A NEW METHOD TO GENERATE FUZZY RULES FROM TRAINING INSTANCES FOR HANDLING CLASSIFICATION PROBLEMS

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A NEW METHOD TO GENERATEFUZZY RULES FROM TRAININGINSTANCES FOR HANDLINGCLASSIFICATION PROBLEMSSHYI-MING CHEN a & CHENG-HAO YU ba Department of Computer Science and InformationEngineering, National Taiwan University of Scienceand Technology Taipei, Taiwan, R.O.C.b Department of Electronic Engineering, NationalTaiwan University of Science and Technology, Taipei,Taiwan, R.O.C.Published online: 30 Nov 2010.

To cite this article: SHYI-MING CHEN & CHENG-HAO YU (2003) A NEW METHOD TOGENERATE FUZZY RULES FROM TRAINING INSTANCES FOR HANDLING CLASSIFICATIONPROBLEMS, Cybernetics and Systems: An International Journal, 34:3, 217-232, DOI:10.1080/01969720302837

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ANEWMETHODTOGENERATEFUZZYRULESFROMTRAINING INSTANCESFORHANDLINGCLASSIFICATIONPROBLEMS

SHYI-MINGCHEN

DepartmentofComputerScienceand InformationEngineering,NationalTaiwan University of Science andTechnology Taipei,Taiwan, R.O.C.

CHENG-HAOYU

Department of Electronic Engineering, NationalTaiwanUniversity of Science andTechnology, Taipei,Taiwan, R.O.C.

A major task in developing a fuzzy classification system is to generate a set of

fuzzy rules from training instances to deal with a specific classification prob-

lem. In recent years, many methods have been developed to generate fuzzy

rules from training instances. We present a new method to generate fuzzy

rules from training instances to deal with the Iris data classification problem.

The proposed method can discard some useless input attributes to improve

the average classification accuracy rate. It can obtain a higher average clas-

sification accuracy rate and it generates fewer fuzzy rules and fewer input

fuzzy sets in the generated fuzzy rules than the existing methods.

This work was supported in part by the National Science Council, Republic of China,

under Grant NSC 90-2213-E-011-053.

Address correspondence to Professor Shyi-Ming Chen, Ph.D., Department of Compu-

ter Science and Information Engineering, National Taiwan University of Science and Tech-

nology, Taipei, Taiwan, R.O.C.

Cybernetics and Systems: An InternationalJournal, 34: 217�232, 2003Copyright# 2003 Taylor & Francis

0196-9722/03 $12.00+ .00

DOI: 10.1080/01969720390184399

217

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INTRODUCTION

In 1965, Zadeh proposed the theory of fuzzy sets (Zadeh 1965). It has

been used to deal with uncertain and imprecise data. Based on the fuzzy

set theory, we can design a fuzzy classification system to generate fuzzy

rules from training data. We usually have two approaches to obtain a set

of fuzzy rules in a fuzzy classification system. One approach is to obtain

the knowledge from human experts and then transfer it into fuzzy rules,

but this is time consuming. The other way is to use a machine learning

method to automatically generate fuzzy rules from the training data

(Hong and Lee 1996, 1999; Castro et al. 1999; Chen and Chen 2000;

Hong and Chen 1999, 2000; Kao and Chen 2000; Wang et al. 1999; Wu

and Chen 1999). Castro et al. (1999) presented a method to generate

fuzzy rules in expert systems. Chang and Chen (2000) presented a method

to generate fuzzy rules from numerical data based on the exclusion of

attribute terms. Chen and Yeh (1998) presented a method for generating

fuzzy rules from relational database systems for estimating null values.

