A vague-rough set approach for uncertain knowledge acquisition

A vague-rough set approach for uncertain knowledge acquisition

Lin Feng a ,b ,?,Tianrui Li b ,Da Ruan c ,d ,Shirong Gou a

a

College of Computer Science,Sichuan Normal University,Chengdu 610101,PR China

b

School of Information Science and Technology,Southwest Jiaotong University,Chengdu 610031,PR China c

Belgian Nuclear Research Centre (SCK áEN),Boeretang 200,2400Mol,Belgium d

Department of Applied Mathematics &Computer Science,Ghent University,Krijgslaan 281(S9),9000Gent,Belgium

a r t i c l e i n f o Article history:

Received 5February 2010

Received in revised form 15March 2011Accepted 20March 2011

Available online 3April 2011Keywords:

Knowledge acquisition Rough sets Vague sets

Vague rough sets Attribute reduction

Uncertain information system

a b s t r a c t

By combining both vague sets and rough sets in fuzzy data processing,we propose a vague-rough set approach for extracting knowledge under uncertain environments.We compute all attribute reductions using the vague-rough lower approximation distribution,concepts of attribute reduction and the discern-ibility matrix in a vague decision information system (VDIS).Research results for extracting decision rules from the VDIS show the proposed approaches extend the corresponding method in classical rough set theory and provide a new avenue to uncertain vague knowledge acquisition.

ó2011Elsevier B.V.All rights reserved.

1.Introduction

Rough set theory by Pawlak [1]is a suitable mathematical ap-proach for handling imprecision,incompletion and uncertainty in data analysis.Its ef?ciency has been successfully demonstrated by many related applications such as attribute reduction,pattern recognition,data mining,and fault detection [2–6].The key ele-ments of rough set theory as cited from [7]are as follows:(1)Only the facts that hidden in the data are analyzed;

(2)No additional information about the data such as thresholds

or expert knowledge is required;

(3)Given a data set with discretized attribute values,it is possi-ble to ?nd a subset (termed as reduct)within the original attributes that are the most informative.Attribute values in real-world applications are often both sym-bolic and real-valued in data sets.The classical rough set theory is mainly based on the indiscernibility relation (equivalence relation)and cannot handle effectively such real-valued attributes.One pos-sible way to solve this problem is to discretize the attribute values beforehand,but it may cause information loss [8,9].Another feasi-ble approach to overcome this shortcoming is to use the fuzzy-rough set theory,in which fuzzy and rough sets encapsulate the re-lated but distinct concepts of fuzziness and indiscernibility,both occurring as a result of uncertainty existed in knowledge [10].In a fuzzy-rough set,a fuzzy similarity relation characterizes the de-gree of similarity between two objects instead of the equivalence relation used in the classical rough sets [11].To date,many re-search results have been obtained in the ?eld of fuzzy-rough sets.For example,Dubois and Prade [12]combined fuzzy and rough sets as a conception of rough fuzzy sets and fuzzy rough sets.Tsang et al.[13]established solid mathematical foundations for an attri-bute reduction in fuzzy rough sets.Pal and Mitra [14]proposed a rough-fuzzy hybridization scheme for case generations.Shen and Jensen [15]developed a fuzzy rule induction based on fuzzy rough sets for feature selection.Wu and Zhang [16]presented construc-tive and axiomatic approaches of fuzzy approximation operators.Lingras and Jensen [17]reviewed the fuzzy and rough hybridiza-tion in supervised learning,information retrieval,feature selection,and neural and evolutionary computing.

Roughly speaking,a fuzzy set F of a universe of discourse U ,U ={x 1,x 2,...,x n },can be represented by F =l F (x 1)/x 1+l F (x 2)/x 2+ááá+l F (x n )/x n ,where l F is a membership function of the fuzzy set F ,l F :U ?[0,1],and l F (x i )indicates the grade of membership of x i in the fuzzy set F .Obviously,for any x i 2U ,the membership va-lue l F (x i )is a single value between 0and 1.On the other hand,if there exist some uncertainties,then a fuzzy set may be inadequate to some models as follows:

0950-7051/$-see front matter ó2011Elsevier B.V.All rights reserved.doi:10.1016/j.knosys.2011.03.005

?Corresponding author at:College of Computer Science,Sichuan Normal University,Chengdu 610101,PR China.

E-mail addresses:scfengyc@https://www.360docs.net/doc/be10011057.html, (L.Feng),trli30@https://www.360docs.net/doc/be10011057.html, (T.Li),druan@sckcen.be ,da.ruan@ugent.be (D.Ruan).

(1)The exact grade of membership l F (x i )of x i may be uncertain

due to limitations or errors of measuring devices;

(2)The exact grades of l F (x i )of x i may be unknown due to the

limitation of human cognitive ability;

(3)Consequents may be bounded by a subinterval,especially

when knowledge is acquired from a group of experts that do not collectively agree with each other,such as a vote model.Gau and Buehrer hence introduced the concept termed vague sets [18].They used a truth-membership function t F and a false-membership function f F to characterize the lower bound on l F .The lower bounds are used to create a subinterval on [0,1],namely [t F (x i ),1àf F (x i )],to generalize the l F (x i )of a fuzzy set.The interval-based membership function generalization in a vague set is more expressive in capturing vagueness of data,and has the ability to model the above-mentioned uncertainties.Vague sets are different from ordinary sets,fuzzy sets,and interval-valued sets [18].The major advantage of vague sets over fuzzy sets is that vague sets separate the positive and negative evidence for membership of an element in the universe [18].In the past decades,investigations of vague set theory and their mathematical properties have been carried out by many researchers [19–24].On the other hand,the existing theories and approaches of knowledge acquisition based on fuzzy-rough sets could not be directly applied to vague data sets [25].It is thus necessary to develop an ef?cient and appropriate ap-proach for knowledge acquisition from vague data sets based on rough sets.

