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A Multi-level Ant-colony Mining Algorithm for Membership

Functions

Tzung-Pei Hong1, 2, Ya-Fang Tung3, Shyue-Liang Wang4, Yu-Lung Wu3, Min-Thai Wu2

1

Department of Computer Science and Information Engineering National University of Kaohsiung, Kaohsiung, 811, Taiwan.

2

Department of Computer Science and Engineering National Sun Yat-sen University, Kaohsiung, 804, Taiwan

3

Institute of Information Management I-Shou University, Kaohsiung, 840, Taiwan

4

Department of Information Management

National University of Kaohsiung, Kaohsiung, 811, Taiwan.

e-mail: tphong@nuk.edu.tw, m9522030@stmail.isu.edu.tw, slwang@nuk.edu.tw, wuyulung@isu.edu.tw, d953040015@student.nsysu.edu.tw

Abstract

Fuzzy data mining is used to extract fuzzy knowledge from linguistic or quantitative data. It is an extension of traditional data mining and the derived knowledge is relatively meaningful to human beings. In the past, we proposed a mining algorithm to find suitable membership functions for fuzzy association rules based on ant colony systems. In that approach, precision was limited by the use of binary bits to encode the membership functions. This paper elaborates on the original approach to increase the accuracy of results by adding multi-level processing. A multi-level ant colony framework is thus designed and an algorithm based on the structure is proposed to achieve the purpose. The proposed approach first transforms the fuzzy mining problem into a multi-stage graph, with each route representing a possible set of membership functions. The new approach then extends the previous one, using multi-level processing to solve the problem in which the maximum quantities of item values in the transactions may be large. The membership functions derived in a given level will be refined in the subsequent level. The final membership functions in the last level are then outputted to the rule-mining phase to find fuzzy association rules. Experiments are also performed to show the performance of the proposed approach. The experimental results show that the proposed multi-level ant colony systems mining approach can obtain improved results.

Keywords: data mining, ant system, ant colony system, fuzzy set, membership function, multi-stage graph.

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1. INTRODUCTION

Knowledge Discovery and Data Mining (KDD) refers to the application of

nontrivial procedures for identifying effective, coherent, potentially useful, and

previously unknown patterns in large databases [11]. The fast growth of KDD has

spurred the development of many related techniques and applications based on

different approaches including classification rules, clustering, association rules, and

others. Since 1993, the practice of inducing association rules from transaction data has

been commonly used in KDD [1]. An association rule is an expression XY, where X

is a set of items and Y is usually a single item. In the set of transactions, if all the

items in X exist in a transaction, then there is a high probability of Y also being in the

transaction. For example, assuming that transactions including the purchase of bread

are usually accompanied by the purchase of milk, then the association rule “bread ->

milk” will be induced.

Most previous studies focused on binary valued transaction data. Transaction

data in real-world applications, however, usually consist of quantitative values. Hong

et al. [13] thus proposed a mining approach that integrated fuzzy-set concepts with the

apriori mining algorithm to find interesting fuzzy itemsets and association rules in

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membership functions which need to be defined in advance, prompting the

development of approaches for automatically finding appropriate membership

functions in fuzzy data mining [12][14][18].

The Ant Colony System (ACS) algorithm is an Ant Colony Optimization (ACO)

technique which has recently shown promising for finding nearly optimal problem

solutions [5][9]. In the past, we proposed a mining algorithm to find suitable

membership functions for fuzzy association rules based on ACS [14]. In that approach,

precision was limited by the use of binary bits to encode the membership functions. A

large maximum quantity of an item value would result in the code growing too long.

Thus, the single-level ACS could not easily obtain a good solution. This paper extends

our previous approach through the addition of multi-level processing to increase

accuracy. In particular, we propose a multi-layered ant colony algorithm to address

large maximum quantities of an item value in the transactions. In the multi-level ACS,

the maximum coding length for a feasible solution is set and the approach derives a

rough set of membership functions at the first level. The set of membership functions

is then refined at the subsequent levels by reducing the search space, resulting in an

improved set of membership functions. Thus, different levels have different search

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previous level but with a smaller search space for adjusting membership functions.

Regardless of the size of the maximum item value, the coding length can thus be kept

fixed without reducing the quality of the derived membership functions. Experimental

results show that the proposed approach can generate better membership functions

than the previous one, though with greater computational time due to the multi-level

processing.

