Abstract—In the past, the choices of β values to be applied to find theβ -reducts in VPRS for an information system are somewhat arbitrary. In this study, a systematic approach to determine the threshold value β of VPRS applied to information systems with continuous attributes is presented. The β value is directly connected to fuzzy membership functions by Implication Relations and Fuzzy Algorithms, in which the membership functions were obtained by the standard Fuzzy C-means method. The argument is that errors of system classification would occur in the fuzzy-clustering phase prior to information classification, therefore the threshold value β should be constrained by the probability of belongingness of an object to the fuzzy clusters, i.e., through the values of membership functions.

I. INTRODUCTION OUGH Sets (RS) was introduced more than twenty

years ago [1] and had emerged as a powerful technique for the automatic classification of objects [2]. However, for RS to be capable of performing a complete classification requires that the collected data must be fully correct or certain. The classification with a controlled degree of uncertainty or misclassification error is outside the realm of RS approach [3]. To overcome these problems, the theory of Variable Precision Rough Sets (VPRS) was first introduced by Ziarko [3]. The VPRS model is an extension of the original RS model [3], which was proposed to analyze and identify data patterns that represent statistical trends rather than being functional. VPRS deals with partial classifications by introducing a precision parameter β. The β value represents a threshold value of the portion of objects in a particular conditional class being classified into the same decision class (in RS the β value is one). Ziarko [3] defined the β value as a classification error and it was defined to be in the domain [ )5.0,0.0 . However, An et al. [4] and Beynon [5] used β to denote the proportion of correct classifications, in which case the appropriate range is ( ]0.1,5.0 . They referred this technique as enhanced RST. The same definition ofβ is used in the presented study. Because the VPRS model has no formal historical background of having empirical evidence to support any particular method of β-reducts selection [6],

Kuang Yu Huang is now with the Department of Management

Information, Ling Tung University, Taichung, Taiwan (corresponding author to provide phone: +886-9-22621030; e-mail: kyhuang@mail.ltu.edu.tw).

Ting-Cheng Chang is now with the Department of Commercial Technology and Management, Ling Tung University, Taichung, Taiwan (e-mail: E-mail: tcchang@mail.ltu.edu.tw).

VPRS-related research studies do not focus on details of the choice of the precision parameter (β) value. Although the threshold value β has been chosen by authors in using β-reducts skills, the choice of β is somewhat arbitrary. For example, Su [7] determined the precision parameter β value based on the least upper bound of the misclassification error of the data set. However, in such a method, the value of β could only be determined after the classification results were obtained, and the origin of uncertainty in classification has never been addressed in corporation with the constraint of β value specifically.

In this study, a systematic method to determine the upper bound of precision parameter value β for information systems with at least one continuous attribute is presented. This method based on the Fuzzy C-Means (FCM) [8, 9, 10] clustering method, Implication Relations and Fuzzy Algorithms [11] on fuzzy set theory. Although other fuzzy cluster methods could be used, we simply chose FCM since the technique has been well established and widely accepted. The argument is that errors of system classification would occur in the fuzzy-clustering phase prior to information classification, therefore the threshold value β should be constrained by the probability of belongingness of an object to the fuzzy clusters, i.e., through the values of membership functions.

The remaining of this paper is organized as follows. Section 2 presents the fundamental principles of VPRS. Section 3 describes the methods that used to determine the threshold value β . Section 4 gives two examples to compare the consequences of applying our method with other β -selecting methods. Finally, Section 5 presents some concluding remarks.

II. FUNDAMENTAL PRINCIPLES OF THE VPRS MODEL VPRS operates on what may be described as a

knowledge-representation system or an information system. The basic principles of information system ( S ) and the application of VPRS theory to the processing of information systems are described in the following sections.

