- Email: [email protected]
On the normalization of fuzzy belief structures
NORTH
- HOILAND
On the Normalization of Fuzzy Belief Structures Ronald R. Yager Machine Intelligence Institute, lona College, N e w Rochelle, N e w York
ABSTRACT The issue of normalization in the fuzzy Dempster-Shafer theory o f evidence is investigated. We suggest a normalization procedure called smooth normalization. It is shown that this procedure is a generalization of the usual Dempster normalization procedure. We also show that the usual process o f normalizing an individual subnormal fuzzy subset by proportionally increasing the membership grades until the maximum membership grade is one is a special case of this smooth normalization process and in turn closely related to the Dempster normalization process. We look an alternative normalization process in the fuzzy Dempster-Shafer environment based on adding to the membership grade of subnormal focal elements the amount by which the fuzzy subset is subnormal.
K E Y W O R D S : fuzzy sets, theory o f evidence, normalization, subnormal,
Dempster's rule 1. I N T R O D U C T I O N The Dempster-Shafer theory of evidence [1, 2] is one of the tools used to model and manipulate uncertain information. The basic representational structure in this theory is a belief structure which consists of a collection of subsets, called focal elements, each having an associated nonnegative weight, the total of which must sum to one. An issue of considerable interest in this field is the problem of normalization: of dealing with nonzero weights that may be assigned to an empty set as a result of the combination of multiple belief structures. The original procedure, as suggested by Sharer [1], is to use Dempster's rule to normalize such a belief structure by reallocating any weight assigned to a null set to nonnull focal elements in a manner proportional to the weights already assigned to those elements. A number of authors have suggested alternative proceAddress correspondence to Professor Ronald Yager, lona CoLlege, 715 North Avenue, New Rochelle, NY 10801-1890. Received October 1993; accepted March 1995.
International Journal of Approximate Reasoning 1996; 14:127 153 © 1996 Elsevier Science Inc. 0888-613X/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0888-613X(96)00092-5
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dures for this normalization [3]. Other authors [4-8] have questioned the process of normalization, indicating that one should leave the weight with the null set, and thus allow null focal elements. Zadeh [4, 5], for example, has shown that normalization can lead to aggregated beliefs in which a possibility very weakly supported by all constituents in the aggregation can end up with very high support. Furthermore, he suggested that normalization washes away a lot of information in a given situation, especially regarding conflicts between pieces of evidence, which would be apparent if no normalization were used. On the other hand, not imposing normalization may lead to technical difficulties with this technology, such as situations in which an upper bounding value may end up being less than a lower bounding value. The issues behind this controversy are very deep and involve subtle questions on the important topic of conflict resolution. It is not our purpose here to enter into this debate, one in which we feel both sides have merit, but rather to accept normalization and investigate its implementation in the fuzzy domain. Thus in this work we are concerned with the issue of normalization in the theory of evidence in the case in which the focal elements are fuzzy sets. We recall that in the crisp environment normalization is used in situations in which one of the focal elements is the null set, whereas in the fuzzy environment normalization is required when a focal element is subnormal (has maximal membership grade less than one). We suggest a normalization procedure, which can be used in the fuzzy environment, called smooth normalization. We show, using the fact that fuzzy subsets can be expressed as consonant belief structures [9], that this new procedure is a generalization of the D e m p s t e r normalization procedure [10, 11]. We also show that the usual process of normalizing a subnormal fuzzy subset by proportionally increasing the membership grades until the maximum membership grade is one is a special case of this smooth normalization and in turn closely related to the D e m p s t e r normalization. We then consider an alternative normalization procedure in the D e m p ster-Shafer environment suggested by Yager [12], in which normalization is accomplished by reallocating the mass assigned to a null focal element to the universe of discourse. We extend this to the fuzzy Dempster-Shafer environment. We also show that in the case of an individual subnormal fuzzy set it is related to a normalization procedure where we add to each membership grade the amount by which the fuzzy subset is subnormal.
2. NORMALIZATION OF BELIEF STRUCTURES
A Dempster-Shafer belief structure defined on the finitc set X is a mapping [1, 2, 13] m : 2 X ~ [0,1]
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129
such that r e ( O ) = O,
(1)
Y'~ r n ( A ) = 1.
(2)
AcX
T h e subsets of X for which m ( A ) 4 : 0 are called the focal elements of m. W e shall d e n o t e these a s Ai, i = 1 . . . . , n. A n essential feature of the belief structure as implied by (1) is that the null set is not a focal element. Two important measures associated with these structures are the plausibility measure P1 and the belief measure Bel. A s s u m e m is a belief structure and let B be any subset of X. T h e n PI(B) =
~
m(A,),
AiNB~-~
BeI(B) =
~
rn(Ai).
AicB
A n equivalent definition of plausibility and belief can be obtained using the ideas of possibility and certainty introduced by Z a d e h [14]. A s s u m e A and B are two subsets of X. W e define Poss(BIA) = Max~[D(x)], where
D=ANB and Cert[B[A] = 1 - Poss(BIA). It is easy to show that P I ( B ) = ~ P o s s ( B [ A i) rn( A i ) , i
B e l ( B ) = ~ C e r t ( B I A i) r n ( A i ) . i
Thus the plausibility is the expected possibility, and the belief is the expected certainty. A m a p p i n g m* is called a pseudo belief structure if we withdraw the condition (1). Every belief structure is also a p s e u d o belief structure. Pseudo belief structures can have some undesirable properties. Consider a p s e u d o belief structure m*, and assume that m * ( O ) = a > 0. In this case
PI(X) = 1 -a
v~ 1.
Thus the u p p e r probability of the whole space is not 1.
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A s e c o n d p r o b l e m that can arise if the weight a s s o c i a t e d with the null set is not z e r o is that the b e l i e f can be larger than the plausibility. Let m* be such that m(O)
= .6,
m(A)
= .4.
L e t B c A. T h e n P I ( B ) - .6, B e I ( B ) = 1. P s e u d o b e l i e f structures can arise when we a g g r e g a t e b e l i e f structures. L e t ,, be any set o p e r a t i o n . A s s u m e m 1 and m 2 are two b e l i e f s t r u c t u r e s with focal e l e m e n t s A i a n d Bj. W e define the structure m* = m l zx m 2 as the m a p p i n g m*:2 x-o
[0,1]
such that m*(D)
=
~
ml(Ai)m2(Bj).
M i, Bj D =AiAB s
E x a m p l e s o f ~ a r e union a n d intersection. W e shall say that ~ is a n o n - n u l l - p r o d u c i n g o p e r a t i o n if for all A i 4= O and Bj 4= O it is the case that Aiz~Bj
4=O.
W e see that union is a n o n - n u l l - p r o d u c i n g o p e r a t i o n , while i n t e r s e c t i o n can p r o d u c e a null. THEOREM
If
~x is" a n o n - n u l l - p r o d u c i n g
operation,
m* = m 1 ~ m 2 is a l w a y s a b e l i e f s t r u c t u r e i f m I a n d m 2 a r e b e l i e f s t r u c t u r e s .
P r o o f I f ~ is n o n - n u l l - p r o d u c i n g , t h e n A ~ B since m j ( Q ) = m 2 ( O ) = 0, we get m * ( D ) = O.
= OiffA
=B
=O,
and
O n the o t h e r hand, if ~ is not n o n - n u l l - p r o d u c i n g , t h e n it is possible for m* to be a p s e u d o b e l i e f structure. In p a r t i c u l a r , the i n t e r s e c t i o n o f b e l i e f s t r u c t u r e s can result in a p s e u d o b e l i e f s t r u c t u r e if t h e r e exist focal e l e m e n t s A i a n d Bj such that A i ~ B j = 0 . In o r d e r to p r o v i d e an a g g r e g a t i o n p r o c e s s which e n a b l e s even null-producing o p e r a t i o n s to result in b e l i e f structures. D e m p s t e r [10, 11] introd u c e d a p r o c e d u r e of n o r m a l i z a t i o n .
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DEFINITION ( D e m p s t e r N o r m a l i z a t i o n P r o c e d u r e ) Assume m* is a pseudo belief structure. Then we can conwert m* into a belief structure m as follows: m : 2 x ~ [0,1] where 1. m(@) = 0;
2. for all A --/: Q, m*(A) m(A)-
1 -
m*(@)"
W e n o t e that if m* is a belief structure t h e n m* = m. If ,, is the i n t e r s e c t i o n o p e r a t i o n a n d if we use the above n o r m a l i z a t i o n p r o c e d u r e , we o b t a i n the D e m p s t e r rule of aggregation [10, 11]. T h e D e m p s t e r n o r m a l i z a t i o n p r o c e d u r e ( D N P ) has a very specific effect o n the m e a s u r e s of belief a n d plausibility. A s s u m e m* is a p s e u d o belief structure. Let P I * ( B ) a n d B e l * ( B ) b e the plausibility a n d belief associated with some n o n n u l l subset B. Let m be o b t a i n e d from m* using the D N P . Then PI(B) =
m(Ai) =
~ i
i
Aif~Bf:D
>
~
m * ( A i) 1 - m*(O)
~ AiNB4-O
m * ( A i) > P I * ( B )
i
Ai~B#:Q In p a r t i c u l a r we see that Pl(B) -
PI* ( B ) - - , q
where q = Y~A,~@m*(Ai). Let E be the set of subsets of X that are c o n t a i n e d in B, excluding the null set; thus
m * ( A i) Bel(B)= BeI*(B) =
~ m(Ai)= AicE
~
1-m*(Q3)'
AidE
Y'. m * ( A i) + m*(Q3). AiEE
Let K ~A~EEm*(Ai ), a n d let d = 1 - K - m*(@). W e n o t e that 0 < d < 1. T h e n =
K BeI(B) -
1 -m*(£5)
1 - m* ( Q ) - d -
1 -m*(@)
B e l * ( B ) = K + m * ( @ ) = 1 - d.
