Extracting Membership Functions Using ACS Method via Multiple Minimum Supports

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					                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                       Vol. 8, No. 9, December 2010




          Extracting Membership Functions Using ACS
            Method via Multiple Minimum Supports

    Ehsan Vejdani Mahmoudi                             Masood Niazi Torshiz                                      Mehrdad Jalali
 Islamic Azad University, Mashhad                     Department of computer                               Department of computer
 Branch, Young Researchers Club,                     Engineering, Islamic Azad                            Engineering, Islamic Azad
           Mashhad, Iran                            University - Mashhad Branch,                         University - Mashhad Branch,
      e.vejdani@mshdiau.ac.ir                              Mashhad, Iran                                         Mashhad, Iran
                                                        niazi@mshdiau.ac.ir                                  jalali@mshdiau.ac.ir



Abstract— Ant Colony Systems (ACS) have been successfully                  approach inspired from the behavior of social insects. Ants
applied to different optimization issues in recent years. However,         deposit their chemical trails called “Pheromone” on the ground
only few works have been done by employing ACS method to                   for communicating with others. According to the pheromone,
data mining. This paper addresses the lack of investigations on            ants can find the shortest path between the source and the
this study by proposing an ACS-based algorithm to extract
                                                                           destination. Recently, Ant Colony Systems (ACS) has been
membership functions in fuzzy data mining. In this paper, the
membership functions were encoded into binary bits, and then               successfully applied to several difficult NP-hard problems,
they have given to the ACS method to discover the optimum set              such as the quadratic assignment [5], communication
of membership functions. By considering this approach, a                   strategies [6], production sequencing problem [7]. Job
comprehensive exploration can be executed to implement the                 Schedule Problem (JSP) [8], the traveling salesman problems
system automation. Therefore, it is a new frontier, since the              [9], [10], Vehicle Routing Problems (VRP) [11], etc.
proposed model does not require any user-specified threshold of
minimum support. Hence, we evaluated our approach                              Basically, fuzzy mining algorithms first used membership
experimentally and could reveal this approach by significant               functions to transform each quantitative value into a fuzzy set
improving of membership functions.                                         in linguistic terms and then used a fuzzy mining process to find
                                                                           fuzzy association rules. Items have their own characteristics,
    Keywords- fuzzy data mining; multiple minimum supports;                different minimum supports specified for different items. Han,
association rule; membership functions; ant colony system.                 Wang, Lu, and Tzvetkov [12] have pointed out that setting the
                                                                           minimum support is quite subtle, which can hinder the
                       I.    INTRODUCTION                                  widespread applications of these algorithms. Our own
    Recently, the fuzzy set theory has been used more and more             experiences of mining transaction databases also tell us that the
frequently in intelligent systems because of its simplicity and            setting is by no means an easy task. Therefore, our approach
similarity to human reasoning [1]. As to fuzzy data mining,                proposed method for computing minimum supports for each
Hong and Kuo proposed a mining approach that integrated                    item in database with own features. This approach leads to
fuzzy-set concepts with the Apriori mining algorithm [2].                  effectiveness, efficiency for global search and system
                                                                           automation, because our model does not require the user
ACO is a branch of a larger field referred to as Swarm                     specified threshold of minimum support.
Intelligence (SI). SI is the property of a system whereby the
collective behaviors of simple agents interacting locally with                Numerical experiments on the proposed algorithm are also
their environment cause coherent functional global patterns to             performed to show its effectiveness. The remaining parts of the
emerge [3]. It is the behavioral simulation of social insects              paper are organized as follows. Section II presents An ACS-
such as bees, ants, wasps and termites. This behavioral                    based mining framework. The proposed algorithm based on the
simulation came about for many reasons—optimization of                     above framework is described in Section III. Numerical
                                                                           simulations are shown in Section IV. Conclusions are given in
systems and learning about self-organization are two of many
                                                                           Section V.
reasons why scientists are interested in simulating these
insects. More specifically, ACO simulates the collective                         II.    THE ACS- BASED FUZZY MINING FRAMEWORK
foraging habits of ants—ants venturing out for food, and
                                                                              In this section, the ACS based fuzzy mining framework
bringing their discovered food back to the nest. Ants have                 [13] is shown in Fig. 1 where each item has its own
poor vision and poor communication skills, and a single ant                membership function set .These membership function sets are
faces a poor probability of longevity. However, a large group,             then fed into the ant colony system to search for the final
or swarm, of ants can collectively perform complex tasks with              proper sets .When the termination condition is reached, the best
proven effectiveness, such as gathering food, sorting corpses              membership function set (with the highest fitness value) can
or performing division of labor [4]. They are a heuristic                  then be used to mine fuzzy association rules from a database.