Chen et al. (1999) presented a method for generating fuzzy rules from

numerical data for handling fuzzy classification problems. Chen and Lin

(2000) presented a method for constructing fuzzy decision trees and

generating fuzzy classification rules from training examples. Chen and

Chen (2000) presented a method to generate fuzzy rules for fuzzy classi-

fication systems. Hong and Chen (1999) presented a method for gen-

erating fuzzy rules based on finding relevant attributes and membership

functions. Hong and Chen (2000) also presented a method for inducing

fuzzy rules based on processing individual fuzzy attributes. Hong and Lee

(1996) presented a method for induction of fuzzy rules and membership

functions from training examples. Hong and Lee (1999) investigated the

effect of the merging order on performance of fuzzy induction. Kao and

Chen (2000) presented a method to generate fuzzy rules from training

data containing noise for handling classification problems. Lin and Chen

(2000) presented a method to generate weighted fuzzy rules from training

data for handling fuzzy classification problems. Wang et al. (1999) pre-

sented a fuzzy inductive strategy for modular rules. Wu and Chen (1999)

presented a method for constructing membership functions and fuzzy

rules from training examples.

In this article, we present a new method to generate fuzzy rules from

training instances to deal with the Iris data (Fisher 1936) classification

problem. The proposed method can discard some useless input attributes

218 S.-M. CHEN AND C.-H. YU

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to improve the average classification accuracy rate. It can obtain a higher

average classification accuracy rate and it generates fewer fuzzy rules and

fewer input fuzzy sets in the generated fuzzy rules than the existing

methods.

BASICCONCEPTSOFFUZZYSETS

In 1965, Zadeh proposed the theory of fuzzy sets (Zadeh 1965). Let A be

a triangular fuzzy set in the universe of discourse U, where

A ¼Z b

a

u� a

b� a

� �=xþ

Z c

b

c� u

c� b

� �=u; 8u 2 U: ð1Þ

The membership function of the triangular fuzzy set A is shown in

Figure 1, where the membership function of the triangular fuzzy set A

also can be represented by a triplet (a, b, c), where b is called the center of

the triangular fuzzy set A; a and c are called the left vertex and the right

vertex of the triangular fuzzy set A, respectively.

In the following, we briefly review the union and the intersection

operations of fuzzy sets and review a similarity measure of fuzzy sets

from Klir and Yuan (1995) and Zadeh (1965).

Definition 1: Let A and B be fuzzy sets of the universe of discourse U

characterized by the membership functions mA and mB, respectively, wheremA :U! [0, 1] and mB :U! [0, 1]. The union of the fuzzy sets A and B,

denoted as A[B, is defined as follows:

mA[BðuiÞ ¼ maxðmAðuiÞ;mBðuiÞÞ; 8ui 2 U: ð2Þ

Figure 1. A triangular membership function.

NEW METHOD TO GENERATE FUZZY RULES 219

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The intersection of the fuzzy sets A and B, denoted as A\B, is defined as

follows:

mA\BðuiÞ ¼ minðmAðuiÞ; mBðuiÞÞ; 8ui 2 U: ð3Þ

Definition 2: Let A and B be fuzzy sets of the universe of discourse U

characterized by the membership functions mA and mB, respectively, wheremA :U! [0, 1] mB :U! [0, 1]. The degree of similarity between the fuzzy

sets A and B, denoted as S(A, B), is defined as follows:

SðA;BÞ ¼ The Area of A\BThe Area of A[B ; ð4Þ

where SðA;BÞ 2 ½0; 1�. The larger the value of S(A, B), the more the

similarity between the fuzzy sets A and B.

ANEWALGORITHMFORFUZZYRULESGENERATION

In this section, we present a new algorithm for constructing membership

functions and generating fuzzy rules from training data to deal with the

Iris data classification problem. The algorithm is presented as follows.

Step1: Choose m instances from the Iris data as the training data set,

and let the rest of the instances of the Iris data be the testing data set.

Step 2: Find the maximum attribute value, the minimum attribute

value, and the average attribute value for each input attribute (i.e., sepal

length (SL), sepal width (SW), petal length (PL), and petal width (PW))

of each species of flowers (i.e., Iris-Setosa, Iris-Versicolor, and Iris-

Virginica) from the training data set to form the membership function of

each attribute for each species.

Step 3: Based on Eq. (4), calculate the average degree of similarity

between each species for each input attribute.