- 1
d
1-m*(•)'
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Since d <_ d / [ 1 - m*(O)], then B e I * ( B ) > Bel(B). T h u s the D N P essentially i n c r e a s e s the plausibility a n d d e c r e a s e s the belief. Since d BeI(B) = 1 1 -
rn* (0)
'
then 1 - m*(O) + d
Bel*(B) - m*(O)
Bel(B) = 1 -
m*(O)
1 -
m*(O)
T h e p r o c e s s o f n o r m a l i z a t i o n i n t r o d u c e d by D e m p s t e r can b e seen as a kind of c o n d i t i o n i n g o p e r a t i o n . In o r d e r to a p p r e c i a t e this fact, we m u s t p r o v i d e an a l t e r n a t i v e view of the p s e u d o b e l i e f structure. Since any b e l i e f s t r u c t u r e is also a p s e u d o b e l i e f structure, we shall also be p r o v i d i n g an a l t e r n a t i v e view o f t h e b e l i e f structure. This a l t e r n a t i v e view is the o n e used by D e m p s t e r in his original w o r k [10, 11]. C o n s i d e r a p s e u d o b e l i e f s t r u c t u r e m* on X with m focal e l e m e n t s , A 1 , A 2 . . . . . A,,. N o w c o n s i d e r a n o t h e r structure, which we shall call a Dernpster structure, consisting o f a set Y = {Yt . . . . . Yn}, a p r o b a b i l i t y distrib u t i o n P* on Y, a n d a r e l a t i o n R on Y X X. H e r e R is d e f i n e d such that (Yi, X) ~ R if x ~ A i, and P* is d e f i n e d such that P * ( Y i ) = m * ( A i ) . A s shown by D e m p s t e r , this new s t r u c t u r e is the s a m e as rn* in the sense that for any subset B o f X it is the case that the u p p e r b o u n d on the p r o b a b i l i t y o f B is P I ( B ) a n d the lower p r o b a b i l i t y o f B is BeI(B). Essentially, in this new s t r u c t u r e we have a r a n d o m e x p e r i m e n t on the space Y such that if the o u t c o m e is the e l e m e n t Yi, then we o b t a i n as the o u t c o m e the subset A i of the space X. This r a n d o m set a p p r o a c h has b e e n extensively s t u d i e d by G o o d m a n a n d N g u y e n [15]. T h u s o n e possible s e m a n t i c s that can be a s s o c i a t e d with a b e l i e f structure is a p r o b a b i l i s t i c s e m a n t i c s b a s e d on the i d e a o f r a n d o m sets. In this s e m a n t i c s the focal e l e m e n t s a r e v i e w e d as o u t c o m e s of an e x p e r i m e n t , that is, the o u t c o m e of the e x p e r i m e n t is a set f r o m X r a t h e r t h a n an e l e m e n t from X, a n d the weights a r e viewed as the p r o b a b i l i t i e s a s s o c i a t e d with these sets. In this view the m e a s u r e s o f b e l i e f a n d plausibility of a set B a r e closely r e l a t e d to the i d e a s o f lower and u p p e r p r o b a b i l i t y o f B [10]. In p a r t i c u l a r B e I ( B ) N P r o b ( B ) _< P I ( B ) . W e now i n t r o d u c e the c o n d i t i o n i n g view o f the n o r m a l i z a t i o n process. W i t h o u t loss of generality, a s s u m e A I - 0 . W e shall now c o n d i t i o n the
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probability distribution P* on the set G = {Y2 . . . . . Y,} = n°t{Yl}. We emphasize that yl is the outcome in the space Y associated with the null focal element Aj. Using this conditioning, we obtain the conditional probability
P(yiln°t{Yl}) = P(Yi[G) =
P*({yi} c3 G)
P*(G)
For i =g 1 we get
P(yi[G)
P*(Yi) 1 - P*(Yl) '
and for i = 1 we get P*(O) O.
P(y~[G)- 1 - P * ( y ~ )
This conditioned probability distribution consists of the new weights in the DNP:
m(Ai ) --
P*(Yi) 1 -
P*(Yl)
m*(Ai) 1 -
m*(A
1) "
Thus, we see that the D N P can be viewed as a conditioning on the original probability distribution P, where the conditioning is obtained by assuming that the outcome must occur in the set G. Since the set G corresponds to elements in Y which have nonnull focal elements in X, we have essentially enforced a condition in which we have a belief structure.
3. NORMALIZATION IN THE FUZZY E N V I R O N M E N T
In this section we consider the process of normalization in the case where the focal elements are fuzzy subsets. We first introduce some concepts from the theory of fuzzy sets. Let A be a fuzzy subset of some space X. We shall let H e i g h t ( A ) = maxxA(x); thus Height(A) is the maximum membership grade of A. If Height(A) = 1, we say that A is normal, and if Height(A) ~ 1, we say A is subnormal. We next recall that the c~-level sets of the fuzzy subset A are crisp subsets of X defined such that
A~ : { x l A ( x ) >_ ~}.
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It is also well established that if c~ > c~' then A~ ~ A s, and that A ( x ) = a * , where ~* is the largest ~ such that x e A~. F u r t h e r m o r e if H e i g h t ( A ) = /3 :g 1, then A~=Q
for
~>/3.
A s s u m e A is a subnormal fuzzy subset of X with H e i g h t ( A ) = K. In some applications of fuzzy set theory there is a need to convert A into a fuzzy subset with height of one. This process is also called normalization: it converts a subnormal fuzzy set into a normal one. The usual normalization [16, 17] process for converting A into a normal fuzzy subset B is to let 1
B(x) = -~A(x). O n e can view this normalization process in terms of level sets. A s s u m e A is a subnormal fuzzy subset with H e i g h t ( A ) = K. T h e n its level sets are A s. If we consider a new fuzzy subset B whose level sets are B~ where B~ = A a K
for
a ~ [0, 1],
then 1
B(x) =~A(x). We now introduce the idea of a fuzzy belief structure and the related idea of a p s e u d o fuzzy belief structure where the focal elements are fuzzy subsets. A f u z z y belief structure m defined on the finite set X is a m a p p i n g from fuzzy subsets of X into the unit interval, m:IX
~ [0,1]
such that 1. F o r A subnormal, r e ( A ) = O. 2. ~ m ( A ) = 1 The fuzzy subsets A 1. . . . . A n, for which m ( A i) 4 : 0 are again called the focal elements. We call a m a p p i n g m* a pseudo f u z z y belief structure if m* is a m a p p i n g m* : I x --* [0, 1]
such that ~_~rn*(A) = 1.
Thus a p s e u d o structure doesn't require the focal elements to be normal.
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T h e measures of plausibility and belief can be easily extended to this environment: P I ( B ) = ~_,Poss[BIA i] m ( A i ) , i B e l ( B ) = Y'~ Cert[B[ A i] m ( A i ) . i
As discussed by Y a g e r [18], if ~ is any fuzzy set operation, we can extend it to work on fuzzy belief structures. For example, if • is the intersection operation, then if m 1 and m 2 are two fuzzy belief structures with focal elements A i and Bj respectively, we can obtain m I zx m 2 :
m,
where m has focal elements FK =Ai~B
j
with weights
m ( FK ) = m ( A i ) m ( Bj). In some cases these aggregations o f fuzzy belief structures may lead to pseudo fuzzy belief structures. In the following we shall suggest a procedure, called the smooth normalization procedure (SNP), for converting p s e u d o fuzzy belief structures into fuzzy belief structures. A s s u m e m* is a p s e u d o fuzzy belief structure with focal elements F i and associated weights m*(Fi). Let H e i g h t ( F i) = h i. T h e following algorithm converts m* into a fuzzy belief structure in a m a n n e r consistent with the D e m p s t e r normalization procedure. SMOOTH
NORMALIZATION
PROCEDURE
1. Calculate u i = m * ( F i ) h i. 2. Calculate T = ~,im*(Fi)(1 - h i) = 1 - Y[im*(Fi)hi . 3. Calculate ui
v,-l_ T 4. Introduce new fuzzy subsets E i with membership grades Fi(x) Ei(x ) -
hi
We call the fuzzy belief structure m with focal elements E i and associated weights m ( E i) = Vii the smooth normalized belief structure.
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First let us assure ourselves that m is a fuzzy belief structure. It has as focal e l e m e n t s E i, since 1 hi H e i g h t ( E i ) = ~ H e i g h t ( F i ) = h i = 1, the E i
are
n o r m a l . Next, y,m(Ei ) = ~ i
i
ui 1-T
_ __1 1 -T
~m*(Fi)hi
= 1.
W e see that m*(F,)h i Vii = y~)~=l m , ( F j ) h j . Let us investigate some p r o p e r t i e s associated with this t r a n s f o r m a t i o n . T h e first p r o p e r t y assures us that if the original structure is a fuzzy belief structure, the s m o o t h t r a n s f o r m a t i o n has n o effect. THEOREM
I f m is a f u z z y belief structure, then m * = m .
P r o o f If m* is a fuzzy belief structure then h i = 1 for all i. In this case E i = F i. F u r t h e r m o r e , u i = m * ( F i) a n d T = 0, a n d h e n c e Vi = m * ( F i ) . T h e next t h e o r e m shows that this p r o c e d u r e is i n d e e d a g e n e r a l i z a t i o n of the DNP. THEOREM I f the F i in m* are crisp subsets, then m is the s a m e as that obtained by the D e m p s t e r normalization procedure. Proof F o r i = 1 to k let F,. b e n o n n u l l , h i = 1. F o r i = k + 1 t o n let F i be null, h i = 0. F o r i = 1 to k we have u i = m * ( ~ ) , a n d for the o t h e r i's, u i = 0. W e see that T = Y?~ m * ( F i ) . F o r i = 1 to k, i~k=l
v~=
m * ( F i) 1 - E7 k+ j m ( F i ) '
a n d for i = k
+ 1 to n, Vi = 0 . F o r i =
1 t o k, E i = F i.
THEOREM I f h i is the s a m e f o r all i, h i = c~, then f o r all i, El(X) = ( 1 / c ~ ) F / ( x ) a n d m ( E i) = m * ( F i ) . P r o o f F o r all i, m * ( Fi)o~/c~ = m * ( F i ) .
u i = m*(F,)=
c~ a n d
T = 1-
c~; thus
~--
It should be n o t e d that the SNP can also be applied to a fuzzy subset a n d leads to the usual n o r m a l i z a t i o n . A s s u m e F is a fuzzy subset of X
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137
with H e i g h t ( F ) = c~. W e can r e p r e s e n t this as a structure m* with o n e focal e l e m e n t F having weight 1. If we apply o u r SNP to this, we o b t a i n u 1 = 1 a n d V l = 1, a n d E ( x ) = ( 1 / a ) F ( x ) . T h u s the s m o o t h n o r m a l i z a tion leads to the usual n o r m a l i z a t i o n used in fuzzy subset theory.
4. C O N S O N A N T BELIEF STRUCTURES T h e r e exists a special class of belief structures called c o n s o n a n t belief structures [1]. A belief structure is called c o n s o n a n t if its focal e l e m e n t s are nested. T h u s a belief structure m is a c o n s o n a n t belief structure if its focal e l e m e n t s are crisp subsets that can be indexed so that A 1 D A 2 D A 3 D ..- D A n . As shown by Sharer [1], the c o n s o n a n t belief structure has th'e following properties: 1. BeI(A C~ B) = m i n [ B e l ( A ) , Bel(B)], 2. P l ( A U B) = m a x [ P l ( A ) , P l ( B ) ] , 3. P I ( A ) = m a x , ~ A Pi({x}). As shown by D u b o i s a n d P r a d e [9], for any x ~ X Pl(x) =
~
rn(Ai).
i xEA
i
F u r t h e r m o r e , they have shown that any c o n s o n a n t belief structure m can be associated with a fuzzy subset F where F(x)
= Pl(x).
Conversely, we can r e p r e s e n t any n o r m a l fuzzy subset F as a c o n s o n a n t belief structure m. A s s u m e F is a fuzzy subset defined o n X = {x, . . . . . x,,}. A s s u m e the e l e m e n t s of X are indexed so that F ( x i ) > F ( x i) if i > j . T h e n we o b t a i n the focal e l e m e n t of m as follows:
A,, = ( x , ) , a,,-i
= {xn,Xn
A i = {x . . . . . . Aj = X ,
m(A,,)
I},
xi},
m(An_l) m(Ai)
= F ( x n)
-
= F(Xn_l) = f(xi)
F(Xn_l) ,
- F(Xn_2) ,
- F(xi_l) ,
m ( A i) = F ( x ] ) .