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                                                                                                     ISSN 1947-5500
                                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                                              Vol. 8, No. 9, December 2010




    The proposed framework modified the ACS-based                              process. In this work, we use the fitness function proposed by
Framework for Fuzzy Data Mining in [13]. The framework is                      Chen et al. [16] to obtain a good set of membership functions.
divided into two phases. The first phase searches for an
appropriate set of membership functions for the items by the                   B. ACS-based fuzzy data mining algorithm
ACS mining algorithm. Having searched for the solutions in                         Although, the proposed algorithm as considered in [15] and
the first phase, we use the best membership functions for fuzzy                [16] concerns one constant minimum support for all items, we
data mining in the second phase.                                               applied the determined minimum support for each item. As a
                                                                               matter of fact, in real world applications such as work on
     The ACS algorithm plays an important role in extracting                   transactional data of chain stores, the items have different
the membership functions. In the past, Parpinelli et al. proposed              quantities. Hence, using different minimum supports for each
the AntMiner to discover association rules [14]. They worked                   item in order to extracting membership functions is an efficient
on categorical attributes and discrete values. They proved that                idea. However, the previous ones that user specified minimum
the ACS algorithm performed well on handling discrete values                   supports, the new approach proposes the minimum supports are
in a solution space. In this work, we assume the parameters of                 achieved by a preprocessing on all items. On the other hand,
membership functions as discrete values and thus try to use the                minimum support for each item is automatically set as a value
ACS algorithm to find them. We transform the extraction of                     correspond with the quantity of the item. We considered a
membership functions into a route-search problem. A route                      method for computing minimum support for each item with its
then represents a possible set of membership functions. The                    characteristics in databases. There are significant criteria for
artificial ants, which refer to virtual ants that are used to solve            computing minimum support like, the number that each item
this problem, can then be used to find a nearly optimal solution.              happened in database and sum of values for each item in
                                                                               database. For example, suppose the number that item A
                                                                               happened in database is 10 and sum values is 20 and also the
                                                                               number that item B happened in database is 2 and sum values
                                                                               is 20. Clearly in mining process item A valuable than item B.
                                                                               We computing minimum support for item B until this item
                                                                               can`t satisfying minimum support. As mentioned above, we
                                                                               suggested in (1) as below:

                                                                                                      ∑
                                                                                   min _Sup(I ) =         ∗ ∗
                                                                                                                                                       (1)

                                                                                   Let I = {i1, i2, ..., im} be a set of items and D = {t1, t2, ... , tn}
                                                                               be a set of transactions. N is total number of transaction data.
                                                                               T is the number that each item happened in database. Si is sum
                                                                               values of an item in database D. P is constant digit with respect
                                                                               to the interval [0, 1].
                                                                                   In addition, as we investigated the parameters defined in
                                                                               [13], the following parameters performed: The number of
                                                                               artificial ants, the minimum pheromone ratio of an ant, the
                                                                               evaporation ratio of pheromone, the local updating ratio, and
                                                                               the global updating ratio. The proposed ACS-based algorithm
                                                                               for mining membership functions and fuzzy association rules
          Figure 1. The ACS-based framework for fuzzy data mining              are given as follow.
                                                                                   INPUT :
   III.     THE ACS_BASED FUZZY DATA MINING ALGORITHM
                                                                                     a) quantitative transaction data,
A. Initializations
                                                                                     b) a set of m items, which is with l predefined linguistic
    As revealing membership functions of all items result in a                 terms,
long code, we will encode the membership function of each
                                                                                     c) a maximum number of iterations G,
item into a binary code. We use the coding algorithm which
was represented in [13]. Furthermore, we utilize some rules                          d) P is constant digit with respect to the interval [0, 1].
called State transition rule, Pheromone updating rule, Local                        OUTPUT: An appropriate set of membership functions for
updating rule, Global updating rule which were defined in [15].                all items in fuzzy data mining.
    In this work, each item will have a set of isosceles-                        step 1) Let = 1 , where is used to keep the identity
triangular membership functions. The membership function                       number of the items to be processed.
stands for the linguistic terms such as low, middle, high.                       step 2) Let the multi- stage graph for the fuzzy mining
Transforming these quantitative values into linguistic terms                   problem be ( , ), where is the set of nodes and is the set
requires a feasible population of database. Therefore, we need                 of edges. Also denote the j- node in the i- th stage as , and
to initialize and update a population during the evolution