Step 4: For each input attribute, if the average degree of similarity

between each species of the input attribute is greater than the threshold

value a, where a 2 [0, 1], then discard this input attribute. In this article,


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we let a ¼ 0:05. Then, the rest of the input attributes (i.e., X1; X2; . . . ;

and Xn) are used as the input attributes of the generated fuzzy rules

shown as follows:

IF X1 is A1 AND X2 is A2 AND � � � AND Xn is An

THEN the flower is B;

where B denotes the species of the flower, B 2 {Iris-Setosa, Iris-Versicolor,

Iris-Virginica}, A1, A2, . . . , and An are input fuzzy sets formed by the

attribute values of the input attributes X1; X2; . . . ; andXn, respectively,

with respect to the species B derived in step 2, where 1 � n � 4. Because

there are three species of the flower, three fuzzy rules will be generated

from the training data set.

If the testing instance is (y1; y2; . . . ; yn), where yi denotes the attribute

value of the input attribute Xi and 1 � i � n, then the degree of possibility

mRðy1; y2; . . . ; ynÞ that the flower is B can be evaluated by mRðy1,y2; . . . ; ynÞ ¼ mA1

ðy1Þ � mA2ðy2Þ � � � � � mAn

ðynÞ, where y1; y2; . . . ; and

yn denote the attribute values of the input attributes A1; A2; . . . ; and An of

the testing instance and mRðy1; y2; . . . ; ynÞ2½0; 1�.

ANEXAMPLE

In the following, we apply the proposed algorithm to deal with the Iris

data (Fisher 1936) classification problem. Table 1 shows the Iris data.

There are three species of flowers in the Iris data (i.e., ‘‘Iris-Setosa,’’ ‘‘Iris-

Versicolor,’’ and ‘‘Iris-Virginica’’) and there are 150 instances in the Iris

data, with 50 instances for each species and each species with four

attributes (i.e., SL, SW, PL, and PW).

The step-by-step illustration of the proposed algorithm is shown as

follows:

Step1: The computer randomly chooses 75 instances of the Iris data as

the training data set and lets the rest of the instances of the Iris data be

the testing data set. Assume that the chosen training instances are as

shown in Table 2.

Step2: Based on Table 2, we can find the maximum attribute value, the

minimum attribute value, and the average attribute value of each input

attribute of each species of flower from the training data set as shown in

Table 3.


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Table 1. Iris data (Fisher 1936)