W e can extend this idea to r e p r e s e n t s u b n o r m a l fuzzy subsets if we can allow c o n s o n a n t p s e u d o belief structures. C o n s i d e r the c o n s o n a n t p s e u d o belief structure m* where A 1 D A 2 D A 3 D ... D A n D A n + 1
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Ronald R. Yager
with An+ 1 = O and rn(A,+ 1) = 1 - a. This leads to a subnormal fuzzy subset F with Height(F) = a and
F(x) = ~
m*(Ai).
i xEA~
Consider now the application of the D N P to this pseudo c o n s o n a n t belief structure. W e obtain a new consonant belief structure with focal elements A 1 DA
2
D A 3 D ... A,,
with weights
m(Ai) =
m * ( A i)
m * ( A i)
m * ( A i)
1 - r n * ( A , + 1)
1 - (1 - a )
a
This can be associated with a fuzzy subset E, where 1 E(x)
=
E m(Ai)=i Ax~A ~
E OL x~A,
1
m*(Ai)=--F(x). Og
T h e classical normalization used in fuzzy subset theory can be viewed as a D N P normalization process applied to the associated consonant belief structure. Thus the normalization is obtained by a conditioning of the focal elements in the underlying consonant structure by eliminating any outc o m e which gives a null focal element. Thus, saying that a fuzzy subset must have an element with m e m b e r s h i p grade one means saying no null focal elements are allowed. Let us now consider a p s e u d o fuzzy belief structure m* with focal elements F 1. . . . . F n and associated weights m*(Fi). E a c h of these fuzzy focal elements can be represented in turn by a consonant p s e u d o belief structure m* with crisp focal elements ~ j , j = 1 . . . . . tTi, with weights m*(F,j). W i t h o u t loss of generality we consider Fi, ' = O for all rn*. If Fi is normal, then rn*(Fi, ,) = 0; otherwise it is nonzero. Using the D e m p s t e r view, we can consider the pseudo fuzzy belief structure as equivalent to the p e r f o r m a n c e of a c o m p o u n d (two part) i n d e p e n d e n t experiment. In the first experiment, guided by the probability distribution Pi = m*(F,), we decide which second experiment to p e r f o r m (see Figure 1). T h e second experiments are p e r f o r m e d using the appropriate consonant representation of the focal element F i. Thus second experiment i has as
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139
PI = m1(Fl)
Second Exp - I
= m*(Fi)
Second Exp - i
~
Pn
=
m
Second Exp - n
Figure 1. Pseudo fuzzy belief structure viewed as a compound experiment.
its outcomes the focal elements of the consonant belief structure {Fil, F, 2. . . . . Fi, ,} representing F i, and the probabilities are rn*(Fij) (see Figure 2). Using this compound model, the probability p~ of any set Fq is
p*j = m * ( F i ) m * ( F q ) . We can now view this compound shown in Figure 3. Using the DNP on this compound ing by eliminating all Fij = ~ , that conditioned probability Pq for each
Pij =
experiment as a single experiment as structure (which consists in conditionis, eliminating all Fi. ), we get a new nonnull Fq, where
P*({Fij) A B ) P*(B) '
where B is the set of nonnull focal elements. Hence
P* (Fq) Pij
-
1 -
P*(B)
Pij --
1 -
Y~p,*.,
,
m* ( Fi)m* (Fij) Pij = 1 - S,i=,m" * ( F i ) m * ( F i , ) " We can now view this probability distribution on the collection of nonnull focal elements as being generated by some fuzzy belief structure
~
F| 1
Fij Fin]
Figure 2. Second experiment i.
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Ronald R. Yager
F~j
Figure 3. Compound experiment.
m. We d e n o t e the focal elements of this structure a s i~ 1. . . . . /~n with weights m ( ~ ) . F u r t h e r m o r e , each fii can be seen as a normal fuzzy subset generated by a c o n s o n a n t belief structure m i which has focal element {FiI,F+2 . . . . .
and weights m i ( F i j ) . given by
F i.... }
In this structure each of the F/j has probability fiij Pij = m ( l ~ i ) m i ( F i j ).
For this structure to be the same as the one obtained by the conditioning, we must have Pij = Pij ;
thus for all ij
m*(F~)mT(Cj) = 1 _ ~ v'ni - , m * ( f,) m *i ( F i n ) "
m(l~i)mi(Fij)
W e must solve this for m ( ~ ) and m i ( F i ) ,
where
re(P,/= 1, i t/i
1 i
Y'~ m,(Fij)
1.
j-1 T h e solution to this is obtained as follows. Let
roT(F,) rns(Fij)
= 1 - r o T ( F i n ,) '
we recall that 1 - r n * ( ~ , ~ ) = hi, the height of F i. F u r t h e r m o r e , we see that n, ~ mi(Fij)
j=l
1 -
1 - m*(Fi,,)
n, , E mi(Fij) j==1
1 - rn* ( F~,,,) -
1
m~[ ( Fin )
-
1.
Normalization of Fuzzy Belief Structures
141
Thus m~
mi't~j','i,
=
"
*(Sj) hi
Using this, we get /3ij = m ( ffi )
m*(Fij) hi
hence /~ij = Pij then m*(Fij) --
, ^,
m[~/ \
/
1-
hi
m*(Fi)mi(Fij) n m * ( F i ) m i ( F i,n) ' Y~i=l
therefore h i m * ( F i) m(Fi)
=
1-
Z { ~' =
i m*
(Fi)m*(ei,,)
Since m*(Fi,,,) = 1 - h i, we get h i m * ( F i)
{E,~ =
1 -
him*(F i)
.-,="n l m * ( F , ) ( 1
-
hi)
1 - ,-.i--v'" lm*¢F~, i, + E'i'=Lm*(Fi)hi
Since
~_,m*(F i) = l, i=1
we get m(fii) =
him*(Fi) E~,=lhim,(Fi) "
Furthermore, we note that fii is associated with the consonant belief structure m i with focal elements {FiL, Fi2 . . . . . Fi,,, ~} whose weights are
mf(Fu) mi(Fij )
hi
The associated fuzzy subset /?i is seen to have membership function 1
1 J
xcFij
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Ronald R. Yager
Thus the resulting fuzzy belief structure is exactly the same as that obtained using the smooth normalization method. In the preceeding construction we have shown that the SNP is effectively the D N P when viewed in the appropriate way.
5. N O R M A L I Z A T I O N A N D SPECIFICITY
The concept of the measure of specificity of a fuzzy subset was introduced by Yager [19] as an indication of the degree to which the fuzzy subset contains one and only one element. Assume F is a fuzzy subset of X. In [20] Yager introduced a linear measure of specificity defined as Sp(F) =b i-
~ wjbj, j-2
where bj is the jth largest membership grade in F, and tlie w i are a collection of weights such that w i ~ [0, 1]
~wj= j 2
1, if
w i > wj
i < j.
It can be easily shown that for any F, S p ( F ) ~ [0, 1]. We also see that Sp(F) = 1 iff F is a singleton fuzzy subset. One semantics that can be associated with the measure of specificity of a set F regards the selection of an element from the set F. In particular, as discussed in [19], the bigger the specificity of a set, the less difficult it is to choose an element from the set as F as the optimal member. Consider now the process of normalizing a fuzzy subset. Starting with the subnormal fuzzy subset F, we normalize it to obtain the fuzzy subset E, 1 E(x)
= --F(x),
where H e i g h t ( F ) = a. If we calculate
Sp(E) = d l -
Y'. wjdj, j- 2
Normalization of Fuzzy Belief Structures
143
where dj is the jth largest of the E(x), then, since d i = (1/ct)bi, we get
1
S p ( E ) = - - b a - ~_,wj---dbj = O~
-
J
= ~- S p ( F ) . ./
Thus the process of normalization changes the specificity by a factor of 1 / a . Thus normalization increases the specificity when a < 1. Consider now a pseudo fuzzy belief structure m* with focal elements A~ . . . . . A n. In [21] Yager suggested a measure of specificity associated with this structure as Sp(rn* ) = ~ S p ( A
i)
m*(Ai).
i
Consider now the fuzzy belief structure m obtained from m* by the SNP. In this case rn has focal elements Bg, where
Bi(x ) = --Ai(x), hi and weights
m*(Ai)h i
m ( B i)
Em* ( A i ) h i '
where h i = Height(Ai). If we now calculate Sp(m) = Y'~Sp(A i) m ( A i ) , l
we get
1
m*(Ai)h i
Sp(m) = ~ ~i Sp(Ai) S . m , ( A i ) h i
~,m* ( A i ) h i
Sp(m* ).
Thus, we see that the specificity is again increased, this time by a factor ~ i m * ( A i ) h i . We can view ~ i m * ( A i ) h i as the average height of the focal elements. Let us try to provide some intuition for these results. In order to do this we must first provide some insight into what we are measuring with Sp. A belief structure can be viewed as having two sources of uncertainty, one being related to the determination of the focal element and the other to the selection of the element within the selected focal element. A very
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Ronald R. Yager
i m p o r t a n t class o f b e l i e f structures a r e o n e s in which the focal e l e m e n t s are just sets with o n e e l e m e n t ; these a r e called Bayesian b e l i e f structures. W e n o t e t h a t in Bayesian b e l i e f s t r u c t u r e s t h e r e exists no u n c e r t a i n t y r e g a r d i n g the selection o f an e l e m e n t f r o m a d e t e r m i n e d focal set. F u r t h e r m o r e , we n o t e that for a focal set with o n e e l e m e n t the specificity is one. F r o m this we see that a B a y e s i a n b e l i e f s t r u c t u r e m has S p ( m ) = 1, which is an i n d i c a t i o n that t h e r e is no difficulty r e g a r d i n g the selection of an e l e m e n t f r o m a d e t e r m i n e d focal set. A s o u r result has shown, the s m o o t h n o r m a l i z a t i o n p r o c e s s t e n d s to increase the specificity. T h u s we can view the S N P as o n e which turns a p s e u d o b e l i e f s t r u c t u r e into a b e l i e f s t r u c t u r e in a m a n n e r that tries to m a k e it m o r e Bayesian.
6. SMOOTH NORMALIZATION, PLAUSIBILITY, AND BELIEF W e have previously shown that the D N P results in a w i d e n i n g of the p l a u s i b i l i t y - b e l i e f interval in the crisp d o m a i n . W e now show the s a m e effect occurs in the fuzzy d o m a i n w h e n we use the SNP. THEOREM Assume m* is a pseudo fuzzy belief structure. Let m be the associated fuzzy belief structure obtained by the smooth normalization process. Then for any fuzzy subset B PI*(B) < Pl(B)
and B e l * ( B ) _> B e l ( B ) . P r o o f A s s u m e A 1. . . . . A , a r e the focal e l e m e n t s o f m*. A s we s h o w e d in the p r e v i o u s section, each o f the A i c a n be v i e w e d as b e i n g a c o n s o n a n t p s e u d o b e l i e f s t r u c t u r e m* with focal e l e m e n t s A i l . . . . . Ain ,. F r o m this p e r s p e c t i v e we can view m* as a p s e u d o crisp b e l i e f s t r u c t u r e with focal element
M = {AI~, Aj2
. . . . .
Ani,},
w h e r e t h e weight a s s o c i a t e d with Aij is m * ( A i ) m * ( A i j ) = q(Aij). In this case ~ q*(Aij). 0 AiycIBT-Q
el* ( B ) =
L e t t i n g D be the subset of focal e l e m e n t s for which A i j :~ B =~ Q~, we have Pl*(B)
=
~] AijG D
W e n o t e t h a t o n e Aij = 0 is not in D.
q*(Aij).
