Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
554
Generalized Interval-Valued Fuzzy Variable Precision Rough Sets Bao Qing Hu and Heung Wong Abstract1 For interval-valued fuzzy datasets, people have started to do research on interval-valued fuzzy rough sets and relevant models. However, these models could not be effectively applied to handle the real-valued datasets such as interval-valued fuzzy datasets as variable precision problems were not considered in interval-valued fuzzy rough sets. In this paper, fuzzy variable precision rough sets are generalized to interval-valued fuzzy sets, with consideration of interval precisions and interval-valued fuzzy relations. Combining interval-valued fuzzy set with rough sets and variable precision rough sets, this paper develops generalized interval-valued fuzzy variable precision rough sets (GIVF-VPRSs) based on triangular norms and fuzzy logical operators respectively. using the basic properties of GIVF-VPRSs, this paper gives granule representation form of the upper approximation operator in GIVF-VPRSs based on triangular norms and granule representation form of the lower approximation operator in GIVF-VPRSs, which is based on fuzzy logical operators. The conclusions show that some existing models, such as interval-valued fuzzy rough sets, variable precision rough sets and fuzzy variable precision rough sets, are special examples of GIVF-VPRSs proposed in this paper. Keywords: Interval-valued fuzzy sets, fuzzy logical operators, interval-valued fuzzy rough sets, interval-valued fuzzy variable precision rough sets.
1. Introduction The theory of fuzzy sets, proposed by Zadeh in 1965 [23], is an extension of set theory for the study of intelligent systems characterized by fuzzy information [18, 25]. The theory of rough sets (RSs), proposed by Pawlak in 1982 [14], is another extension of set theory for the Corresponding Author: Bao Qing Hu is with School of Mathematics and Statistics, Wuhan University, Wuhan, P.R. China. E-mail: [email protected] Heung Wong is with Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. E-mail: [email protected] Manuscript received 23 Sep. 2013; revised 20 June 2014; accepted 05 Aug. 2014.
study of intelligent systems characterized by insufficient and incomplete information. It made RSs work well on some problems, but it also limited the further applications of RSs. For example, RSs could not work effectively on the datasets with real values. Fuzzy rough sets (FRSs) were proposed by Dubois and Prade to deal with fuzzy information systems [4]. RSs and FRSs were generalized to two universes of discourse, i.e. generalized fuzzy rough sets (GFRSs) [19-21]. Operators are extended from max and min to triangular norm (t-norm, t-conorm) or fuzzy logical operators. For example, (, T) -GFRSs [7, 11], ( T , ) -GFRSs [12] and ( I , T) -GFRSs [15, 19]. Variable precision rough sets (VPRSs) were extended by Ziarko in 1993 [30] to overcome the shortcomings on misclassification and/or perturbation [31-32]. For interval-valued fuzzy (IVF) datasets, one has started to research into interval-valued fuzzy rough sets (IVF-RSs) [5-6, 10, 17, 26-27]. However, these models could not be effectively applied to handle the real-valued datasets such as IVF datasets as variable precision problems were not considered in IVF-RSs. By considering the characteristics of FRSs and VPRSs, several flexible RSs based generalizations were proposed which can handle the numerical problems with misclassification and/or perturbation. For example, variable precision rough set-fuzzy rough set (VPRS-FRSs) [16], variable precision fuzzy rough set (VPFRSs, [13]), vaguely quantified rough set (VQRSs, [1]) and fuzzy variable precision rough set (FVPRSs, [28-29]). Zhao, Tsang and Chen [28] pointed out various shortcomings in these generalized models, mainly on misclassification, perturbation and attribute reduction, and introduced FVPRSs and gave a corresponding method on attribute reduction. For IVF information systems, however, there is no work on the generalization of RS with due consideration of variable precision. In order to do this, this paper firstly extends IVF-RSs to two universes of discourse, as GIVF-RSs, and then extends FVPRSs to IVF and two universes of discourse, as GIVF-VPRSs, based on IVF triangular norms and IVF logical operators. This paper gives definitions of GIVF-VPRSs based on triangle norms and logical operators on two universes of discourse and discusses their properties. These properties mainly include granule representation form of the upper
© 2014 TFSA
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
approximation operator in GIVF-VPRSs based on triangular norms and granule representation form of the lower approximation operator in GIVF-VPRSs based on fuzzy logical operators.
lower and upper approximation operators with variable precision [0,1) , also called the lower and upper approximation operators of GFVPRS, are respectively defined as follows. x X :
R ( A)( x) inf N ( R( x, u )),
2. Preliminaries
Define
i
i
T ( x, y ) sup [0,1] T( x, ) y ,
( x, y ) inf [0,1] ( x, ) y .
A ( u )
(6)
R T ( A)( x) sup T R( x, u ), N ( ) A ( u ) N ( ) sup T R( x, u ), A(u ) , A ( u ) N ( )
R T ( A)( x) inf T R( x, u ),
A ( u )
(7)
inf T R( x, u ), A(u ) , A ( u )
(8)
R ( A)( x) sup N ( R( x, u )), N ( ) A ( u ) N ( ) sup N ( R( x, u )), A(u ) . A ( u ) N ( )
(2) (3) (4)
A negator N is a decreasing function N :[0,1] [0,1] satisfying N (0) 1 and N (1) 0 . A negator N is referred to as involutive iff N ( N ( x)) x for all x [0,1] . Specially N s ( x) 1 x is an involutive negator. Given a negator N, t-norm and t-conorm are called dual with regard to N if T( N ( x), N ( y )) N ( ( x, y )) , and under such condition, N ( T ( x, y )) ( N ( x), N ( y )) . (5) Let us always assume that T and are continuous and dual w.r.t. the involutive negator N in this paper. In this paper Y ( X ) denotes the class of all fuzzy sets of the universe X. The union, intersection and complement of fuzzy sets are denoted by , and Ac respectively. A fuzzy subset R Y ( X Y ) is referred to as a fuzzy binary relation from X to Y. Let X and Y be two finite nonempty universes and R a fuzzy relation from X to Y. The triple ( X , Y , R) is referred to as a generalized fuzzy approximation space in the following discussions. There exist many generalized models for variable precision rough sets. To solve the problems on misclassification and perturbation, Zhao, Tsang and Chen [28] introduced FVPRSs and gave the corresponding attribute reduction. We give the following definition in generalized form with two universes of discourse. Definition 2.1: In the generalized fuzzy approximation space (X, Y, R), for A Y (Y ) , 0 1 , the fuzzy
A ( u )
inf N ( R( x, u )), A(u ) ,
Rough sets (RSs) theory proposed by Pawlak [14] is an extension of set theory. Since then, fuzzy sets are introduced into the RS, universes of discourse are generalized to two universes which are not necessarily the same and operators are extended to fuzzy logical operators. We first review fuzzy logical operators. For a continuous t-norm T and a continuous t-conorm on [0, 1], we have T x,sup yi sup T x, yi , (1) i i x,inf yi inf x, yi ( is any index set).
555
(9)
In [28-29], X=Y was considered in four fuzzy approximation operators with variable precision Eq. (6)-Eq.(9). The four fuzzy approximation operators are extended to interval numbers in [0, 1] as follows. In this paper, I (2) [a , a ] : 0 a a 1 , specially for a [0,1] , a [a, a] . An order relation on I (2) is defined as [a , a ] [b , b ] a b , a b
for [a , a ],[b , b ] I (2) . Two
incomparable,
[a , a ],[b , b ] I (2)
intervals
if
neither
i
[b , b ] [a , a ] holds. If [a , a ] I define sup[ ai , ai ] [sup ai ,sup ai ] , i
i i
called
[a , a ] [b , b ] i
i
are
(2)
, i , we
(10)
inf[ai , ai ] [inf a ,inf ai ] . i
i
nor
(11)
i
We define [a , a ],[b , b ] I (2) ,
T [a , a ],[b , b ] [T(a , b ), T(a , b )] , [2] [a , a ],[b , b ] [ (a , b ), (a , b )] , N [a , a ] = [ N (a ), N ( a )] .
T [a , a ],[b , b ]
[2, 3]
(12) (13) (14)
sup [ , ] I (2) T [a , a ],[ , ] [b , b ] , [2]
[a , a ],[b , b ]
(15)
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
556
inf [ , ] I (2) [a , a ],[ , ] [b , b ] . (16)
A B
It is easy to obtain the following properties. Lemma 2.1 [10]: (1) T [a , a ],[b , b ]
( x) 0 and X ( x) 1 , x X , respectively. A binary IVF subset R of X Y is called an IVF relation from X to Y. For the IVF relation R ( R Y I ( 2 ) ( X X ) ) of the universe X, then (1) R is re-
flexive, if R ( x, x) 1 , for any x X . (2) R is symmetric, if R ( x, y ) R ( y , x) , for any x, y X . (3) R is
N [a , a ] , N [b , b ] .
Lemma 2.2 [10]: (1) If [a , a ] [b , b ] , then
T -transitive, if R ( x, z ) T R ( x, y ), R( y, z ) , for any x, y, z X . (4) R is serial if x X , y Y , such that
T [a , a ],[c , c ] T [b , b ],[c , c ] and
R x, y 1 . An IVF relation R on X is referred to as an IVF T -equivalence relation on X if it is reflexive, symmetric and T -transitive. It follows from Lemma 2.2(2) that an IVF relation R is T -transitive iff R ( y , z ) T R ( x, y ), R ( x, z ) , for any x, y, z X .
T ([c , c ],[a , a ]) T ([c , c ],[b , b ]) .
(2) T [a , a ],[b , b ] [c , c ]
T [a , a ],[c , c ] [b , b ] .
T [a , a ], [b , b ],[c , c ] .
3. Generalized Interval-valued Fuzzy Variable Precision Rough Sets Based on Triangle Norms
(4) [a , a ] [b , b ] T [a , a ],[b , b ] 1 . Lemma 2.3 [10]: Let T and be continuous t-norm and t-conorm respectively. Then [ai , ai ] I (2) , i ,[b , b ] I (2) , (1) sup[ai , ai ],[b , b ] inf [ai , ai ],[b , b ] . i i
(2) [b , b ],inf[ai , ai ] inf [b , b ],[ai , ai ] . i
i
Y I ( 2) ( X ) is the class of all IVF sets [22, 24] over the
In this section FVPRSs are extended to IVF sets and two universes of discourse. Definition 3.1: In the generalized IVF approximation space ( X , Y , R ) , for an IVF set A Y I ( 2 ) (Y ) , the generalized IVF lower and generalized IVF upper approximation operators with variable precision [ , ] ( 0 1 ) based on triangle norms, are respectively defined as follows. x X : [ , ]
universe X. For [ , ] I (2) and M X , we define M [ , ] Y I ( 2) ( X ) with membership function M [ , ] ( x) [ , ]
while
xM ,
0
and
with membership functions
( 2)
(3) N T ([a , a ],[b , b ])
A ( x ) B ( x )
And , X Y I ( X )
(a , b ), (a , b ) (a , b ) ,
iff
(19)
A ( x) B ( x) .
(2) [a , a ],[b , b ]
A( x) B ( x)
iff
T (a , b ) T (a , b ), T (a , b ) ,
(3) T T [a , a ],[b , b ] ,[c , c ]
( AN s )( x) 1 A( x) [1 A ( x),1 A ( x)] .
R
otherwise. If [ , ]
M {x} , then {x}[ , ] is tersely denoted by x[ , ] .
RT
M [ , ] is also denoted by [ , ]M . If [ , ] I (2) and
A( x) [ , ] for all x X , then A is simply denoted by [ , ] , i.e. A [ , ] . For a crisp set A X , A may be viewed as a special kind of IVF sets with membership function A( x) 1 for x A , otherwise 0 . We define union, intersection, complement and order relation pointwise by the following formulas. For A [ A , A ], B [ B , B ] Y I ( 2 ) ( X ) , ( A B)( x) [ A ( x) B ( x), A ( x) B ( x)] ,
(17)
( A B)( x) [ A ( x) B ( x), A ( x) B ( x)] ,
(18)
( A)( x)
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u ) ,
(20)
( A)( x) sup T R( x, u ), N ([ , ]) A(u ) N ([ , ]) sup T R( x, u ), A(u ) . (21) A ( u ) N ([ , ])
The pair of IVF sets
R
[ , ]
[ , ]
( A), R T
( A)
is called a
(, T) -generalized IVF variable precision rough set ( (, T) -GIVF-VPRS).