Iris-Setosa Iris-Versicolor Iris-Virginica

SL SW PL PW SL SW PL PW SL SW PL PW

5.1 3.5 1.4 0.2 7.0 3.2 4.7 1.4 6.3 3.3 6.0 2.5

4.9 3.0 1.4 0.2 6.4 3.2 4.5 1.5 5.8 2.7 5.1 1.9

4.7 3.2 1.3 0.2 6.9 3.1 4.9 1.5 7.1 3.0 5.9 2.1

4.6 3.1 1.5 0.2 5.5 2.3 4.0 1.3 6.3 2.9 5.6 1.8

5.0 3.6 1.4 0.2 6.5 2.8 4.6 1.5 6.5 3.0 5.8 2.2

5.4 3.9 1.7 0.4 5.7 2.8 4.5 1.3 7.6 3.0 6.6 2.1

4.6 3.4 1.4 0.3 6.3 3.3 4.7 1.6 4.9 2.5 4.5 1.7

5.0 3.4 1.5 0.2 4.9 2.4 3.3 1.0 7.3 2.9 6.3 1.8

4.4 2.9 1.4 0.2 6.6 2.9 4.6 1.3 6.7 2.5 5.8 1.8

4.9 3.1 1.5 0.1 5.2 2.7 3.9 1.4 7.2 3.6 6.1 2.5

5.4 3.7 1.5 0.2 5.0 2.0 3.5 1.0 6.5 3.2 5.1 2.0

4.8 3.4 1.6 0.2 5.9 3.0 4.2 1.5 6.4 2.7 5.3 1.9

4.8 3.0 1.4 0.1 6.0 2.2 4.0 1.0 6.8 3.0 5.5 2.1

4.3 3.0 1.1 0.1 6.1 2.9 4.7 1.4 5.7 2.5 5.0 2.0

5.8 4.0 1.2 0.2 5.6 2.9 3.6 1.3 5.8 2.8 5.1 2.4

5.7 4.4 1.5 0.4 6.7 3.1 4.4 1.4 6.4 3.2 5.3 2.3

5.4 3.9 1.3 0.4 5.6 3.0 4.5 1.5 6.5 3.0 5.5 1.8

5.1 3.5 1.4 0.3 5.8 2.7 4.1 1.0 7.7 3.8 6.7 2.2

5.7 3.8 1.7 0.3 6.2 2.2 4.5 1.5 7.7 2.6 6.9 2.3

5.1 3.8 1.5 0.3 5.6 2.5 3.9 1.1 6.0 2.2 5.0 1.5

5.4 3.4 1.7 0.2 5.9 3.2 4.8 1.8 6.9 3.2 5.7 2.3

5.1 3.7 1.5 0.4 6.1 2.8 4.0 1.3 5.6 2.8 4.9 2.0

4.6 3.6 1.0 0.2 6.3 2.5 4.9 1.5 7.7 2.8 6.7 2.0

5.1 3.3 1.7 0.5 6.1 2.8 4.7 1.2 6.3 2.7 4.9 1.8

4.8 3.4 1.9 0.2 6.4 2.9 4.3 1.3 6.7 3.3 5.7 2.1

5.0 3.0 1.6 0.2 6.6 3.0 4.4 1.4 7.2 3.2 6.0 1.8

5.0 3.4 1.6 0.4 6.8 2.8 4.8 1.4 6.2 2.8 4.8 1.8

5.2 3.5 1.5 0.2 6.7 3.0 5.0 1.7 6.1 3.0 4.9 1.8

5.2 3.4 1.4 0.2 6.0 2.9 4.5 1.5 6.4 2.8 5.6 2.1

4.7 3.2 1.6 0.2 5.7 2.6 3.5 1.0 7.2 3.0 5.8 1.6

4.8 3.1 1.6 0.2 5.5 2.4 3.8 1.1 7.4 2.8 6.1 1.9

5.4 3.4 1.5 0.4 5.5 2.4 3.7 1.0 7.9 3.8 6.4 2.0

5.2 4.1 1.5 0.1 5.8 2.7 3.9 1.2 6.4 2.8 5.6 2.2

5.5 4.2 1.4 0.2 6.0 2.7 5.1 1.6 6.3 2.8 5.1 1.5

4.9 3.1 1.5 0.2 5.4 3.0 4.5 1.5 6.1 2.6 5.6 1.4

5.0 3.2 1.2 0.2 6.0 3.4 4.5 1.6 7.7 3.0 6.1 2.3

5.5 3.5 1.3 0.2 6.7 3.1 4.7 1.5 6.3 3.4 5.6 2.4

4.9 3.6 1.4 0.1 6.3 2.3 4.4 1.3 6.4 3.1 5.5 1.8

(Continued )


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For each species Y of the flowers, where Y 2 {Iris-Setosa, Iris-

Versicolor, Iris-Virginica}, the minimum attribute value, the maximum

attribute value, and the average attribute value of each input attribute X,

where X 2 {SL, SW, PL, PW}, are defined as the left vertex, the right

vertex, and the center of the triangular fuzzy set X(Y ) of the linguistic

value of the input attribute X, where we let the membership degree of the

left vertex and the right vertex of the fuzzy set X(Y) be equal to 1/the

number of training data for each species, as shown in Figures 2 through

5. For example, from Table 3, we can see that the minimum attribute

value, the maximum attribute value, and the average attribute value of

the input attribute SL of the species Iris-Setosa of the training instances

are 4.3 cm, 5.8 cm, and 5.028 cm, respectively. We can construct the

membership function SL(Iris-Setosa) for the input attribute SL of the

species Iris-Setosa as shown in Figure 2. Because the number of instances

for each species in the training data set is 25, we let the membership

degree of the left vertex (i.e., 4.3 cm) and the right vertex (i.e., 5.8 cm)

of the fuzzy set SL(Iris-Setosa) be equal to 125 ¼ 0:04, as shown in

Figure 2. In the same way, we can construct the membership functions

SL(Iris-Versicolor), SL(Iris-Virginica), SW(Iris-Setosa), SW(Iris-

Versicolor), SW(Iris-Virginica) PL(Iris-Setosa), PL(Iris-Versicolor),

PL(Iris-Virginica), PW(Iris-Setosa), PW(Iris-Versicolor), and PW(Iris-

Virginica), respectively, as shown in Figures 2 to 5.