The remaining parts of this paper are organized as follows. Some related work is

briefly reviewed in Section 2. A multi-level ACS-based mining framework is

presented in Section 3. The proposed multiple-level ACS-mining approach based on

the framework is described in detail in Section 4. An example to illustrate the

proposed approach is given in Section 5. Experimental results are presented in Section

6, and conclusions and suggestions for future work are given in Section 7.

2. RELATED WORKS

Fuzzy set theory has been increasingly used in intelligent systems  due to its simplicity and similarity to human reasoning [17]. The role of fuzzy sets in data

mining may help transform quantitative values into linguistic terms, thus generating

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as manufacturing, engineering, medical diagnostics, economics, and others. It can also

be easily combined with other techniques. Examples include a fuzzy rule-based

back-propagation method for training binary multilayer perceptrons proposed by

Delgado et al. [8] and a genetic algorithm to find useful fuzzy concepts for pattern

classification proposed by Hu [15].

As for fuzzy data mining, Hong et al. integrated the fuzzy-set concepts and the

apriori mining algorithm to obtain fuzzy association rules [13]. Defining an

appropriate set of membership functions is important in that the set may have a

critical impact on the final results of fuzzy data mining. Thus, some GA-based fuzzy

data-mining methods for extracting both association rules and membership functions

from quantitative transactions have been proposed [12].

Swarm intelligence studies the collective behavior of unsophisticated agents that

interact locally through their environment [2], and is inspired by the behavior of social

insects and animals, such as ants, fish and birds. The ant system was first introduced

by Colorni et al. in [5][9]. Ants are capable of cooperating to solve complex problems

such as searching for foods, carrying food and so on. Without using vision, they can

find the shortest path between their nests and a food source. As they search, they

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crosses these paths, it will select the path with high pheromone density, which thus

determines the next node on an ant’s route. Once all ants have ended their tours, the

amount of pheromone on the tours will have been modified. Ant algorithms have been

designed to simulate this type of behavior for solving optimization problems.

The ant colony system (ACS) algorithm [10] is based on the ant system [5][9]

and, in this paper, is used to extract membership functions. ACS has been successfully

used to obtain nearly optimal solutions for difficult NP-hard problems, such as the

Traveling Salesman Problem (TSP), Job Schedule Problem (JSP), Vehicle Routing

Problem (VRP) [6][10][23] and other applications [4][22]. The ACS uses three rules:

the state transition rule (pseudo random proportional rule), pheromone global

updating rule and local updating rule. The ant system is normally adapted according

to the different characteristics of the problems to be solved, resulting in many variants

of the ant system algorithm, such as the Ant Colony System, the Ant System Elitist,

the Rank-Based Version of the Ant System and the Max-Min Ant System, etc. These

algorithms differ in several respects, including how they update the pheromone, the

state transition, and the heuristic functions used, etc. Stützle et al. organized these

variants into an integrated system named the Ant Colony Optimization (ACO) [21],

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Colony Optimization. The Ant System, the Ant Colony System and the Max-Min Ant

System differ in some aspects of state transaction, local update and global update

methods (see Table 1). For example, the Ant system and the Max-Min Ant System use

the same methods for state transition and local update, but use different methods for

global update.

Table 1. Comparison of different ACO algorithms Algorithm

Rules

Ant System [5][9]

Ant Colony System [10]

Max-Min Ant System [21] State Transition Rule Random proportional rule Pseudo random proportional rule Random proportional rule

Local Update No Yes No

Global Update

Update all routes passed

by all ants

Update the best route passed by an ant: 1. Iteration Best 2. Global Best

Update the best route passed by an ant: 1. Iteration Best 2. Global Best

ACO techniques have also shown promise for discovering useful and interesting

information from databases, such as classification rules [16][19]. However, research

on data mining based on the ant colony system is still rare. Parpinelli et al. proposed

an ACS-based approach to find rules from medical data [20]. Cordon and Herrera also

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systems [7]. Very little research, however, has investigated the mining of association

rules.

3. MULTI-LEVEL ACS-BASED MINING FRAMWORK

The proposed multi-level ACS-based framework for fuzzy data mining is shown

in Figure 1, where each item has its own set of membership functions and there are

totally m items. In fuzzy data mining, the membership functions are used to convert

item values into memberships ranging from zero to one. Each set of membership

functions is then fed into the ant colony system to search for the appropriate final set.