For any IVF set f, we define inf f (u ) 1 and u
sup f (u ) 0 . u
It follows from Definition 3.1 the following special
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[ N ( R ( x, u )), N ( R ( x, u ))],[ A (u ), A (u )] [ A ( u ), Ainf ( u )] [ , ]
cases. (1) For the arbitrary crisp subset A of Y, i.e. A( y ) 1
inf N ( R ( x, u )), inf N ( R ( x, u )), A (u ) , A ( u ) A ( u )
for y Y , otherwise 0 , then it follows from [ , ] 1 iff N ([ , ]) 0 and A(u ) [ , ] ( 0 1 ) iff [ , ]
inf N ( R ( x, u )), inf N ( R ( x, u ))], A (u ) A (u )
A ( u )
A(u ) 0 that
R
( A)( x) inf N ( R( x, u )),[ , ]
A( u ) 0
inf N ( R( x, u )), A(u ) A( u ) 1
Then [ , ]
A(u ) 0
R
Proof: R
(2) If , A Y (Y ) and R Y ( X Y ) , then Eq. (20) and Eq. (21) are simplified to the two equations in Eq. (6) and Eq. (7) respectively, which were discussed in [29] . (3) If 0 , then
A(u ) 0
And R
[0,0] T
( A)( x) sup T R( x, u ), N ([0]) A( u ) N ( 0) sup T R( x, u ), A(u ) A(u ) N ([ 0])
sup T R ( x, u ), 1 sup T R ( x, u ), A(u ) A(u ) 1 A(u ) 1 R T ( A)( x) .
R
[0,0]
[0,0]
( A), R T ( A)
( A)( x)
sup T R( x, u ), A(u ) R T ( A)( x) . uY
[ , ]
R
is a (, T) -GIVF-RS discussed in
( A)( x)
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u )
inf
A ( u ) [ , ]
A ( u ) [ , ]
uY
Proposition 3.3: Let 0 1 and A, B Y I ( 2) (Y ) . Then [ , ]
(1) R T
[ , ]
(2) R T
[ , ]
() , R
[ , ]
( AN ) R
( A)
[ , ]
(3) R is serial R RT
(Y ) X ;
N
[ , ]
, R
[ , ]
( AN ) R T
sup T R( x, u ), N ([ , ]) 0 , [ , ]
[ , ]
RT
Proof: R
[ , ]
sup T R ( x, u ), N ([ , ])
Y ( u ) [ , ]
u
( A) ( R ) ( A ),( R ) ( A ) ,
(22)
( A) ( R )T ( A ), ( R )T ( A ) .
(23)
inf
Y ( u ) [ , ]
inf [ N ( R ( x, u )), N ( R ( x, u ))],[ , ] [ A (u ), A (u )][ , ]
N ( R( x, u )),[ , ]
inf N ( R( x, u )),[ , ] 1 . u
R
( A)( x)
N
Proof: (1) Since [ , ] 1 is not true, we have sup T R ( x, u ), N ([ , ])
Then R
(Y ) N ([ , ]) for any 0 1 .
( u ) N ([ , ])
( A)
() [ , ]
[10]. Proposition 3.1: Let 0 1 and A Y I ( 2 ) (Y ) .
inf N ( R( x, u )), A(u ) N ( R( x, u )), A(u ) inf N ( R( x, u )), A(u ) .
[ , ]
[0,0]
That is R ( A) R ( A) and R T ( A) R T ( A) , i.e. [0,0]
( A) R ( A) .
sup T R( x, u ), A(u ) sup T R( x, u ), A(u ) A(u ) N ([ , ]) A(u ) N ([ , ])
inf N ( R( x, u )), A(u ) R ( A)( x) A(u ) 0
[ , ] T
sup T R(x, u), N([, ]) sup T R(x, u), A(u) A(u)N([, ]) A(u)N([, ])
A(u ) 1
[0,0]
[ , ]
( A) R T ( A) , R
RT
( A)( x) sup T R( x, u ), N ([ , ]) A u ( ) 1 sup T R( x, u ), A(u ) A(u ) 0 sup T R( x, u ), N ([ , ]) .
R ( A)( x) inf N ( R( x, u )), 0
( R ) ( A ),( R ) ( A ) ( x) . The second one is proven in a similar way. Proposition 3.2: Let 0 1 and A Y I ( 2 ) (Y ) .
inf N ( R( x, u )),[ , ] , [ , ] T
557
()( x) sup T R( x, u ), N ([ , ]) u N ( ) ([ , ]) sup T R( x, u ), (u ) 0 0 0 . (u ) N ([ , ])
[ , ] T
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
558
[ , ]
R
(2)
(Y )( x) inf
Y ( u ) [ , ]
[ , ]
R
( A)
N
N
inf
Y ( u ) [ , ]
[ , ]
R
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
[ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u )
[ , ] R
[ , ] T
[ , ]
[ , ]
R
()
(Y ) N ([ , ]) . If R is serial, then for any
()( x)
inf
0 ( u ) [ , ]
inf
0 ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), 0)
[ , ]
[ , ]
If N ([ , ]) A R T 0 1 , then
inf
0 ( u ) [ , ]
inf
0 ( u ) [ , ]
N ( R( x, u )), 0
N ( R( x, u )), 0)
[ , ]
R
R
(1{ x} )( y ) R
[ , ] T
( A) , A Y I ( 2) ( X ) , for
(1{ y} )( x), ( x, y ) X X .
(3) R is T -transitive
[ , ]
(1X \{ y} )( x) R
(1X \{ x} )( y ), ( x, y ) X X
[ , ]
(1{ x} )( y ) R T
(1{ y} )( x), ( x, y ) X X .
N ( R ( x, u )),[ , ] (1X \{ y} )( x) inf 1 ( ) [ , ] u X \{ y } N ( R ( x, u )), 1X \{ y} (u ) inf 1X \{ y } ( u ) [ , ] N ( R ( x, y )),[ , ] N ( R ( y , x)),[ , ]
[ , ]
R
[ , ]
R [ , ]
If R T
[ , ]
(1{ x} )( y ) R T
(1X \{ x} )( y ) .
(1{ y} )( x), ( x, y ) X X , then it
follows from taking [ , ] 0 that [0,0]
R T (1{ y} )( x) sup T R( x, u ), 1 sup T R( x, u ), 1{ y} (u ) 1{ y } ( u ) 1 1{ y } (u ) 1
uY
[ , ] T
If R is symmetric, then ( x, y ) X X ,
Proposition 3.4: Given X=Y, then the following is true. [ , ] (1) R ( A) [ , ] A , A Y I ( 2) ( X ) , any 0 1 R is reflexive. (2) R is symmetric [ , ] [ , ] R (1X \{ y} )( x) R (1X \{ x} )( y ), ( x, y ) X X ,
(2) By the duality,
uY
N ([ , ]) A R
A 1{ x} and
x X , taking
i.e. R ( x, x) 1 .
Thus sup R( x, u ) 1 , i.e. R is serial.
[ , ] T
( A) , A Y I ( 2) ( X ) , for any
R( x, x) N ([0,0]) 1{ x} ( x) 1 .
[ , ]
inf N ( R( x, u )), 0 N (sup R( x, u )) . uY
( A) , A Y I ( 2) ( X ) , for any
() [ , ] .
()( x)
( A) [ , ] A ,
[0,0] RT (1{x} )(x) sup T R(x, u),1 sup T R(x, u),1{x} (u) 1{x} (u)1 1{x} (u)1
[ , ]
0R
[ , ]
R
[ , ] 0 , we have
Conversely, if R () [ , ] , for any 0 1 , then for 0 , [0,0]
( A) , A Y I ( 2) ( X ) .
0 1.
N (sup R ( x, u )),[ , ] [ , ] . uY
And then R
[ , ]
( A)) R T
Proof: (1) By the duality, A Y I ( 2) ( X ) ,
RT
inf N ( R( x, u )),[ , ] 1 uY
( A)) , A Y I ( 2) ( X ) ,
[ , ]
x X , there is a y0 Y such that R ( x, y0 ) 1 . Thus R
[ , ]
( RT
[ , ]
(R
N ([ , ]) A R T
sup T R(x, u), N ([ , ]) sup T R(x, u), N( A(u)) A ( u ) [ , ] A ( u ) [ , ] sup T R( x, u ), N ([ , ]) N A (u ) N ([ , ]) [ , ] sup T R( x, u ), N ( A(u )) R T ( A N )( x) . N A (u ) N ([ , ])
(3) It follows from the duality that
[ , ]
( A) R
RT
N ( R( x, u )), Y (u ) 1 1 1 .
( x) N
N ( R( x, u )),[ , ]
[0,0]
R( x, y ) , R T (1{ x} )( y ) sup T R ( y, u ), 1 sup T R( y, u ), 1{ x} (u ) 1{ x } (u ) 1 1{ x } (u ) 1 R( y, x) . Thus R( x, y ) R( y, x) .
(3) If R is T -transitive, then A Y I ( X ) , ( 2)
[ , ]
R
( A))( x) [ , ] inf N ( R ( x, u )),[ , ] R ( A)(u ) [ , ] [ , ] [ , ] inf N ( R( x, u )), R ( A)(u ) R ( A)( u ) [ , ] [ , ]
(R
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[ , ]
inf N ( R( x, u )),[ , ] R u X
inf N ( R ( x, u )),[ , ] u X
A ( v ) [ , ]
u X
inf
u X
A ( v ) [ , ]
A ( v ) [ , ]
inf
A ( v ) [ , ]
sup T R ( x, u ), y[ , ] (u ) T R ( x, y ),[ , ] ,
uY
we have RT ( y[ , ] ) y[ , ] AN ([ , ]) [ , ]I ( 2) , yY
N ( R( x, u )), N ( R(u , v)),[ , ]
N ( R( x, u )), N ( R(u, v)), A(v)
N (R( x, v)),[ , ]
inf
A( v ) [ , ]
N ( R( x, v)), A(v)
R
If R
[ , ] T
(R
(R
[ , ] T
( A)) R
( A)) R
[ , ] T
[ , ] T
T R( x, y),[ , ] y
( x)
AN ([ , ])
[ , ]
T R( x, y),[ , ] [ , ] A
T R( x, y),[ , ] [ , ] A
N ([ , ])
sup [ , ]I ( 2 ) , yY
R ( y T
[ , ]
)( x) [ , ] AN ([ , ])
sup
( A) , A Y I ( 2 ) ( X ) .
( A) , A Y I ( X ) , for any ( 2)
0 1 , then for all x, y X , taking A 1{ y} and [ , ] 0 , we have R T R( x, y ) R( x, y ) . Thus R is
T -transitive. It is follows from Definition 3.1 the following property. Proposition 3.5: For two generalized IVF approximation space ( X , Y , R) and ( X , Y , P) , if R P , then for an IVF set A Y I (Y ) , the following hold.
N ([ , ])
[ , ]I ( 2 ) , yY
yY
( x) ( y )
T R( x, y),[ , ] [ , ] A sup T R ( x, y ), A ( y )
( y)
( x) .
(2) R T ( y[ , ] ) y[ , ] AN ([ , ]) [ , ]I ( 2) , yY
( y )
N ([ , ])
RT ( y[ , ] ) [ , ] AN ([ , ]) ( y) [ , ]I ( 2 ) , yY
[ , ]
[ , ] T
sup [ , ]I ( 2) , yY
R ( A)( x) . By the duality, we have [ , ] [ , ] [ , ] R ( A) R ( R ( A)) [ , ] T
sup [ , ]I ( 2) , yY
u X
u X
sup [ , ]I ( 2) , yY
inf N ( R( x, u )), N ( R(u, v)) , A(v)
inf
inf
N ( R( x, v)), A(v)
Proof: (1) Since R T ( y[ , ] )( x)
inf N ( R( x, u )), N ( R(u, v)) ,[ , ]
A ( v ) [ , ]
A( v)[ , ]
A ( v ) [ , ]
N ( R(u , v)),[ , ]
inf
A ( v ) [ , ]
inf
inf
inf
N ( R(u , v)),[ , ]
N ( R( x, v)), A(v)
inf inf N ( R( x, u )),
inf
A ( v ) [ , ]
( A)(u )
559
( y)
N ([ , ])
sup T R( x, y ), N ([ , ]) A( y ) N ([ , ]) [ , ] sup T R( x, y ), A( y ) R T ( A)( x) . A ( y ) N ([ , ]) Theorem 3.1 shows that the membership function [ , ]
representation and granular representation of R T are equivalent. The following gives a simple example on the generalized IVF lower approximation and generalized IVF upper approximation based on triangle norm T and with two sets of variable precision. [ , ] [ , ] (1) R ( A) P ( A) . (24) Example 3.1: Let X Y {x1 , x2 , , x6 } , T (ordinary multiplication) and ˆ ( a ˆ b a b ab ), [ , ] (25) N N s ( N s ( x) 1 x ). (2) R T ( A) PT[ , ] ( A) . Theorem 3.1: Let 0 1 and A, y[ , ] Y I ( 2 ) (Y ) . An interval-valued fuzzy relation R is shown in Table 1. An IVF set A and its the IVF lower approximations Then (2)
(1)
R ( y T
[ , ]I ( 2 ) , yY
[ , ]
) y[ , ] AN ([ , ])
[ , ]I ( 2 ) , yY
[ , ]
(2) R T
( A)
[ , ]I ( 2 ) , yY
R ( y R ( y
[ , ]
R
T
[ , ]
) [ , ] AN ([ , ]) ( y ) .
