict attribute domains and then combining adjacent partitions [8]. boundary for separating early morning from morning. To Although this method can solve problems introduced by finite represent such vague terms, fuzzy sets are required. In this paper domain, it causes the sharp boundary problem. To soften the we discuss the problem of mining dense intervals using a fuzzy affect of soft boundaries, fuzzy sets are used. Here each concept. The objective of this paper is three fold. First we quantitative attribute is associated with several fuzzy sets. A propose the definition of density of a fuzzy interval over a fuzzy association rule looks like if X is A then Y is B, where X transactional (where each transaction is associated with a time and Y are attributes and A and B are fuzzy sets which describe X duration) dataset. Secondly, we propose to define a join and Y respectively. Prade et al [6] defined support and operation on the fuzzy intervals and lastly we propose an confidence of a fuzzy association rule. 102 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 In [2], Rossi and Ale extended the well-known A-priori A(x) for all x ∈[a, b] is known as left reference function and A(x) algorithm for mining association rules to temporal data and for x ∈ [c, d] is known as the right reference function. The left described a technique to find interesting patterns on the data that reference function is non-decreasing and the right reference are time bounded. function is non-increasing [see e.g. [4]]. The area of a fuzzy In [5], the problem of mining maximal frequent intervals is interval is defined as the area bounded by the membership discussed. They define a maximal frequent interval as an interval function of the fuzzy interval and the real line. that is frequent which means that it is present in sufficient number of transactions and no other frequent interval contains it. B. Contribution (vote) of a transaction to a fuzzy interval Using a pre-fix traversal algorithm, the maximal frequent We define vote of a transaction t associated with the time intervals have been found and it was also found experimentally interval [t/, t//] for the fuzzy interval A= [a, b, c, d] as follows: that pre-order traversal algorithm outperforms the A-priori based t // algorithm. Our approach is different from the above approaches. We are votet A = ∫t/ A( x)dx d taking into account the fact that the intervals of time are of fuzzy nature. By calculating density of the fuzzy intervals in a ∫a A( x)dx particular transactional dataset where transactions are associated where A(x) is the membership function associated with the fuzzy with time intervals (non-fuzzy) as mentioned in the next section, interval. we first compute the dense fuzzy time intervals by using some t // user defined minimum density value and then apply a join Here ∫t/ A( x)dx is the portion of the area bounded by A(x) and operation to join neighboring intervals to find maximal dense d fuzzy intervals. The fuzzy intervals and their membership functions are provided by domain experts. the real line included in the time interval [t/, t//]. ∫a A( x)dx is the total area bounded by A(x) and the real line. III PROBLEM DEFINITION Obviously votet A lies in [0,1] and if A⊆[t/, t//], then votet A = / // 1 and if A∩[t , t ] =Φ, then votet A =0. A. Some basic definitions related to fuzziness Let E be the universe of discourse. A fuzzy set A in E is C. Density of a fuzzy time interval in a data set characterized by a membership function A(x) lying in [0,1]. A(x) The density of a fuzzy interval over a given temporal interval for x ∈E represents the grade of membership of x in A. Thus a dataset D is computed by summing up the votes of all the fuzzy set A is defined as transactions of D for the corresponding fuzzy time interval and A={(x, A(x)), x ∈ E } dividing it by the total number of transactions in D. Each record A Fuzzy set A is said to be normal if A(x) =1 for at least one x contributes a vote, which falls in [0, 1]. ∈ E. density D A = ∑ votet A / | D | An