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                                                                                                                ISSN 1947-5500
                                                      (IJCSIS) International Journal of Computer Science and Information Security,
                                                      Vol. 8, No. 9, December 2010




the edge from           to ( ) as           . Initially set the                             IV. NUMERICAL SIMULATION
pheromone on every edge          as 0.5.                                      We experimentally evaluated our approach to expose the
   step 3) Let the initial generation = 1.                                performance of the proposed algorithm. The experiments were
   step 4) Sets up the complete route for each artificial ant             implemented in C/C++ on a computer with Intel Core(TM) 2
       by the following sub steps.                                        Duo Processor 2.66GHz and 4 GB main memory, running the
      a) Selects the edges from start to end according to the             Microsoft Windows 7 operating system. We used two datasets
state transaction rule.                                                   to present results: Dataset [13] with a total of 64 items and
                                                                          10,000 transactions. In addition, a real dataset called
      b) Update the pheromone of the edges passed through                 FOODMART from an anonymous chain store was used in the
by        according to the local updating rule.                           experiments [17]. The FOODMART dataset contained
   step 5) Evaluate the fitness value of the solution                     quantitative transactions about the products sold in the chain
(membership functions) obtained by each artificial ant                    store. There were totally 21,556 transactions with 1600 items in
according to the following sub steps.                                     the dataset Used in the experiments. The initial count of ants
                                                                          was set at 10. The parameters in the ACS algorithm were set as
      a) For each transaction datum , = 1 to n, transfer                  follows: the initial ratio of pheromone was 0.05, the minimum
its quantitative value      for item  into a fuzzy set                    pheromone of ants was 0.2, the evaporation ration was 0.9, the
according to the membership functions obtained from the ant               local updating ratio was 0.1 and the global updating ratio was
in (2). That is, is represented as :                                      0.9, minimum support for FOODMART dataset was set to
                                                                          0.0015 and for dataset [13] was set to 0.04. We considered the
             +                 +⋯+        +⋯+                (2)          value of constant P, as mentioned in (1) for FOODMART
                                                                          dataset equal to 0.05 and for dataset [13] equal to 0.02.
                                                                              The average fitness values of the artificial ants along with
    Where Region is the k-th fuzzy term of itemI , f                      different numbers of generations for two datasets are shown in
is v ’s fuzzy membership value in the region, and l is the                Fig. 2 and Fig. 3.
number of fuzzy membership functions.

    b) The scalar cardinality of each region in the                                                              ACS with constant minimum support
transactions is calculated in (3):                                                                               ACS with multiple minimum supports
                               ()                                                                   2.5
                  =∑                                         (3)
                                                                           Avrage fitness values




                                                                                                       2
             ()
  Where f      is the fuzzy membership value of region
R from the i-th datum.                                                                              1.5
     c) Check for each        whether its       /      is larger
                                                                                                       1
than or equal to the minimum support threshold         . If
satisfies the above condition, put it in the set of    large 1 -
                                                                                                    0.5
itemsets ( ).
     d) Calculate the fitness value of the solution    from the
                                                                                                       0
ant by dividing the number of large itemsets in        over the
suitability. That is Equation(4),                                                                          0          1000          2000          3000          4000
                                                                                                                                Generations
                       |   |
              =                                              (4)                                   Figure 2. The average fitness values along with different numbers of
                                                                                                                      generations with dataset [13]

   step 6) Once all the artificial ants find their entire routes,             It can be vividly seen from Fig. 2 and Fig. 3 that in our
the one holding the highest fitness value will be used to update          approach, the average fitness values increased by an offset
the pheromone according to the global updating rule.                      compared with the previous one. Thus, became stable within
   step 7) If the generation g is equal to G, output the current          less number of generations. In addition, we used smaller
best set of membership functions of item I for fuzzy data                 numbers for generation with the aim of comparing the
                                                p
mining; otherwise, g =g +1 and go to s 4.                                 difference between our model and the existing one that has
   step 8) If p ≠ m, set p =p +1 and go to Step 2 for another             static constant minimum support in Fig. 4. It is obviously
item; otherwise, stop the algorithm.                                      represents that our model achieved the best fitness at 300
                                                                          numbers of generations, whereas the existing one reached its
    The final set of membership functions output in step 7 and            best fitness at 500 numbers of generations.
the 1-itemsets obtained are then used to mine fuzzy association
rules from the given database.