Table 1. Continued



4.4 3.0 1.3 0.2 5.6 3.0 4.1 1.3 6.0 3.0 4.8 1.8

5.1 3.4 1.5 0.2 5.5 2.5 4.0 1.3 6.9 3.1 5.4 2.1

5.0 3.5 1.3 0.3 5.5 2.6 4.4 1.2 6.7 3.1 5.6 2.4

4.5 2.3 1.3 0.3 6.1 3.0 4.6 1.4 6.9 3.1 5.1 2.3

4.4 3.2 1.3 0.2 5.8 2.6 4.0 1.2 5.8 2.7 5.1 1.9

5.0 3.5 1.6 0.6 5.0 2.3 3.3 1.0 6.8 3.2 5.9 2.3

5.1 3.8 1.9 0.4 5.6 2.7 4.2 1.3 6.7 3.3 5.7 2.5

4.8 3.0 1.4 0.3 5.7 3.0 4.2 1.2 6.7 3.0 5.2 2.3

5.1 3.8 1.6 0.2 5.7 2.9 4.2 1.3 6.3 2.5 5.0 1.9

4.6 3.2 1.4 0.2 6.2 2.9 4.3 1.3 6.5 3.0 5.2 2.0

5.3 3.7 1.5 0.2 5.1 2.5 3.0 1.1 6.2 3.4 5.4 2.3

5.0 3.3 1.4 0.2 5.7 2.8 4.1 1.3 5.9 3.0 5.1 1.8


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Step 3: Based on Eq. (4), we can calculate the average degree of simi-

larity of each pair of species for each input attribute as follows. First, we

calculate the degree of similarity of each pair of the species for the

attributes SL:

i. Based on Eq. (4), the degree of similarity S(SL(Iris-Setosa), SL(Iris-

Versicolor)) between the species Iris-Setosa and Iris-Versicolor for the

attribute SL can be evaluated as follows:

S(SL(Iris-Setosa), SL(Iris-Versicolor))¼ 0.1356.