In this paper,we attempt to extend the approaches of attribute reduction by combing rough sets and vague sets from [26].First,we introduce basic ideas of rough sets given in the form of the low and upper approximations in Vague Approximate Space (VAS).Then,we develop a discernibility-matrix approach to com-pute all the attributes reductions in a Vague Decision Information System (VDIS).Finally,we propose an extracting for decision rules from VDIS.

The rest of the paper is organized as follows.We review basic notions and operators of vague sets in Section 2.We discuss vague rough approximations sets in VAS in Section 3.We develop ap-proaches for knowledge reduction and knowledge acquisition in VDIS in Section 4.We summarize the study and outline the further research work in Section 5.2.Preliminaries

2.1.Basic notions of vague sets

Basic notions of vague sets are mainly cited from [18].Let U be a space of points (objects),with a generic element of U denoted by x .A vague set V in U is characterized by a true-membership function t V and a false-membership function f V .t V (x )is a lower bound on the grade of membership of x derived from the evidence for x ,and f V (x )is a lower bound on the negation of x derived from the evidence against x .Both t V (x )and f V (x )associate with a real number in the interval [0,1]with each point in U ,where t V (x )+f V (x )61.That is,

t V :U !?0;1 ;f V :U !?0;1 :

e1T

This approach bounds the membership grade of x to a subinterval [t V (x ),1àf V (x )]of [0,1].In other words,the exact membership grade l V (x )of x may be unknown,but is bounded by t V (x )6l V (x )61àf V (x ),where t V (x )+f V (x )61.

The precision of human knowledge about x is immediately clear,with the uncertainty described by the difference

D (x )(D (x )=1àf V (x )àt V (x )).If D (x )is small,human knowledge about x is relatively precise;if it is large,the knowledge about x is relatively little.If 1àf V (x )is equal to t V (x ),i.e.,D (x )=0,the knowledge about x is exact,and the theory reverts back to that of fuzzy sets.If 1àf V (x )and t V (x )are both equal to 1or 0,i.e.,D (x )=[0,0]or D (x )=[1,1],depending on whether x does or does not belong to V ,the knowledge about x is exact and the theory re-verts back to that of ordinary sets (i.e.,sets with two-value charac-teristic functions)[16].

When U is continuous,a vague set is written as V ?Z

U

?t V ex T;1àf V ex T =x x 2U :e2T

When U is discrete,a vague set is written as

V ?

X

n i ?1

?t V ex i T;1àf V ex i T =x i x i 2U :e3T

The vague set theory can be interpreted by a voting model.Suppose that V is a vague set in U ,x 2U ,and the vague value is [0.6,0.8],that is,t V (x )=0.6,f V (x )=1à0.8=0.2.Then,the degree of x 2V is 0.6,and the degree of x R V is 0.2.The vague value [0.6,0.8]can be inter-preted as ‘‘The vote for a resolution is 6in favor,2against,and 2abstentions (The number of the total voting people is assumed to be 10)’’.

2.2.Algebraic operations

We list basic operations of vague sets,including containment,equal,union,intersection,and complement.Let x be an element of U .

De?nition 1(Containment ).A vague set X is contained in another vague set Y ,written as X #Y ,if and only if

t X ex T6t Y ex T;1àf X ex T61àf Y ex T:

e4T

De?nition 2(Equal ).Vague sets X and Y are equal,written as X =Y ,if and only if X #Y and X Y ;that is

t X ex T?t Y ex T;

1àf X ex T?1àf Y ex T:

e5T

De?nition 3(Union ).The union of two vague sets X and Y with respective truth-membership and false-membership functions t X ,f X ,t Y and f Y is a vague set Z ,written as Z =X [Y ,whose truth-mem-bership and false-membership functions are related to those of X and Y by

t Z ex T?max et X ex T;t Y ex TT;

1àf Z ex T?max e1àf X ex T;1àf Y ex TT?1àmin ef X ex T;f Y ex TT:

e6T

De?nition 4(Intersection ).The intersection of two vague sets X and Y with respective truth-membership and false-membership functions t X ,f X ,t Y and f Y is a vague set Z ,written as Z =X \Y ,whose truth-membership and false-membership functions are related to those of X and Y by

t Z ex T?min et X ex T;t Y ex TT;

1àf Z ex T?min e1àf X ex T;1àf Y ex TT?1àmax ef X ex T;f Y ex TT:

e7T

838L.Feng et al./Knowledge-Based Systems 24(2011)837–843

De?nition 5(Complement ).The complement of a vague set X denoted by X c is de?ned by

t X c ex T?f X ex T;1àf X c ex T?1àt X ex T:

e8T

3.Generalized vague rough approximations

Denote (U ,R )a Pawlak approximation space,where U is a ?nite

and non-empty set of objects called a universe,and R is an equiv-alence relation on U .We know a set X (X #U ),can be represented by rough sets (R à(X ),R à(X )).We also know a set X can be repre-sented by fuzzy rough sets eb R

àeX T;b R àeX TTwhen R is a fuzzy rela-tion b R

on U .However,if R is a vague relation e R ,an extension of a fuzzy relation on U ,how to describe X in eU ;e R

T?To solve this prob-lem,we develop here a vague-rough set.To begin with,we intro-duce a method for measuring the degree of containment and intersection between two vague sets based on fuzzy rough sets in [27,28].