T
[ , ]
) y[ , ] AN ([ , ]) .
[ , ]
( A) and the upper approximation R T
( A) are
[ , ] T
[ , ]
shown in Table 2, where R ( A) and R ( A) are computed by Eq. (20) and Eq. (21) respectively.
4. Generalized Interval-Valued Fuzzy Variable Precision Rough Sets Based on Logic Operators
where AN ([ , ]) ( y ) N ([ , ]) A( y ), N ([ , ]) A( y ) or A( y ) N ([ , ]), N ([ , ]) and A( y ) are incomparable 0,
Definition 4.1: In the generalized IVF approximation space ( X , Y , R) , for an IVF set A Y I (Y ) , the (2)
560
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
Table 1. IVF relation R. R
x1
x2
x3
x4
x5
x6
x1
1
[0.1,0.2]
[0.3,0.5]
[0,0.1]
[0.15,0.2]
[0.7,0.9]
x2
[0.1,0.2]
1
[0.45,0.6]
[0.8,0.9]
[0.75,0.8]
[0.05,0.1]
x3
[0.3,0.5]
[0.45,0.6]
1
[0.45,0.7]
[0.55,0.65]
[0.3,0.45]
x4
[0,0.1]
[0.8,0.9]
[0.45,0.7]
1
[0.3,0.45]
[0.8,0.9]
x5
[0.15,0.2]
[0.75,0.8]
[0.55,0.65]
[0.3,0.45]
1
[0.5,0.7]
x6
[0.7,0.9]
[0.05,0.1]
[0.3,0.45]
[0.8,0.9]
[0.5,0.7]
1
[0.2,0.4]
Table 2. IVF set A and its the IVF lower approximations R ( A) , R
R
[0.2,0.4] T
( A) and upper approximations R T ( A) ,
( A) .
0 X
A
0.2, 0.4
R ( A)
R T ( A)
R
[0.2,0.4]
[0.2,0.4]
( A)
RT
( A)
x1
[0.15, 0.2]
[0.15, 0.2]
[0.525,0.765]
[0.2, 0.4]
[0.42, 0.72]
x2
[0.7, 0.85]
[0.19, 0.26]
[0.7, 0.85]
[0.28, 0.52]
[0.6, 0.8]
x3
[0.5, 0.6]
[0.37, 0.39]
[0.5, 0.6]
[0.44, 0.6]
[0.5, 0.6]
x4
[0.1, 0.2]
[0.1, 0.2]
[0.6, 0.765]
[0.2, 0.4]
[0.48, 0.72]
x5
[0.4, 0.55]
[0.4, 0.55]
[0.525, 0.68]
[0.4, 0.55]
[0.45, 0.64]
x6
[0.75, 0.85]
[0.19, 0.24]
[0.75, 0.85]
[0.28, 0.52]
[0.6, 0.8]
generalized IVF lower and generalized IVF upper approximation operators with variable precision [ , ] ( 0 1 ) based on logic operators T and , are, respectively, defined as follows. x X : [ , ]
R T ( A)( x)
inf
A ( u ) [ , ]
[ , ]
R
T R( x, u ),[ , ]
inf
A ( u ) [ , ]
[ , ] T
[ , ]
iff N ([ , ]) 0 and A(u ) [ , ] ( 0 1 ) iff A(u ) 0 . It follows from Lemma 2.2(4) that [ , ]
T R( x, u ), A(u ) ,
R
for y Y , otherwise 0 , then it follows from [ , ] 1
RT
(26)
( A)( x) sup N ( R( x, u )), N ([ , ]) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) . (27) A(u ) N ([ , ])
The pair of IVF sets
(1) For the arbitrary crisp subset A of Y, i.e. A( y ) 1
( A), R ( A)
is called a
( T , ) -generalized IVF variable precision rough sets ( ( T , ) -GIVF-VPRS). From Definition 4.1, it follows the following special cases.
( A)( x) inf T R( x, u ),[ , ] , u A
[ , ]
R ( A)( x) sup N ( R ( x, u )), N ([ , ]) . u A
(2) If , A Y (Y ) and R Y ( X Y ) , then Eq. (26) and Eq. (27) are simplified to the two equations in Eq. (8) and Eq. (9), respectively, which were discussed in [28]. (3) If 0 , then R[0,0] ( A)( x)
T
inf T R( x, u ), 0 inf T R( x, u ), A(u ) A( u ) 0
A(u ) 0
RT ( A)( x)
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
Proposition 4.2: Let 0 1 and A Y I (Y ) .
[0,0] R ( A)( x) sup N ( R( x, u )), N ([0]) A u N ( ) ( 0) sup N ( R( x, u )), A(u ) A(u ) N ([ 0]) sup N ( R( x, u )), 1 A( u ) 1 sup N ( R( x, u )), A(u ) R ( A)( x) . A(u ) 1
(2)
Then [ , ]
[0,0]
T
[0,0]
( A), R ( A)
T
is a (T , ) -GIVF-RS discussed in
[10]. Proposition 4.1: Let 0 1 and A Y I (Y ) .
sup N ( R( x, u )), A(u ) R ( A)( x) .
Then
uY
[ , ]
RT
( A) ( R ) ( A ) ( R ) ( A ),( R ) ( A ) , T
T
[ , ]
RT
[ , ] R ( A) ( R ) ( A ), ( R ) ( A ) ( R ) ( A ) .
( A)( x)
Proof: x X , R[ , ] ( A)( x) T
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
T R( x, u ),[ , ]
T R( x, u ), A(u )
T R( x, u ), A(u )
( 2)
Then [ , ]
(1) R () , R[ , ] (Y ) X ;
[ T ( R ( x, u ), A (u )) T ( R ( x, u ), A (u )), T ( R ( x, u ), A (u ))] inf [ A ( u ), A (u )][ , ]
(2) R ( A ) R
inf T R ( x, u ), T ( R ( x, u ), ) [ A ( u ), A (u )][ , ]
inf
T R ( x, u ), A (u )
inf R ( x, u ), A (u ) inf ( R ( x, u ), A (u )) , A ( u ) T T A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A ( u ) inf T ( R ( x, u ), ) inf T ( R ( x, u ), A (u )) , A (u ) A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A (u ) ( R ) ( A ) ( R ) ( A ), ( R ) ( A ) ( x) . T T T The second one is proven in a similar way.
N
[ , ]
inf T R ( x, u ), inf T ( R ( x, u ), ) A (u ) A (u )
T
[ , ]
RT
T R ( x, u ), A (u ) T ( R ( x, u ), A (u )) , inf [ A (u ), A (u )][ , ] [ A ( u ), A ( u )][ , ]
T R( x, u ), A(u )
Proposition 4.3: Let 0 1 and A, B Y I (Y ) .
inf [ T ( R ( x, u ), ) T ( R ( x, u ), ), T ( R ( x, u ), )] [ A ( u ), A (u )][ , ]
T R ( x, u ),
inf
A ( u ) [ , ]
uY
T [ R ( x, u ), R ( x, u )],[ A (u ), A (u )] inf [ A ( u ), A ( u )] [ , ]
inf
inf T R ( x, u ), A(u ) .
T [ R ( x, u ), R ( x, u )],[ , ] inf [ A ( u ), A ( u )] [ , ]
[ A ( u ), A ( u )][ , ]
(28)
[ , ] R ( A)( x) sup N ( R( x, u )), N ([ , ]) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A ( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A(u ) N ([ , ])
(2)
T
( A) RT ( A) .
Proof:
[0,0]
R
[ , ]
R ( A) R ( A) , RT
That is R[0,0] ( A) R ( A) and R T ( A) R T ( A) , i.e. T
561
[ , ] T
( A)
N
,
[ , ]
( AN ) R ( A)
N
;
(3) R is serial R[ , ] () [ , ] T
[ , ]
R (Y ) N ([ , ]) for any 0 1 .
Proof: It is easily proven similar to Proposition 4.3 (1), (2) and (3). Proposition 4.4. Given X=Y, then (1) R[ , ] ( A) [ , ] A , A Y I ( X ) , ( 2)
T
[ , ]
N ([ , ]) A R ( A) , A Y I ( 2 ) ( X ) , for any
0 1, R is reflexive. (2) R is symmetric [ , ]
RT
[ , ]
(1X \{ y} )( x) RT
[ , ]
(1X \{ x} )( y ), ( x, y ) X X ,
[ , ]
R (1{ x} )( y ) R (1{ y} )( x), ( x, y ) X X .
(3) R is T -transitive [ , ] [ , ] [ , ] R ( A) R R ( A) , A Y I ( X ) , T
[ , ]
R
R
T
[ , ]
T
[ , ]
( 2)
( A) R ( A) , A Y I ( 2 ) ( X ) .
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
562
Proof: (1) and (2) can be proven analogously to Proposition 4.4. (3) If R is T -transitive, then A Y I ( X ) , ( 2)
[ , ]
RT
[ , ] RT ( A) ( x) [ , ] inf T N ( R( x, u )),[ , ] RT ( A)(u ) [ , ]
[ , ] inf T N ( R( x, u )), R[T , ] ( A)(u ) R ( A )( u ) [ , ] T
[ , ]
inf T N ( R( x, u )),[ , ] RT u X
inf
T N ( R(u, v)),[ , ]
A ( v ) [ , ]
T N ( R( x, v)), A(v)
inf T N ( R( x, u )), inf
u X
A ( v ) [ , ]
inf
u X
inf
A ( v ) [ , ]
inf
A ( v ) [ , ]
inf
inf
A ( v ) [ , ]
(2)
T -equivalence relation, 0 1 and A Y I ( X ) . Then we have the following the granule
R T ( y[ , ] ) R T ( y[ , ] )(u ) A(u ), u A[ , ] [ , ]I ( 2) , y A [ , ]
,
T N ( R(u, v)),[ , ]
T N ( R( x, v)), A(v)
R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) , y A [ , ]
T (( N ( R( x, u )), T N ( R(u, v)), A(v)
inf
inf
T T N ( R( x, u )), N ( R (u , v)) ,[ , ]
T T N ( R( x, u )), N ( R(u, v)) , A(v)
inf T sup T( N ( R( x, u )), N ( R(u, v))), A(v) A ( v ) [ , ] uX (by Lemm 2.3(1))
inf
A ( v ) [ , ]
N ( R( x, v)),[ , ]
inf
A ( v ) [ , ]
T
T N ( R( x, v)), A(v)
(by T –transitivity and Lemma 2.2(1)) [ , ] R ( A)( x) . T
By the duality, we have [ , ] [ , ] [ , ] R ( A) R R ( A) [ , ]
R
T
[ , ]
T
T
[ , ]
( A) R ( A) , A Y I ( 2 ) ( X ) .
From Definition 4.2, it follows the following property. Proposition 4.5: For two generalized IVF approximation space ( X , Y , R) and ( X , Y , P) , if R P , then for an IVF set A Y I (Y ) , we have the following statements: (2)
inf
A ( u ) [ , ]
sup [ , ] I (2) T( R( x, u ),[ , ]) [ , ]
inf
A ( u ) [ , ]
sup [ , ] I (2) ( R( x, u ),[ , ]) A(u )
inf sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] u A [ , ]
inf T sup T( N ( R( x, u )), N ( R(u , v))),[ , ] A( v ) [ , ] uX
where A[ , ] {u A(u) [, ]} and A[ , ] {u A(u) [, ]} . Proof: For all x X , we have [ , ] RT ( A)( x)
(by Lemma 2.2.(3))
(30)
Theorem 4.1: Let X Y , R Y I ( X X ) be an IVF
(by Lemma 2.3.(2))
A ( v ) [ , ]
R
[ , ]
(2) R ( A) P[ , ] ( A) .
representation form of the lower approximation operator. [ , ] R ( A)
T N ( R( x, u )), T N ( R(u, v)),[ , ]
A ( v ) [ , ]
(29)
T
T
u X
A ( v ) [ , ]
T
(2)
inf T N ( R ( x, u )),[ , ]
inf u X
( A)(u )
(1) R[ , ] ( A) P[ , ] ( A) .
inf sup [ , ] I (2) R T ( x[ , ] )(u ) A(u ) u A [ , ]
Let [ u ,u ] = sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] and [ , ] = inf [u ,u ] , u A [ , ]
[ u , u ] = sup [ , ] I (2) R T ( x[ , ] )(u ) A(u )
and
[ , ] = inf [ u , u ] . u A [ , ]
Since for all u X , R T ( x[ , ] )(u ) T( R ( x, u ),[u ,u ])
u
u
= T R (u , x),sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] = sup [ , ]I ( 2 )
T( R(u, x),[ , ]) T( R(u, x),[ , ]) [ , ]
[ , ] . R T ( x[ , ] )(u ) R T ( x[ u ,u ] )(u ) [ , ]
Thus
for
all
u A[ , ] . For any [ , ] [ , ] , there exists at least a u A[ , ]
T R ( x, u ),[ , ] [ , ] .
such that
T R ( x, u ),[ , ] [ , ]
[ , ] [u ,u ] for
all
for
all [ , ]
u A
u A[ , ] ,
which
(If then
means
[ , ] inf [u ,u ] [ , ] , a contradiction.) Thus it u A [ , ]1
follows [ , ]
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[0.2,0.4]
Table 3. IVF set A and its the IVF lower approximations RT ( A) , RT [0.2,0.4]
R 0 X
A
563
( A) and the upper approximations R ( A) ,
( A) .