A. Information systems A typical information system has the

form ),,,( qq fVAUS = , where U is a non-empty finite set

of objects and A is a non-empty finite set of attributes describing each object. Assume that the attributes in set A

A Novel Approach to Establishing the VPRS Model with Threshold Parameter Selection Mechanism Based on Fuzzy Algorithms

Kuang Yu Huang, and Ting-Cheng Chang

R

2009 10th International Symposium on Pervasive Systems, Algorithms, and Networks

978-0-7695-3908-9/09 $26.00 © 2009 IEEE

DOI 10.1109/I-SPAN.2009.75

432

can be partitioned into a set of conditional attributes φ≠C and another set of decisional attributes φ≠D , A = DC ∪ and φ=DC ∩ . For each attribute qVAq ,∈ represents the

domain of q , i.e. ∪ qVV = . Finally, VAUf q →×: is the

information function such that qVqxf ∈),( for Aq ∈∀ and

Ux ∈∀ . For an attribute subset P , CP ⊆ and φ≠P , the

equivalence relation is denoted by { }PqqyfqxfUUyxI P ∈∀=×∈= ),(),(:),( .

PI partitions U into a family of disjoint subsets PIU / called a quotient set of U : [ ]{ }UxxIU PP ∈= :/

where [ ]Px denotes the equivalence class determined by x with respect to (w.r.t.) P , i.e., [ ] ( ){ }PP IyxUyx ∈∈= ,:

B. β-lower and β-upper approximation In VPRS, the β value represents a threshold value of the

portion of objects in a particular conditional class being classified into the same decision class. In processing the information system ),,,( qq fVAUS = , DCA ∪= , UX ⊆ ,

CP ⊆ using a VPRS model with 15.0 ≤< β , the data analysis procedure hinges on two basic concepts, namely the β-lower and β-upper approximations of a set. The β-lower approximation of sets UX ⊆ and CP ⊆ can be expressed

as follows: )(XRPβ ( ){ }β≥∈= PxXPUx ]/[: =

[ ] ( ){ }β≥PP xXPx ]/[:∪ , Similarly, the β-upper

approximation of sets UX ⊆ and CP ⊆ is given by )(XRPβ

= ( ){ }β−>∈ 1]/[: PxXPUx = [ ] ( ){ }β−> 1]/[: PP xXPx∪ .

Where ( )Y

YXYXP

∩=/ if 0>Y , and ( ) 1/ =YXP

otherwise. X is the cardinality of the set X . Beynon [6]

defined the following expressions for the β-negative region and β-boundary region of [ ]DxX ∈ in S :

( ){ } [ ] ( ){ }βββ −≤=−≤∈= 1]/[:1]/[:)( PPPP xXPxxXPUxXNEG ∪

( ){ } [ ] ( ){ }βββββ <<−=<<−∈= PPPP xXPxxXPUxXBND ]/[1:]/[1:)( ∪

In VPRS theory, the quality of the classification result is quantified using the metric

[ ] ( ){ } UxXPxDP PP βγ β ≥= ]/[:),( ∪ .

Essentially, the value ),( DPβγ indicates the proportion

of objects in the universe U for which a classification based on decision attribute D is possible at the specified value of β. In other words, it involves combining all β-positive regions and summing up the number of objects involved in such a combination. The measurement (quality of classification) is used operationally to define and extract reducts, which is the kernel part of RS theory (and VPRS) in

the application to data mining and rule construction.

III. DETERMINATION OF SUITABLE THRESHOLD VALUE β

This study proposes a mechanism for determining a suitable value of the VPRS precision parameter β based on the Fuzzy C-Means clustering method and the general principles of fuzzy algorithms. The various components of the proposed mechanism are introduced in the sections below.

A. Fuzzy C-Means (FCM) FCM, first developed by Dunn [12] in 1973 and later

refined by Bezdek [13] in 1981, is an unsupervised clustering algorithm with multiple applications, ranging from feature analysis, to clustering and classifier design. In the present work, Fuzzy C-means (FCM) clustering was applied in the initialization of the antecedent membership functions of the fuzzy sets for its simplicity and acceptable correctness on pattern recognition among various Fuzzy clustering techniques.

FCM clustering consists of two processes. The first step is to calculate the cluster centers and the assignment of points to these centers using a form of Euclidean distance. This process is repeated until the cluster centers have stabilized. Before the first iteration, an initial set of membership values must be chosen. FCM imposes a direct constraint on fuzzy membership function associated with each point as the

following: ∑ ===

p

jij kix

1...,3,2,1,1)(μ

where p is the number of specified clusters, k is the number of objects, ix is the i -th object, and )( ij xμ returns

the membership value of ix in the j -th cluster. Clearly, the sum of the cluster membership values of each object from one specific attribute must be equal to one. The objective of FCM is to minimize a standard loss function expressed as the weighted sum of squared errors within clusters, i.e.