Let X and Y be two vague sets of the universe of the discourse U .By De?nition 1,X is contained in Y

X #Y ()For any x 2U ;t X ex T6t Y ex Tand

1àf X ex T61àf Y ex T:

e9T

If the condition (9)is satis?ed,then X is in Y .However,we intend to evaluate the containment degree of X in Y about each object of U .Therefore,for all x 2U ,we obtain a new vague set,which is called

as the vague containment of X in Y and denoted by X ~#

Y .By using an implication operator ‘‘?’’,we have:

l X ~#Y ex T?l X ex T!l Y ex T??l X ex T c

[l Y ex T:

e10T

By De?nitions 3and 5,(10)is rewritten as

l X ~#Y ex T?max ef X ex T;t Y ex TT;max e1àt X ex T;1àf Y ex TT? :

e11T

Applying (11),the containment degree of X in Y ,denoted by I (X ,Y ),

is a vague value,which is de?ned as

I eX ;Y T?inf x 2U

max ef X ex T;t Y ex TT;inf x 2U

max e1àt X ex T;1àf Y ex TT

:

e12T

At the same time,by De?nition 4,the intersection degree of be-tween X and Y ,denoted by T (X ,Y ),is a vague value,which is de?ned

as

T eX ;Y T?sup x 2U

min et X ex T;t Y ex TT;sup x 2U

min e1àf X ex T;1àf Y ex TT

:

e13T

Example 1.Assume that U ={x 1,x 2,x 3,x 4}.Let

X ??0:7;0:8 =x 1t?0:5;0:9 =x 2t?0:6;0:8 =x 3t?0:3;0:5 =x 4;

Y ??0:2;0:3 =x 1t?0:7;0:9 =x 2t?0:6;0:8 =x 3t?0:3;0:8 =x 4:

By (12)and (13),we have

I eX ;Y T?inf x i 2U

e0:2;0:7;0:6;0:5T;inf x i 2U

e0:3;0:9;0:8;0:8T

??0:2;0:3 ;

T eX ;Y T?sup x i 2U

e0:2;0:5;0:6;0:3T;sup x i 2U

e0:3;0:9;0:8;0:5T

"

#

??0:6;0:9 :

De?nition 6(Vague relation ).Let U be a ?nite and nonempty set of objects and U ?U be the product set of U and U ."x ,y ,z 2U ,any

vague subset e R

of U ?U is a vague relation on U ,which denotes e R ex ;y T??t e R

ex ;y T;1àf e

R

ex ;y T ,if e R satis?es:(1)Re?exivity:e R

ex ;x T??1;1 ;(2)Symmetry:e R

ex ;y T?e R ey ;x T;(3)Transitivity:t R ex ;z TP sup y 2U

min et R ex ;y T;t R ey ;z TTand

1àf R ex ;z TP sup y 2U

min e1àf R ex ;y T;1àf R ey ;z TT:

Clearly,e R

is a vague equivalence relation.If e R satis?es condi-tions (1)and (2),e R

is a vague similarity relation.As mentioned in Section 2,for any x ,y 2U ,if e R

ex ;y T??0;0 or [1,1],e R degener-ates a classical equivalence relation.If t e R ex ;y T?1àf e R

ex ;y T;e R

degenerates a fuzzy equivalence relation.

Because U is discrete,by De?nition 6,the e R -vague classes ?z e R

containing object z can be written as a vague set ?z e R

?P y 2U ?t e R ez ;y T;1àf e R

ez ;y T =ez ;y T.On the other hand,for all x 2U ,the vague rough approximations need to calculate the containment

and the intersection degree of X in e R -vague classes ?x e R

.By (12)and (13),the vague rough approximations could be de?ned as follows.De?nition 7(Vague rough approximations ).Let eU ;e R

Tbe a Vague Approximation Space (VAS),where U is a ?nite and non-empty set

of objects called a universe,and e R

is a vague relation.For any x 2U ,X #U ,the vague lower approximation of X with respect to eU ;e R

T,denoted by e R

àeX T,is a vague set of U ,whose membership function is de?ned by

l e R àeX Tex T?inf y 2U max ef e R ex ;y T;t X ey TT;inf y 2U

max e1àt e R ex ;y T;1àf X ey TT

:e14T

The vague upper approximation of X with respect to eU ;e R

T,denoted by e R àeX T,is a vague set of U ,whose membership function is de?ned by

l e R àeX Tex T?sup y 2U

min et e R ex ;y T;t X ey TT;sup y 2U

min e1àf e R

ex ;y T;1àf X ey TT"#

:e15T

The pair ee R

àeX T;e R àeX TTis called the generalized vague-rough sets of X with respect to eU ;e R

T.Since X is a crisp set,we have ?t X ey T;1àf X ey T ?

?1;1 ;y 2X ;?0;0 ;

y R X :

e16T

We thus easily derive (17)from (14)–(16),

l e R àeX Tex T??inf y R X f e R ex ;y T;inf y R X

e1àt e R ex ;y TT ;l e R à

eX T

ex T??sup y 2X

t e R

ex ;y T;sup y 2X

e1àf e R

ex ;y TT :

8

<:e17T

Proposition 1.Let eU ;e R

Tbe a VAS.If e R degenerates into an equiva-lence relation,vague rough approximation sets,e R

àeX Tand e R àeX T,are equivalent to Pawlak approximation sets R à(X)and R à(X),respectively.

Proof.If e R

is an equivalence relation,then for any x ;y 2U ;l e R ex ;y T??1;1 or l e R ex ;y T??0;0 ,i.e.,?t ?x e R ey T;1àf ?x e R ey T ??1;1 y 2?x e

R ?0;0 y R ?x e

R

(

.