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                                                                                                                             ISSN 1947-5500
                                                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                     Vol. 8, No. 9, December 2010




                                      ACS with constant minimum support                                                                           ACS with constant minimum support
                                      ACS with multiple minimum supports                                                                          ACS with multiple minimum supports
                         2.5
                                                                                                                                    2.5
 Avrage fitness values




                           2




                                                                                                       Avrage fitness values
                                                                                                                                      2
                         1.5
                                                                                                                                    1.5
                           1
                                                                                                                                      1
                         0.5
                                                                                                                                    0.5
                           0
                                                                                                                                      0
                               0           1000           2000         3000           4000
                                                     Generations                                                                           0        200        400         600        800        1000

                         Figure 3. The average fitness values along with different numbers of                                                                   Generations
                                        generations with FOODMART dataset
                                                                                                                                    Figure 5. The average fitness values along with different numbers of
                                                                                                                                         generations with FOODMART dataset (in smaller scale)

                                      ACS with constant minimum support
                                      ACS with multiple minimum supports                                                                          ACS with constant minimum support
                         2.5
                                                                                                                                                  ACS with multiple minimum supports
 Avrage fitness values




                           2                                                                                                        140
                                                                                                       Number of Large 1-itemsets




                                                                                                                                    120
                         1.5
                                                                                                                                    100
                           1                                                                                                          80

                         0.5                                                                                                          60

                                                                                                                                      40
                           0
                                                                                                                                      20
                               0         200        400          600       800        1000
                                                     Generations                                                                       0
                                                                                                                                           0           1000          2000           3000          4000
                         Figure 4. The average fitness values along with different numbers of
                                    generations with dataset [13] (in smaller scale)                                                                                 Generations

    As shown in Fig.5, the result of executing ACS algorithm                                               Figure 6. The numbers of large 1-itemsets along with different numbers of
with multiple minimum supports on FOODMART dataset is                                                                           generations with dataset [13]
much better than ACS algorithm with constant minimum
support since it has higher average of fitness values. The                                                Fig.7 illustrates the numbers of large 1-itemsets along with
number of items in FOODMART dataset is too many.                                                      different generations for FOODMART dataset. The proposed
Therefore artificial ants have been through difficulty for                                            ACS algorithm could increase large 1-itemsets in interval 50 to
optimizing membership functions. Meanwhile, ACS algorithm                                             500 generations, and stabilize after about 500 generations while
with multiple minimum supports could easily pass this test, and                                       the existing method with increasing generation had no changes,
extracting membership functions with high average of fitness                                          since the existing algorithm cannot work with FOODMART
values.                                                                                               dataset which have a lot of items.
    The numbers of large 1-itemsets along with different
generations are shown in Fig. 6. The curve of the existing
method stabilized after about three thousand generations while
the curve of our approach remained constant after one thousand
generations. Besides, the number of large 1-itemsets of our
approach is clearly much higher.




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                                                                                                                                                              ISSN 1947-5500
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                                           ACS with constant minimum support                                                                  ACS with constant minimum support
                                           ACS with multiple minimum supports                                                                 ACS with multiple minimum supports
                                3500                                                                                          35000
   Number of large 1-itemsets




                                3000                                                                                          30000
                                                                                                                              25000




                                                                                                    Time (second)
                                2500
                                2000                                                                                          20000
                                1500                                                                                          15000
                                1000                                                                                          10000
                                 500                                                                                               5000
                                   0                                                                                                 0
                                       0        1000      2000        3000      4000                                                      0          1000      2000         3000         4000
                                                        Generations                                                                                          Generatons
   Figure 7. The numbers of large 1-itemsets along with different numbers of                    Figure 9. The execution time of the ACS mining algorithm with FOODMART
                    generations with FOODMART dataset                                                                               dataset

    Fig. 8 and Fig. 9 reveal the execution time of the ACS
algorithms for different numbers of generations. Although,                                                                                    ACS with constant minimum support
execution time increased along with the generations within
both line graphs. Therefore, our approach represents the same                                                                                 ACS with multiple minimum supports
execution time for smaller number of generations, but increases                                                              2.5
for high number of generations, slightly.
                                                                                                  Avrage fitnees values



                                                                                                                              2

                                           ACS with constant minimum support                                                 1.5

                                           ACS with multiple minimum support                                                  1
                                8000
                                7000                                                                                         0.5
                                6000
 Time (second)




                                                                                                                              0
                                5000                                                                                               0%         20%      40%      60%         80%       100%
                                4000                                                                                                                    Size of dataset
                                3000
                                                                                                Figure 10. The average fitness values along with different size of dataset [13]
                                2000
                                1000
                                                                                                                                              ACS with constant minimum support
                                   0
                                       0       1000      2000        3000    4000                                                             ACS with multiple minimum supports
                                                       Generations                                                           2.5
                                                                                                     Avrage fitness values