[ ] ∞<′<−∑ ∑=′

= =mcxxl ji

mp

j

n

iij 1,)(

2

1 1μ

where l is the minimized loss value, p is the number of specified clusters, n is the number of objects, )( ij xμ is the

membership value of object ix in the j -th cluster, ix is the i -th object, m′ is the fuzzification parameter, and jc is the

center of j -th cluster. It has been shown by Bezdek [13] that

if 0>− ji cx for all i and j , then the loss function could

be minimized when 1>′m . Under this condition, the corresponding cluster center value can be computed in

accordance with [ ][ ] pjfor

x

xxc

im

ij

i im

ijj ≤≤

∑

∑= ′

′

1)(

)(

μ

μ

Having calculated the cluster centers, the second step in the FCM procedure is to determine the cluster memberships of a

433

sample point. To do so, it is necessary to determine the distance from each point ix to each of the cluster centers ( pccc ,...,, 21 ). In practice, this is achieved by computing the

Euclidean distance between the point and the cluster center in

accordance with 2

jiji cxd −=

where jid is the distance of ix from the center of

cluster jc . Tthe membership in the j -th cluster is calculated

by

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

−′

=

−′−′ p

k

m

ki

jip

k

m

ki

m

jiij d

ddd

x1

11

1

11

11

111)(μ

nipjfor ≤≤≤≤ 1,1 . Here, jid is the distance metric for

ix from the center of cluster jc , m′ is the fuzzification

parameter, p is the number of specified clusters, and kid is the distance metric for ix in cluster kc . Having computed the value of )( ij xμ , it is used in place of the original value of

)( ij xμ in the first step of the FCM procedure. This two-step

procedure is repeated iteratively until the centers of all the clusters within the dataset converge.

B. Index function maxI

Suppose that every object has m conditional attributes and the l -th attribute la can be divided into lp clusters, then

( )il xC gives the index of cluster to which the l -th attribute

la of the object ix belongs. Here ( )il xC is given by ( ) ( )( ))(max lijil axIxC μ= =Index ( )( )( ))(max lij axμ

nimlfor ≤≤≤≤ 1,1 where ( )( ))(max lij axI μ index of the cluster

corresponding to the maximum value of the membership functions ix associated with the l -th attribute.

C. Implication Relations and Fuzzy Algorithms Fuzzy thenif / rules are conditional statements that

describe the dependence of one (or more) linguistic variable(s) on another variable. The underlying analytical form of an thenif / rule is a fuzzy relation known as an implication relation. Typically, these implication relations are acquired by inputting the left-hand side (LHS) and right-hand side (RHS) of a rule into an implication operator,. φ The choice of an appropriate implication operator is a significant step in the overall development of a fuzzy linguistic description and reflects both the application-specific criteria and the logical and intuitive considerations relating to the interpretation of the connectives AND , OR , and ELSE .Consider a generic thenif / rule involving two linguistic variables, one on each side of the rule expression, i.e. if x is A then y is B

in which the linguistic variables x and y take the values of A and B , respectively. The underlying analytical form of this rule is given by the following implication relation [14]

∫= ),( ),/(),(),( yx yxyxyxR μ where ( )yx,μ is the

membership function of the implication relation ),( yxR . Several options exist for acquiring the membership function of an implication relation. For the rule given above, the implication operator φ takes the membership functions of antecedent and consequent parts, i.e. )(xAμ and )(yBμ , respectively, and generates an output ),( yxμ , i.e.