By (14),we have l e R àeX Tex T??t X e?x e R T;1àf X e?x e R

T ??1;1 ,which implies e R àeX T?f x 2U j?x e R

#X g .Thus,e R àeX Tis equivalent to R à(X ).

Similarly,e R

àeX Tis equivalent to R à(X ).h L.Feng et al./Knowledge-Based Systems 24(2011)837–843839

Theorem1.LeteU;e RTbe a VAS.X,Y#U.If e R is a vague similarity relation,a vague-rough set has the following algebra properties:

(1)e RàeX\YT?e RàeXT\e RàeYT;

(2)e RàeX[YT?e RàeXT[e RàeYT;

(3)X#Y)e RàeXT#e RàeYT;

(4)X#Y)e RàeXT#e RàeYT;

(5)e RàeX[YT e RàeXT[e RàeYT;

(6)e RàeX\YT#e RàeXT\e RàeYT.

Theorem2.LeteU;e RTbe a VAS.X,Y#U.If e R is a vague equivalence relation,a vague-rough set has the following algebra properties:

(1)e RàeX\YT?e RàeXT\e RàeYT;

(2)e RàeX[YT?e RàeXT[e RàeYT;

(3)X#Y)e RàeXT#e RàeYT;

(4)X#Y)e RàeXT#e RàeYT;

(5)e RàeX[YT e RàeXT[e RàeYT;

(6)e RàeX\YT#e RàeXT\e RàeYT;

(7)e RàeXT#X#e RàeXT;

(8)e Ràee RàeXTT?e RàeXT;

(9)e Ràee RàeXTT?e RàeXT.

From Theorems1and2,we conclude that the operations of intersection,union and containment between vague rough sets are not the same as the corresponding operations of vague sets in Section2.Because a vague-rough set is de?ned on the special knowledge space(approximation space),it has many properties that may contribute to knowledge acquisition whereas a pure va-gue set does not own.Next,we will use the vague-rough set model to study approaches of attribute reduction and knowledge discov-ery in VDIS.

4.Attribute reductions and knowledge acquisition in VDIS

4.1.Vague rough approximations in VDIS

Denote S=(U,A)a Vague Information System(VIS),where U is a ?nite nonempty set of objects called a discourse,A is a?nite non-

empty set of attributes.For any a i2A,a i:U!V a

i ,where V a

i

is

called the value domain of a i,which can be represented as a set of linguistic terms V a

i

?f v i1;v i2;...;v im g.For any v ij2V a i,v ij is a vague set of U.For any x i2U,a i(x i)indicates the values of x i with respect to a i,which can be represented as a vague set of a uni-

verse of the discourse V a

i ,such that a iex iT?l v

i1

ex iT=v i1t

l v

i2ex iT=v i2tááátl v imex iT=v im,where l v ij is a grade of membership

of x i belonging to v ij.

If A=C[{d},C\{d}=;,where C is a?nite non-empty set of conditional attributes,d is a decision attribute,VIS is called a VDIS. Obviously,in a VDIS,for any v i12V a i and v i22V a iàf v i1g,if a(x i)(v i1)=[1,1]and a(x i)(v i2)=[0,0],then VDIS degenerates into a Pawlak information system.

We now study the approximation sets in VDIS based on the dis-cussions of the fuzzy approximations in[31].

De?nition8.Approximations in VDISLet S=(U,C[{d})be a VDIS. For any x,y2U,a2C,a vague relation~a on a is de?ned by

~aex;yT?sup

v2V a minet vexT;t veyTT;sup

v2V a

mine1àf vexT;1àf veyTT

"#

:

e18TTherefore,for any B#A,a vague relation e B on B can be induced

e Bex;yT?inf

a2B sup

v2V a

minet vexT;t veyTT;inf

a2B

sup

v2V a

mine1àf vexT;1àf veyTT

"#

:

e19TExample2.Table1is a VDIS,which is extended by a fuzzy deci-sion table in[29],while some instances are modi?ed according to[30].It has four condition attributes C={Outlook(a1),Tempera-ture(a2),Humidity(a3),Wind(a4)},and a decision attribute Play Tennis(d).Each condition attribute has linguistic terms,i.e., V a1={Sunny(v11),Cloudy(v12),Rain(v13)},V a2={Hot(v21),Mild (v22),Cool(v23)},V a3={Humid(v31),Normal(v32)},and V a4={Wind

(v41),Not windy(v42)}.Decision attribute d has two values,i.e., V d={No(0),Yes(1)}.The a1(x1)can be represented as a vague set of a universe of the discourse V a1,i.e.,a1(x1)=[0.8,0.9]/v11+ [0.1,0.3]/v12+[0,0]/v13.The v11is a vague set of U,which can be represented as v11=[0.8,0.9]/x1+[0.8,1]/x2+ááá+[0,0]/x14.

At the same time,in Table1,for attribute a3,we have

~a

3

ex1;x2T?sup f min f0:8;0:8g;min f0:2;0:3gg;

?

sup f min f1;1g;min f0:3;0:4gg ??0:8;1 :

Also for attribute a4,we have

~a

4

ex1;x2T?sup f min f0:4;0:7g;min f0:6;0:4gg;

?

sup f min f0:6;0:9g;min f0:7;0:5gg ??0:4;0:5 : Therefore,for attribute sets M={a3,a4},we obtain

e Mex1;x2T?in

f f0:8;0:4g;inf f1;0:5g

? ??0:4;0:5 :

Let S=(U,C[{d})be a VDIS.For any B#A,X#U,e B is a vague rela-tion on https://www.360docs.net/doc/be10011057.html,ing(17),the membership functions of the lower approximation e BàeXTand the upper approximation e BàeXTin VDIS, denoted as l e

BàeXT

exTand l e

BàeXT

exTrespectively,are de?ned as

l e

BàeXT

exT?inf

y R X

f e

B

ex;yT;inf

y R X

e1àt e

B

ex;yTT

;

l e

BàeXT

exT?sup

y2X

t e

B

ex;yT;sup

y2X

e1àf e

B

ex;yTT

"#

:

8

>>>

><

>>>

>:

e20T

Proposition2.Let S=(U,C[{d})be a VDIS,a2C and x,y2U.If a(x)#a(y),then e Cex;zT#e Cey;zT,for all z2U.