0.2, 0.4
RT ( A)
R ( A)
[0.2,0.4]
RT
[0.2,0.4]
( A)
R
( A)
x1
[0.15, 0.2]
[0.15, 0.2]
[0.722, 0.786]
[0.2, 0.4]
[0.556, 0.714]
x2
[0.7, 0.85]
[0.125, 0.222]
[0.7, 0.85]
[0.25, 0.44]
[0.6, 0.8]
x3
[0.5, 0.6]
[0.222, 0.286]
[0.5, 0.667]
[0.44, 0.57]
[0.5, 0.6]
x4
[0.1, 0.2]
[0.1, 0.2]
[0.722, 0.813]
[0.2, 0.4]
[0.556, 0.75]
x5
[0.4, 0.55]
[0.333, 0.444]
[0.643, 0.8]
[0.4, 0.55]
[0.5, 0.733]
x6
[0.75, 0.85]
[0.125, 0.222]
[0.75, 0.85]
[0.25, 0.44]
[0.6, 0.8]
5. Conclusions = R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) y A , [ , ]
It is similar to prove [ , ]
=
[ , ]I ( 2 ) , y A [ , ]
R ( y T
[ , ]
( x) .
) R T ( y[ , ] )(u ) A(u ), u A[ , ] ( x) .
For all x X , we have [ , ] R ( A)( x) [ , ] [ , ] T
R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) , y A [ , ]
R T ( y[ , ] ) R T ( y[ , ] )(u ) A(u ), u A[ , ] [ , ]I ( 2) y A , [ , ]
.
Theorem 4.1 shows that the membership function representation and granular representation of R[ , ] are T
equivalent. The following gives a simple example on the generalized IVF lower approximation and generalized IVF upper approximation based on fuzzy logic operators T and with two sets of variable precision. Example 4.1: Consider Example 3.1 again. Then ab 1, a b 0, T ( a, b) b , ( a, b) b a , 1 a , a b a , a b An IVF set A and its the IVF lower approximations [ , ]
RT
[ , ]
( A) and the upper approximation R ( A) are [ , ]
shown in Table 3, where R[ , ] ( A) and R T
computed by Eq. (26) and Eq. (27) respectively.
( A) are
In this paper, the concepts of rough set are generalized to interval-valued fuzzy sets, and generalized interval-valued fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy triangular norms and fuzzy logical operators are developed by combining interval-valued fuzzy sets and rough sets/variable precision rough sets. Some developments and future work are: (1) This paper generalizes fuzzy variable precision rough sets [28] to interval-valued fuzzy sets, interval precisions and interval-valued fuzzy relations on two universes of discourse. If interval-valued fuzzy set, interval-valued fuzzy relation and interval precision are degenerated to fuzzy set, fuzzy relation and point precision respectively, and two universes of discourse are the same, then generalized interval-valued fuzzy variable precision rough sets are reduced to fuzzy variable precision rough sets proposed in [28]. And more if fuzzy set and fuzzy relation are degenerated to crisp set and crisp equivalence relation respectively, then generalized interval-valued fuzzy variable precision rough sets are reduced to variable precision rough sets proposed by Ziarko in [30]. (2) This paper sets up a foundation to deal with interval-valued fuzzy datasets by introducing new concepts of generalized interval-valued fuzzy variable precision rough sets and studying their properties. (3) The discernibility matrix approach to investigate the structure of attribute reductions in interval-valued fuzzy variable precision rough sets and the algorithms to find all reductions will be studied, and the experimental comparisons to show the feasibility and effectiveness of interval-valued fuzzy variable precision rough sets will be made in another paper.
564
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
norm,” Inf. Sci., vol. 178, pp. 3203-3213, 2008. [12] J.-S. Mi and W.-X. Zhang, “An axiomatic characterization of a fuzzy generalization of rough sets,” Inf. Sci., vol. 160, pp. 235-249, 2004. [13] A. Mieszkowicz-Rolka and L. Rolka, “Variable precision fuzzy rough sets model in the analysis of Acknowledgment process data,” in Proc. Int. Workshop Rough Sets, Fuzzy Sets, Data Mining, Granular Computing This work was jointly supported by grants from The (Lecture Notes in Artificial Intelligence), vol. 3641, Hong Kong Polytechnic University Research Committee Regina, SK, Canada, 2005, pp. 354-363. and The National Natural Science Foundation of China [14] Z. Pawlak, “Rough sets,” Int. J. Comput. Inf. Sci., (Grant No. 61179038). vol. 11, no. 5, pp. 341-356, 1982. A. M, Radzikowska and E. E. Kerre, “A compara[15] References tive study of fuzzy rough sets,” Fuzzy Sets Syst., vol. 126, pp. 137-155, 2002. [1] C. Cornelis, M. D. Cock, and A. M. Radzikowska, J. M. F. Salido and S. Murakami, “Rough set analy[16] “Vaguely quantified rough sets,” in Proc. Rough sis of a general type of fuzzy data using transitive Sets, Fuzzy Sets, Data Mining Granular Computing aggregations of fuzzy similarity relations,” Fuzzy (Lecture Notes in Artificial Intelligence), vol. 4482, Sets Syst., vol. 139, pp. 635-660, 2003. pp. 87-94, 2007. B. Sun, Z. Gong, and D. Chen, “Fuzzy rough set [17] [2] G. Deschrijver, “Characterizations of (weakly) theory for the interval-valued fuzzy information Archimedean t-norms in interval-valued fuzzy set systems,” Inf. Sci., vol. 178, pp. 2794-2815, 2008. theory,” Fuzzy Sets Syst., vol. 160, pp. 778-801, L. Wang, Q.-L. Fu, and Y.-R. Zeng, “Continuous [18] 2009. review inventory models with a mixture of backor[3] G. Deschrijver, C. Cornelis, and E. E. Kerre, “On ders and lost sales under fuzzy demand and differthe representation of intuitionistic fuzzy t-norms ent decision situations,” Expert Systems with Appliand t-conorms,” IEEE Trans. on Fuzzy Syst., vol. 12, cations, vol. 39, pp. 4181-4189, 2012. no. 1, pp. 45-61, 2004. W.-Z. Wu, J.-S. Mi, and W.-X. Zhang, “General[19] [4] D. Dubois and H. Prade, “Rough fuzzy sets and ized fuzzy rough sets,” Inf. Sci., vol. 151, pp. fuzzy rough sets,” Int. J. Gen. Syst., vol. 17, pp. 263-282, 2003. 191-209, 1990. W.-Z. Wu and W.-X. Zhang, “Constructive and [20] [5] W. S. Du and B. Q. Hu, “Approximate distribution axiomatic approaches of fuzzy approximation opreducts in inconsistent interval-valued ordered decierators,” Inf. Sci., vol. 159, pp. 233-254, 2004. sion tables,” Inf. Sci., vol. 271, pp. 93-114, 2014. [6] Z. Gong, B. Sun, and D. Chen, “Rough set theory [21] H. Xie and B. Q. Hu, “New extended patterns of fuzzy rough set models on two universes,” Int. J. for the interval-valued fuzzy information systems,” Gen. Syst., vol. 43, no. 6, pp. 570-585, 2014. Inf. Sci., vol. 178, pp. 1968-1985, 2008. R. R. Yager, “Level sets and the extension principle [22] [7] B. Q. Hu and Z. H. Huang, “( , T )-Generalized for interval valued fuzzy sets and its application to fuzzy rough sets based on fuzzy composition operauncertainty measures”, Inf. Sci., vol. 178, pp. tions,” B.-Y. Cao, C.-Y. Zhang, and T.-F. Li (Eds.): 3565-3576, 2008. Fuzzy Information and Engineering, ASC 54, pp. [23] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, pp. 647-659. Springer-Verlag, Berlin Heidelberg, 2009. 338-353, 1965. [8] B. Q. Hu and C. K. Kwong, “On type-2 fuzzy sets [24] L. A. Zadeh, “The concept of a linguistic variable and their t-norm operations,” Inf. Sci., vol. 255, pp. and its applications to approximate reasoning-I, II, 58-81, 2014. III,” Inf. Sci., vol. 8, pp.199-249, 1975; vol. 8, pp. [9] B. Q. Hu and C. Y. Wang, “On type-2 fuzzy rela301-357, 1975; vol. 9, pp. 43-80, 1975. tions and interval-valued type-2 fuzzy sets,” Fuzzy [25] Y.-R. Zeng, L. Wang, and J. He, “A novel approach Sets Syst., vol. 236, pp. 1-32, 2014. for evaluating control criticality of spare parts using [10] B. Q. Hu and H. Wong, “Generalized interfuzzy comprehensive evaluation and GRA,” Int. J. val-valued fuzzy rough sets based on interFuzzy Syst., vol. 14, no. 3, pp. 392-401, 2012. val-valued fuzzy logical operators,” Int. J. Fuzzy [26] H.-Y. Zhang, Y. Leung, and L. Zhou, “VariSyst., vol. 15, no. 4, pp. 381-391, 2013. able-precision-dominance-based rough set approach [11] J.-S. Mi, Y. Leung, H.-Y. Zhao, and T. Feng, “Gento interval-valued information systems,” Inf. Sci., eralized fuzzy rough sets determined by a triangular vol. 244, pp.75-91, 2013.
(4) These models can be extended to type-2 fuzzy sets and interval-valued type-2 fuzzy sets [8-9]. These extension models can be applied to the type-2 fuzzy information system and granular computing etc.
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[27] H.-Y. Zhang, W.-X. Zhang, and W.-Z. Wu, “On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse,” Int. J. Approx. Reason, vol. 51, pp. 56-70, 2009. [28] S. Zhao, E. C. C. Tsang, and D. Chen, “The model of fuzzy variable precision rough sets,” IEEE Trans. on Fuzzy Syst., vol. 17, no. 2, pp. 451-467, 2009. [29] S. Zhao, E. C. C. Tsang, D. Chen, and X. Wang, “Building a rule-based classifier- A fuzzy-rough set approach,” IEEE Trans. on Knowl. Data Eng., vol. 22, no. 5, pp. 624-638, 2010. [30] W. Ziarko, “Variable precision rough set model,” J. Comput. Syst. Sci., vol. 46, no. 1, pp. 39-59, 1993. [31] W. Ziarko, “Probabilistic decision tables in the variable precision rough set model,” Comput. Intell., vol. 17, pp. 593-603, 2001. [32] W. Ziarko, “Probabilistic approach to rough sets,” Int. J. Approx. Reason, vol. 49, pp. 272-284, 2008. Bao Qing Hu is a Professor at the School of Mathematics and Statistics, Wuhan University (China). He obtained his BSc, MSc, and PhD from Wuhan University. His current research interests include Fuzzy Mathematics, Rough Set Theory and Soft Computing.
Heung Wong is a Professor at the Department of Applied Mathematics, The Hong Kong Polytechnic University. He obtained his BSc (Mathematics), MSc (Statistics), and PhD (Statistics) from The Chinese University of Hong Kong, Newcastle University (UK) and The University of Hong Kong respectively. His current research interests include time series analysis, environmental statistics and stochastic hydrology.