When the termination condition is reached, the best sets of membership functions

(with the highest fitness values) are then used to mine fuzzy association rules from a

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d th 2 nd 1 st Linguistic terms Large Transaction Database ……... ACS Fuzzy MF

Acquisition Process Fuzzy Mining for Large 1-itemsets

Minimum confidence Phase 1: Mining

Membership Functions

Initial Graph Constructed by Ants

Membership Function Set1 Membership Function Set2 Membership Function Setm …….. Minimum support Membership Function Set1 Membership Function Set2 Membership Function Setm ……….. … .. ….. ….. ….. Membership Function Set1 Membership Function Set2 Membership Function Setm ……….. … .. Final Membership Function Sets (1~m) Fuzzy Mining Phase 2: Mining Fuzzy Association Rules

Figure 1. Multi‐level ACS-based framework for fuzzy data mining

The framework in Figure 1 can be divided into two phases. The first phase uses

the ACS mining algorithm to search for appropriate sets of membership functions. In

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membership functions is transformed into a route-search problem, represented by a

multi-stage graph. The detailed transformation process will be explained in the next

section. A route then represents a possible set of membership functions. The artificial

ants are then used to find an appropriate set of membership functions for each item.

The difference between the proposed approach and the previous one is that the

proposed approach uses multiple-level searching to obtain the final membership

function. The search process is executed for several (d) levels, and the membership

functions obtained at a given level are refined in the next level by changing the search

scales. The ants at later levels thus search smaller solution spaces than those at earlier

levels, generating more accurate results. As shown in Figure 1, the membership

functions are polished several times to meet the precision criteria. After the search for

the solutions in the first phase is completed, the best sets of membership functions are

used for finding fuzzy association rules in the second phase. The process will be

illustrated in greater detail in the next section.

In this paper, the ACS algorithm plays an important role in extracting the

membership functions in the first phase. In the past, Parpinelli et al. proposed the AntMiner for discovering association rules [20]. Accommodating categorical

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handling discrete values in a solution space. In this paper, we further investigate the

handling of quantitative and fuzzy values.

4. MULTI-LEVEL ACS-BASED FUZZY MINING

This section describes the proposed multi-level ACS-based mining algorithm,

beginning with the encoding scheme and the fitness function are first described,

followed by and explanation of the design concept. A flow chart for the proposed

algorithm is then designed and explained, followed by the presentation of the entire

algorithm.

4.1 ENCODING REPRESENTATION

With the same approach as used in [14], the membership functions of each item

are encoded into a pair of binary strings. Each item has a set of membership functions

which, for the sake of simplicity, are assumed to be the shape of an isosceles triangle.

Each membership function thus has two parameters, center and half the spread (called

span). First, we use n binary-bits to encode the center and the span of a membership

function for an item according to the quantity range of the item in the database. For

example, if the quantity range of an item is among 0 to 15, we may use four bits to

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(membership functions) for an item Ij, let Cj1, Cj2, Cj3 denote the three centers of the linguistic terms and Sj1, Sj2, Sj3 represent their spans. The pair of binary strings for the

centers of the item will thus be represented by 12 bits, such as {(0, 0, 1, 1) (0, 1, 1, 1)

(1, 1, 0, 1)}, as in Figure 2. Similarly, the spans of each linguistic term will be

encoded as {(0, 0, 1, 1) (0, 1, 1, 0) (0, 1, 1, 1)}. From the coding scheme, the three

centers are {3, 7, 13} and the three spans are {3, 6, 7}.

Figure 2. String representation of membership functions for an item

After the membership functions are encoded, the ACS algorithm can then be

applied to obtain a (nearly) optimal solution. As can be observed in Figure 3, each

position of a string includes two bits, one for the center and one for the span, resulting

in four cases, namely (0, 0), (0, 1), (1, 0), (1, 1). If the decision of each pair of bits is

thought of as a node, then the problem can be transformed into a multi-stage decision

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NP-hard.

Figure 3. Multi-stage graph transformed for the proposed ACS-based mining algorithm

Since the problem has been converted into an optimization problem in a

multi-stage graph, the ACS algorithm can thus be used to solve it.