Proof.For all z2U,we have

~aex;zT?sup

v2V a

minet vexT;t vezTT;sup

v2V a

mine1àf vexT;1àf vezTT"#

and

~aey;zT?sup

v2V a

minet veyT;t vezTT;sup

v2V a

mine1àf veyT;1àf vezTT"#

:

Since a(x)#a(y),for any v2V a,we obtain t v(x)6t v(y)and 1àf v(x)61àf v(y).Therefore,we have~aex;zT#~aey;zT.And since e Cex;zT?inf

a2C

~aex;zTand e Cey;zT?inf

a2C

~aey;zT,we have e Cex;zT#

e Cey;zT.h

4.2.Attribute reduction in VDIS

One fundamental aspect of rough sets for knowledge acquisi-tion involves the searching for some particular subsets of condition attributes.One subset can provide the same quality of classi?ca-tion as the original.Such subsets are called reducts.To facilitate our discussion,we?rst introduce the notion of attribute reduction in VDIS.

Let S=(U,C[{d})be a VDIS,U=f d g?f D1;D2;...;D j V

d

j

g.We denote

n CexT?el e

CàeD1T

exT;l e

CàeD2T

exT;...;l e

CàeD j V

d j

T

exTT;

c

C

exT?f D k j maxet D

k

exTàf D

k

exTT;k?1;2;...;j V d jg:

(

e21T

840L.Feng et al./Knowledge-Based Systems24(2011)837–843

n C(x)and c C(x)are called the lower approximation distribution, maximum lower approximation distribution of x with respect to d of C,respectively.

De?nition9(Reductions in VDIS).Let S=(U,C[{d})be a VDIS.For an attribute subset B#C and x2U,B is referred to as an attribute reduction in VDIS if and only if n B(x)=n C(x)and"b2B(n B n{b}(x)–n C(x)).

By De?nition9,a reduct B is a minimum attribute subset of C in VDIS,which enables us to reduce condition attributes C in such a way that for any x2U,the lower approximation distribution of x with respect to decision d of C is preserved.

In real situations,it might not be convenient to obtain n B (x)=n C(x)and"b2B(n B n{b}(x)–n C(x))for any x in U.Hence, we introduce a notion of the discernibility matrix to serve as a tool for discussing and analyzing attribute reductions in the VDIS.

De?nition10(Discernibility matrix in VDIS).Let M d be a discern-ibility matrix in a VDIS,and U={x1,x2,...,x n}.An element of M d in row i of column j,denoted by M d(i,j),is de?ned as

M dei;jT?

f a i j a i2C^ge~a iex i;x jTT?;and xex i;x jTg;

;;otherwise:

e22T

x(x i,x j)satis?es one of the following conditions:

(1)jen e

C ex iTT–;and jen e

C

ex jTT?;;

(2)jen e

C ex iTT?;and jen e

C

ex jTT–;;

(3)jen e

C ex iTT–;;jen e

C

ex jTT–;and jen e

C

ex iTT–jen e

C

ex jTT.

where jen e

C ex iTT?f

D k jet D

k

ex iTàf D

k

ex iTT>0g,l e

CàeD kT

ex iT?

?t D

k ex iT;1àf D

k

ex iT ek?1;2;...;j V d jT,

ge~a iex i;x jTT?f a i2C jet~a

i ex i;x jTàf~a

i

ex i;x jTT60g:

Obviously,the discernibility matrix is an n?n matrix,where n rep-resents the number of objects in the VDIS.Its properties are:

(1)M d(i,j)=M d(j,i);

(2)M d(i,i)=;.

Proposition3.Let S=(U,C[{d})be a VDIS.If the vague conditional-attribute space degenerates into the crisp data space,the discernibility matrix in the VDIS is equivalent to the Skowron’s discernibility matrix in Pawlak decision information systems.Proof.If the vague conditional-attribute space degenerates into the crisp data space,a i2C,for any v ik2V a i,v il2V a iàf v ik g, l v

ik

exT??1;1 and l v

il

exT??0;0 ,the VDIS degenerates the Pawlak decision information system.And x(x i,x j)degenerates into the fol-lowing conditions:

(1)x i2POS C({d})and x j R POS C({d});

(2)x i R POS C({d})and x j2POS C({d});

(3)x i,x j2POS C({d})and(x i,x j)R IND({d}).

According to De?nition of the discernibility matrix in[27],if x(x i,x j) satis?es one of the above conditions,it is a Skowron’s discernibility matrix.h

De?nition11.Discernibility functionLet S=(U,C[{d})be a VDIS. For any x i,x j2U,L=^(_{a i j a i2M d(i,j)})is called the discernibility function in the VDIS.

Theorem3.Each minimal disjunction normal formula of L is a reduc-tion set in a VDIS.