565
554
Generalized Interval-Valued Fuzzy Variable Precision Rough Sets Bao Qing Hu and Heung Wong Abstract1 For interval-valued fuzzy datasets, people have started to do research on interval-valued fuzzy rough sets and relevant models. However, these models could not be effectively applied to handle the real-valued datasets such as interval-valued fuzzy datasets as variable precision problems were not considered in interval-valued fuzzy rough sets. In this paper, fuzzy variable precision rough sets are generalized to interval-valued fuzzy sets, with consideration of interval precisions and interval-valued fuzzy relations. Combining interval-valued fuzzy set with rough sets and variable precision rough sets, this paper develops generalized interval-valued fuzzy variable precision rough sets (GIVF-VPRSs) based on triangular norms and fuzzy logical operators respectively. using the basic properties of GIVF-VPRSs, this paper gives granule representation form of the upper approximation operator in GIVF-VPRSs based on triangular norms and granule representation form of the lower approximation operator in GIVF-VPRSs, which is based on fuzzy logical operators. The conclusions show that some existing models, such as interval-valued fuzzy rough sets, variable precision rough sets and fuzzy variable precision rough sets, are special examples of GIVF-VPRSs proposed in this paper. Keywords: Interval-valued fuzzy sets, fuzzy logical operators, interval-valued fuzzy rough sets, interval-valued fuzzy variable precision rough sets.
1. Introduction The theory of fuzzy sets, proposed by Zadeh in 1965 [23], is an extension of set theory for the study of intelligent systems characterized by fuzzy information [18, 25]. The theory of rough sets (RSs), proposed by Pawlak in 1982 [14], is another extension of set theory for the Corresponding Author: Bao Qing Hu is with School of Mathematics and Statistics, Wuhan University, Wuhan, P.R. China. E-mail: [email protected] Heung Wong is with Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. E-mail: [email protected] Manuscript received 23 Sep. 2013; revised 20 June 2014; accepted 05 Aug. 2014.
study of intelligent systems characterized by insufficient and incomplete information. It made RSs work well on some problems, but it also limited the further applications of RSs. For example, RSs could not work effectively on the datasets with real values. Fuzzy rough sets (FRSs) were proposed by Dubois and Prade to deal with fuzzy information systems [4]. RSs and FRSs were generalized to two universes of discourse, i.e. generalized fuzzy rough sets (GFRSs) [19-21]. Operators are extended from max and min to triangular norm (t-norm, t-conorm) or fuzzy logical operators. For example, (, T) -GFRSs [7, 11], ( T , ) -GFRSs [12] and ( I , T) -GFRSs [15, 19]. Variable precision rough sets (VPRSs) were extended by Ziarko in 1993 [30] to overcome the shortcomings on misclassification and/or perturbation [31-32]. For interval-valued fuzzy (IVF) datasets, one has started to research into interval-valued fuzzy rough sets (IVF-RSs) [5-6, 10, 17, 26-27]. However, these models could not be effectively applied to handle the real-valued datasets such as IVF datasets as variable precision problems were not considered in IVF-RSs. By considering the characteristics of FRSs and VPRSs, several flexible RSs based generalizations were proposed which can handle the numerical problems with misclassification and/or perturbation. For example, variable precision rough set-fuzzy rough set (VPRS-FRSs) [16], variable precision fuzzy rough set (VPFRSs, [13]), vaguely quantified rough set (VQRSs, [1]) and fuzzy variable precision rough set (FVPRSs, [28-29]). Zhao, Tsang and Chen [28] pointed out various shortcomings in these generalized models, mainly on misclassification, perturbation and attribute reduction, and introduced FVPRSs and gave a corresponding method on attribute reduction. For IVF information systems, however, there is no work on the generalization of RS with due consideration of variable precision. In order to do this, this paper firstly extends IVF-RSs to two universes of discourse, as GIVF-RSs, and then extends FVPRSs to IVF and two universes of discourse, as GIVF-VPRSs, based on IVF triangular norms and IVF logical operators. This paper gives definitions of GIVF-VPRSs based on triangle norms and logical operators on two universes of discourse and discusses their properties. These properties mainly include granule representation form of the upper
© 2014 TFSA
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
approximation operator in GIVF-VPRSs based on triangular norms and granule representation form of the lower approximation operator in GIVF-VPRSs based on fuzzy logical operators.
lower and upper approximation operators with variable precision [0,1) , also called the lower and upper approximation operators of GFVPRS, are respectively defined as follows. x X :
R ( A)( x) inf N ( R( x, u )),
2. Preliminaries
Define
i
i
T ( x, y ) sup [0,1] T( x, ) y ,
( x, y ) inf [0,1] ( x, ) y .
A ( u )
(6)
R T ( A)( x) sup T R( x, u ), N ( ) A ( u ) N ( ) sup T R( x, u ), A(u ) , A ( u ) N ( )
R T ( A)( x) inf T R( x, u ),
A ( u )
(7)
inf T R( x, u ), A(u ) , A ( u )
(8)
R ( A)( x) sup N ( R( x, u )), N ( ) A ( u ) N ( ) sup N ( R( x, u )), A(u ) . A ( u ) N ( )
(2) (3) (4)
A negator N is a decreasing function N :[0,1] [0,1] satisfying N (0) 1 and N (1) 0 . A negator N is referred to as involutive iff N ( N ( x)) x for all x [0,1] . Specially N s ( x) 1 x is an involutive negator. Given a negator N, t-norm and t-conorm are called dual with regard to N if T( N ( x), N ( y )) N ( ( x, y )) , and under such condition, N ( T ( x, y )) ( N ( x), N ( y )) . (5) Let us always assume that T and are continuous and dual w.r.t. the involutive negator N in this paper. In this paper Y ( X ) denotes the class of all fuzzy sets of the universe X. The union, intersection and complement of fuzzy sets are denoted by , and Ac respectively. A fuzzy subset R Y ( X Y ) is referred to as a fuzzy binary relation from X to Y. Let X and Y be two finite nonempty universes and R a fuzzy relation from X to Y. The triple ( X , Y , R) is referred to as a generalized fuzzy approximation space in the following discussions. There exist many generalized models for variable precision rough sets. To solve the problems on misclassification and perturbation, Zhao, Tsang and Chen [28] introduced FVPRSs and gave the corresponding attribute reduction. We give the following definition in generalized form with two universes of discourse. Definition 2.1: In the generalized fuzzy approximation space (X, Y, R), for A Y (Y ) , 0 1 , the fuzzy
A ( u )
inf N ( R( x, u )), A(u ) ,
Rough sets (RSs) theory proposed by Pawlak [14] is an extension of set theory. Since then, fuzzy sets are introduced into the RS, universes of discourse are generalized to two universes which are not necessarily the same and operators are extended to fuzzy logical operators. We first review fuzzy logical operators. For a continuous t-norm T and a continuous t-conorm on [0, 1], we have T x,sup yi sup T x, yi , (1) i i x,inf yi inf x, yi ( is any index set).
555
(9)
In [28-29], X=Y was considered in four fuzzy approximation operators with variable precision Eq. (6)-Eq.(9). The four fuzzy approximation operators are extended to interval numbers in [0, 1] as follows. In this paper, I (2) [a , a ] : 0 a a 1 , specially for a [0,1] , a [a, a] . An order relation on I (2) is defined as [a , a ] [b , b ] a b , a b
for [a , a ],[b , b ] I (2) . Two
incomparable,
[a , a ],[b , b ] I (2)
intervals
if
neither
i
[b , b ] [a , a ] holds. If [a , a ] I define sup[ ai , ai ] [sup ai ,sup ai ] , i
i i
called
[a , a ] [b , b ] i
i
are
(2)
, i , we
(10)
inf[ai , ai ] [inf a ,inf ai ] . i
i
nor
(11)
i
We define [a , a ],[b , b ] I (2) ,
T [a , a ],[b , b ] [T(a , b ), T(a , b )] , [2] [a , a ],[b , b ] [ (a , b ), (a , b )] , N [a , a ] = [ N (a ), N ( a )] .
T [a , a ],[b , b ]
[2, 3]
(12) (13) (14)
sup [ , ] I (2) T [a , a ],[ , ] [b , b ] , [2]
[a , a ],[b , b ]
(15)
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
556
inf [ , ] I (2) [a , a ],[ , ] [b , b ] . (16)
A B
It is easy to obtain the following properties. Lemma 2.1 [10]: (1) T [a , a ],[b , b ]
( x) 0 and X ( x) 1 , x X , respectively. A binary IVF subset R of X Y is called an IVF relation from X to Y. For the IVF relation R ( R Y I ( 2 ) ( X X ) ) of the universe X, then (1) R is re-
flexive, if R ( x, x) 1 , for any x X . (2) R is symmetric, if R ( x, y ) R ( y , x) , for any x, y X . (3) R is
N [a , a ] , N [b , b ] .
Lemma 2.2 [10]: (1) If [a , a ] [b , b ] , then
T -transitive, if R ( x, z ) T R ( x, y ), R( y, z ) , for any x, y, z X . (4) R is serial if x X , y Y , such that
T [a , a ],[c , c ] T [b , b ],[c , c ] and
R x, y 1 . An IVF relation R on X is referred to as an IVF T -equivalence relation on X if it is reflexive, symmetric and T -transitive. It follows from Lemma 2.2(2) that an IVF relation R is T -transitive iff R ( y , z ) T R ( x, y ), R ( x, z ) , for any x, y, z X .
T ([c , c ],[a , a ]) T ([c , c ],[b , b ]) .
(2) T [a , a ],[b , b ] [c , c ]
T [a , a ],[c , c ] [b , b ] .
T [a , a ], [b , b ],[c , c ] .
3. Generalized Interval-valued Fuzzy Variable Precision Rough Sets Based on Triangle Norms
(4) [a , a ] [b , b ] T [a , a ],[b , b ] 1 . Lemma 2.3 [10]: Let T and be continuous t-norm and t-conorm respectively. Then [ai , ai ] I (2) , i ,[b , b ] I (2) , (1) sup[ai , ai ],[b , b ] inf [ai , ai ],[b , b ] . i i
(2) [b , b ],inf[ai , ai ] inf [b , b ],[ai , ai ] . i
i
Y I ( 2) ( X ) is the class of all IVF sets [22, 24] over the
In this section FVPRSs are extended to IVF sets and two universes of discourse. Definition 3.1: In the generalized IVF approximation space ( X , Y , R ) , for an IVF set A Y I ( 2 ) (Y ) , the generalized IVF lower and generalized IVF upper approximation operators with variable precision [ , ] ( 0 1 ) based on triangle norms, are respectively defined as follows. x X : [ , ]
universe X. For [ , ] I (2) and M X , we define M [ , ] Y I ( 2) ( X ) with membership function M [ , ] ( x) [ , ]
while
xM ,
0
and
with membership functions
( 2)
(3) N T ([a , a ],[b , b ])
A ( x ) B ( x )
And , X Y I ( X )
(a , b ), (a , b ) (a , b ) ,
iff
(19)
A ( x) B ( x) .
(2) [a , a ],[b , b ]
A( x) B ( x)
iff
T (a , b ) T (a , b ), T (a , b ) ,
(3) T T [a , a ],[b , b ] ,[c , c ]
( AN s )( x) 1 A( x) [1 A ( x),1 A ( x)] .
R
otherwise. If [ , ]
M {x} , then {x}[ , ] is tersely denoted by x[ , ] .
RT
M [ , ] is also denoted by [ , ]M . If [ , ] I (2) and
A( x) [ , ] for all x X , then A is simply denoted by [ , ] , i.e. A [ , ] . For a crisp set A X , A may be viewed as a special kind of IVF sets with membership function A( x) 1 for x A , otherwise 0 . We define union, intersection, complement and order relation pointwise by the following formulas. For A [ A , A ], B [ B , B ] Y I ( 2 ) ( X ) , ( A B)( x) [ A ( x) B ( x), A ( x) B ( x)] ,
(17)
( A B)( x) [ A ( x) B ( x), A ( x) B ( x)] ,
(18)
( A)( x)
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u ) ,
(20)
( A)( x) sup T R( x, u ), N ([ , ]) A(u ) N ([ , ]) sup T R( x, u ), A(u ) . (21) A ( u ) N ([ , ])
The pair of IVF sets
R
[ , ]
[ , ]
( A), R T
( A)
is called a
(, T) -generalized IVF variable precision rough set ( (, T) -GIVF-VPRS).