4.2 FITNESS FUNCTIONS

The fitness value of a possible solution is obtained according to the criteria

proposed by Hong et al. [12], which is defined as follows:

y suitabilit

L

f  | 1| , 

where |L1| is the number of large 1-itemsets obtained by using the set of membership functions for a given ant. The suitability factor is designed to reduce the occurrence of

the two bad kinds of membership functions shown in Figure 4, where 4(a) is too

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Figure 4. The two bad types of membership functions

The suitability of the membership functions considers two factors: the overlap

factor designed for avoiding the 4(a) case and the coverage factor designed for

avoiding the 4(b) case. The calculation for the suitability for Item Ij is thus designed

as follows: ). ( _ ) ( _ )

(Ij overlap factor Ij coverage factor Ij

y

suitabilit    

The overlap factor is defined as follows:

. ] 1 ) 1 ), ) , ( ) , ( (( [ _ 1 1 1

   k j jk jk j R R min R R overlap max factor overlap  

The term overlap(Rjk, Rji) represents the overlap ratio of two membership functions

Rjk and Rji, which is defined as the overlap length divided by the minimum span (half the spread) of the two functions. If the overlap length is longer than the span, then

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follows: . ) ( ) ,... ( _ j jk j1 I max R R range 1 factor coverage

The term range(Rj1, Rj2, …, Rjk) is the coverage range of the membership functions, and max (Ij) is the maximum quantity of Ij in the transactions. The coverage

factor of a set of membership functions for an item Ij is thus defined as the coverage

range of the membership functions divided by the maximum quantity of that item in

the transactions. The larger the coverage ratio is, the better the derived membership

functions are.

4.3 DESIGN CONCEPT

In this paper, a fixed number of k bits are used to encode membership functions.

The maximum quantity value M of each item in the transactions is first found to

determine how many levels need to be run. The scale d, used as the unit for the

membership functions of an item in a level, is calculated as follows:

     Mk d 2 .

For example, assume the best membership functions obtained after a level are

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  Figure 5. An example for membership functions obtained after a level

The membership functions obtained at this level are then refined in the next level.

A new scale d’ is then formed to refine the center and the span as follows:

     dk d 2 2 ' .

The membership functions are then tuned using the new scale to obtain more

precise values of each center and span of a given fuzzy region in the ACS algorithm.

The adjusting scope is 2d from the previous solutions, with the concept shown in

Figure 6. The center of a membership function is adjusted between d and 3d, and the

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Figure 6. The refinement process for the center

The adjusting process for the span of a membership function is similar to that for

the center. Assume the original span is s, then the search range for the next level will

be from s-d to s+d using the new scale d’. An example is shown in Figure 7, where

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Figure 7. Refinement process for the span

The scale d’ is then further shortened in the next level and the same steps are

repeated until some termination criterion for precision is reached.

The proposed algorithm starts with the first item and applies the operations of the

ACS algorithm to extract membership functions. The ACS algorithm is repeatedly

executed until the maximum number G of generations is reached. Applying the ACS

algorithm with local and global pheromone updates gradually forms the best route,

obtaining rough membership functions at the outset and then refining them in the

following levels. The new scale d is then calculated and used in the process of

refining the membership functions. After several levels of execution, the final refined

membership functions could be obtained.

4.4 PROPOSED MULTI-LEVEL ACS-BASED MINING ALGORITHM

As mentioned above, the proposed approach applies the ACS algorithm to

initially extract rough membership functions. The algorithm is then repeated to obtain

more precise functions. The following parameters are used in this approach: the

number of artificial ants, an ant’s minimum pheromone ratio, the pheromone

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algorithm is described as follows.

The proposed algorithm:

INPUT:

(1) n quantitative transaction data,

(2) a set of m items, each with l predefined linguistic terms (different items may have

different l values),

(3) a support threshold α,

(4) a confidence threshold λ,

(5) a maximum number G of iterations, and

(6) a number k of encoding bits for a membership function

OUTPUT:

A set of fuzzy association rules with its associated set of membership functions.

STEP 1: Let p = 1, where p is used to hold the identity number of the item to be

processed.

STEP 2: Let the multi-stage graph for the fuzzy mining problem be (N, E), where N is

the set of nodes and E is the set of edges (Figure 8). Also denote the j-node

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pheromone on every edge Eijk as τ0 (usually set as 0.5).

  Figure 8. Multi-stage graph for the fuzzy mining problem

STEP 3: Find the maximum quantitative value M of the p-th item in the transactions.

STEP 4: Find the scale d for the searching process as follows:

     Mk d 2 ,

where k is the number of encoding bits for a membership function of the p-th

item. The unit used for the membership functions of the p-th item in this

phase is then d (Figure 9).