Normalization of Fuzzy Belief Structures
145
A f t e r doing the normalization, we obtain a belief structure with focal elements M' =M-
U,
where U is the set of elements in M that are null sets. F u r t h e r m o r e , the weights assigned to an element Aij in M ' is q(Aij) where
q*(Aij) q(Aij) = EAij ~ M'q*(Aij) " Hence PI(B) = Since
q(Aij) >_q*(Aij),
~ q(Aij). AijcD
we get PI(B) > Pl*(B).
Let E be the subset of M ' consisting of all the subsets which contain B. Then
Let K =
Bel(B) =
Y'~ q(A~j), Aij~E
Bel*(B) =
~ q*(Aij) Ai: EE
+
~
~Ai~Eq*(Aij), q) = ~'A~ ~¢t*(Aij), K BeI(B) - 1-q~
q*(Aig).
Aij=G and d = 1 - K - q~. T h e n
1-d-q~ 1-q~
d 1
1-~o
and Bel*(B) =K+
q~= 1 - d .
Since d d < - then B e I * ( B ) > Bel(B). In proving the above t h e o r e m we have actually developed a precise relationship between P I * ( B ) and PI(B): PI* ( B ) PI(B) =
EA,, ~ M'q* (Aij)
'
146
Ronald R. Yager
since ~_.
q*(Aij)
q*(Aij) = 1 -
= 1 -
AoEM'
Ai]=U
~q*(Ai~,). i= 1
Furthermore, since
q*(Ain~) = m * ( A i ) ,
m*(Ai,, )
and mi(Ain ,) = 1 - h i , where h i is the height of A i, then
q*( Ain ) = m* ( A i ) - h i m * ( A i ) ' and therefore
~ q * ( A i n ~) = ~ m * ( A i=1
i)-
Y'~him*(A i) = 1 - ~_,him*(Ai).
i=1
Thus PI*(B) PI(B)
=
Eni
1h i m
* '£A i3""
In a similar manner we can show that BeI(B) =
BeI*(B) + E h i m * ( A ~) - 1
1 - BeI*(B) 1
E~z= i h i m * ( A i )
Y~him*(A i )
The basic result of the theorem provided in this section is that the process of smooth normalization essentially results in a decrease in belief and an increase in plausibility.
7. N O R M A L I Z A T I O N
AND DEFUZZIFICATION
The process of defuzzification plays an important role in the applications of fuzzy logic controllers [22]. This operation allows us to select from a fuzzy subset of the real line a typical element. Assume F is a fuzzy subset of the real line. The center of area (COA) defuzzification technique selects as a prototypical element EiF(xi)Y
i
EiF(xi) Another often used technique is called the mean of maximal (MOM) method. Let G be the subset of elements for which F ( x ) attains its
Normalization of Fuzzy Belief Structures
147
maximal value. Let g = Card(G). Then the M O M defuzzified value is 1
Exi.
g icG
In [23] Filev and Yager unified these approaches using the B A D D transformation. The B A D D defuzzified value Y is obtained as
~" = F-,[F(xi)]'~xi F~[F(xi)] ~ ' where a ~ [1, oc]. We note that if a = 1 we obtain the C O A method, and if a = zc we obtain the M O M method. Let F be a fuzzy subset such that Height(F) = h. Let E be the fuzzy subset obtained by the normalization E ( x ) = ( 1 / h ) F ( x ) of F. Consider the B A D D defuzzification of E:
~'E = E i [ E ( x i ) ] = x i F_,iE(xi) ~
= Ei[(1/h)F(xi)]~xi ~,i[(1/n)F(xi)] ~
-- ~-'i[F(xi)]~xi
--YF"
~_,i[f(xi)]axi
Thus the defuzzified value of E and F are the same, and hence the normalization does not affect the defuzzified value under any B A D D defuzzification. Consider now a fuzzy belief structure m with n focal elements Aj. In [24] Yager and Filev suggest an approach to defuzzifying this kind of structure. Let 33j be the defuzzified value of Aj using any B A D D defuzzification. Then the defuzzified value of m is
fP =
~
f~jm(Aj).
j=l
Consider now a pseudo fuzzy belief structure m* with focal elements Aj and where hj = max x A j ( x ) . The defuzzified value of this structure is
y* = ~ ~[m*(Aj), j=l
where 13~ is the defuzzified value of As. Let m be the fuzzy belief structure obtained from m* by the smooth normalization procedure. The focal elements of m are Bj, where B j ( x ) =
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Ronald R. Yager
(1/hj)Aj(x), and the weights are
m*(Bj)hj rn( Bj) = Ens=lm* ( Bj)hi " Consider now the defuzzified value 1) of m,
~" : ~ 5m(Bj), j=l
where )3j are the defuzzified values of the Aj. Since then, as we have already shown, )3j = )3f and thus
Bj(x) = (1/hj)Aj(x),
tl
j- lm*( Aj)hi
j=t
Thus in general )3 ~ )3*. Consider next the special case w h e n all the hj are the s a m e , hj = K. All the focal elements have the same height in this case,
)3-
Zy*m*(Aj) Em* ( A j)
However, since 32~' lm*(Aj) : 1, we get
)3 = Ey[m*(Aj) =)3". Thus in this special case where all focal elements have the same height, the normalization process does not affect the defuzzification process for any B A D D defuzzification. In [24] Yager and Filev have investigated the use of fuzzy belief structures to represent the consequent of rules in fuzzy systems models. They have also shown that the output of such systems are pseudo fuzzy belief structures in which the heights of all the focal elements are the same. Thus in this environment the defuzzification process is unaffected by any normalization.
8. A L T E R N A T I V E N O R M A L I Z A T I O N
PROCEDURES
In [12] Yager suggested an alternative approach to the normalization of a pseudo belief structure. Assume rn* is a belief structure on X with focal elements A j , . . . , A s. Assume A n = Q and m*(A,,) = c~. Yager suggested that the normalization of m* be accomplished by transforming m* into a
Normalization of Fuzzy Belief Structures
149
b e l i e f s t r u c t u r e m with focal e l e m e n t s B~ . . . . . B n w h e r e B i=A
i
for
i=
1..... n-
1,
B n = X, a n d w h e r e for all i m(Ai)
= m*(Ai).
Thus, this m e t h o d essentially r e p l a c e s t h e null focal e l e m e n t by space X. L e t us investigate the use o f this t y p e o f n o r m a l i z a t i o n in e n v i r o n m e n t . A s s u m e F is a s u b n o r m a l fuzzy subset o f H e i g h t ( F ) = 1 - ~. C o n s i d e r now t h e r e p r e s e n t a t i o n o f F as a p s e u d o b e l i e f s t r u c t u r e m* with focal e l e m e n t s
the w h o l e the fuzzy X where consonant
F 1 D F 2 D F 3 D ... D F , . B e c a u s e F is s u b n o r m a l , t h e n F, = Q5 a n d m * ( F , ) ment
= c~. In this e n v i r o n -
FIx
F(x)
= Pl({x}) = ' ~ m * ( F / ) , i-I
w h e r e n x is the largest i such that x ~ F i. C o n s i d e r now t h e n o r m a l i z a t i o n o f m* using the t e c h n i q u e d e s c r i b e d above. In this case we get a b e l i e f s t r u c t u r e m with focal e l e m e n t s A0 . . . . . An m, w h e r e A i = F i f o r i = 1. . . . , n land A 0 =X, andwhere r n ( A i) = m * ( F i) for i = 1 . . . . . n - 1 a n d m ( A o) = c~. W e see t h a t m is still a c o n s o n a n t b e l i e f structure, since A o D A 1 2DA 2 D "" D A n _ l , a n d thus induces a fuzzy s u b s e t E. W e also see t h a t the m e m b e r s h i p function o f E is r/x
E(x)
r/x
= ~ _ , m ( A i) = m ( A
o) + ~ r n ( A
i=0
i)
i=1 i"/x
= m(Ao)
+ ~_, m * ( F i ) i=1
= ol + F ( x ) . Thus, t h e n o r m a l i z a t i o n a p p r o a c h s u g g e s t e d by Y a g e r in [12] c o r r e s p o n d s to the n o r m a l i z a t i o n o f the fuzzy s u b s e t F by a d d i n g c~, the a m o u n t o f which t h e e l e m e n t with the m a x i m a l m e m b e r s h i p g r a d e d e v i a t e s f r o m one, to all the m e m b e r s h i p grades.
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Ronald R. Yager
It is interesting to note that the above normalization p r o c e d u r e does not affect the specificity of the fuzzy subset. Consider first the subnormal fuzzy subset F. W i t h o u t loss of generality we shall assume F ( x i) > F ( x j ) for i < j. T h e n if C a r d ( X ) = q, q
S p ( F ) = F ( x l)
Y'~ w j F ( x j ) . j=2
Consider now E where E(x)
= F(x)
+ ol,
ol = 1 - F ( x l ) . T h e n again E ( x i) >_ E ( x j ) for i < j ; hence q
Sp(E~
=
F(x,) + ~ -
~ w j [ F ( x j ) + o~] j=2 q
= F ( x 1) + a -
q
o~ ~_, wj + ~ , w j F ( x j ) . j=2
j=2
Since Ejwj = 1, we get q
S p ( E ) = F ( x 1) + c~ - o~ + Y', w j F ( x j ) j-2
= Sp(F). W e note that this type of normalization, when viewed from the perspective of a D e m p s t e r structure, essentially corresponds to a change in the relation R relating the spaces Y and X. Namely, we transform R into R', where R and R' are the same except for the row corresponding to the null set. If Yn corresponds to the o u t c o m e in the space Y which generates the null set, then we change R into R' by letting R'(y~,x)
= 1
for all x, whereas
R ( y . , x) = 0 for all x. T h e extension of this normalization process to pseudo belief structures can be easily accomplished. A s s u m e m* is a p s e u d o belief structure whose focal elements are the fuzzy subsets F 1. . . . . Fq, where H e i g h t ( ~ ) = 1 - ~i. T h e n we can f o r m the normalized belief structure m with focal elements
Normalization of Fuzzy Belief Structures
151
E 1. . . . . Eq and where E i ( x ) = Fi(x) + ~i-
The weights associated with the E i are m(Ei) = m*(Fi).
In this section we have investigated a normalization procedure different in spirit from the one suggested by Dempster. In this method, instead of proportionally allocating the weight associated with nonnormal focal elements, we convert the nonnormal focal elements into normal focal elements by increasing their membership grades by adding to each focal element the fixed amount needed to raise the maximum membership to one.
9. C O N C L U S I O N In this work we have considered the problem of normalization of belief structures which have fuzzy focal elements. We first noted that normalization is required in this environment when we have nonnormal focal elements. A normalization procedure, called the smooth normalization procedure, was introduced. This procedure was shown to be an extension of the D e m p s t e r normalization as well as an extension of the process used for normalizing fuzzy subsets. An interpretation of this procedure as a conditioning operation was provided. We looked at the effect of this normalization on the specificity, belief, and plausibility measures. We showed that the process of normalization has no effect on the defuzzification process. This has important implications for the use of fuzzy belief structures in fuzzy system modeling as discussed in [24]. An alternative normalization procedure involving the addition of membership weight to nonnormal focal elements was also studied.
References
1. Shafer, G., A Mathematical Theory of Euidence, Princeton U.P., Princeton, N.J., 1976. 2. Smets, P., Belief functions, in Non-standard Logics for Automated Reasoning (P. Smets, E. H. Mamdani, D. Dubois, and H. Prade, Eds.), Academic, London, 253-277, 1988. 3. Yager, R. R., Kacprzyk, J., and Fedrizzi, M., Advances in the Dempster-Shafer Theory of Euidence, Wiley, New York, 1994.