α-cut of a fuzzy set is an ordinary set of elements with t∈D membership grade greater than or equal to a threshold α, 0≤α≤1. A fuzzy interval is dense if its density is more than a user Thus an α-cut Aα of a fuzzy set A is characterized by specified threshold called min_density. Aα={x ∈E; A(x) ≥ α} [see e.g. [3]] A fuzzy set is said to be convex if all its α-cuts are convex sets. D. Join of two fuzzy intervals The fuzzy intervals are given by the user as input. Two fuzzy A fuzzy number is a convex normalized fuzzy set A defined intervals A and B are called neighbors or adjacent to each other on the real line R such that if supp(A ∩ B) ≠Φ where supp(A ∩ B) ={x; (A ∩ B)(x) > 0 }[see 1. there exists an x0 ∈ R such that A(x0) =1, and e.g.[4]]. We assume that the input fuzzy intervals are such that if 2. A(x) is piecewise continuous. the intervals are arranged in the ascending order according to Thus a fuzzy number can be thought of as containing the real their starting time then each fuzzy interval has a unique left numbers within some interval to varying degrees. neighbor and a unique right neighbor. Let A = [a1, b1, c1, d1] and Fuzzy intervals are special fuzzy numbers satisfying the B = [a2, b2, c2, d2] be two adjacent fuzzy intervals. Without loss following. of generality we can assume that a1 < a2. Also we assume that for 1. there exists an interval [a, b] ⊂ R such that A(x0) =1 for any two adjacent fuzzy intervals such as A and B above c1 = a2 all x0∈ [a, b], and and d1 = b2 and for c1 ≤ x ≤ d1 A(x) = 1 – B(x). Our assumption is 2. A(x) is piecewise continuous. natural since otherwise some points will be given more emphasis and some less emphasis. We define the join of A and B denoted A fuzzy interval can be thought of as a fuzzy number with a flat by A∧ B is defined as region. A fuzzy interval A is denoted by A = [a, b, c, d] with a < A∧ B = [a1, b1, c2, d2] b < c < d where A(a) = A(d) = 0 and A(x) = 1 for all x ∈[b, c]. 103 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 0, x ≤ 4 and x ≥ 9 Where (A∧ B)(x) = A(x), a1 ≤ x ≤ b1 B(x) = (x – 4)/2, 4≤ x ≤ 6 A(x) + B(x)=1,b1 ≤ x ≤ c2 1, 6≤x≤7 B(x) for c2 ≤ x ≤ d2 (9-x)/2, 7≤ x ≤ 9 To explain the joining operation we again consider two fuzzy 3 intervals [a1,b1,c1,d1] and [a2,b2,c2,d2] whose membership ∫ A( x)dx =1/3 votet1 A = 1 functions are shown in the figure1. Here c1 = a2 and b2 = d1. Any 6 point in between c1and d1 will have a membership value of A(x) corresponding to A and corresponding to B it will have a ∫ A( x)dx1 6 membership value of B(x) = 1 – A(x) so that A(x) + B(x) = 1. Thus our joined fuzzy interval will be [a1, b1, c2, d2] (shown in vote A = ∫ A( x)dx = 1 1 t2 6 fig.2). B C F G ∫ A( x)dx1 6 a1 b1 c1=a2 d1=b2 c2 d2 vote A = ∫ A( x)dx =2/3 3 t3 6 A E D H ∫ A( x)dx1 Fig 1: Join of two fuzzy intervals 6 B G vote A = ∫ A( x)dx = 2.75/3 2 t4 6 a1 b1 c2 d2 ∫ A( x)dx1 A H 7 Fig 2: Joined interval vote A = ∫ A( x)dx =.25/3 5 t5 6 A dense fuzzy interval is maximal if no super set of it is dense. ∫ A( x)dx1 However a subset of it may not be dense because the downward 7 and upward closure property for dense sets may not hold in this case. vote A = ∫ A( x)dx = 0 6 t6 6 E. Theorem ∫ A( x)dx 1 2 The join of two fuzzy intervals is not dense if both of the fuzzy intervals are not dense and dense if at least one of the fuzzy vote A = ∫ A( x)dx =.25/3 1 t7 6 intervals is dense. ∫ A( x)dx1 7 Proof. To prove the above result we consider a data set D with 8 transactions. The time-intervals associated with the transactions vote A = ∫ A( x)dx = .25/3 5 are shown below. t8 6 ∫ A( x)dx1 Transac Therefore, tion id t1 t2 t3 t4 t5 t6 t7 t8 votet1 A+ votet 2 A+ votet 3 A+ votet 4 A+ votet 5 A+ votet 6 A+ votet 7 A+ votet 8 A Time- Density ( A) = 8 interval [1,3] [1,6] [3,6] [2,6] [5,7] [6,7] [1,2] [5,7] [ti , tj] =3.1666666/8 Table1: Transaction datasets = 0.395833325 Similarly Consider the fuzzy intervals A = [1, 3, 4, 6] and B = [4, 6, 7, 9] 3 where the membership functions of A and B are respectively votet1 B= ∫ B( x)dx =0 1 9 0, x ≤ 1 and x ≥ 6 ∫ B( x)dx 4 A(x) = (x – 1)/2, 1≤ x ≤ 3 6 1, 3≤x≤4 votet2 B= ∫ B( x)dx = 1/3 1 (6-x)/2, 4≤ x ≤ 6 9 ∫ B( x)dx 4 and 104 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 6 7 ∫ B( x)dx = 1/3 ∫ ( A B)( x)dx =2/6 ^ vote B = t3 3 9 votet5 ( A B) = ^ 5 9 ∫ B( x)dx ∫ ( A B)( x)dx ^ 4 1 6 7 ∫ B( x)dx = 1/3 ∫ ( A B)( x)dx = 1/6 ^ vote B = t4 2 9 votet6 ( A B) = ^ 6 9 ∫ B( x)dx ∫ ( A B)( x)dx ^ 4 1 7 2 ∫ B( x)dx =1.75 ∫ ( A B)( x)dx =.25/6 ^ vote B = t5 5 9 votet7 ( A B) = ^ 1 9 ∫ B( x)dx ∫ ( A B)( x)dx ^ 4 1 7 7 ∫ B( x)dx = 1/3 ∫ ( A B)( x)dx = 2/6 ^ vote B = t6 6 9 votet8 ( A B) = ^ 5 9 ∫ B( x)dx ∫ ( A B)( x)dx ^ 4 1 2 vote B = ∫ B( x)dx =01 Therefore, t7 9 ∫ B( x)dx 4 Density ( A ^ B ) = votet1 A+votet 2 A+ votet 3 A+ votet 4 A+ votet 5 A+ votet 6 A+ votet 7 A+ votet 8 A 8 7 vote B = ∫ B( x)dx = 1.75/3 5 Therefore ^ Density ( A B ) = 2.83333/8 t8 9 = 0.35416625 ∫ B( x)dx 4 So if we take min_dense = 0.35 then we see that A is dense but B Therefore, is not dense whereas (A^B) is dense. This establishes that the downward as well as upward closure property is not satisfied for votet1 B + votet 2 B + votet 3 B + votet 4 B + votet 5 B + votet 6 B + votet 7 B + votet 8 B dense fuzzy intervals. Density ( B ) = 8 = 2.5/8 = 0.3125 IV. PROPOSED ALGORITH ^ Now, ( A B ) = [1, 3, 7, 9] The algorithm is a level wise algorithm similar to the A-priori algorithm used for frequent item set mining [1]. Input to the 0, x ≤ 1 and x ≥ 9 algorithm is a temporal interval data set say D, n fuzzy intervals ^ ( A B ) (x)= (x–1)/2, 1≤ x ≤ 3 (called basic fuzzy intervals here) satisfying both the assumptions made in definition of join of fuzzy intervals defined 1, 3≤x≤7 on the time period covered by the dataset and with a value of (9-x)/2, 7≤ x ≤ 9 min_density (minimum density value). The algorithm first finds 3 the dense basic fuzzy intervals by going through the dataset once ∫ ( A B)( x)dx =1/6 ^ and using the definition C given in section III. They are dense votet1 ( A B) = ^ 1 9 fuzzy intervals at level 1 we denote this set of dense intervals by ∫ ( A B)( x)dx L1. Next each dense fuzzy interval at level 1 is joined with its left ^ 1 neighbour and right neighbour both of which are basic intervals 6 (may not be dense) using the join operation defined definition D ∫ ( A B)( x)dx = 4/6 ^ in section III. They are the candidates C2 at level 2. Using the votet2 ( A B) = ^ 1 9 same technique, going through the data set once more the dense ∫ ( A B)( x)dx ^ 1 fuzzy intervals at level 2 say L2 are obtained. These are kept and 6 the others removed. If any of the intervals obtained by joining a ∫ ( A B)( x)dx = 3/6 ^ dense interval say A with its neighbours turn out to be dense then votet3 ( A B) = ^ 3 9 A is removed from the list of dense intervals maintained at the ∫ ( A B)( x)dx ^ previous level. This level wise extraction goes on till a particular 1 6 level becomes empty. Then the intervals kept at each level are ∫ ( A B)( x)dx =2.75/6 ^ the maximal dense fuzzy intervals. It is mentioned here that at votet4 ( A B) = ^ 2 9 any level the dense intervals are joined with their neighbors from ∫ ( A B)( x)dx the basic fuzzy intervals only. This is done because two new ^ 1 fuzzy intervals obtained by joining basic intervals although 105 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 neighbors may not satisfy our second assumption (Definition D) Thus the set of first level dense fuzzy number is for being conformable for the join operation. When two intervals L1= {D, E} A and B are joined where A is the left neighbor of B, then the left Candidates for the second pass are neighbor of A becomes the left neighbor of A^B and the right C2 = {C^D, D^E, E^F} neighbor of B becomes the right neighbor of A ^B. where each members of C2 are formed by joining the members of L1 with their left right neighbor of C1 using the definition of join and C^D = [3, 4, 5, 6], D^E = [4, 5, 6, 7]. E^F = [5, 6, 7, 8] • Algorithm 1 After the second pass, we get Density(C^D) = 0.4375, Input C1 = { Ai ; i = 1, 2,…n} /* set of fuzzy intervals */ Density(D^E) = 