For any IVF set f, we define inf f (u ) 1 and u
sup f (u ) 0 . u
It follows from Definition 3.1 the following special
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[ N ( R ( x, u )), N ( R ( x, u ))],[ A (u ), A (u )] [ A ( u ), Ainf ( u )] [ , ]
cases. (1) For the arbitrary crisp subset A of Y, i.e. A( y ) 1
inf N ( R ( x, u )), inf N ( R ( x, u )), A (u ) , A ( u ) A ( u )
for y Y , otherwise 0 , then it follows from [ , ] 1 iff N ([ , ]) 0 and A(u ) [ , ] ( 0 1 ) iff [ , ]
inf N ( R ( x, u )), inf N ( R ( x, u ))], A (u ) A (u )
A ( u )
A(u ) 0 that
R
( A)( x) inf N ( R( x, u )),[ , ]
A( u ) 0
inf N ( R( x, u )), A(u ) A( u ) 1
Then [ , ]
A(u ) 0
R
Proof: R
(2) If , A Y (Y ) and R Y ( X Y ) , then Eq. (20) and Eq. (21) are simplified to the two equations in Eq. (6) and Eq. (7) respectively, which were discussed in [29] . (3) If 0 , then
A(u ) 0
And R
[0,0] T
( A)( x) sup T R( x, u ), N ([0]) A( u ) N ( 0) sup T R( x, u ), A(u ) A(u ) N ([ 0])
sup T R ( x, u ), 1 sup T R ( x, u ), A(u ) A(u ) 1 A(u ) 1 R T ( A)( x) .
R
[0,0]
[0,0]
( A), R T ( A)
( A)( x)
sup T R( x, u ), A(u ) R T ( A)( x) . uY
[ , ]
R
is a (, T) -GIVF-RS discussed in
( A)( x)
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u )
inf
A ( u ) [ , ]
A ( u ) [ , ]
uY
Proposition 3.3: Let 0 1 and A, B Y I ( 2) (Y ) . Then [ , ]
(1) R T
[ , ]
(2) R T
[ , ]
() , R
[ , ]
( AN ) R
( A)
[ , ]
(3) R is serial R RT
(Y ) X ;
N
[ , ]
, R
[ , ]
( AN ) R T
sup T R( x, u ), N ([ , ]) 0 , [ , ]
[ , ]
RT
Proof: R
[ , ]
sup T R ( x, u ), N ([ , ])
Y ( u ) [ , ]
u
( A) ( R ) ( A ),( R ) ( A ) ,
(22)
( A) ( R )T ( A ), ( R )T ( A ) .
(23)
inf
Y ( u ) [ , ]
inf [ N ( R ( x, u )), N ( R ( x, u ))],[ , ] [ A (u ), A (u )][ , ]
N ( R( x, u )),[ , ]
inf N ( R( x, u )),[ , ] 1 . u
R
( A)( x)
N
Proof: (1) Since [ , ] 1 is not true, we have sup T R ( x, u ), N ([ , ])
Then R
(Y ) N ([ , ]) for any 0 1 .
( u ) N ([ , ])
( A)
() [ , ]
[10]. Proposition 3.1: Let 0 1 and A Y I ( 2 ) (Y ) .
inf N ( R( x, u )), A(u ) N ( R( x, u )), A(u ) inf N ( R( x, u )), A(u ) .
[ , ]
[0,0]
That is R ( A) R ( A) and R T ( A) R T ( A) , i.e. [0,0]
( A) R ( A) .
sup T R( x, u ), A(u ) sup T R( x, u ), A(u ) A(u ) N ([ , ]) A(u ) N ([ , ])
inf N ( R( x, u )), A(u ) R ( A)( x) A(u ) 0
[ , ] T
sup T R(x, u), N([, ]) sup T R(x, u), A(u) A(u)N([, ]) A(u)N([, ])
A(u ) 1
[0,0]
[ , ]
( A) R T ( A) , R
RT
( A)( x) sup T R( x, u ), N ([ , ]) A u ( ) 1 sup T R( x, u ), A(u ) A(u ) 0 sup T R( x, u ), N ([ , ]) .
R ( A)( x) inf N ( R( x, u )), 0
( R ) ( A ),( R ) ( A ) ( x) . The second one is proven in a similar way. Proposition 3.2: Let 0 1 and A Y I ( 2 ) (Y ) .
inf N ( R( x, u )),[ , ] , [ , ] T
557
()( x) sup T R( x, u ), N ([ , ]) u N ( ) ([ , ]) sup T R( x, u ), (u ) 0 0 0 . (u ) N ([ , ])
[ , ] T
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
558
[ , ]
R
(2)
(Y )( x) inf
Y ( u ) [ , ]
[ , ]
R
( A)
N
N
inf
Y ( u ) [ , ]
[ , ]
R
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
[ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), A(u )
[ , ] R
[ , ] T
[ , ]
[ , ]
R
()
(Y ) N ([ , ]) . If R is serial, then for any
()( x)
inf
0 ( u ) [ , ]
inf
0 ( u ) [ , ]
N ( R( x, u )),[ , ]
N ( R( x, u )), 0)
[ , ]
[ , ]
If N ([ , ]) A R T 0 1 , then
inf
0 ( u ) [ , ]
inf
0 ( u ) [ , ]
N ( R( x, u )), 0
N ( R( x, u )), 0)
[ , ]
R
R
(1{ x} )( y ) R
[ , ] T
( A) , A Y I ( 2) ( X ) , for
(1{ y} )( x), ( x, y ) X X .
(3) R is T -transitive
[ , ]
(1X \{ y} )( x) R
(1X \{ x} )( y ), ( x, y ) X X
[ , ]
(1{ x} )( y ) R T
(1{ y} )( x), ( x, y ) X X .
N ( R ( x, u )),[ , ] (1X \{ y} )( x) inf 1 ( ) [ , ] u X \{ y } N ( R ( x, u )), 1X \{ y} (u ) inf 1X \{ y } ( u ) [ , ] N ( R ( x, y )),[ , ] N ( R ( y , x)),[ , ]
[ , ]
R
[ , ]
R [ , ]
If R T
[ , ]
(1{ x} )( y ) R T
(1X \{ x} )( y ) .
(1{ y} )( x), ( x, y ) X X , then it
follows from taking [ , ] 0 that [0,0]
R T (1{ y} )( x) sup T R( x, u ), 1 sup T R( x, u ), 1{ y} (u ) 1{ y } ( u ) 1 1{ y } (u ) 1
uY
[ , ] T
If R is symmetric, then ( x, y ) X X ,
Proposition 3.4: Given X=Y, then the following is true. [ , ] (1) R ( A) [ , ] A , A Y I ( 2) ( X ) , any 0 1 R is reflexive. (2) R is symmetric [ , ] [ , ] R (1X \{ y} )( x) R (1X \{ x} )( y ), ( x, y ) X X ,
(2) By the duality,
uY
N ([ , ]) A R
A 1{ x} and
x X , taking
i.e. R ( x, x) 1 .
Thus sup R( x, u ) 1 , i.e. R is serial.
[ , ] T
( A) , A Y I ( 2) ( X ) , for any
R( x, x) N ([0,0]) 1{ x} ( x) 1 .
[ , ]
inf N ( R( x, u )), 0 N (sup R( x, u )) . uY
( A) , A Y I ( 2) ( X ) , for any
() [ , ] .
()( x)
( A) [ , ] A ,
[0,0] RT (1{x} )(x) sup T R(x, u),1 sup T R(x, u),1{x} (u) 1{x} (u)1 1{x} (u)1
[ , ]
0R
[ , ]
R
[ , ] 0 , we have
Conversely, if R () [ , ] , for any 0 1 , then for 0 , [0,0]
( A) , A Y I ( 2) ( X ) .
0 1.
N (sup R ( x, u )),[ , ] [ , ] . uY
And then R
[ , ]
( A)) R T
Proof: (1) By the duality, A Y I ( 2) ( X ) ,
RT
inf N ( R( x, u )),[ , ] 1 uY
( A)) , A Y I ( 2) ( X ) ,
[ , ]
x X , there is a y0 Y such that R ( x, y0 ) 1 . Thus R
[ , ]
( RT
[ , ]
(R
N ([ , ]) A R T
sup T R(x, u), N ([ , ]) sup T R(x, u), N( A(u)) A ( u ) [ , ] A ( u ) [ , ] sup T R( x, u ), N ([ , ]) N A (u ) N ([ , ]) [ , ] sup T R( x, u ), N ( A(u )) R T ( A N )( x) . N A (u ) N ([ , ])
(3) It follows from the duality that
[ , ]
( A) R
RT
N ( R( x, u )), Y (u ) 1 1 1 .
( x) N
N ( R( x, u )),[ , ]
[0,0]
R( x, y ) , R T (1{ x} )( y ) sup T R ( y, u ), 1 sup T R( y, u ), 1{ x} (u ) 1{ x } (u ) 1 1{ x } (u ) 1 R( y, x) . Thus R( x, y ) R( y, x) .
(3) If R is T -transitive, then A Y I ( X ) , ( 2)
[ , ]
R
( A))( x) [ , ] inf N ( R ( x, u )),[ , ] R ( A)(u ) [ , ] [ , ] [ , ] inf N ( R( x, u )), R ( A)(u ) R ( A)( u ) [ , ] [ , ]
(R
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[ , ]
inf N ( R( x, u )),[ , ] R u X
inf N ( R ( x, u )),[ , ] u X
A ( v ) [ , ]
u X
inf
u X
A ( v ) [ , ]
A ( v ) [ , ]
inf
A ( v ) [ , ]
sup T R ( x, u ), y[ , ] (u ) T R ( x, y ),[ , ] ,
uY
we have RT ( y[ , ] ) y[ , ] AN ([ , ]) [ , ]I ( 2) , yY
N ( R( x, u )), N ( R(u , v)),[ , ]
N ( R( x, u )), N ( R(u, v)), A(v)
N (R( x, v)),[ , ]
inf
A( v ) [ , ]
N ( R( x, v)), A(v)
R
If R
[ , ] T
(R
(R
[ , ] T
( A)) R
( A)) R
[ , ] T
[ , ] T
T R( x, y),[ , ] y
( x)
AN ([ , ])
[ , ]
T R( x, y),[ , ] [ , ] A
T R( x, y),[ , ] [ , ] A
N ([ , ])
sup [ , ]I ( 2 ) , yY
R ( y T
[ , ]
)( x) [ , ] AN ([ , ])
sup
( A) , A Y I ( 2 ) ( X ) .
( A) , A Y I ( X ) , for any ( 2)
0 1 , then for all x, y X , taking A 1{ y} and [ , ] 0 , we have R T R( x, y ) R( x, y ) . Thus R is
T -transitive. It is follows from Definition 3.1 the following property. Proposition 3.5: For two generalized IVF approximation space ( X , Y , R) and ( X , Y , P) , if R P , then for an IVF set A Y I (Y ) , the following hold.
N ([ , ])
[ , ]I ( 2 ) , yY
yY
( x) ( y )
T R( x, y),[ , ] [ , ] A sup T R ( x, y ), A ( y )
( y)
( x) .
(2) R T ( y[ , ] ) y[ , ] AN ([ , ]) [ , ]I ( 2) , yY
( y )
N ([ , ])
RT ( y[ , ] ) [ , ] AN ([ , ]) ( y) [ , ]I ( 2 ) , yY
[ , ]
[ , ] T
sup [ , ]I ( 2) , yY
R ( A)( x) . By the duality, we have [ , ] [ , ] [ , ] R ( A) R ( R ( A)) [ , ] T
sup [ , ]I ( 2) , yY
u X
u X
sup [ , ]I ( 2) , yY
inf N ( R( x, u )), N ( R(u, v)) , A(v)
inf
inf
N ( R( x, v)), A(v)
Proof: (1) Since R T ( y[ , ] )( x)
inf N ( R( x, u )), N ( R(u, v)) ,[ , ]
A ( v ) [ , ]
A( v)[ , ]
A ( v ) [ , ]
N ( R(u , v)),[ , ]
inf
A ( v ) [ , ]
inf
inf
inf
N ( R(u , v)),[ , ]
N ( R( x, v)), A(v)
inf inf N ( R( x, u )),
inf
A ( v ) [ , ]
( A)(u )
559
( y)
N ([ , ])
sup T R( x, y ), N ([ , ]) A( y ) N ([ , ]) [ , ] sup T R( x, y ), A( y ) R T ( A)( x) . A ( y ) N ([ , ]) Theorem 3.1 shows that the membership function [ , ]
representation and granular representation of R T are equivalent. The following gives a simple example on the generalized IVF lower approximation and generalized IVF upper approximation based on triangle norm T and with two sets of variable precision. [ , ] [ , ] (1) R ( A) P ( A) . (24) Example 3.1: Let X Y {x1 , x2 , , x6 } , T (ordinary multiplication) and ˆ ( a ˆ b a b ab ), [ , ] (25) N N s ( N s ( x) 1 x ). (2) R T ( A) PT[ , ] ( A) . Theorem 3.1: Let 0 1 and A, y[ , ] Y I ( 2 ) (Y ) . An interval-valued fuzzy relation R is shown in Table 1. An IVF set A and its the IVF lower approximations Then (2)
(1)
R ( y T
[ , ]I ( 2 ) , yY
[ , ]
) y[ , ] AN ([ , ])
[ , ]I ( 2 ) , yY
[ , ]
(2) R T
( A)
[ , ]I ( 2 ) , yY
R ( y R ( y
[ , ]
R
T
[ , ]
) [ , ] AN ([ , ]) ( y ) .