Table 2. Training data set



5.1 3.5 1.4 0.2 7.0 3.2 4.7 1.4 6.3 3.3 6.0 2.5

4.9 3.0 1.4 0.2 6.4 3.2 4.5 1.5 5.8 2.7 5.1 1.9

4.7 3.2 1.3 0.2 6.9 3.1 4.9 1.5 7.1 3.0 5.9 2.1

4.6 3.1 1.5 0.2 5.5 2.3 4.0 1.3 6.3 2.9 5.6 1.8

5.0 3.6 1.4 0.2 6.5 2.8 4.6 1.5 6.5 3.0 5.8 2.2

5.4 3.9 1.7 0.4 5.7 2.8 4.5 1.3 7.6 3.0 6.6 2.1

4.6 3.4 1.4 0.3 6.3 3.3 4.7 1.6 4.9 2.5 4.5 1.7

5.0 3.4 1.5 0.2 4.9 2.4 3.3 1.0 7.3 2.9 6.3 1.8

4.4 2.9 1.4 0.2 6.6 2.9 4.6 1.3 6.7 2.5 5.8 1.8

4.9 3.1 1.5 0.1 5.2 2.7 3.9 1.4 7.2 3.6 6.1 2.5

5.4 3.7 1.5 0.2 5.0 2.0 3.5 1.0 6.5 3.2 5.1 2.0

4.8 3.4 1.6 0.2 5.9 3.0 4.2 1.5 6.4 2.7 5.3 1.9

4.8 3.0 1.4 0.1 6.0 2.2 4.0 1.0 6.8 3.0 5.5 2.1

4.3 3.0 1.1 0.1 6.1 2.9 4.7 1.4 5.7 2.5 5.0 2.0

5.8 4.0 1.2 0.2 5.6 2.9 3.6 1.3 5.8 2.8 5.1 2.4

5.7 4.4 1.5 0.4 6.7 3.1 4.4 1.4 6.4 3.2 5.3 2.3

5.4 3.9 1.3 0.4 5.6 3.0 4.5 1.5 6.5 3.0 5.5 1.8

5.1 3.5 1.4 0.3 5.8 2.7 4.1 1.0 7.7 3.8 6.7 2.2

5.7 3.8 1.7 0.3 6.2 2.2 4.5 1.5 7.7 2.6 6.9 2.3

5.1 3.8 1.5 0.3 5.6 2.5 3.9 1.1 6.0 2.2 5.0 1.5

5.4 3.4 1.7 0.2 5.9 3.2 4.8 1.8 6.9 3.2 5.7 2.3

5.1 3.7 1.5 0.4 6.1 2.8 4.0 1.3 5.6 2.8 4.9 2.0

4.6 3.6 1.0 0.2 6.3 2.5 4.9 1.5 7.7 2.8 6.7 2.0

5.1 3.3 1.7 0.5 6.1 2.8 4.7 1.2 6.3 2.7 4.9 1.8

4.8 3.4 1.9 0.2 6.4 2.9 4.3 1.3 6.7 3.3 5.7 2.1


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ii. Based on Eq. (4), the degree of similarity S(SL(Iris-Setosa), SL(Iris-

Virginica)) between the species Iris-Setosa and Iris-Virginica for the

attribute SL can be evaluated as follows:

S(SL(Iris-Setosa), SL(Iris-Virginica)) ¼ 0:083:

iii. Based on Eq. (4), the degree of similarity S(SL(Iris-Versicolor),

SL(Iris-Virginica)) between the species Iris-Versicolor and Iris-

Virginica for the attribute SL can be evaluated as follows:

S(SL(Iris-Versicolor), SL(Iris-Virginica)) ¼ 0:510:

Thus, the average degree of similarity AVE(SL) of each pair of the

species for the attribute SL can be evaluated as follows:

Table 3. Minimum attribute value, maximum attribute value, and average attribute

value for the training data set

Species Input attribute

Minimum

attribute

value (cm)

Maximum

attribute

value (cm)

Average

attribute

value (cm)

Iris-Setosa Sepal Length (SL) 4.3 5.8 5.028

Sepal Width (SW) 2.9 4.4 3.480

Petal Length (PL) 1.0 1.9 1.460

Petal Width (PW) 0.1 0.5 0.248

Iris-Versicolor Sepal Length (SL) 4.9 7.0 6.012




Iris-Virginica Sepal Length (SL) 4.9 7.7 6.576




Figure 2. Membership functions of the attribute sepal length (SL) for the species.


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AVE(SL) ¼ ½S(SL(Iris-Setosa), SL(Iris-Versicolor))þ S(SL(Iris-Setosa), SL(Iris-Virginica))

þ S(SL(Iris-Versicolor), SL(Iris-Virginica))�=3¼ 0:243:

In the same way, we can calculate the average degrees of similarity

AVE(SW ), AVE(PL), and AVE(PW ) of each pair of the species for the

attributes SW, PL, and PW, respectively. The results are as follows:

S(SW(Iris-Setosa), SW(Iris-Versicolor)) ¼ 0:054;

S(SW(Iris-Setosa), SW(Iris-Virginica)) ¼ 0:2194;

S(SW(Iris-Versicolor), SW(Iris-Virginica)) ¼ 0:492;

AVE(SW) ¼ ðS(SW(Iris-Setosa), SW(Iris-Versicolor))

þ S(SW(Iris-Setosa), SW(Iris-Virginica))

þ S(SW(Iris-Versicolor), SW(Iris-Virginica))Þ=3¼ 0:255;

Figure 3. Membership functions of the attribute sepal width (SW) for the species.

Figure 4. Membership functions of the attribute petal length (PL) for the species.