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4. Zadeh, L. A., On the validity of Dempster's rule of combination of evidence, Memo U C B / E R L , M79/32, Univ. of California, Berkeley, 1979. 5. Zadeh, L. A., A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination, AI Mag., Summer, 85-90, 1986. 6. Smets, P., and Kennes, R., The transferable belief model, Artif. Intell. 66, 191-234, 1994. 7. Kohlas, J., and Monney, P. A., Theory of evidence: A survey of its mathematical foundations, applications and computations, ZOR Math. Method Oper. Res. 39, 35-68, 1994. 8. Smets, P., "Nonstandard probabilistic and nonprobabilistic representations of uncertainty," in Advances in Intelligent Computing--IPMU'94 (B. BouchonMeunier, R. R. Yager, and L. A. Zadeh, Eds.), Springer-Verlag, Berlin, 13-38, 1995. 9. Dubois, D., Prade, H., "On several representations of an uncertain body of evidence," in Fuzzy Information and Decision Processes (M. M. Gupta and E. Sanchez, Eds.), North-Holland, Amsterdam, 309-322, 1982. 10. Dempster, A. P., Upper and lower probabilities induced by a multi-valued mapping, Ann. Math. Statist. 38, 325-339, 1967. 11. Dempster, A. P., A generalization of Bayesian inference, J. Roy. Statist. Soc., 205-247, 1968. 12. Yager, R. R., On the Dempster-Shafer framework and new combination rules, Inform. Sci. 41, 93-137, 1987. 13. Yen, J., Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Trans. Systems Man Cybernet. 20, 559-570, 1990. 14. Zadeh, L. A., Fuzzy sets and information granularity, in Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, Eds.), North-Holland, Amsterdam, 3-18, 1979. 15. Goodman, I. R., and Nguyen, H. T., Uncertainty Models for Knowledge-Based Systems, North-Holland, Amsterdam, 1985. 16. Negoita, C. V., and Ralescu, D., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. 17. Dubois, D., and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic, New York, 1980. 18. Yager, R. R., "Arithmetic and other operations on Dempster-Shafer structures," Internat. J. Man-Machine Stud. 25, 357-366, 1986. 19. Yager, R. R., "Measuring tranquility and anxiety in decision making: An application of fuzzy sets," Internat. J. Gen. Systems 8, 139-146, 1982. 20. Yager, R. R., "On the specificity of a possibility distribution," Fuzzy Sets and Systems 50, 279-292, 1992.
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21. Yager, R. R., "Entropy and specificity in a mathematical theory of evidence," Internat. J. Gen. Systems 9, 249-260, 1983. 22. Yager, R. R., and Filev, D. P., On the issue of defuzzification and selection based on a fuzzy set, Fuzzy Sets and Systems 55, 255-272, 1993. 23. Filev, D., and Yager, R. R., A generalized defuzzification method under BAD distributions, Internat. J. Intell. Systems 6, 687-697, 1991. 24. Yager, R. R., and Filev, D. P., Including probabilistic uncertainty in fuzzy logic controller modeling using Dempster-Shafer theory, IEEE Trans. Systems Man Cybernet. 25, 1221-1230, 1995.
- HOILAND
On the Normalization of Fuzzy Belief Structures Ronald R. Yager Machine Intelligence Institute, lona College, N e w Rochelle, N e w York
ABSTRACT The issue of normalization in the fuzzy Dempster-Shafer theory o f evidence is investigated. We suggest a normalization procedure called smooth normalization. It is shown that this procedure is a generalization of the usual Dempster normalization procedure. We also show that the usual process o f normalizing an individual subnormal fuzzy subset by proportionally increasing the membership grades until the maximum membership grade is one is a special case of this smooth normalization process and in turn closely related to the Dempster normalization process. We look an alternative normalization process in the fuzzy Dempster-Shafer environment based on adding to the membership grade of subnormal focal elements the amount by which the fuzzy subset is subnormal.
K E Y W O R D S : fuzzy sets, theory o f evidence, normalization, subnormal,
Dempster's rule 1. I N T R O D U C T I O N The Dempster-Shafer theory of evidence [1, 2] is one of the tools used to model and manipulate uncertain information. The basic representational structure in this theory is a belief structure which consists of a collection of subsets, called focal elements, each having an associated nonnegative weight, the total of which must sum to one. An issue of considerable interest in this field is the problem of normalization: of dealing with nonzero weights that may be assigned to an empty set as a result of the combination of multiple belief structures. The original procedure, as suggested by Sharer [1], is to use Dempster's rule to normalize such a belief structure by reallocating any weight assigned to a null set to nonnull focal elements in a manner proportional to the weights already assigned to those elements. A number of authors have suggested alternative proceAddress correspondence to Professor Ronald Yager, lona CoLlege, 715 North Avenue, New Rochelle, NY 10801-1890. Received October 1993; accepted March 1995.
International Journal of Approximate Reasoning 1996; 14:127 153 © 1996 Elsevier Science Inc. 0888-613X/96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0888-613X(96)00092-5
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dures for this normalization [3]. Other authors [4-8] have questioned the process of normalization, indicating that one should leave the weight with the null set, and thus allow null focal elements. Zadeh [4, 5], for example, has shown that normalization can lead to aggregated beliefs in which a possibility very weakly supported by all constituents in the aggregation can end up with very high support. Furthermore, he suggested that normalization washes away a lot of information in a given situation, especially regarding conflicts between pieces of evidence, which would be apparent if no normalization were used. On the other hand, not imposing normalization may lead to technical difficulties with this technology, such as situations in which an upper bounding value may end up being less than a lower bounding value. The issues behind this controversy are very deep and involve subtle questions on the important topic of conflict resolution. It is not our purpose here to enter into this debate, one in which we feel both sides have merit, but rather to accept normalization and investigate its implementation in the fuzzy domain. Thus in this work we are concerned with the issue of normalization in the theory of evidence in the case in which the focal elements are fuzzy sets. We recall that in the crisp environment normalization is used in situations in which one of the focal elements is the null set, whereas in the fuzzy environment normalization is required when a focal element is subnormal (has maximal membership grade less than one). We suggest a normalization procedure, which can be used in the fuzzy environment, called smooth normalization. We show, using the fact that fuzzy subsets can be expressed as consonant belief structures [9], that this new procedure is a generalization of the D e m p s t e r normalization procedure [10, 11]. We also show that the usual process of normalizing a subnormal fuzzy subset by proportionally increasing the membership grades until the maximum membership grade is one is a special case of this smooth normalization and in turn closely related to the D e m p s t e r normalization. We then consider an alternative normalization procedure in the D e m p ster-Shafer environment suggested by Yager [12], in which normalization is accomplished by reallocating the mass assigned to a null focal element to the universe of discourse. We extend this to the fuzzy Dempster-Shafer environment. We also show that in the case of an individual subnormal fuzzy set it is related to a normalization procedure where we add to each membership grade the amount by which the fuzzy subset is subnormal.
2. NORMALIZATION OF BELIEF STRUCTURES
A Dempster-Shafer belief structure defined on the finitc set X is a mapping [1, 2, 13] m : 2 X ~ [0,1]
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129
such that r e ( O ) = O,
(1)
Y'~ r n ( A ) = 1.
(2)
AcX
T h e subsets of X for which m ( A ) 4 : 0 are called the focal elements of m. W e shall d e n o t e these a s Ai, i = 1 . . . . , n. A n essential feature of the belief structure as implied by (1) is that the null set is not a focal element. Two important measures associated with these structures are the plausibility measure P1 and the belief measure Bel. A s s u m e m is a belief structure and let B be any subset of X. T h e n PI(B) =
~
m(A,),
AiNB~-~
BeI(B) =
~
rn(Ai).
AicB
A n equivalent definition of plausibility and belief can be obtained using the ideas of possibility and certainty introduced by Z a d e h [14]. A s s u m e A and B are two subsets of X. W e define Poss(BIA) = Max~[D(x)], where
D=ANB and Cert[B[A] = 1 - Poss(BIA). It is easy to show that P I ( B ) = ~ P o s s ( B [ A i) rn( A i ) , i
B e l ( B ) = ~ C e r t ( B I A i) r n ( A i ) . i
Thus the plausibility is the expected possibility, and the belief is the expected certainty. A m a p p i n g m* is called a pseudo belief structure if we withdraw the condition (1). Every belief structure is also a p s e u d o belief structure. Pseudo belief structures can have some undesirable properties. Consider a p s e u d o belief structure m*, and assume that m * ( O ) = a > 0. In this case
PI(X) = 1 -a
v~ 1.
Thus the u p p e r probability of the whole space is not 1.
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A s e c o n d p r o b l e m that can arise if the weight a s s o c i a t e d with the null set is not z e r o is that the b e l i e f can be larger than the plausibility. Let m* be such that m(O)
= .6,
m(A)
= .4.
L e t B c A. T h e n P I ( B ) - .6, B e I ( B ) = 1. P s e u d o b e l i e f structures can arise when we a g g r e g a t e b e l i e f structures. L e t ,, be any set o p e r a t i o n . A s s u m e m 1 and m 2 are two b e l i e f s t r u c t u r e s with focal e l e m e n t s A i a n d Bj. W e define the structure m* = m l zx m 2 as the m a p p i n g m*:2 x-o
[0,1]
such that m*(D)
=
~
ml(Ai)m2(Bj).
M i, Bj D =AiAB s
E x a m p l e s o f ~ a r e union a n d intersection. W e shall say that ~ is a n o n - n u l l - p r o d u c i n g o p e r a t i o n if for all A i 4= O and Bj 4= O it is the case that Aiz~Bj
4=O.
W e see that union is a n o n - n u l l - p r o d u c i n g o p e r a t i o n , while i n t e r s e c t i o n can p r o d u c e a null. THEOREM
If
~x is" a n o n - n u l l - p r o d u c i n g
operation,
m* = m 1 ~ m 2 is a l w a y s a b e l i e f s t r u c t u r e i f m I a n d m 2 a r e b e l i e f s t r u c t u r e s .
P r o o f I f ~ is n o n - n u l l - p r o d u c i n g , t h e n A ~ B since m j ( Q ) = m 2 ( O ) = 0, we get m * ( D ) = O.
= OiffA
=B
=O,
and
O n the o t h e r hand, if ~ is not n o n - n u l l - p r o d u c i n g , t h e n it is possible for m* to be a p s e u d o b e l i e f structure. In p a r t i c u l a r , the i n t e r s e c t i o n o f b e l i e f s t r u c t u r e s can result in a p s e u d o b e l i e f s t r u c t u r e if t h e r e exist focal e l e m e n t s A i a n d Bj such that A i ~ B j = 0 . In o r d e r to p r o v i d e an a g g r e g a t i o n p r o c e s s which e n a b l e s even null-producing o p e r a t i o n s to result in b e l i e f structures. D e m p s t e r [10, 11] introd u c e d a p r o c e d u r e of n o r m a l i z a t i o n .