T
[ , ]
) y[ , ] AN ([ , ]) .
[ , ]
( A) and the upper approximation R T
( A) are
[ , ] T
[ , ]
shown in Table 2, where R ( A) and R ( A) are computed by Eq. (20) and Eq. (21) respectively.
4. Generalized Interval-Valued Fuzzy Variable Precision Rough Sets Based on Logic Operators
where AN ([ , ]) ( y ) N ([ , ]) A( y ), N ([ , ]) A( y ) or A( y ) N ([ , ]), N ([ , ]) and A( y ) are incomparable 0,
Definition 4.1: In the generalized IVF approximation space ( X , Y , R) , for an IVF set A Y I (Y ) , the (2)
560
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
Table 1. IVF relation R. R
x1
x2
x3
x4
x5
x6
x1
1
[0.1,0.2]
[0.3,0.5]
[0,0.1]
[0.15,0.2]
[0.7,0.9]
x2
[0.1,0.2]
1
[0.45,0.6]
[0.8,0.9]
[0.75,0.8]
[0.05,0.1]
x3
[0.3,0.5]
[0.45,0.6]
1
[0.45,0.7]
[0.55,0.65]
[0.3,0.45]
x4
[0,0.1]
[0.8,0.9]
[0.45,0.7]
1
[0.3,0.45]
[0.8,0.9]
x5
[0.15,0.2]
[0.75,0.8]
[0.55,0.65]
[0.3,0.45]
1
[0.5,0.7]
x6
[0.7,0.9]
[0.05,0.1]
[0.3,0.45]
[0.8,0.9]
[0.5,0.7]
1
[0.2,0.4]
Table 2. IVF set A and its the IVF lower approximations R ( A) , R
R
[0.2,0.4] T
( A) and upper approximations R T ( A) ,
( A) .
0 X
A
0.2, 0.4
R ( A)
R T ( A)
R
[0.2,0.4]
[0.2,0.4]
( A)
RT
( A)
x1
[0.15, 0.2]
[0.15, 0.2]
[0.525,0.765]
[0.2, 0.4]
[0.42, 0.72]
x2
[0.7, 0.85]
[0.19, 0.26]
[0.7, 0.85]
[0.28, 0.52]
[0.6, 0.8]
x3
[0.5, 0.6]
[0.37, 0.39]
[0.5, 0.6]
[0.44, 0.6]
[0.5, 0.6]
x4
[0.1, 0.2]
[0.1, 0.2]
[0.6, 0.765]
[0.2, 0.4]
[0.48, 0.72]
x5
[0.4, 0.55]
[0.4, 0.55]
[0.525, 0.68]
[0.4, 0.55]
[0.45, 0.64]
x6
[0.75, 0.85]
[0.19, 0.24]
[0.75, 0.85]
[0.28, 0.52]
[0.6, 0.8]
generalized IVF lower and generalized IVF upper approximation operators with variable precision [ , ] ( 0 1 ) based on logic operators T and , are, respectively, defined as follows. x X : [ , ]
R T ( A)( x)
inf
A ( u ) [ , ]
[ , ]
R
T R( x, u ),[ , ]
inf
A ( u ) [ , ]
[ , ] T
[ , ]
iff N ([ , ]) 0 and A(u ) [ , ] ( 0 1 ) iff A(u ) 0 . It follows from Lemma 2.2(4) that [ , ]
T R( x, u ), A(u ) ,
R
for y Y , otherwise 0 , then it follows from [ , ] 1
RT
(26)
( A)( x) sup N ( R( x, u )), N ([ , ]) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) . (27) A(u ) N ([ , ])
The pair of IVF sets
(1) For the arbitrary crisp subset A of Y, i.e. A( y ) 1
( A), R ( A)
is called a
( T , ) -generalized IVF variable precision rough sets ( ( T , ) -GIVF-VPRS). From Definition 4.1, it follows the following special cases.
( A)( x) inf T R( x, u ),[ , ] , u A
[ , ]
R ( A)( x) sup N ( R ( x, u )), N ([ , ]) . u A
(2) If , A Y (Y ) and R Y ( X Y ) , then Eq. (26) and Eq. (27) are simplified to the two equations in Eq. (8) and Eq. (9), respectively, which were discussed in [28]. (3) If 0 , then R[0,0] ( A)( x)
T
inf T R( x, u ), 0 inf T R( x, u ), A(u ) A( u ) 0
A(u ) 0
RT ( A)( x)
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
Proposition 4.2: Let 0 1 and A Y I (Y ) .
[0,0] R ( A)( x) sup N ( R( x, u )), N ([0]) A u N ( ) ( 0) sup N ( R( x, u )), A(u ) A(u ) N ([ 0]) sup N ( R( x, u )), 1 A( u ) 1 sup N ( R( x, u )), A(u ) R ( A)( x) . A(u ) 1
(2)
Then [ , ]
[0,0]
T
[0,0]
( A), R ( A)
T
is a (T , ) -GIVF-RS discussed in
[10]. Proposition 4.1: Let 0 1 and A Y I (Y ) .
sup N ( R( x, u )), A(u ) R ( A)( x) .
Then
uY
[ , ]
RT
( A) ( R ) ( A ) ( R ) ( A ),( R ) ( A ) , T
T
[ , ]
RT
[ , ] R ( A) ( R ) ( A ), ( R ) ( A ) ( R ) ( A ) .
( A)( x)
Proof: x X , R[ , ] ( A)( x) T
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
inf
A ( u ) [ , ]
T R( x, u ),[ , ]
T R( x, u ), A(u )
T R( x, u ), A(u )
( 2)
Then [ , ]
(1) R () , R[ , ] (Y ) X ;
[ T ( R ( x, u ), A (u )) T ( R ( x, u ), A (u )), T ( R ( x, u ), A (u ))] inf [ A ( u ), A (u )][ , ]
(2) R ( A ) R
inf T R ( x, u ), T ( R ( x, u ), ) [ A ( u ), A (u )][ , ]
inf
T R ( x, u ), A (u )
inf R ( x, u ), A (u ) inf ( R ( x, u ), A (u )) , A ( u ) T T A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A ( u ) inf T ( R ( x, u ), ) inf T ( R ( x, u ), A (u )) , A (u ) A (u ) inf T R ( x, u ), inf T R ( x, u ), A (u ) A ( u ) A (u ) ( R ) ( A ) ( R ) ( A ), ( R ) ( A ) ( x) . T T T The second one is proven in a similar way.
N
[ , ]
inf T R ( x, u ), inf T ( R ( x, u ), ) A (u ) A (u )
T
[ , ]
RT
T R ( x, u ), A (u ) T ( R ( x, u ), A (u )) , inf [ A (u ), A (u )][ , ] [ A ( u ), A ( u )][ , ]
T R( x, u ), A(u )
Proposition 4.3: Let 0 1 and A, B Y I (Y ) .
inf [ T ( R ( x, u ), ) T ( R ( x, u ), ), T ( R ( x, u ), )] [ A ( u ), A (u )][ , ]
T R ( x, u ),
inf
A ( u ) [ , ]
uY
T [ R ( x, u ), R ( x, u )],[ A (u ), A (u )] inf [ A ( u ), A ( u )] [ , ]
inf
inf T R ( x, u ), A(u ) .
T [ R ( x, u ), R ( x, u )],[ , ] inf [ A ( u ), A ( u )] [ , ]
[ A ( u ), A ( u )][ , ]
(28)
[ , ] R ( A)( x) sup N ( R( x, u )), N ([ , ]) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A ( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A( u ) N ([ , ]) sup N ( R( x, u )), A(u ) A(u ) N ([ , ])
(2)
T
( A) RT ( A) .
Proof:
[0,0]
R
[ , ]
R ( A) R ( A) , RT
That is R[0,0] ( A) R ( A) and R T ( A) R T ( A) , i.e. T
561
[ , ] T
( A)
N
,
[ , ]
( AN ) R ( A)
N
;
(3) R is serial R[ , ] () [ , ] T
[ , ]
R (Y ) N ([ , ]) for any 0 1 .
Proof: It is easily proven similar to Proposition 4.3 (1), (2) and (3). Proposition 4.4. Given X=Y, then (1) R[ , ] ( A) [ , ] A , A Y I ( X ) , ( 2)
T
[ , ]
N ([ , ]) A R ( A) , A Y I ( 2 ) ( X ) , for any
0 1, R is reflexive. (2) R is symmetric [ , ]
RT
[ , ]
(1X \{ y} )( x) RT
[ , ]
(1X \{ x} )( y ), ( x, y ) X X ,
[ , ]
R (1{ x} )( y ) R (1{ y} )( x), ( x, y ) X X .
(3) R is T -transitive [ , ] [ , ] [ , ] R ( A) R R ( A) , A Y I ( X ) , T
[ , ]
R
R
T
[ , ]
T
[ , ]
( 2)
( A) R ( A) , A Y I ( 2 ) ( X ) .
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
562
Proof: (1) and (2) can be proven analogously to Proposition 4.4. (3) If R is T -transitive, then A Y I ( X ) , ( 2)
[ , ]
RT
[ , ] RT ( A) ( x) [ , ] inf T N ( R( x, u )),[ , ] RT ( A)(u ) [ , ]
[ , ] inf T N ( R( x, u )), R[T , ] ( A)(u ) R ( A )( u ) [ , ] T
[ , ]
inf T N ( R( x, u )),[ , ] RT u X
inf
T N ( R(u, v)),[ , ]
A ( v ) [ , ]
T N ( R( x, v)), A(v)
inf T N ( R( x, u )), inf
u X
A ( v ) [ , ]
inf
u X
inf
A ( v ) [ , ]
inf
A ( v ) [ , ]
inf
inf
A ( v ) [ , ]
(2)
T -equivalence relation, 0 1 and A Y I ( X ) . Then we have the following the granule
R T ( y[ , ] ) R T ( y[ , ] )(u ) A(u ), u A[ , ] [ , ]I ( 2) , y A [ , ]
,
T N ( R(u, v)),[ , ]
T N ( R( x, v)), A(v)
R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) , y A [ , ]
T (( N ( R( x, u )), T N ( R(u, v)), A(v)
inf
inf
T T N ( R( x, u )), N ( R (u , v)) ,[ , ]
T T N ( R( x, u )), N ( R(u, v)) , A(v)
inf T sup T( N ( R( x, u )), N ( R(u, v))), A(v) A ( v ) [ , ] uX (by Lemm 2.3(1))
inf
A ( v ) [ , ]
N ( R( x, v)),[ , ]
inf
A ( v ) [ , ]
T
T N ( R( x, v)), A(v)
(by T –transitivity and Lemma 2.2(1)) [ , ] R ( A)( x) . T
By the duality, we have [ , ] [ , ] [ , ] R ( A) R R ( A) [ , ]
R
T
[ , ]
T
T
[ , ]
( A) R ( A) , A Y I ( 2 ) ( X ) .
From Definition 4.2, it follows the following property. Proposition 4.5: For two generalized IVF approximation space ( X , Y , R) and ( X , Y , P) , if R P , then for an IVF set A Y I (Y ) , we have the following statements: (2)
inf
A ( u ) [ , ]
sup [ , ] I (2) T( R( x, u ),[ , ]) [ , ]
inf
A ( u ) [ , ]
sup [ , ] I (2) ( R( x, u ),[ , ]) A(u )
inf sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] u A [ , ]
inf T sup T( N ( R( x, u )), N ( R(u , v))),[ , ] A( v ) [ , ] uX
where A[ , ] {u A(u) [, ]} and A[ , ] {u A(u) [, ]} . Proof: For all x X , we have [ , ] RT ( A)( x)
(by Lemma 2.2.(3))
(30)
Theorem 4.1: Let X Y , R Y I ( X X ) be an IVF
(by Lemma 2.3.(2))
A ( v ) [ , ]
R
[ , ]
(2) R ( A) P[ , ] ( A) .
representation form of the lower approximation operator. [ , ] R ( A)
T N ( R( x, u )), T N ( R(u, v)),[ , ]
A ( v ) [ , ]
(29)
T
T
u X
A ( v ) [ , ]
T
(2)
inf T N ( R ( x, u )),[ , ]
inf u X
( A)(u )
(1) R[ , ] ( A) P[ , ] ( A) .
inf sup [ , ] I (2) R T ( x[ , ] )(u ) A(u ) u A [ , ]
Let [ u ,u ] = sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] and [ , ] = inf [u ,u ] , u A [ , ]
[ u , u ] = sup [ , ] I (2) R T ( x[ , ] )(u ) A(u )
and
[ , ] = inf [ u , u ] . u A [ , ]
Since for all u X , R T ( x[ , ] )(u ) T( R ( x, u ),[u ,u ])
u
u
= T R (u , x),sup [ , ] I (2) R T ( x[ , ] )(u ) [ , ] = sup [ , ]I ( 2 )
T( R(u, x),[ , ]) T( R(u, x),[ , ]) [ , ]
[ , ] . R T ( x[ , ] )(u ) R T ( x[ u ,u ] )(u ) [ , ]
Thus
for
all
u A[ , ] . For any [ , ] [ , ] , there exists at least a u A[ , ]
T R ( x, u ),[ , ] [ , ] .
such that
T R ( x, u ),[ , ] [ , ]
[ , ] [u ,u ] for
all
for
all [ , ]
u A
u A[ , ] ,
which
(If then
means
[ , ] inf [u ,u ] [ , ] , a contradiction.) Thus it u A [ , ]1
follows [ , ]
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[0.2,0.4]
Table 3. IVF set A and its the IVF lower approximations RT ( A) , RT [0.2,0.4]
R 0 X
A
563
( A) and the upper approximations R ( A) ,
( A) .