Proof.Suppose one of the disjuncts in the discernibility function is a1_a2,while another disjunct is a1_a2_a3.Obviously,the second disjunct will be satis?ed by either a1or a2,and a3is not required, i.e.,the second disjunct a1_a2_a3could be removed.We may obtain all reductions when this equation is reduced to a disjunction of conjunctions,i.e.,each conjunctive item of the disjunction nor-mal formula corresponds to a reduct set.h

Proposition4.Let S=(U,C[{d})be a VDIS.If the vague conditional-attribute space degenerates into the crisp data space,the reducts in the VDIS are equivalent to those in Pawlak information systems.

Proof.If the vague conditional-attribute space degenerates into the crisp data space,then by Proposition3and De?nition11,we have that the discernibility function L is equivalent to the Skowron discernibility function in[32].Thus,each minimal disjunction nor-mal formula of L in the VDIS is equivalent to in Pawlak information systems.By Theorem3,the conclusion holds.h

Algorithm 1.Attribute reductions based on the discernibility matrix in the VDIS.

Input:VDIS=(U,C[{d}),where C={a i j i=1, 2,...,j C j};

Output:Attribute reduction in the VDIS.

Table1

A VDIS for the Saturday morning play tennis problem.

U Outlook(a1)Temperature(a2)Humidity(a3)Wind(a4)Play tennis(d) Sunny(v11)Cloudy(v12)Rain(v13)Hot(v21)Mild(v22)Cool(v23)Humid(v31)Normal(v32)Windy(v41)Not-windy(v42)

x1[0.8,0.9][0.1,0.3][0,0][1,1][0,0][0,0][0.8,1][0.2,0.3][0.4,0.5][0.6,0.7]No(0) x2[0.8,1][0.2,0.3][0,0][0.5,0.7][0.3,0.4][0.1,0.2][0.8,1][0.3,0.4][0.7,0.9][0.4,0.5]No(0) x3[0.1,0.2][0.7,0.9][0.2,0.4][0.9,1][0.2,0.3][0,0][0.7,0.9][0.4,0.5][0.2,0.3][0.8,0.9]Yes(1) x4[0,0][0.1,0.2][0.9,1][0.2,0.4][0.8,0.9][0,0][0.6,0.9][0.2,0.5][0.3,0.4][0.7,0.9]Yes(1) x5[0,0][0.2,0.4][0.7,0.9][0,0][0.3,0.5][0.7,0.9][0.5,0.6][0.5,0.7][0.5,0.5][0.5,0.6]Yes(1) x6[0,0][0.3,0.5][0.7,0.8][0,0][0.3,0.4][0.8,0.9][0.3,0.5][0.7,0.9][0.6,0.7][0.4,0.5]No(0) x7[0.1,0.2][0.7,0.8][0.1,0.3][0,0][0.2,0.3][0.7,0.9][0.3,0.5][0.7,0.9][0.9,1][0.1,0.2]Yes(1) x8[0.7,0.9][0.2,0.3][0,0][0.1,0.3][0.7,0.9][0.1,0.2][0.8,1][0.2,0.4][0.2,0.4][0.8,0.9]No(0) x9[0.9,1][0.1,0.5][0,0][0,0][0.1,0.3][0.9,1][0.3,0.5][0.7,0.9][0.3,0.4][0.7,0.9]Yes(1) x10[0,0][0.2,0.3][0.7,0.9][0.1,0.3][0.7,0.9][0.1,0.2][0.4,0.6][0.6,0.9][0.3,0.5][0.7,0.9]Yes(1) x11[0.8,1][0.2,0.2][0,0][0.1,0.3][0.9,1][0,0][0.2,0.4][0.6,0.8][0.8,0.9][0.2,0.4]Yes(1) x12[0.2,0.3][0.6,0.9][0.1,0.3][0.2,0.5][0.8,0.9][0,0][0.7,0.8][0.3,0.4][0.7,0.8][0.3,0.6]Yes(1) x13[0.2,0.3][0.8,1][0.1,0.1][0.8,0.9][0.2,0.3][0,0][0.2,0.5][0.8,0.9][0.2,0.4][0.8,1]Yes(1) x14[0,0][0.1,0.2][0.9,1][0.5,0.5][0.5,0.6][0,0][0.9,1][0.1,0.2][0.8,0.9][0.2,0.3]No(0)

L.Feng et al./Knowledge-Based Systems24(2011)837–843841

Step https://www.360docs.net/doc/be10011057.html,pute the discernibility matrix M d in the VDIS;Step https://www.360docs.net/doc/be10011057.html,pute L ij ?_a i 2M d ei ;j Ta i ,where M d (i ,j )–;;Step https://www.360docs.net/doc/be10011057.html,pute the discernibility function L =^i ,j L ij ;Step 4.Convert L into a disjunctive normal form L ?_i L 0i ;

Step 5.

Each item L 0i in L corresponds to an attribute reduction.

Through Algorithm 1,the all attribute reduction sets can be found,which keep the lower approximation distribution preserved.

4.3.Vague decision rules in a VDIS

Let S =(U ,C [{d })be a VDIS,and B (B #C )be an attribute reduc-tion.For any x 2U ,assume that the maximum lower approxima-tion distribution c B (x )={D 1,D 2,...,D r }(16r 6j V d j ).Then,the knowledge hidden in the VDIS may be extracted in the form of decision rules

l !d econf ;supp T;

e23T

where l ?^a 2B

ea ?t ev TT;t ev T?f v 2V a jet v e_N

B ex TTàf v e_N B ex TTP 0g ,_N B ex T?f y 2U jet e B

ex ;y Tàf e

B

ex ;y TT>0g ;d ?c B ex T:

conf is the con?dence degree of the rule,and is de?ned as

conf ??inf v 2B

t v e_N

B ex TT;inf v 2B

e1àf v e_N B ex TTT ;e24T

supp is the support degree of the rule,and is de?ned as

supp ?

j _N

B ex Tj :e25T

Proposition 5.Let S =(U,C [{d })be a VDIS.If the vague conditional-attribute space degenerates into the crisp data space,the decision rules in the VDIS are equivalent to the positive region decision rules in Pawlak information systems.