0.5, Density(E^F) = 0.34375. Set Density[i]=0;for i = 1,2,…,n /* Density[i] stores the Thus the second level dense sets are Density of Ai */ L2 = {C^D, D^E} for each transaction t in D Joining with their left and right neighbors from the basic fuzzy { numbers we obtain the candidates for the third pass as Compute votet(Ai) for i = 1, 2, ….n C3 = {B^C^D, C^D^E, D^E^F} Density[i] += votet(Ai) After third pass, we get Density(B^C^D) = 0.458333333, } Density(C^D^E) = 0.458333333, Density (D^E^F) = for(i = 1, 2,….,n) do 0.3958333333. { Thus the third level dense sets are if( ( Density[i])/D ≥ min_density ) L3= {B^C^D, C^D^E} Add Ai to L1 Similarly candidates for the fourth pass as } C4 = {A^B^C^D, B^C^D^E, C^D^E^F} k=1 After the fourth pass, we get Density(A^B^C^D) = 0.40625, L1= [Dense fuzzy intervals at level 1] Density(B^C^D^E) = 0.0.4375, Density(C^D^E^F) = 0.390625. for (k = 2 ; Lk ≠ φ ; k++) Thus the fourth level dense sets are { L4 = {A^B^C^D, C^D^E^F} do Candidates for the fifth pass as { C5 = {A^B^C^D^E, B^C^D^E^F} Ck = candidate-gen (Lk-1) After the fifth pass, we get Density(A^B^C^D^E) = 0.425, Compute Lk by going through the transactions Density(B^C^D^E^F) = 0.3875. in the dataset Thus the fifth level frequent sets are k=k+1 L5 = {A^B^C^D^E} } Candidates for the sixth pass are } C6 = {A^B^C^D^E^F} After the sixth pass Density(A^B^C^D^E^F) = 0.385416666, which is less than min_ density. Candidate-gen(Lk-1, Ck) Thus the sixth level is empty which is empty. So the algorithm { terminates giving the following maximal dense sets A^B^C^D^E. for all A∈ Lk-1 form A^L and A^R where L and R are the left and right neighbours of A respetively in case CONCLUSIONS these exists. /* For the extreme intervals both the In this paper, we have introduced the concept of fuzziness in neighbours may not exist */ mining maximal dense intervals. In our datasets each transaction Ck = Ck ∪ {A^L, A^R} has associated with it a time interval of the form [start_time, } end_time]. It is a level-wise method of generating dense fuzzy intervals. At the bottom level we have the basic dense fuzzy To illustrate the above algorithm we again consider the example intervals. In subsequent levels the already obtained dense fuzzy given in the section-III. For the sake of convenience, consider the intervals are expanded by joining them with their neignbours basic fuzzy interval as fuzzy number with triangular membership from the basic fuzzy intervals and their density counted by going function, which will be the input intervals for the first level i.e. through the dataset to check whether they are frequent or not. C1 = {A, B, C, D, E, F}, where A = [1, 2, 3], B = [2, 3, 4], C = [3, The process continues till no candidate is generated or some 4, 5], D = [4, 5, 6], E = [5, 6, 7] and F = [6, 7, 8] and min_density level is empty. The algorithm finally gives only the maximal = 0.4. dense fuzzy intervals. This algorithm although looks like A- After the first pass we have, Density(A) = 0.375, Density(B) = priori algorithm, has a slight variation in the sense that it has to 0.375, Density(C) = 0.375, Density(D) = 0.5, Density(E) = 0.5, take into account the fact that the downward and upward closure Density(F) = 0.1875. properties of dense interval do not hold here. 106 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 2, February 2011 Mohammed Abdul Khaleel received B.Sc. degree in Mathematics from Osmania University, India and M.C.A degree from Osmania REFERENCES University, India. After that worked in Global [1] Agrawal, R., Imielinski, T. and Swami, A(1993), Mining Suhaimi Company Dammam Saudi Arabia as association rules between sets of items in large databases, Senior Software Developer.Since 2008 serving as Proceedings of the ACM SIGMOD ’93, Washington, USA. Lecturer at College of Computer Science, King [2] Ale, Juan M and Rossi, G. H.(2000), An approach to Khalid University, Abha, kingdom of Saudi Arabia. His research discovering temporal association rules; Proceedings of the interest includes Data Mining, Software Engineering. 