0.2, 0.4
RT ( A)
R ( A)
[0.2,0.4]
RT
[0.2,0.4]
( A)
R
( A)
x1
[0.15, 0.2]
[0.15, 0.2]
[0.722, 0.786]
[0.2, 0.4]
[0.556, 0.714]
x2
[0.7, 0.85]
[0.125, 0.222]
[0.7, 0.85]
[0.25, 0.44]
[0.6, 0.8]
x3
[0.5, 0.6]
[0.222, 0.286]
[0.5, 0.667]
[0.44, 0.57]
[0.5, 0.6]
x4
[0.1, 0.2]
[0.1, 0.2]
[0.722, 0.813]
[0.2, 0.4]
[0.556, 0.75]
x5
[0.4, 0.55]
[0.333, 0.444]
[0.643, 0.8]
[0.4, 0.55]
[0.5, 0.733]
x6
[0.75, 0.85]
[0.125, 0.222]
[0.75, 0.85]
[0.25, 0.44]
[0.6, 0.8]
5. Conclusions = R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) y A , [ , ]
It is similar to prove [ , ]
=
[ , ]I ( 2 ) , y A [ , ]
R ( y T
[ , ]
( x) .
) R T ( y[ , ] )(u ) A(u ), u A[ , ] ( x) .
For all x X , we have [ , ] R ( A)( x) [ , ] [ , ] T
R T ( y[ , ] ) R T ( y[ , ] )(u ) [ , ], u A[ , ] [ , ]I ( 2) , y A [ , ]
R T ( y[ , ] ) R T ( y[ , ] )(u ) A(u ), u A[ , ] [ , ]I ( 2) y A , [ , ]
.
Theorem 4.1 shows that the membership function representation and granular representation of R[ , ] are T
equivalent. The following gives a simple example on the generalized IVF lower approximation and generalized IVF upper approximation based on fuzzy logic operators T and with two sets of variable precision. Example 4.1: Consider Example 3.1 again. Then ab 1, a b 0, T ( a, b) b , ( a, b) b a , 1 a , a b a , a b An IVF set A and its the IVF lower approximations [ , ]
RT
[ , ]
( A) and the upper approximation R ( A) are [ , ]
shown in Table 3, where R[ , ] ( A) and R T
computed by Eq. (26) and Eq. (27) respectively.
( A) are
In this paper, the concepts of rough set are generalized to interval-valued fuzzy sets, and generalized interval-valued fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy triangular norms and fuzzy logical operators are developed by combining interval-valued fuzzy sets and rough sets/variable precision rough sets. Some developments and future work are: (1) This paper generalizes fuzzy variable precision rough sets [28] to interval-valued fuzzy sets, interval precisions and interval-valued fuzzy relations on two universes of discourse. If interval-valued fuzzy set, interval-valued fuzzy relation and interval precision are degenerated to fuzzy set, fuzzy relation and point precision respectively, and two universes of discourse are the same, then generalized interval-valued fuzzy variable precision rough sets are reduced to fuzzy variable precision rough sets proposed in [28]. And more if fuzzy set and fuzzy relation are degenerated to crisp set and crisp equivalence relation respectively, then generalized interval-valued fuzzy variable precision rough sets are reduced to variable precision rough sets proposed by Ziarko in [30]. (2) This paper sets up a foundation to deal with interval-valued fuzzy datasets by introducing new concepts of generalized interval-valued fuzzy variable precision rough sets and studying their properties. (3) The discernibility matrix approach to investigate the structure of attribute reductions in interval-valued fuzzy variable precision rough sets and the algorithms to find all reductions will be studied, and the experimental comparisons to show the feasibility and effectiveness of interval-valued fuzzy variable precision rough sets will be made in another paper.
564
International Journal of Fuzzy Systems, Vol. 16, No. 4, December 2014
norm,” Inf. Sci., vol. 178, pp. 3203-3213, 2008. [12] J.-S. Mi and W.-X. Zhang, “An axiomatic characterization of a fuzzy generalization of rough sets,” Inf. Sci., vol. 160, pp. 235-249, 2004. [13] A. Mieszkowicz-Rolka and L. Rolka, “Variable precision fuzzy rough sets model in the analysis of Acknowledgment process data,” in Proc. Int. Workshop Rough Sets, Fuzzy Sets, Data Mining, Granular Computing This work was jointly supported by grants from The (Lecture Notes in Artificial Intelligence), vol. 3641, Hong Kong Polytechnic University Research Committee Regina, SK, Canada, 2005, pp. 354-363. and The National Natural Science Foundation of China [14] Z. Pawlak, “Rough sets,” Int. J. Comput. Inf. Sci., (Grant No. 61179038). vol. 11, no. 5, pp. 341-356, 1982. A. M, Radzikowska and E. E. Kerre, “A compara[15] References tive study of fuzzy rough sets,” Fuzzy Sets Syst., vol. 126, pp. 137-155, 2002. [1] C. Cornelis, M. D. Cock, and A. M. Radzikowska, J. M. F. Salido and S. Murakami, “Rough set analy[16] “Vaguely quantified rough sets,” in Proc. Rough sis of a general type of fuzzy data using transitive Sets, Fuzzy Sets, Data Mining Granular Computing aggregations of fuzzy similarity relations,” Fuzzy (Lecture Notes in Artificial Intelligence), vol. 4482, Sets Syst., vol. 139, pp. 635-660, 2003. pp. 87-94, 2007. B. Sun, Z. Gong, and D. Chen, “Fuzzy rough set [17] [2] G. Deschrijver, “Characterizations of (weakly) theory for the interval-valued fuzzy information Archimedean t-norms in interval-valued fuzzy set systems,” Inf. Sci., vol. 178, pp. 2794-2815, 2008. theory,” Fuzzy Sets Syst., vol. 160, pp. 778-801, L. Wang, Q.-L. Fu, and Y.-R. Zeng, “Continuous [18] 2009. review inventory models with a mixture of backor[3] G. Deschrijver, C. Cornelis, and E. E. Kerre, “On ders and lost sales under fuzzy demand and differthe representation of intuitionistic fuzzy t-norms ent decision situations,” Expert Systems with Appliand t-conorms,” IEEE Trans. on Fuzzy Syst., vol. 12, cations, vol. 39, pp. 4181-4189, 2012. no. 1, pp. 45-61, 2004. W.-Z. Wu, J.-S. Mi, and W.-X. Zhang, “General[19] [4] D. Dubois and H. Prade, “Rough fuzzy sets and ized fuzzy rough sets,” Inf. Sci., vol. 151, pp. fuzzy rough sets,” Int. J. Gen. Syst., vol. 17, pp. 263-282, 2003. 191-209, 1990. W.-Z. Wu and W.-X. Zhang, “Constructive and [20] [5] W. S. Du and B. Q. Hu, “Approximate distribution axiomatic approaches of fuzzy approximation opreducts in inconsistent interval-valued ordered decierators,” Inf. Sci., vol. 159, pp. 233-254, 2004. sion tables,” Inf. Sci., vol. 271, pp. 93-114, 2014. [6] Z. Gong, B. Sun, and D. Chen, “Rough set theory [21] H. Xie and B. Q. Hu, “New extended patterns of fuzzy rough set models on two universes,” Int. J. for the interval-valued fuzzy information systems,” Gen. Syst., vol. 43, no. 6, pp. 570-585, 2014. Inf. Sci., vol. 178, pp. 1968-1985, 2008. R. R. Yager, “Level sets and the extension principle [22] [7] B. Q. Hu and Z. H. Huang, “( , T )-Generalized for interval valued fuzzy sets and its application to fuzzy rough sets based on fuzzy composition operauncertainty measures”, Inf. Sci., vol. 178, pp. tions,” B.-Y. Cao, C.-Y. Zhang, and T.-F. Li (Eds.): 3565-3576, 2008. Fuzzy Information and Engineering, ASC 54, pp. [23] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, pp. 647-659. Springer-Verlag, Berlin Heidelberg, 2009. 338-353, 1965. [8] B. Q. Hu and C. K. Kwong, “On type-2 fuzzy sets [24] L. A. Zadeh, “The concept of a linguistic variable and their t-norm operations,” Inf. Sci., vol. 255, pp. and its applications to approximate reasoning-I, II, 58-81, 2014. III,” Inf. Sci., vol. 8, pp.199-249, 1975; vol. 8, pp. [9] B. Q. Hu and C. Y. Wang, “On type-2 fuzzy rela301-357, 1975; vol. 9, pp. 43-80, 1975. tions and interval-valued type-2 fuzzy sets,” Fuzzy [25] Y.-R. Zeng, L. Wang, and J. He, “A novel approach Sets Syst., vol. 236, pp. 1-32, 2014. for evaluating control criticality of spare parts using [10] B. Q. Hu and H. Wong, “Generalized interfuzzy comprehensive evaluation and GRA,” Int. J. val-valued fuzzy rough sets based on interFuzzy Syst., vol. 14, no. 3, pp. 392-401, 2012. val-valued fuzzy logical operators,” Int. J. Fuzzy [26] H.-Y. Zhang, Y. Leung, and L. Zhou, “VariSyst., vol. 15, no. 4, pp. 381-391, 2013. able-precision-dominance-based rough set approach [11] J.-S. Mi, Y. Leung, H.-Y. Zhao, and T. Feng, “Gento interval-valued information systems,” Inf. Sci., eralized fuzzy rough sets determined by a triangular vol. 244, pp.75-91, 2013.
(4) These models can be extended to type-2 fuzzy sets and interval-valued type-2 fuzzy sets [8-9]. These extension models can be applied to the type-2 fuzzy information system and granular computing etc.
B. Qing and H. Wong: Generalized Interval-Valued Fuzzy Variable Precision Rough Sets
[27] H.-Y. Zhang, W.-X. Zhang, and W.-Z. Wu, “On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse,” Int. J. Approx. Reason, vol. 51, pp. 56-70, 2009. [28] S. Zhao, E. C. C. Tsang, and D. Chen, “The model of fuzzy variable precision rough sets,” IEEE Trans. on Fuzzy Syst., vol. 17, no. 2, pp. 451-467, 2009. [29] S. Zhao, E. C. C. Tsang, D. Chen, and X. Wang, “Building a rule-based classifier- A fuzzy-rough set approach,” IEEE Trans. on Knowl. Data Eng., vol. 22, no. 5, pp. 624-638, 2010. [30] W. Ziarko, “Variable precision rough set model,” J. Comput. Syst. Sci., vol. 46, no. 1, pp. 39-59, 1993. [31] W. Ziarko, “Probabilistic decision tables in the variable precision rough set model,” Comput. Intell., vol. 17, pp. 593-603, 2001. [32] W. Ziarko, “Probabilistic approach to rough sets,” Int. J. Approx. Reason, vol. 49, pp. 272-284, 2008. Bao Qing Hu is a Professor at the School of Mathematics and Statistics, Wuhan University (China). He obtained his BSc, MSc, and PhD from Wuhan University. His current research interests include Fuzzy Mathematics, Rough Set Theory and Soft Computing.
Heung Wong is a Professor at the Department of Applied Mathematics, The Hong Kong Polytechnic University. He obtained his BSc (Mathematics), MSc (Statistics), and PhD (Statistics) from The Chinese University of Hong Kong, Newcastle University (UK) and The University of Hong Kong respectively. His current research interests include time series analysis, environmental statistics and stochastic hydrology.
565