Proof.For a vague decision rule l ?d ,if the vague conditional-attribute space degenerates into the crisp data space in VDIS,then for any x 2U ,l denotes the logic formula deducing a crisp equiva-lence class [x ]l ,and d derives a decision class [x ]d .Then,we obtain [x ]l #[x ]d .By the basic de?nitions of rough decision rules in [33],we have l ?d is a positive decision rule.The conclusion holds.h

5.An illustration to the Saturday morning tennis-play problem Through a Saturday morning play tennis problem (see Table 1),we illustrate the knowledge acquisition process of the proposed approaches and its application.

In Table 1,there are 14instances x i (i =1,2,...,14)to be consid-ered,which are evaluated by vague values.Decision is classi?ed into two classes.We now apply the proposed approaches for knowledge acquisition.

From Table 1,U /{d }={{x 1,x 2,x 6,x 8,x 14},{x 3,x 4,x 5,x 7,x 9,x 10,x 11,x 12,x 13}}.Suppose D 1={x 1,x 2,x 6,x 8,x 14},D 2={x 3,x 4,x 5,x 7,x 9,x 10,x 11,x 12,x 13}.Using (21),we compute the maximum lower approximation distribution of Table 1and list the results in Table 2.By De?nition 10,we obtain the discernibility matrix in Fig.1.Then,we calculate the disjunctive normal form of L ,i.e.,L =a 1^a 4^(a 2_a 3).Thus,the disjunctive normal form of L is (a 1^a 2^a 4)_(a 1^a 3^a 4).Through using Algorithm 1,we ob-tain all of the reducts from Table 1as a 1^a 2^a 4or a 1^a 3^a 4.Let B ={a 1^a 2^a 4}be an attribute reduction.The main deci-sion rules from Table 1could be induced as follows:

R 1:If Outlook =‘‘Cloudy’’^Temperature =‘‘Hot’’^Wind =‘‘Not-windy’’Then Play tennis =‘‘Yes’’with conf =[0.7,0.9]and supp =0.14;

R 2:If Outlook =‘‘Rain’’^Temperature =‘‘Mild’’^Wind =‘‘Not-windy’’Then Play tennis =‘‘Yes’’with conf =[0.7,0.9]and supp =0.14;

R 3:If Outlook =‘‘Sunny’’^Temperature =‘‘Hot’’^Wind =‘‘Not-windy’’Then Play tennis =‘‘No’’with conf =[0.6,0.7]and supp =0.07;R 4:If Outlook =‘‘Sunny’’^Temperature =‘‘Hot’’^Wind =‘‘Windy’’Then Play tennis =‘‘No’’with conf =[0.5,0.7]and supp =0.07;R 5:If Outlook =‘‘Sunny’’^Temperature =‘‘Cool’’^Wind =‘‘Not-windy’’Then Play tennis =‘‘Yes’’with conf =[0.7,0.9]and supp =0.07.

With the arbitrary vague decision rule given above,it induced from a vague or imprecise data set using the proposed vague rough sets theory,and could be used for description of the Saturday morning tennis-play decision policies.On the other hand,there are good grounds for keeping a subinterval of [0,1]for the con?-

Table 2

The maximum lower approximation distribution of Table 1.

x 1

x 2x 3x 4x 5x 6x 7x 8x 9x 10x 11x 12x 13x 14c C (x i )D 1

D 1

D 2

D 2

D 2

D 1

D 2

D 1

D 2

D 2

D 2

D 2

D 2

D

1

842L.Feng et al./Knowledge-Based Systems 24(2011)837–843

dence degree of each vague decision rule.According to vague sets theory[18],this subinterval keeps track of both the favoring evi-dence and the opposing evidence.Therefore,the lower/upper bound of con?dence degree for each vague decision rule can be used to perform constraint decisions.We not only have an estimate of how likely the con?dence degree of each vague decision rule is, but we also have a lower and upper bound on this likelihood.

6.Conclusions and future work

In this paper,we presented a new concept of a vague-rough set, as a generalization of a rough set by combing the vague set and rough https://www.360docs.net/doc/be10011057.html,ing the vague-rough approximation sets,we proposed the concept of attribute reduction and an approach for attribute reduction based on the discernibility matrix in a VDIS.Thus,the knowledge hidden in the VDIS could be unraveled in the form of decision rules.The research results contribute to uncertain knowl-edge acquisition in vague information systems.We will focus on the simpli?cation of vague decision rules and application of the proposed approaches in real-life vague information systems as our future research task.

Acknowledgments

The authors thank the anonymous referees for their valuable comments.This paper is in part supported by the National Natural Science Foundation of PR China under Grants No.60873108,the Scienti?c Research Fund of Sichuan Provincial Education Depart-ment under Grants No.09ZC079,the Scienti?c Research Fund of Sichuan Key Laboratory of Visualization Computing and Virtual Reality under Grants No.J2010N01,and the Key Research Founda-tion of Sichuan Normal University,respectively.

References

[1]Z.Pawlak,Rough sets,International Journal Computer Information Science11

(5)(1982)341–356.

[2]T.R.Li,D.Ruan,W.Geert,J.Song,Y.Xu,A rough set based characteristic

relation approach for dynamic attribute generalization in data mining, Knowledge-Based Systems20(5)(2007)485–494.