2000 ACM symposium on Applied Computing. [3] Chen, G. Q., Samuel C. Lee and Eden S.H.Yu (1983), Anjana Kakoti Mahanta received her B.Sc. degree in Application of fuzzy set theory to Economics, in Advances Mathematics and M.Sc. degree in Mathematics from Gauhati in Fuzzy Sets, Possibility Theory, and Applications, Ed. Paul University, India. After that she received her PGDSA from the P. Wang, 277-305, (Plenum Press, N.Y.). same University. Then she joined in Assam Engineering College, [4] Klir, J. and Yuan, B.; Fuzzy Sets and Logic Theory and India as a Lecturer. After this she received her Ph. D. in Application, Prentice Hill Pvt. Ltd.(2002) Computer Science from Gauhati University, India. Currently she [5] Lin, J.,L.(2002), Mining maximal frequent intervals. working as a Professor and Head in the Department of Computer Technical report, Department information management, Science, Gauhati University. She has a good number of Yuan Ze University. publications in defferent National/ international Journals has [6] Prade, H., Hullermeir, E. and Dubois, D.(2003), A Note on produced a couple of Ph.D.s till today. Her research interest Quality Measures for Fuzzy Association Rules, In includes Data mining, Soft Computing, Optimization, Automata, Proceedings IFSA-03, 10th International Fuzzy Systems and Fuzzy Logic. Asssociation World Congress. LNAI 2715, Istambul, 677- 684. Hemanta K. Baruah received his B.Sc. degree in Mathematics [7] Roddick, J. F., Spillopoulou, M. (1999), A Biblography of and M.Sc. degree in Statistics from Gauhati University, India. Temporal, Spatial and Spatio-Temporal Data Mining After that he received Ph. D. in Mathematics from IIT Research, ACM SIGKDD. Kharagpur, India. He worked as a Lecturer in Mathematics in [8] Srikant, R. and Agrawal, R.(1996), Mining quantitative Jawarlal Nehru University, Manipur Campus, India. He is former association rules in large relational tables; Proceedings of Dean of faculty of Science, Gauhati University, India. Currently the 1996 ACM SIGMOD Conference on management of he is working as a Professor in the Department of Statistics, data, Montreal, Canada. Gauhati University. He has a good number of publications in [9] Wong, M., H., Ada, F. and Kuok, C., M.(1998), Mining defferent National/ international Journals has produced a couple fuzzy association Rules in Databases, SIGMOD Record 27; of Ph.D.s till today. His research interest includes Fuzzy 41- 46. Mathematics, Data mining, Soft Computing, Optimization, and [10] Zimbrao, G., Moreira de Souza, J., Teixeira de Almeida V. Fuzzy Logic. and Araujo da Silva, W.(2002), An Algorithm to Discover Calendar-based Temporal Association Rules with Item’s Lifespan Restriction, Proc. of the 8th ACM SIGKDD Int’l Conf. on Knowledge Discovery and Data Mining (2002) Canada, 2nd Workshop on Temporal Data Mining, v. 8 (2002) 701-70 AUTHOR’S PROFILE Fokrul Alom Mazarbhuiya received B.Sc. degree in Mathematics from Assam University, India and M.Sc. degree in Mathematics from Aligarh Muslim University, India. After this he obtained the Ph.D. degree in Computer Science from Gauhati University, India. Since 2008 he has been serving as an Assistant Professor in College of Computer Science, King Khalid University, Abha, kingdom of Saudi Arabia. His research interest includes Data Mining, Information security, Fuzzy Mathematics and Fuzzy logic. 107 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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The International Journal of Computer Science and Information Security (IJCSIS Vol. 9 No. 2) is a reputable venue for publishing novel ideas, state-of-the-art research results and fundamental advances in all aspects of computer science and information & communication security. IJCSIS is a peer reviewed international journal with a key objective to provide the academic and industrial community a medium for presenting original research and applications related to Computer Science and Information Security.
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