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S(PL(Iris-Setosa), PL(Iris-Versicolor)) ¼ 0;

S(PL(Iris-Setosa), PL(Iris-Virginica)) ¼ 0;

S(PL(Iris-Versicolor), PL(Iris-Virginica)) ¼ 0:024;

AVE(PL) ¼ ðS(PL(Iris-Setosa), PL(Iris-Versicolor))þ S(PL(Iris-Setosa), PL(Iris-Virginica))

þ S(PL(Iris-Versicolor), PL(Iris-Virginica))Þ=3¼ 0:0079;

S(PW(Iris-Setosa), PW(Iris-Versicolor)) ¼ 0;

S(PW(Iris-Setosa), PW(Iris-Virginica)) ¼ 0;

S(PW(Iris-Versicolor), PW(Iris-Virginica)) ¼ 0:053;

AVE(PW) ¼ ðS(PW(Iris-Setosa), PW(Iris-Versicolor))

þ S(PW(Iris-Setosa), PW(Iris-Virginica))

þ S(PW(Iris-Versicolor), PW(Iris-Virginica))Þ=3¼ 0:0018:

Step 4: Because the threshold value a given by the user is 0.05, if the

average degree of similarity of each pair of the species of an input at-

tribute is greater than 0.05, then discard this input attribute. From the

calculation results of Step 3, we can see that the average degrees of si-

milarity of each pair of the species for the attributes SL and SW are

greater than 0.05. Thus, the attributes SL and SW are discarded. Finally,

we use the rest of the input attributes (i.e., PL and PW) as the input

attributes of the generated fuzzy rules. Because there are three species of

Figure 5. Membership functions of the attribute petal width (PW) for the species.


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flowers in the Iris data, there are three fuzzy rules to be generated, shown

as follows:

Rule R1: IF Petal Length is PL(Iris-Setosa) AND

Petal Width is PW(Iris-Setosa)

THEN the flower is Iris-Setosa,

Rule R2: IF Petal Length is PL(Iris-Versicolor) AND

Petal Width is PW(Iris-Versicolor)

THEN the flower is Iris-Versicolor,

Rule R3: IF Petal Length is PL(Iris-Virginica) AND

Petal Width is PW(Iris-Virginica)

THEN the flower is Iris-Virginica,

where the membership functions of PL(Iris-Setosa), PW(Iris-Setosa),

PL(Iris-Versicolor), PW(Iris-Versicolor), PL(Iris-Virginica), and PW(Iris-

Virginica) are shown in Figures 4 and 5, respectively. In the following, we

use the generated fuzzy rules to classify the testing instance (5.0 cm,

3.0 cm, 1.6 cm, 0.2 cm) in Table 1 as follows:

i. Let’s consider the generated fuzzy rule R1:

Rule R1: IF Petal Length is PL(Iris-Setosa) AND

Petal Width is PW(Iris-Setosa)

THEN the flower is Iris-Setosa.

From Figures 4 and 5, we can see mPLðIris-SetosaÞð1:6 cmÞ ¼ 0:682 and

mPWðIris-SetosaÞð0:2 cmÞ ¼ 0:676. Then, from fuzzy rule R1, we can see

that the degree of possibility that the flower is Iris-Setosa is equal to

mPLðIris-SetosaÞð1:6 cmÞ � mPWðIris-SetosaÞð0:2 cmÞ � 0:461:

ii. Let’s consider the generated fuzzy rule R2:

Rule R2: IF Petal Length is PL(Iris-Versicolor) AND

Petal Width is PW(Iris-Versicolor)

THEN the flower is Iris-Versicolor.

From Figures 4 and 5, we can see that mPLðIris-VersicolorÞð1:6 cmÞ ¼ 0 and

mPWðIris-VersicolorÞð0:2 cmÞ ¼ 0. Then, from fuzzy rule R2, we can see that

the degree of possibility that the flower is Iris-Versicolor is equal to

mPLðIris-VersicolorÞð1:6 cmÞ � mPWðIris-VersicolorÞ ð0:2 cmÞ ¼ 0.


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iii. Let’s consider the generated fuzzy rule R3:

Rule R3: IF Petal Length is PL(Iris-Virginica) AND

Petal Width is PW(Iris-Virginica)

THEN the flower is Iris-Virginica.

From Figures 4 and 5, we can see mPLðIris-VirginicaÞð1:6 cmÞ ¼ 0

and mPWðIris-VirginicaÞð0:2 cmÞ ¼ 0. Then, from fuzzy rule R3, we can see

that the degree of possibility that the flower is Iris-Virginica is equal to

mPLðIris-VirginicaÞð1:6 cmÞ � mPWðIris-VirginicaÞð0:2 cmÞ ¼ 0.