 Figure 8. The execution time of the ACS mining algorithm with dataset [13]
                                                                                                                               2
    In the following study, we expressed the ACS algorithms
efficiency with scalability test on two datasets. The generation                                                             1.5
parameter among execution of algorithms is considered with
constant value of 500. The average of fitness values of the                                                                    1
artificial ants along with different size of dataset [13] is shown
in Fig. 10. By increasing the size of dataset, the accurate                                                                  0.5
membership functions are extracted, and the artificial ant can
learn more and find proper solutions. While in existing                                                                        0
algorithm with increasing the size of dataset has no changes.                                                                       0%         20%     40%       60%        80%       100%
Fig. 11 which is executed on FOODMART dataset, is as the                                                                                                Size of dataset
same as Fig.10 mentioned before.
                                                                                                                             Figure 11. The average fitness values along with different size of
                                                                                                                                                  FOODMART dataset




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                                                                                                                                                       ISSN 1947-5500
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    The large 1-itemsets of the artificial ants along with
different size of datasets is shown in Fig. 12 and Fig. 13. By                                                                        ACS with constant minimum support
increasing the size of dataset, the number of large 1-itemsets                                                                        ACS with multiple minimum supports
increased as well. However, at the first existing algorithm had
high values. Nevertheless, the ACS with constant minimum                                                                  1000
support had remained steady.                                                                                               900
                                                                                                                           800




                                                                                                          Time (second)
                                                                                                                           700
                                                  ACS with constant minimum support                                        600
                                                                                                                           500
                                                  ACS with multiple minimum supports                                       400
  Number of large 1-itemsets




                                                                                                                           300
                                   140                                                                                     200
                                   120                                                                                     100
                                                                                                                             0
                                   100
                                                                                                                                 0%      20%    40%       60%       80%      100%
                                   80
                                                                                                                                                 Size of dataset
                                   60
                                   40                                                                       Figure 14. The execution time of the ACS mining algorithm along with
                                                                                                                                different size of dataset [13]
                                   20
                                     0
                                                                                                                                      ACS with constant minimum support
                                         0%        20%     40%    60%        80%    100%
                                                           Size of dataset                                                            ACS with multiple minimum supports
                                                                                                                          4000
Figure 12. The numbers of large 1-itemsets along with different size of dataset
                                     [13]
                                                                                                                          3600
                                                                                                         Time (second)




                                                  ACS with constant minimum support                                       3200
                                                  ACS with multiple minimum supports
                                   3500                                                                                   2800
      Number of large 1-itemsets




                                   3000
                                                                                                                          2400
                                   2500
                                   2000                                                                                   2000
                                                                                                                                 0%      20%    40%       60%       80%      100%
                                   1500
                                                                                                                                                 Size of dataset
                                   1000
                                                                                                            Figure 15. The execution time of the ACS mining algorithm along with
                                    500                                                                                     different size of FOODMART dataset
                                         0
                                                                                                                               V. CONCLUSIONS
                                             0%      20%    40%     60%       80%    100%
                                                            Size of dataset                                In this paper, we could seek for the issues of applying the
                                                                                                       ACS algorithm to extract membership functions for fuzzy data
             Figure 13. The numbers of large 1-itemsets along with different size of                   mining and have proposed an algorithm to address this aim. As
                                    FOODMART dataset                                                   a matter of fact, in this approach we could deliver two benefits
                                                                                                       including the usage of multiple minimum supports, and system
    Fig. 14 and Fig. 15 reveal the execution time of the ACS                                           automation. On the other hand, computation results illustrated
algorithms for different size of datasets. As can be observed in                                       our work can be given as an alternative for effective association
Fig. 14 and 15, the execution time of both algorithms is nearly                                        rule mining.
equal. It therefore proves that proposed algorithm does not
                                                                                                           Meanwhile, the most significant difference between our
increase the execution time as well as improving efficiency                                            algorithm and older ACS algorithms to extract membership
encourages us       to employ the proposed algorithm for                                               functions concerns the independency of minimum support
extracting membership functions.                                                                       threshold. The experimental results of this new approach
                                                                                                       encouraged us to improve the system and utilize this strategy in
                                                                                                       real world applications, magnificently.




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                                                                                                                                               ISSN 1947-5500
                                                                   (IJCSIS) International Journal of Computer Science and Information Security,
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