Figure 9. Unit used for the membership functions

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STEP 6: Find a complete route for each artificial ant Antq by selecting the edges from

start to end according to the state transaction rule.

STEP 7: Update the pheromone of the edges through which an ant just passed

according to the local updating rule.

STEP 8: Evaluate the fitness value of the solution (membership functions) obtained

by each artificial ant according to the criteria stated above.

STEP 9: Find the ant with the highest fitness value among all ants and update the

pheromone of the best route according to the global updating rule.

STEP 10: If the generation g is equal to the termination generation G, keep the

current best membership functions of the p-th item and go to the next step;

otherwise, g = g + 1 and go to STEP 6.

STEP 11: Find the new scale d’, which is more precise than the original one as

follows:      dk d 2 2 '

STEP 12: If 2d is less than 2k and the previous d’’ is not 1, go to the next step.

Otherwise, set d = d’, d’’ = d’, and go to Step 5 to obtain more precise

membership functions.

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stop the algorithm and output the membership functions.

The final set of membership functions output in STEP 13 and the 1-itemsets

obtained from the best ant are then used to mine fuzzy association rules from the

given database.

5. AN EXAMPLE

In this section, an example is given to illustrate the proposed method for finding

membership functions from transactions with large quantitative values. Assume there

are four items A, B, C and D, in a transaction data base. Also assume the data set

includes the five transactions shown in Table 2, with the values in the parentheses

representing the purchased quantities for the corresponding items.

Table 2. Transactions in the example

TID Transaction T1 A(144), B(84), C(55) T2 A(64), B(70), D(114) T3 A(124), B(140) T4 C(131), D(100) T5 A(90), C(108)

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Assume each item has three fuzzy regions: Low, Middle, and High. Thus, three

fuzzy membership functions are to be derived for each item. For the data shown in

Table 2, the proposed multi-layer ant-mining algorithm proceeds as follows.

STEP 1: Let the item A be the first item to be processed.

STEP 2: Set the initial pheromone amounts on all the edges at 0.5.

STEP 3: Search for the maximum quantitative value of Item A in the transactions. In

this example, the maximum quantity of Item A is 144.

STEP 4: Assume four bits are used here to encode a membership function. The scale

d for the search process is then calculated as:

9 2 144 4      d .

The search unit of 9 is thus used for the membership functions of Item A in

the first level. The concept is shown in Figure 10.

Figure 10. Search unit for the membership functions of item A

STEP 5: Assume the maximum generation number G is set at 10. The current

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STEP 6: The complete route by each artificial ant is formed by the ACS algorithm

proposed in [14]. Each ant selects the edges according to the state

transaction rule. In this example, since the pheromone on each edge is

initially set at 0.5, the probability for an ant to choose any node at the next

stage for t = 0 is thus the same and is calculated according to the state

transition rule as follows:

25 . 0 4 5 . 0 * 5 . 0 * ) 0 (      

      n jn jn P

STEP 7: Each ant updates the pheromone on the edge which was just selected

according to the local updating rule. Assume the parameter ρ is set at 0.1.

Once one edge is chosen by an ant, its pheromone is updated as: 275 . 0 5 . 0 1 . 0 25 . 0 9 . 0 ) 0 ( ) 1 ( ) 1 (    0      jn jn

In this case for t = 1, the pheromone of this constructed path will be

increased up to the initial pheromone 0.5.

STEP 8: The fitness value of the solution (membership functions) obtained by each

ant is evaluated according to the criteria stated above. For example, the

fitness value of path (jn) is calculated as 2.

STEP 9: Find the ant with the highest fitness value. Set the decay parameter α at

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according to the iteration-best updating rule. The global updating rule is given below: path best the ij if path best the ij if ij ij ij ij ij                   , 025 . 0 25 . 0 * ) 9 . 0 1 ( * ) 1 ( , 65 . 1 ) 2 1 ( 9 . 0 275 . 0 * ) 9 . 0 1 ( * ) 1 (        

STEP 10: Steps 6 to 9 are repeated until the termination generation number (10, in

this example). The final membership functions of Item A derived in this

level are (45, 45) (90, 36) (127, 9), as shown in Figure 11.