[ ])(),(),( yxyx BA μμφμ = The implication operators used in this study are: (a) Zadeh Max-Min Implication Operator [ ])(),( yx BAm μμφ = ),( yxμ = ( ) ( ))(1)()( xyx ABA μμμ −∨∧ (b) Mamdani Min Implication Operator [ ])(),( yx BAc μμφ = ( ))()( yx BA μμ ∧ for these two operators have been widely used. Their

interpretations for the linguistic connective “ELSE” are AND ( ∧ ) and OR ( ∨ ) respectively. Generally, we are

interested in linguistic descriptions that may have more than one variable in either side, to which we refer as multivariate fuzzy algorithm. Considered an thenif / rule of the form:

if 1x is 1A AND 2x is 2A ... AND mx is mA then y isB

where mxx ,...,1 are the antecedent linguistic variables with mAA ,...,1 be their correspondent fuzzy values, and y is the consequent linguistic variable with fuzzy value B . The connectives AND in the LHS of the above rule can be analytically modeled either as min ( ∧ ) or product ( • ). In such case, one could combine the proposition in the LHS through min ( ∧ ) and use an appropriate implication operator φ to acquire the membership function of the implication relation above. Thus one has:

( )yxxx m ,,...,, 21μ = [ ])(),(...)()( 21 21yxxx BmAAA m

μμμμφ ∧∧∧

whichφ is an appropriate implication operator. Fuzzy algorithms are essentially an automated procedure

for interpreting a linguistic statement formulated as a collection of fuzzy thenif / rules, which in this paper is the collection of implication relations. All of the rules within the collection are formed by the same attributes and are connected by the connective ELSE . The final result may be interpreted either as either a union or an intersection of the rules depending on the implication operator used in the individual rules. For example, consider the following set of rules:

if x is 1A then y is 1B ELSE

if x is 2A then y is2B ELSE

... if x is nA then y is nB Recall that each rule above is represented analytically by

434

an implication relation ),( yxR , where the form of ),( yxR depends on the choice of implication operator. In this study, the interpretations of the connective ELSE are given as follows:

Implication: mφ ( Zadeh), Interpretation of ELSE : AND ( ∧ )

Implication: mφ ( Mamdani), Interpretation of ELSE : OR ( ∨ )

The relation describing the entire collection of rules given above is known as the algorithmic relation and has the form:

∫= ),( ),/(),(),( yx yxyxyxR αα μ

Note that αR is either a union( ∨ ) or an intersection( ∧ ) of the implication relations of the individual rules depending on the chosen implication operator. D. The procedure to acquire the β value

This study presents a five-step procedure for determining an appropriate value of the threshold parameter β in the VPRS model.

1) Fuzzify the continuous attributes of the information system using the FCM method

A continuous-value information system can only be converted into an equivalent fuzzy information system under the condition once a classified fuzzy set has been provided. The attribute data of each object are converted to the values of the membership functions in this step. In the FCM clustering method, the l -th attribute la can be divided into lp fuzzy clusters, and ( ))( lij axμ represents the values of the

membership functions associated with the l -th conditional attribute la of the i -th object. If an attribute is crisp, there would be no ambiguity during clustering phase, and then the corresponding membership function value for crisp attributes will be set to 1.

2) Find the corresponding cluster of every continuous attribute in accordance with the index function maxI

Utilizing the index function ( )( ))(max lij axI μ =

Index ( )( )( ))(max lij axμ for niml ≤≤≤≤ 1,1 , identify the

cluster corresponding to the l -th attribute ( la ) of every object )( li ax . The resulting cluster indexes give each object a set of discrete attribute values in the decision table. For crisp attributes (if there are any) in the objects, the assigning of cluster indexes is arbitrary and is simply for identification.

3) Arrange the Implication Relations Considering a generic thenif / rule involving two

linguistic variables, one on each side of the rule expression, i.e: if x is A then y is B

A rule extracted from the decision table actually has the same form. For example, for an object with 2 conditional attributes and on decision attribute having the following values ( 1a , 2a , d ) = (1, 2, 1), the object entry can be translated into the following linguistic expression:

if ( 1a , 2a ) is (1,2) then ( d ) is (1). An implication relation for the above rule could be formed

by using proper implication operators. As discussed above, the uses of different implication operators result in the construction of different implication relations, and consequently the membership function of an Implication Relation might vary. The present study utilizes the Zadeh Max-Min implication operator [15], which yields the following implication relations for two operands )(xAμ and

)(yBμ : [ ] ( ) ( ))(1)()(),()(),( xyxyxyx ABABAM μμμμμμφ −∨∧==

In the multivariate case, suppose that there exist m conditional attributes and one decision attribute for each object. According to Zadeh’s Implication operator for multivariate fuzzy algorithm, the implication relation for the i -th object ( ix ) is therefore given as