From i, ii, and iii, we can see that fuzzy rule R1 gets the highest degree of

possibility (i.e., 0.461) among the fuzzy rules R1, R2, and R3. Therefore,

the testing instance (5.0 cm, 3.0 cm, 1.6 cm, 0.2 cm) is classified as the

species ‘‘Iris-Setosa.’’ It is obvious that this classification result coincides

with the one shown in Table 1. It should be noted that in the generated

fuzzy rules R1, R2, and R3, the membership functions PL(Iris-Setosa),

PW(Iris-Setosa), PL(Iris-Versicolor), PW(Iris-Versicolor), PL(Iris-Virgi-

nica), and PW(Iris-Virginica) are called the input fuzzy sets of the gen-

erated fuzzy rules.

We have implemented the proposed algorithm on a Pentium III PC

by using Visual Basic 6.0. By applying the implemented program to deal

with the Iris data classification problem, we can obtain the following

experimental results:

1. If the training data set contains 150 training instances (i.e., the full Iris

data) and the testing data set is equal to the training data set containing

150 training instances (i.e., the full Iris data), then after executing the

program200 times, the average classification accuracy rate is 97.3300%,

the number of generated fuzzy rules is 3, and the number of input fuzzy

sets in the antecedent portions of the generated fuzzy rules is 6.

2. If the training data set contains 120 training instances randomly chosen

from the Iris data, and the testing data set contains the rest of the in-

stances of the Iris data (i.e., 30 instances), then after executing the

program200 times, the average classification accuracy rate is 96.8250%,

the number of generated fuzzy rules is 3, and the number of input fuzzy

sets in the antecedent portions of the generated fuzzy rules is 6.

3. If the training data set contains 75 training instances randomly chosen

from the Iris data, and the testing data set contains the rest of the

instances of the Iris data (i.e., 75 instances), then after executing the


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program 200 times, the average classification accuracy rate is

96.3427%, the number of generated fuzzy rules is 3, and the number

of input fuzzy sets in the antecedent portions of the generated fuzzy

rules is 6.

In the following, we compare the experimental results of the pro-

posed algorithm with Hong and Lee’s (1996) algorithm, Wu and Chen’s

(1999) algorithm, and Castro et al.’s (1999) algorithm as shown in

Table 4.

Table 4. A comparison of the average classification accuracy rate, the average number of

generated fuzzy rules, and the average number of input fuzzy sets for different

algorithms

Algorithms

Average

classification

accuracy rate

Average

number of

generated

fuzzy rules

Average

number of

input fuzzy

sets

The Proposed Algorithm (Training

Data Set: 150 Instances; Testing

Data Set: 150 Instances; Executing

200 Times)

97.3300% 3 6


Data Set: 120 Instances; Testing

Data Set: 30 Instances; Executing

200 Times)

96.8250% 3 6


Data Set: 75 Instances; Testing data

Set: 75 Instances; Executing 200

Times)

96.3427% 3 6

Wu and Chen’s (1999) Algorithm

(Training Data set: 75 Instances;

Testing Data Set: 75 Instances;

Executing 200 Times)

96.2100% 3 8.21

Hong and Lee’s (1996) Algorithm

(Training Data Set: 75 Instances;


Executing 200 Times)

95.5700% 6.21 8

Castro et al.’s (1999) Algorithm

(Training Data Set: 120 Instances;


Executing 10 Times)

96.6000% 11 25


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CONCLUSIONS

In this article, we have presented a new method for generating fuzzy rules

and constructing membership functions from training instances to deal

with the Iris data classification problem. The proposed algorithm has the

following advantages: (1) it can obtain a higher average classification

accuracy rate; (2) it generates fewer fuzzy rules; and (3) it generates fewer

input fuzzy sets in the antecedent portions of the generated fuzzy rules.

From Table 4, we can see that the proposed method is better than the

existing methods due to the fact that it has a higher average classification

accuracy rate and it generates fewer fuzzy rules and fewer input fuzzy sets

in the antecedent portions of the generated fuzzy rules.

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Documents

A NEW METHOD TO GENERATE FUZZY RULES FROM TRAINING INSTANCES FOR HANDLING CLASSIFICATION PROBLEMS