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DEFINITION ( D e m p s t e r N o r m a l i z a t i o n P r o c e d u r e ) Assume m* is a pseudo belief structure. Then we can conwert m* into a belief structure m as follows: m : 2 x ~ [0,1] where 1. m(@) = 0;
2. for all A --/: Q, m*(A) m(A)-
1 -
m*(@)"
W e n o t e that if m* is a belief structure t h e n m* = m. If ,, is the i n t e r s e c t i o n o p e r a t i o n a n d if we use the above n o r m a l i z a t i o n p r o c e d u r e , we o b t a i n the D e m p s t e r rule of aggregation [10, 11]. T h e D e m p s t e r n o r m a l i z a t i o n p r o c e d u r e ( D N P ) has a very specific effect o n the m e a s u r e s of belief a n d plausibility. A s s u m e m* is a p s e u d o belief structure. Let P I * ( B ) a n d B e l * ( B ) b e the plausibility a n d belief associated with some n o n n u l l subset B. Let m be o b t a i n e d from m* using the D N P . Then PI(B) =
m(Ai) =
~ i
i
Aif~Bf:D
>
~
m * ( A i) 1 - m*(O)
~ AiNB4-O
m * ( A i) > P I * ( B )
i
Ai~B#:Q In p a r t i c u l a r we see that Pl(B) -
PI* ( B ) - - , q
where q = Y~A,~@m*(Ai). Let E be the set of subsets of X that are c o n t a i n e d in B, excluding the null set; thus
m * ( A i) Bel(B)= BeI*(B) =
~ m(Ai)= AicE
~
1-m*(Q3)'
AidE
Y'. m * ( A i) + m*(Q3). AiEE
Let K ~A~EEm*(Ai ), a n d let d = 1 - K - m*(@). W e n o t e that 0 < d < 1. T h e n =
K BeI(B) -
1 -m*(£5)
1 - m* ( Q ) - d -
1 -m*(@)
B e l * ( B ) = K + m * ( @ ) = 1 - d.
- 1
d
1-m*(•)'
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Since d <_ d / [ 1 - m*(O)], then B e I * ( B ) > Bel(B). T h u s the D N P essentially i n c r e a s e s the plausibility a n d d e c r e a s e s the belief. Since d BeI(B) = 1 1 -
rn* (0)
'
then 1 - m*(O) + d
Bel*(B) - m*(O)
Bel(B) = 1 -
m*(O)
1 -
m*(O)
T h e p r o c e s s o f n o r m a l i z a t i o n i n t r o d u c e d by D e m p s t e r can b e seen as a kind of c o n d i t i o n i n g o p e r a t i o n . In o r d e r to a p p r e c i a t e this fact, we m u s t p r o v i d e an a l t e r n a t i v e view of the p s e u d o b e l i e f structure. Since any b e l i e f s t r u c t u r e is also a p s e u d o b e l i e f structure, we shall also be p r o v i d i n g an a l t e r n a t i v e view o f t h e b e l i e f structure. This a l t e r n a t i v e view is the o n e used by D e m p s t e r in his original w o r k [10, 11]. C o n s i d e r a p s e u d o b e l i e f s t r u c t u r e m* on X with m focal e l e m e n t s , A 1 , A 2 . . . . . A,,. N o w c o n s i d e r a n o t h e r structure, which we shall call a Dernpster structure, consisting o f a set Y = {Yt . . . . . Yn}, a p r o b a b i l i t y distrib u t i o n P* on Y, a n d a r e l a t i o n R on Y X X. H e r e R is d e f i n e d such that (Yi, X) ~ R if x ~ A i, and P* is d e f i n e d such that P * ( Y i ) = m * ( A i ) . A s shown by D e m p s t e r , this new s t r u c t u r e is the s a m e as rn* in the sense that for any subset B o f X it is the case that the u p p e r b o u n d on the p r o b a b i l i t y o f B is P I ( B ) a n d the lower p r o b a b i l i t y o f B is BeI(B). Essentially, in this new s t r u c t u r e we have a r a n d o m e x p e r i m e n t on the space Y such that if the o u t c o m e is the e l e m e n t Yi, then we o b t a i n as the o u t c o m e the subset A i of the space X. This r a n d o m set a p p r o a c h has b e e n extensively s t u d i e d by G o o d m a n a n d N g u y e n [15]. T h u s o n e possible s e m a n t i c s that can be a s s o c i a t e d with a b e l i e f structure is a p r o b a b i l i s t i c s e m a n t i c s b a s e d on the i d e a o f r a n d o m sets. In this s e m a n t i c s the focal e l e m e n t s a r e v i e w e d as o u t c o m e s of an e x p e r i m e n t , that is, the o u t c o m e of the e x p e r i m e n t is a set f r o m X r a t h e r t h a n an e l e m e n t from X, a n d the weights a r e viewed as the p r o b a b i l i t i e s a s s o c i a t e d with these sets. In this view the m e a s u r e s o f b e l i e f a n d plausibility of a set B a r e closely r e l a t e d to the i d e a s o f lower and u p p e r p r o b a b i l i t y o f B [10]. In p a r t i c u l a r B e I ( B ) N P r o b ( B ) _< P I ( B ) . W e now i n t r o d u c e the c o n d i t i o n i n g view o f the n o r m a l i z a t i o n process. W i t h o u t loss of generality, a s s u m e A I - 0 . W e shall now c o n d i t i o n the
Normalization of Fuzzy Belief Structures
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probability distribution P* on the set G = {Y2 . . . . . Y,} = n°t{Yl}. We emphasize that yl is the outcome in the space Y associated with the null focal element Aj. Using this conditioning, we obtain the conditional probability
P(yiln°t{Yl}) = P(Yi[G) =
P*({yi} c3 G)
P*(G)
For i =g 1 we get
P(yi[G)
P*(Yi) 1 - P*(Yl) '
and for i = 1 we get P*(O) O.
P(y~[G)- 1 - P * ( y ~ )
This conditioned probability distribution consists of the new weights in the DNP:
m(Ai ) --
P*(Yi) 1 -
P*(Yl)
m*(Ai) 1 -
m*(A
1) "
Thus, we see that the D N P can be viewed as a conditioning on the original probability distribution P, where the conditioning is obtained by assuming that the outcome must occur in the set G. Since the set G corresponds to elements in Y which have nonnull focal elements in X, we have essentially enforced a condition in which we have a belief structure.
3. NORMALIZATION IN THE FUZZY E N V I R O N M E N T
In this section we consider the process of normalization in the case where the focal elements are fuzzy subsets. We first introduce some concepts from the theory of fuzzy sets. Let A be a fuzzy subset of some space X. We shall let H e i g h t ( A ) = maxxA(x); thus Height(A) is the maximum membership grade of A. If Height(A) = 1, we say that A is normal, and if Height(A) ~ 1, we say A is subnormal. We next recall that the c~-level sets of the fuzzy subset A are crisp subsets of X defined such that
A~ : { x l A ( x ) >_ ~}.
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It is also well established that if c~ > c~' then A~ ~ A s, and that A ( x ) = a * , where ~* is the largest ~ such that x e A~. F u r t h e r m o r e if H e i g h t ( A ) = /3 :g 1, then A~=Q
for
~>/3.
A s s u m e A is a subnormal fuzzy subset of X with H e i g h t ( A ) = K. In some applications of fuzzy set theory there is a need to convert A into a fuzzy subset with height of one. This process is also called normalization: it converts a subnormal fuzzy set into a normal one. The usual normalization [16, 17] process for converting A into a normal fuzzy subset B is to let 1
B(x) = -~A(x). O n e can view this normalization process in terms of level sets. A s s u m e A is a subnormal fuzzy subset with H e i g h t ( A ) = K. T h e n its level sets are A s. If we consider a new fuzzy subset B whose level sets are B~ where B~ = A a K
for
a ~ [0, 1],
then 1
B(x) =~A(x). We now introduce the idea of a fuzzy belief structure and the related idea of a p s e u d o fuzzy belief structure where the focal elements are fuzzy subsets. A f u z z y belief structure m defined on the finite set X is a m a p p i n g from fuzzy subsets of X into the unit interval, m:IX
~ [0,1]
such that 1. F o r A subnormal, r e ( A ) = O. 2. ~ m ( A ) = 1 The fuzzy subsets A 1. . . . . A n, for which m ( A i) 4 : 0 are again called the focal elements. We call a m a p p i n g m* a pseudo f u z z y belief structure if m* is a m a p p i n g m* : I x --* [0, 1]
such that ~_~rn*(A) = 1.
Thus a p s e u d o structure doesn't require the focal elements to be normal.
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T h e measures of plausibility and belief can be easily extended to this environment: P I ( B ) = ~_,Poss[BIA i] m ( A i ) , i B e l ( B ) = Y'~ Cert[B[ A i] m ( A i ) . i
As discussed by Y a g e r [18], if ~ is any fuzzy set operation, we can extend it to work on fuzzy belief structures. For example, if • is the intersection operation, then if m 1 and m 2 are two fuzzy belief structures with focal elements A i and Bj respectively, we can obtain m I zx m 2 :
m,
where m has focal elements FK =Ai~B
j
with weights
m ( FK ) = m ( A i ) m ( Bj). In some cases these aggregations o f fuzzy belief structures may lead to pseudo fuzzy belief structures. In the following we shall suggest a procedure, called the smooth normalization procedure (SNP), for converting p s e u d o fuzzy belief structures into fuzzy belief structures. A s s u m e m* is a p s e u d o fuzzy belief structure with focal elements F i and associated weights m*(Fi). Let H e i g h t ( F i) = h i. T h e following algorithm converts m* into a fuzzy belief structure in a m a n n e r consistent with the D e m p s t e r normalization procedure. SMOOTH
NORMALIZATION
PROCEDURE
1. Calculate u i = m * ( F i ) h i. 2. Calculate T = ~,im*(Fi)(1 - h i) = 1 - Y[im*(Fi)hi . 3. Calculate ui
v,-l_ T 4. Introduce new fuzzy subsets E i with membership grades Fi(x) Ei(x ) -
hi
We call the fuzzy belief structure m with focal elements E i and associated weights m ( E i) = Vii the smooth normalized belief structure.
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First let us assure ourselves that m is a fuzzy belief structure. It has as focal e l e m e n t s E i, since 1 hi H e i g h t ( E i ) = ~ H e i g h t ( F i ) = h i = 1, the E i
are
n o r m a l . Next, y,m(Ei ) = ~ i
i
ui 1-T
_ __1 1 -T
~m*(Fi)hi
= 1.
W e see that m*(F,)h i Vii = y~)~=l m , ( F j ) h j . Let us investigate some p r o p e r t i e s associated with this t r a n s f o r m a t i o n . T h e first p r o p e r t y assures us that if the original structure is a fuzzy belief structure, the s m o o t h t r a n s f o r m a t i o n has n o effect. THEOREM
I f m is a f u z z y belief structure, then m * = m .
P r o o f If m* is a fuzzy belief structure then h i = 1 for all i. In this case E i = F i. F u r t h e r m o r e , u i = m * ( F i) a n d T = 0, a n d h e n c e Vi = m * ( F i ) . T h e next t h e o r e m shows that this p r o c e d u r e is i n d e e d a g e n e r a l i z a t i o n of the DNP. THEOREM I f the F i in m* are crisp subsets, then m is the s a m e as that obtained by the D e m p s t e r normalization procedure. Proof F o r i = 1 to k let F,. b e n o n n u l l , h i = 1. F o r i = k + 1 t o n let F i be null, h i = 0. F o r i = 1 to k we have u i = m * ( ~ ) , a n d for the o t h e r i's, u i = 0. W e see that T = Y?~ m * ( F i ) . F o r i = 1 to k, i~k=l
v~=
m * ( F i) 1 - E7 k+ j m ( F i ) '
a n d for i = k
+ 1 to n, Vi = 0 . F o r i =
1 t o k, E i = F i.