[3]L.Feng,G.Y.Wang,T.R.Li,Knowledge acquisition from decision tables

containing continuous-valued attributes,Acta Electronica Sinica37(11) (2009)2432–2438.

[4]J.S.Mi,Y.Leung,W.Z.Wu,Approaches to attribute reduction in concept lattices

induced by axialities,Knowledge-Based Systems23(6)(2010)504–511. [5]G.L.Liu,Rough set theory based on two universal sets and its applications,

Knowledge-Based Systems23(2)(2010)110–115.

[6]P.Guo,Rough set feature extraction by remarkable degrees with real world

decision-making problems,Soft Computing,doi:10.1007/s00500-009-0494-1.

[7]H.Y.Wu,Y.Y.Wu,J.P.Luo,An interval type-2fuzzy rough set model for

attribute reduction,IEEE Transactions on Fuzzy Systems17(2)(2009)301–315.

[8]H.S.Nguyen,Discretization problem for rough sets methods,Lecture Notes in

Arti?cial Intelligence1424(1998)545–552.

[9]X.Z.Wang,S.Y.Zhao,J.H.Wang,Simpli?cation of information table with fuzzy-

valued attributes based on rough sets,Journal of Computer Research and Development41(11)(2004)1974–1981.

[10]Y.Y.Yao,A comparative study of fuzzy sets and rough sets,Information

Sciences109(1998)227–242.

[11]D.S.Yeung,D.Chen,E.C.C.Tsang,J.W.T.Lee,X.Z.Wang,On the generalization

of fuzzy rough sets,IEEE Transactions on Fuzzy Systems13(3)(2005)343–361.

[12]D.Dubois,H.Prade,Rough fuzzy sets and fuzzy rough sets,International

Journal of General Systems17(2)(1990)191–209.

[13]E.C.C.Tsang,D.Chen,D.S.Yeung,X.Z.Wang,J.W.T.Lee,Attributes reduction

using fuzzy rough sets,IEEE Transactions on Fuzzy Systems16(5)(2008) 1130–1141.

[14]S.K.Pal,P.Mitra,Case generation using rough sets with fuzzy representation,

IEEE Transactions on Knowledge and Data Engineering16(3)(2004)292–300.

[15]Q.Shen,R.Jensen,Selecting informative features with fuzzy-rough sets and its

application for complex systems monitoring,Pattern Recognition37(2004) 1351–1363.

[16]W.Z.Wu,W.X.Zhang,Constructive and axiomatic approaches of fuzzy

approximation operators,Information Sciences159(2004)233–254.

[17]P.Lingras,R.Jensen,Survey of rough and fuzzy hybridization,in:Proceeding of

the16th International Conference of Fuzzy Systems2007,pp.125–130. [18]W.L.Gau,D.J.Buehrer,Vague sets,IEEE Transactions on Systems,Man and

Cybernetics23(2)(1993)610–614.

[19]Y.Liu,G.Y.Wang,L.Feng,A general model for transforming vague sets into

fuzzy sets,LNCS Transaction on Computer Science II5150(2008)133–144.

[20]D.H.Hong,K.Chul,A note on similarity measures between vague sets and

between elements,Information Sciences115(1999)83–96.

[21]D.Q.Yan,Z.X.Chi,Similarity measure between vague sets,Pattern Recognition

and Arti?cial Intelligence17(1)(2004)22–26.

[22]C.Y.Xu,The universal approximation of a class of vague systems,Chinese

Journal of Computer28(9)(2005)1508–1513.

[23]S.M.Chen,Similarity measures between vague sets and between elements,

IEEE Transactions on Systems,Man,and Cybernetics,Part B:Cybernetics27(1) (1997)153–158.

[24]S.M.Chen,Measures of similarity between vague sets,Fuzzy Sets and Systems

74(2)(1995)217–223.

[25]L.Feng,G.Y.Wang,X.X.Li,Knowledge acquisition in vague objective

information systems based on rough sets,Expert Systems27(2)(2010) 129–142.

[26]L.Feng,G.Y.Wang,Knowledge acquisition in vague information systems,in:

Proceeding of2006International Conference on Arti?cial Intelligence2006, pp.533–537.

[27]W.X.Zhang,W.Z.Wu,J.Y.Liang,D.Y.Li,Rough Set Theory and Approach,

Science Press,Beijing,2003.pp.158–168.

[28]M.R.Alicja,R.Leszek,Variable precision fuzzy rough sets,LNCS Transactions

on Rough Sets I3100(2004)144–160.

[29]S.M.Chen,S.H.Lee,C.H.Lee,A new method for generating fuzzy rules from

numerical data for handling classi?cation problems,Applied Arti?cial Intelligence15(2001)645–664.

[30]L.Y.Chang,G.Y.Wang,Y.Wu,An approach for attribute reduction and rule

generation based on rough set theory,Journal of Software10(11)(1999) 1206–1211.

[31]W.Z.Wu,W.X.Zhang,H.Z.Li,Knowledge acquisition in incomplete fuzzy

information systems via the rough set approach,Expert Systems20(5)(2004) 280–286.

[32]A.Skowron, C.Rauszer,The discernibility matrices and functions in

information systems,in:R.S?owinski(Ed.),Intelligent Decision Support.

Handbook of Applications and Advances of the Rough Set Theory,Kluwer, Dordrecht,1993.

[33]Y.Y.Yao,Three-way decision:an interpretation of rules in rough set theory,

Lecture Notes in Computer Science5589(2009)642–649,doi:10.1007/978-3-642-02962-2_81.

L.Feng et al./Knowledge-Based Systems24(2011)837–843843