Figure 11. Membership functions of Item A derived in the first level

STEP 11: The new scale d’ in the next level is calculated as follows:

2 2 9 * 2 2 2 'dk 4  d

STEP 12: The new scale 2 is set as the unit for searching for the desired membership

functions in the second level. Steps 5 to 10 are then repeated to obtain more

precise membership functions of Item A. The results in this level are (46, 46)

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Figure 12. Membership function of Item A derived in the second level

The new scale d’ in the next level is calculated as follows: 1 2 2 * 2 2 2 ' 4            dk d .

The new scale 1 is then used as the unit for the next level. Steps 5 to 10 are then

repeated again to obtain yet more precise membership functions of Item A. The

results are (47, 47) (95, 34) (129, 7), as shown in Figure 13.

Figure 13. Membership functions of Item A derived in the third level

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third level are the final results for Item A. The above steps are then repeated

to obtain the membership functions of the next item.

In this example, the membership functions of item A are generated by the

multi-level ant-mining algorithm above. The last membership functions are (47, 47)

(95, 34) (129, 7), which are more precise than the initial functions, (45, 45) (90, 36)

(127, 9).

 

6. EXPERIMENTS

This section describes experiments comparing the multi-level ant-mining

approach and the original approach. The experiments were implemented in C/C++ on

a personal computer, and involved a total of 64 items and 10,000 transactions. The

maximum value of item quantities in the transactions was 128. The proposed

multi-level algorithm used four bits to encode a membership function. Each item had

three linguistic terms and a total of 12 bits was thus used for a given item. Due to the

brevity of the encoding, the proposed approach required three levels of execution to

extract the accurate membership functions. The minimum support for association

rules was set at 0.04. The proposed approach used the ACS algorithm to search for the

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was set at 10, the initial pheromone was 0.5, the evaporation ratio α was 0.9, the

local updating ratio ρ was 0.1. and the parameter β was set at 1. The average fitness

values of the artificial ants, along with the number of generations for both approaches

are shown in Figure 14, which clearly indicates the proposed method performed better

on the average fitness values than the original method in [14].

Figure 14. Average fitness values by the two algorithms

Experiments were then conducted to show the average suitability values of the

two methods for different numbers of generations. The results, shown in Figure 15,

indicate that the proposed approach achieved a smaller suitability value than the

original method, meaning the former approach has a better membership-function

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Figure 15. Average suitability values by the two algorithms

Experiments were then conducted to show the numbers of large 1-itemsets for

different numbers of generations. The results, shown in Figure 16, indicate that the

proposed approach could generate greater amounts of knowledge than the original

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Figure 16. Average number of large 1-itemsets by the two algorithms

The average execution time for different numbers of generations by the two

algorithms is shown in Figure 17. It’s apparent that the proposed approach, which

needs to run multi-level ACS, requires more execution time than the original method.

Figure 17. Average execution time along with different generations by the two algorithms

Experiments were then conducted to compare the average number of large

1-itemsets for different minimal supports, with the results shown in Figure 18. As

expected, the number of large 1-itemsets decreases as the minimum support threshold

increases. The proposed approach also outperforms the previous one with different

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Figure 18. Average number of large 1-itemsets for different minimum supports

7. CONCLUSION AND FUTURE WORK

This paper has described a multi-level ACS-based mining algorithm to extract an

appropriate set of membership functions in fuzzy data mining. It first transforms the

mining problem into a multi-stage graph, with each route representing a possible set

of membership functions. It also adopts multi-layer processing such that the precision

of the final membership functions may be incrementally improved. The approach is

thus suitable for solving problems with potentially large maximum quantities of an

item value in the transactions. As opposed to the single-level approach, the multi-level

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exploitative capability to obtain better search results. However, the proposed approach

requires several iterative ACS runs, one for each level, which may increase the

execution time. Experimental results show that the proposed approach can obtain

better membership functions, but with more computational time due to the multi-level

processing.

More work needs to be done in the future. For example, we may attempt to study

the effects of different encoding methods and other variant ant strategies on the

proposed algorithm. We may also apply the algorithm to solve some real-world

mining problems. Defining more constraints for the center and the span of a

membership function may also be worth studying. The choice of fitness functions can

be further discussed as well.

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數據

Table 1. Comparison of different ACO algorithms  Algorithm
Figure 1. Multi‐level ACS-based framework for fuzzy data mining
Figure 2. String representation of membership functions for an item
Figure 3. Multi-stage graph transformed for the proposed ACS-based mining  algorithm
+7

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