THEOREM I f h i is the s a m e f o r all i, h i = c~, then f o r all i, El(X) = ( 1 / c ~ ) F / ( x ) a n d m ( E i) = m * ( F i ) . P r o o f F o r all i, m * ( Fi)o~/c~ = m * ( F i ) .
u i = m*(F,)=
c~ a n d
T = 1-
c~; thus
~--
It should be n o t e d that the SNP can also be applied to a fuzzy subset a n d leads to the usual n o r m a l i z a t i o n . A s s u m e F is a fuzzy subset of X
Normalization of Fuzzy Belief Structures
137
with H e i g h t ( F ) = c~. W e can r e p r e s e n t this as a structure m* with o n e focal e l e m e n t F having weight 1. If we apply o u r SNP to this, we o b t a i n u 1 = 1 a n d V l = 1, a n d E ( x ) = ( 1 / a ) F ( x ) . T h u s the s m o o t h n o r m a l i z a tion leads to the usual n o r m a l i z a t i o n used in fuzzy subset theory.
4. C O N S O N A N T BELIEF STRUCTURES T h e r e exists a special class of belief structures called c o n s o n a n t belief structures [1]. A belief structure is called c o n s o n a n t if its focal e l e m e n t s are nested. T h u s a belief structure m is a c o n s o n a n t belief structure if its focal e l e m e n t s are crisp subsets that can be indexed so that A 1 D A 2 D A 3 D ..- D A n . As shown by Sharer [1], the c o n s o n a n t belief structure has th'e following properties: 1. BeI(A C~ B) = m i n [ B e l ( A ) , Bel(B)], 2. P l ( A U B) = m a x [ P l ( A ) , P l ( B ) ] , 3. P I ( A ) = m a x , ~ A Pi({x}). As shown by D u b o i s a n d P r a d e [9], for any x ~ X Pl(x) =
~
rn(Ai).
i xEA
i
F u r t h e r m o r e , they have shown that any c o n s o n a n t belief structure m can be associated with a fuzzy subset F where F(x)
= Pl(x).
Conversely, we can r e p r e s e n t any n o r m a l fuzzy subset F as a c o n s o n a n t belief structure m. A s s u m e F is a fuzzy subset defined o n X = {x, . . . . . x,,}. A s s u m e the e l e m e n t s of X are indexed so that F ( x i ) > F ( x i) if i > j . T h e n we o b t a i n the focal e l e m e n t of m as follows:
A,, = ( x , ) , a,,-i
= {xn,Xn
A i = {x . . . . . . Aj = X ,
m(A,,)
I},
xi},
m(An_l) m(Ai)
= F ( x n)
-
= F(Xn_l) = f(xi)
F(Xn_l) ,
- F(Xn_2) ,
- F(xi_l) ,
m ( A i) = F ( x ] ) .
W e can extend this idea to r e p r e s e n t s u b n o r m a l fuzzy subsets if we can allow c o n s o n a n t p s e u d o belief structures. C o n s i d e r the c o n s o n a n t p s e u d o belief structure m* where A 1 D A 2 D A 3 D ... D A n D A n + 1
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Ronald R. Yager
with An+ 1 = O and rn(A,+ 1) = 1 - a. This leads to a subnormal fuzzy subset F with Height(F) = a and
F(x) = ~
m*(Ai).
i xEA~
Consider now the application of the D N P to this pseudo c o n s o n a n t belief structure. W e obtain a new consonant belief structure with focal elements A 1 DA
2
D A 3 D ... A,,
with weights
m(Ai) =
m * ( A i)
m * ( A i)
m * ( A i)
1 - r n * ( A , + 1)
1 - (1 - a )
a
This can be associated with a fuzzy subset E, where 1 E(x)
=
E m(Ai)=i Ax~A ~
E OL x~A,
1
m*(Ai)=--F(x). Og
T h e classical normalization used in fuzzy subset theory can be viewed as a D N P normalization process applied to the associated consonant belief structure. Thus the normalization is obtained by a conditioning of the focal elements in the underlying consonant structure by eliminating any outc o m e which gives a null focal element. Thus, saying that a fuzzy subset must have an element with m e m b e r s h i p grade one means saying no null focal elements are allowed. Let us now consider a p s e u d o fuzzy belief structure m* with focal elements F 1. . . . . F n and associated weights m*(Fi). E a c h of these fuzzy focal elements can be represented in turn by a consonant p s e u d o belief structure m* with crisp focal elements ~ j , j = 1 . . . . . tTi, with weights m*(F,j). W i t h o u t loss of generality we consider Fi, ' = O for all rn*. If Fi is normal, then rn*(Fi, ,) = 0; otherwise it is nonzero. Using the D e m p s t e r view, we can consider the pseudo fuzzy belief structure as equivalent to the p e r f o r m a n c e of a c o m p o u n d (two part) i n d e p e n d e n t experiment. In the first experiment, guided by the probability distribution Pi = m*(F,), we decide which second experiment to p e r f o r m (see Figure 1). T h e second experiments are p e r f o r m e d using the appropriate consonant representation of the focal element F i. Thus second experiment i has as
Normalization of Fuzzy Belief Structures
139
PI = m1(Fl)
Second Exp - I
= m*(Fi)
Second Exp - i
~
Pn
=
m
Second Exp - n
Figure 1. Pseudo fuzzy belief structure viewed as a compound experiment.
its outcomes the focal elements of the consonant belief structure {Fil, F, 2. . . . . Fi, ,} representing F i, and the probabilities are rn*(Fij) (see Figure 2). Using this compound model, the probability p~ of any set Fq is
p*j = m * ( F i ) m * ( F q ) . We can now view this compound shown in Figure 3. Using the DNP on this compound ing by eliminating all Fij = ~ , that conditioned probability Pq for each
Pij =
experiment as a single experiment as structure (which consists in conditionis, eliminating all Fi. ), we get a new nonnull Fq, where
P*({Fij) A B ) P*(B) '
where B is the set of nonnull focal elements. Hence
P* (Fq) Pij
-
1 -
P*(B)
Pij --
1 -
Y~p,*.,
,
m* ( Fi)m* (Fij) Pij = 1 - S,i=,m" * ( F i ) m * ( F i , ) " We can now view this probability distribution on the collection of nonnull focal elements as being generated by some fuzzy belief structure
~
F| 1
Fij Fin]
Figure 2. Second experiment i.
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Ronald R. Yager
F~j
Figure 3. Compound experiment.
m. We d e n o t e the focal elements of this structure a s i~ 1. . . . . /~n with weights m ( ~ ) . F u r t h e r m o r e , each fii can be seen as a normal fuzzy subset generated by a c o n s o n a n t belief structure m i which has focal element {FiI,F+2 . . . . .
and weights m i ( F i j ) . given by
F i.... }
In this structure each of the F/j has probability fiij Pij = m ( l ~ i ) m i ( F i j ).
For this structure to be the same as the one obtained by the conditioning, we must have Pij = Pij ;
thus for all ij
m*(F~)mT(Cj) = 1 _ ~ v'ni - , m * ( f,) m *i ( F i n ) "
m(l~i)mi(Fij)
W e must solve this for m ( ~ ) and m i ( F i ) ,
where
re(P,/= 1, i t/i
1 i
Y'~ m,(Fij)
1.
j-1 T h e solution to this is obtained as follows. Let
roT(F,) rns(Fij)
= 1 - r o T ( F i n ,) '
we recall that 1 - r n * ( ~ , ~ ) = hi, the height of F i. F u r t h e r m o r e , we see that n, ~ mi(Fij)
j=l
1 -
1 - m*(Fi,,)
n, , E mi(Fij) j==1
1 - rn* ( F~,,,) -
1
m~[ ( Fin )
-
1.
Normalization of Fuzzy Belief Structures
141
Thus m~
mi't~j','i,
=
"
*(Sj) hi
Using this, we get /3ij = m ( ffi )
m*(Fij) hi
hence /~ij = Pij then m*(Fij) --
, ^,
m[~/ \
/
1-
hi
m*(Fi)mi(Fij) n m * ( F i ) m i ( F i,n) ' Y~i=l
therefore h i m * ( F i) m(Fi)
=
1-
Z { ~' =
i m*
(Fi)m*(ei,,)
Since m*(Fi,,,) = 1 - h i, we get h i m * ( F i)
{E,~ =
1 -
him*(F i)
.-,="n l m * ( F , ) ( 1
-
hi)
1 - ,-.i--v'" lm*¢F~, i, + E'i'=Lm*(Fi)hi
Since
~_,m*(F i) = l, i=1
we get m(fii) =
him*(Fi) E~,=lhim,(Fi) "
Furthermore, we note that fii is associated with the consonant belief structure m i with focal elements {FiL, Fi2 . . . . . Fi,,, ~} whose weights are
mf(Fu) mi(Fij )
hi
The associated fuzzy subset /?i is seen to have membership function 1
1 J
xcFij
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Ronald R. Yager
Thus the resulting fuzzy belief structure is exactly the same as that obtained using the smooth normalization method. In the preceeding construction we have shown that the SNP is effectively the D N P when viewed in the appropriate way.
5. N O R M A L I Z A T I O N A N D SPECIFICITY
The concept of the measure of specificity of a fuzzy subset was introduced by Yager [19] as an indication of the degree to which the fuzzy subset contains one and only one element. Assume F is a fuzzy subset of X. In [20] Yager introduced a linear measure of specificity defined as Sp(F) =b i-
~ wjbj, j-2
where bj is the jth largest membership grade in F, and tlie w i are a collection of weights such that w i ~ [0, 1]
~wj= j 2
1, if
w i > wj
i < j.
It can be easily shown that for any F, S p ( F ) ~ [0, 1]. We also see that Sp(F) = 1 iff F is a singleton fuzzy subset. One semantics that can be associated with the measure of specificity of a set F regards the selection of an element from the set F. In particular, as discussed in [19], the bigger the specificity of a set, the less difficult it is to choose an element from the set as F as the optimal member. Consider now the process of normalizing a fuzzy subset. Starting with the subnormal fuzzy subset F, we normalize it to obtain the fuzzy subset E, 1 E(x)
= --F(x),
where H e i g h t ( F ) = a. If we calculate
Sp(E) = d l -
Y'. wjdj, j- 2
Normalization of Fuzzy Belief Structures
143
where dj is the jth largest of the E(x), then, since d i = (1/ct)bi, we get
1
S p ( E ) = - - b a - ~_,wj---dbj = O~
-
J
= ~- S p ( F ) . ./
Thus the process of normalization changes the specificity by a factor of 1 / a . Thus normalization increases the specificity when a < 1. Consider now a pseudo fuzzy belief structure m* with focal elements A~ . . . . . A n. In [21] Yager suggested a measure of specificity associated with this structure as Sp(rn* ) = ~ S p ( A
i)
m*(Ai).
i
Consider now the fuzzy belief structure m obtained from m* by the SNP. In this case rn has focal elements Bg, where
Bi(x ) = --Ai(x), hi and weights
m*(Ai)h i
m ( B i)
Em* ( A i ) h i '
where h i = Height(Ai). If we now calculate Sp(m) = Y'~Sp(A i) m ( A i ) , l
we get
1
m*(Ai)h i
Sp(m) = ~ ~i Sp(Ai) S . m , ( A i ) h i
~,m* ( A i ) h i
Sp(m* ).