( ) ( ) ( ) ( )[ ])(,)(...)()( 21 21dxaxaxax idmiaiaiaM m

μμμμφ ∧∧ = ( ) ( ) ( ) ( )( )dxaxaxax imiii ,,...,, 21μ

where ( ))( lij axμ represents the j -th membership

function of the l -th conditional attribute ( la ) of the i -th object ( ix ), and ( ))(dxijμ is the j -th membership function

of the decision attribute ( d ) of the i -th object ( ix ).For example, suppose that there are 2 conditional attributes ( 21, aa ) and 1 decision attribute ( d ) are to be classified using the FCM approach. Suppose further that the objective of the clustering process is to divide the conditional attributes into 3 clusters and the decision attributes into 2 clusters. Finally, suppose that the i -th object’s class is given by (1, 3, 2), where (1,3) denotes the clustering indexes of the 2 conditional attributes ( 21, aa ), respectively, and (2) denotes the clustering index of the decision attribute ( d ). In general, there exist a total m classes for classified objects for which the decision attribute index has a value of (2). Suppose that there exist 3 objects complying with the class (1, 3, 2). Suppose further that the serial numbers of these 3 objects are 4, 7, 9, respectively. The rule obtained from object “4” will be

if ( 21, aa ) is (1,3) then ( d ) is (2). Then the implication relation of the object with a serial

number “4” can be expressed as ( ) ( ) ( )[ ] ( ))(),(),()(,)()( 4241442414 21

dxaxaxdxaxax daaM μμμμφ =∧

( ))(),(),( 42414 dxaxaxμ gives the final membership value of the rule obtained from object “4”. Since ( ))( 14 axμ could be understood as the probability of object 4x belongs to the current cluster in attribute 1a , etc, one could interpret the value of the overall membership function as the reliability of this rule.

4) Determine the β value of VPRS using Fuzzy Algorithm Taking the implication relation example given in Step 3 for

illustration purposes, the objects (4, 7, 9), which belong to the same equivalence class, can be related to the linguistic

435

descriptions in a fuzzy algorithm by means of the connective operator ELSE , i.e.

if 1a is ( ))( 141 axμ and 2a is ( ))( 243 axμ then d is ( ))(42 dxμ ELSE

if 1a is ( ))( 171 axμ and 2a is ( ))( 273 axμ then d is ( ))(72 dxμ ELSE

if 1a is ( ))( 191 axμ and 2a is ( ))( 293 axμ then d is ( ))(92 dxμ

Note that as described above, the interpretation of the connective operator ELSE in the fuzzy algorithm varies in accordance with the particular choice of implication operator, i.e. AND ( ∧ ) for the Zadeh implication operator or OR ( ∨ ) for the Mamdani implication operator. As the result, a final integrated membership function

( ) ( ) ( ) ( )( )dxaxaxax imiiiq ,,...,, 21μ was generated for the

above algorithmic relation. Here ( ) ( ) ( ) ( )( )dxaxaxax imiiiq ,,...,, 21μ is the threshold value

obtained from Fuzzy Algorithm of a certain q classified objects set qX . If the β values are to be interpreted as the

reliabilities of rules, then the Zadeh definition is the desired one. Recall that the membership function of an object is aroused from the fuzzy nature during clustering and the uncertainties may be led to the information system. This uncertainty would limit the accuracy of rule findings. Thus, we argue that a reasonable choice of β value should not exceed the value obtained from the Fuzzy Algorithms:

( ) ( ) ( ) ( )( )dxaxaxax imiiiq ,,...,, 21μβ ≤ where qi Xx ∈

IV. COMPARISON WITH OTHER β -SELECTING METHODS

A. Illustration Case Consider the simple data example given in Table1. All

objects are ambiguously classified. Subsequently, the indiscernible classes of objects grouped by conditional attributes are

{ }4321* ,,, CCCCC = , where { }711 ,ooC = , { }8522 ,, oooC = , { }9633 ,, oooC = , { }44 oC =

Similarly, the decision classes are { }FMD ,* = , where { }654321 ,,,,, ooooooDM = and { }987 ,, oooDF =

