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Mining Maximal Dense Intervals from Temporal Interval Data

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Mining Maximal Dense Intervals from Temporal Interval Data Powered By Docstoc
					                                                                 (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.



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                                                                                                      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].



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                                                                                                                  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




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                                                                                                                                               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



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                                                                                                                                                                 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.




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                                                                                                    ISSN 1947-5500

				
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