(insert Table 1 and caption here) Follow the definitions in Ref [7],

( )( )

( )( )( ) 0

0

0

1,=>

⎪⎩

⎪⎨⎧ −=

XcardifXcardif

XcardYXcard

YXc∩

where card

denotes set cardinality, and the precision parameter is defined as ( )DC,ζ = ( )21,max mm , where

1m = ( )[ ]DCc ,5.0min1 <− , 2m = ( )[ ]5.0,max <DCc . Applying the above definitions to objects in Table 1, it

follows that ( )21

211,1 =−=MDCc , ( )

21

211,1 =−=FDCc ,

( )31

321,2 =−=MDCc , ( )

32

311,2 =−=FDCc ,

( )31

321,3 =−=MDCc , ( )

32

311,3 =−=FDCc ,

( ) 0111,4 =−=MDCc , ( ) 1

101,4 =−=FDCc ,

311,

32,

32min11 =⎟

⎠⎞

⎜⎝⎛−=m ,

310,

31,

31max2 =⎟

⎠⎞

⎜⎝⎛=m .

Thus, ( )31

31,

31max, =⎟

⎠⎞

⎜⎝⎛=DCζ . Therefore, the precision

parameter value is equal to 31 . Since the β value, based on the least upper bound of data misclassification error, is equal

to 31 , then ( )MDC β = 432 CCC ∪∪ =

{ }9865432 ,,,,, , ooooooo and ( ) { }=FDC β . Follow the

analysis in Ref [11], where β denoted the proportion of correct classifications, for β =0.7, then one could

obtain ( ) { }47.0 oDPOS MC = , ( ) { }=MC DNEG 7.0 ,

( ) { }987653217.0 ,,,,,,, ooooooooDBND MC = . And forβ

=0.55, one could obtain ( )MC DPOS 55.0 = { }9865432 ,,,,, , ooooooo ,

( ) { }=MC DNEG 55.0 , ( ) { }7155.0 , ooDBND MC = . It follows

that the quality of classification for the information system in Table 1 for a β value chosen between 0.5 and 0.667 equals to 0.777(i.e. 7/9). However, if the β value were chosen between 0.667and 1, the quality of classification would equal to 0.11(i.e. 1/9). We could not find the β value at which the boundary region will become empty suggested by Ref [8].

In this paper, consider the class { }8522 ,, oooC = of

{ }4321* ,,, CCCCC = , the β value could be obtained from

the chosen Fuzzy Algorithms: ( ))(),2(),1( 222 MDxxxμ ∧ ( ))(),2(),1( 455 MDxxxμ ∧ ( ))(),2(),1( 888 FDxxxμ =

( )1,93.0,71.0μ ∧ ( )1,61.0,91.0μ ∧ ( )1,88.0,97.0μ =0.61. Similarly, consider the 1C , 3C and 4C classes, the β values obtained from the chosen Fuzzy Algorithms are 0.66, 0.56 and 0.86, respectively. So, the β -lower approximation of the

1C , 2C , 3C and 4C classes are { } , { } , { }963 ,, ooo , and { }4o , respectively. Integrate all the 1C , 2C , 3C and 4C classes by the chosen Fuzzy Algorithms, one could use the value 0.56 as the desired β value. If the objects { }5o of

2C class and { }6o of 3C class were deleted, the data misclassification error (or the proportion of correct classifications) of all classes will be equal to 0.5, then the β value could not be obtained by adopting the methods in Ref

436

[7] and [8]. But the β values of 1C , 2C , 3C and 4C classes are still obtainable from the chosen Fuzzy Algorithms, and they will be 0.66, 0.71, 0.56 and 0.86, respectively.

V. CONCLUSIONS AND DISCUSSIONS VPRS is an extension of RS by setting the classification

correctness threshold value β to release the strict definition of approximation boundary in RS. In the past, a β value was selected after the objects have been classified, and the origin of the classification ambiguity has never been addressed to incorporate with the choice of β. This study developed a method to determine the value β and took into account errors during the clustering phase, which is the pre-processing stage of system classification. Given below are the summaries and findings of this paper:

(1) This study utilized the membership function of Fuzzy C-Means as the basis to extract the β value for VPRS. Suppose that there are some attributes for each object, and several clusters in each of the attribute. According to the definition of index function maxI , the value of membership function of a certain attribute was represented by the value of maximum membership function from within all clusters of that attribute. Then this study adopted the Zadeh Implication Operators and Zadeh Fuzzy Algorithm Operators ( AND ) to obtain the value β for VPRS.

(2) Whether using the least upper bound of data misclassification error or the proportion of correct classifications methods, the threshold value β obtained depends on the relative degree of misclassification (or the classification precision) of all the condition classes with respect to all the decision classes. In our method, one acquires for a unique β value for each equivalence class instead. Each group of indiscernible objects has its own β value; this is closer to the reality. The β values obtained from the chosen Fuzzy Algorithms depends on the values of membership function of each equivalence class. And the integrated β value of information system could be obtained by joining allβ values of each equivalent classes through the connective operators in the chosen Fuzzy Algorithms.

(3) It is not possible in the RS to assert whether two attribute values are similar and to what extent they are the same; for example, two close values might only differ as a result of noise, but in RS they are considered to be as different as two values of different orders of magnitude. It is, therefore, desirable to develop the fuzzy C-means and other Fuzzy Algorithms to provide tolerances for ambiguities from real-value attributes of which values are close. This could be achieved through the use of Fuzzy Algorithms in this paper.

REFERENCES [1] Pawlak, Z., Rough sets, International Journal of Information and

Computer Sciences11 (1982) 341-356.

[2] Tsumato, S., Slowinski, S., Komorowsk, J., & Grzymala-Busse, J. W., lecture notes in artificial intelligence, The fourth international conference on rough sets and current trends in computing ( RSCTC’2004) 2004.

[3] Ziarko, W., Variable precision rough set model, Journal of Computer and System Sciences 46 (1993) 39-59.

[4] An, A., Shan, N., Chan, C., Cercone, N., Ziarko, W., Discovering rules for water demand prediction: an enhanced rough-set approach, Engineering Application and Artifical Intelligence 9 (1996) 645-653.

[5] Beynon, M., Reducts within the variable precision rough sets model: a further investigation, European Journal of Operational Research 134 (2001) 592-605.

[6] Beynon, M., The identification of low-paying workplaces: an analysis using the variable precision rough sets model, The Third International Conference on Rough Sets and Current Trend in Computing. Lecture notes in artificial intelligence Series, Berlin: Springer, 2000 530-537.

[7] Su, C.-T., Hsu, J.-H., Precision parameter in the variable precision rough sets model: an application, Omega 34 (2006) 149-157.

[8] Glackina, C., Maguirea, L., McIvorb, R., Humphreysb, P., Hermana, P., A comparison of fuzzy strategies for corporate acquisition analysis, Fuzzy Sets and Systems 158 (2007) 2039-2056.

[9] Cox, E., Fuzzy Modeling and Genetic Algorithms for Data Mining and Exploration, Elsevier Inc., 2005.

[10] Ramze Rezaee, M., Lelieveldt, B.P.F., Reiber, J.H.C., A New Cluster Validity Index for the Fuzzy C-Mean, Pattern Recognition Letters, 19 (1998) 237-246.

[11] Tsoukalas, L.H., Uhrig, R.E., Fuzzy and Neural Approaches in Engineering, John Wiley & Sons, Inc., 1997.

[12] Dunn, J.C., A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cybernet, 3 (1973) 32-57.

[13] Bezdek, J.C., Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981.

[14] Ruan, D. and Kerre, E. E., Fuzzy Implication Operators and Generalized Fuzzy Method of Cases, Fuzzy Sets and Systems, 54 (1993) 23-37.

[15] Zadeh, L. A., Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Transactions on Systems, Man, and Cybernetics, 3(1973)28-44.

Table 1 An illustrative information system objects a b D 1 1(0.82) 1(0.66) M 2 1(0.71) 2(0.93) M 3 2(0.75) 1(0.56) M 4 2(0.86) 2(0.94) M 5 1(0.91) 2(0.61) M 6 2(0.69) 1(0.76) M 7 1(0.84) 1(0.78) F 8 1(0.97) 2(0.88) F 9 2(0.85) 1(0.72) F

437