Thus, we see that the specificity is again increased, this time by a factor ~ i m * ( A i ) h i . We can view ~ i m * ( A i ) h i as the average height of the focal elements. Let us try to provide some intuition for these results. In order to do this we must first provide some insight into what we are measuring with Sp. A belief structure can be viewed as having two sources of uncertainty, one being related to the determination of the focal element and the other to the selection of the element within the selected focal element. A very
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Ronald R. Yager
i m p o r t a n t class o f b e l i e f structures a r e o n e s in which the focal e l e m e n t s are just sets with o n e e l e m e n t ; these a r e called Bayesian b e l i e f structures. W e n o t e t h a t in Bayesian b e l i e f s t r u c t u r e s t h e r e exists no u n c e r t a i n t y r e g a r d i n g the selection o f an e l e m e n t f r o m a d e t e r m i n e d focal set. F u r t h e r m o r e , we n o t e that for a focal set with o n e e l e m e n t the specificity is one. F r o m this we see that a B a y e s i a n b e l i e f s t r u c t u r e m has S p ( m ) = 1, which is an i n d i c a t i o n that t h e r e is no difficulty r e g a r d i n g the selection of an e l e m e n t f r o m a d e t e r m i n e d focal set. A s o u r result has shown, the s m o o t h n o r m a l i z a t i o n p r o c e s s t e n d s to increase the specificity. T h u s we can view the S N P as o n e which turns a p s e u d o b e l i e f s t r u c t u r e into a b e l i e f s t r u c t u r e in a m a n n e r that tries to m a k e it m o r e Bayesian.
6. SMOOTH NORMALIZATION, PLAUSIBILITY, AND BELIEF W e have previously shown that the D N P results in a w i d e n i n g of the p l a u s i b i l i t y - b e l i e f interval in the crisp d o m a i n . W e now show the s a m e effect occurs in the fuzzy d o m a i n w h e n we use the SNP. THEOREM Assume m* is a pseudo fuzzy belief structure. Let m be the associated fuzzy belief structure obtained by the smooth normalization process. Then for any fuzzy subset B PI*(B) < Pl(B)
and B e l * ( B ) _> B e l ( B ) . P r o o f A s s u m e A 1. . . . . A , a r e the focal e l e m e n t s o f m*. A s we s h o w e d in the p r e v i o u s section, each o f the A i c a n be v i e w e d as b e i n g a c o n s o n a n t p s e u d o b e l i e f s t r u c t u r e m* with focal e l e m e n t s A i l . . . . . Ain ,. F r o m this p e r s p e c t i v e we can view m* as a p s e u d o crisp b e l i e f s t r u c t u r e with focal element
M = {AI~, Aj2
. . . . .
Ani,},
w h e r e t h e weight a s s o c i a t e d with Aij is m * ( A i ) m * ( A i j ) = q(Aij). In this case ~ q*(Aij). 0 AiycIBT-Q
el* ( B ) =
L e t t i n g D be the subset of focal e l e m e n t s for which A i j :~ B =~ Q~, we have Pl*(B)
=
~] AijG D
W e n o t e t h a t o n e Aij = 0 is not in D.
q*(Aij).
Normalization of Fuzzy Belief Structures
145
A f t e r doing the normalization, we obtain a belief structure with focal elements M' =M-
U,
where U is the set of elements in M that are null sets. F u r t h e r m o r e , the weights assigned to an element Aij in M ' is q(Aij) where
q*(Aij) q(Aij) = EAij ~ M'q*(Aij) " Hence PI(B) = Since
q(Aij) >_q*(Aij),
~ q(Aij). AijcD
we get PI(B) > Pl*(B).
Let E be the subset of M ' consisting of all the subsets which contain B. Then
Let K =
Bel(B) =
Y'~ q(A~j), Aij~E
Bel*(B) =
~ q*(Aij) Ai: EE
+
~
~Ai~Eq*(Aij), q) = ~'A~ ~¢t*(Aij), K BeI(B) - 1-q~
q*(Aig).
Aij=G and d = 1 - K - q~. T h e n
1-d-q~ 1-q~
d 1
1-~o
and Bel*(B) =K+
q~= 1 - d .
Since d d < - then B e I * ( B ) > Bel(B). In proving the above t h e o r e m we have actually developed a precise relationship between P I * ( B ) and PI(B): PI* ( B ) PI(B) =
EA,, ~ M'q* (Aij)
'
146
Ronald R. Yager
since ~_.
q*(Aij)
q*(Aij) = 1 -
= 1 -
AoEM'
Ai]=U
~q*(Ai~,). i= 1
Furthermore, since
q*(Ain~) = m * ( A i ) ,
m*(Ai,, )
and mi(Ain ,) = 1 - h i , where h i is the height of A i, then
q*( Ain ) = m* ( A i ) - h i m * ( A i ) ' and therefore
~ q * ( A i n ~) = ~ m * ( A i=1
i)-
Y'~him*(A i) = 1 - ~_,him*(Ai).
i=1
Thus PI*(B) PI(B)
=
Eni
1h i m
* '£A i3""
In a similar manner we can show that BeI(B) =
BeI*(B) + E h i m * ( A ~) - 1
1 - BeI*(B) 1
E~z= i h i m * ( A i )
Y~him*(A i )
The basic result of the theorem provided in this section is that the process of smooth normalization essentially results in a decrease in belief and an increase in plausibility.
7. N O R M A L I Z A T I O N
AND DEFUZZIFICATION
The process of defuzzification plays an important role in the applications of fuzzy logic controllers [22]. This operation allows us to select from a fuzzy subset of the real line a typical element. Assume F is a fuzzy subset of the real line. The center of area (COA) defuzzification technique selects as a prototypical element EiF(xi)Y
i
EiF(xi) Another often used technique is called the mean of maximal (MOM) method. Let G be the subset of elements for which F ( x ) attains its
Normalization of Fuzzy Belief Structures
147
maximal value. Let g = Card(G). Then the M O M defuzzified value is 1
Exi.
g icG
In [23] Filev and Yager unified these approaches using the B A D D transformation. The B A D D defuzzified value Y is obtained as
~" = F-,[F(xi)]'~xi F~[F(xi)] ~ ' where a ~ [1, oc]. We note that if a = 1 we obtain the C O A method, and if a = zc we obtain the M O M method. Let F be a fuzzy subset such that Height(F) = h. Let E be the fuzzy subset obtained by the normalization E ( x ) = ( 1 / h ) F ( x ) of F. Consider the B A D D defuzzification of E:
~'E = E i [ E ( x i ) ] = x i F_,iE(xi) ~
= Ei[(1/h)F(xi)]~xi ~,i[(1/n)F(xi)] ~
-- ~-'i[F(xi)]~xi
--YF"
~_,i[f(xi)]axi
Thus the defuzzified value of E and F are the same, and hence the normalization does not affect the defuzzified value under any B A D D defuzzification. Consider now a fuzzy belief structure m with n focal elements Aj. In [24] Yager and Filev suggest an approach to defuzzifying this kind of structure. Let 33j be the defuzzified value of Aj using any B A D D defuzzification. Then the defuzzified value of m is
fP =
~
f~jm(Aj).
j=l
Consider now a pseudo fuzzy belief structure m* with focal elements Aj and where hj = max x A j ( x ) . The defuzzified value of this structure is
y* = ~ ~[m*(Aj), j=l
where 13~ is the defuzzified value of As. Let m be the fuzzy belief structure obtained from m* by the smooth normalization procedure. The focal elements of m are Bj, where B j ( x ) =
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Ronald R. Yager
(1/hj)Aj(x), and the weights are
m*(Bj)hj rn( Bj) = Ens=lm* ( Bj)hi " Consider now the defuzzified value 1) of m,
~" : ~ 5m(Bj), j=l
where )3j are the defuzzified values of the Aj. Since then, as we have already shown, )3j = )3f and thus
Bj(x) = (1/hj)Aj(x),
tl
j- lm*( Aj)hi
j=t
Thus in general )3 ~ )3*. Consider next the special case w h e n all the hj are the s a m e , hj = K. All the focal elements have the same height in this case,
)3-
Zy*m*(Aj) Em* ( A j)
However, since 32~' lm*(Aj) : 1, we get
)3 = Ey[m*(Aj) =)3". Thus in this special case where all focal elements have the same height, the normalization process does not affect the defuzzification process for any B A D D defuzzification. In [24] Yager and Filev have investigated the use of fuzzy belief structures to represent the consequent of rules in fuzzy systems models. They have also shown that the output of such systems are pseudo fuzzy belief structures in which the heights of all the focal elements are the same. Thus in this environment the defuzzification process is unaffected by any normalization.
8. A L T E R N A T I V E N O R M A L I Z A T I O N
PROCEDURES
In [12] Yager suggested an alternative approach to the normalization of a pseudo belief structure. Assume rn* is a belief structure on X with focal elements A j , . . . , A s. Assume A n = Q and m*(A,,) = c~. Yager suggested that the normalization of m* be accomplished by transforming m* into a
Normalization of Fuzzy Belief Structures
149
b e l i e f s t r u c t u r e m with focal e l e m e n t s B~ . . . . . B n w h e r e B i=A
i
for
i=
1..... n-
1,
B n = X, a n d w h e r e for all i m(Ai)
= m*(Ai).
Thus, this m e t h o d essentially r e p l a c e s t h e null focal e l e m e n t by space X. L e t us investigate the use o f this t y p e o f n o r m a l i z a t i o n in e n v i r o n m e n t . A s s u m e F is a s u b n o r m a l fuzzy subset o f H e i g h t ( F ) = 1 - ~. C o n s i d e r now t h e r e p r e s e n t a t i o n o f F as a p s e u d o b e l i e f s t r u c t u r e m* with focal e l e m e n t s
the w h o l e the fuzzy X where consonant
F 1 D F 2 D F 3 D ... D F , . B e c a u s e F is s u b n o r m a l , t h e n F, = Q5 a n d m * ( F , ) ment
= c~. In this e n v i r o n -
FIx
F(x)
= Pl({x}) = ' ~ m * ( F / ) , i-I
w h e r e n x is the largest i such that x ~ F i. C o n s i d e r now t h e n o r m a l i z a t i o n o f m* using the t e c h n i q u e d e s c r i b e d above. In this case we get a b e l i e f s t r u c t u r e m with focal e l e m e n t s A0 . . . . . An m, w h e r e A i = F i f o r i = 1. . . . , n land A 0 =X, andwhere r n ( A i) = m * ( F i) for i = 1 . . . . . n - 1 a n d m ( A o) = c~. W e see t h a t m is still a c o n s o n a n t b e l i e f structure, since A o D A 1 2DA 2 D "" D A n _ l , a n d thus induces a fuzzy s u b s e t E. W e also see t h a t the m e m b e r s h i p function o f E is r/x
E(x)
r/x
= ~ _ , m ( A i) = m ( A
o) + ~ r n ( A
i=0
i)
i=1 i"/x
= m(Ao)
+ ~_, m * ( F i ) i=1
= ol + F ( x ) . Thus, t h e n o r m a l i z a t i o n a p p r o a c h s u g g e s t e d by Y a g e r in [12] c o r r e s p o n d s to the n o r m a l i z a t i o n o f the fuzzy s u b s e t F by a d d i n g c~, the a m o u n t o f which t h e e l e m e n t with the m a x i m a l m e m b e r s h i p g r a d e d e v i a t e s f r o m one, to all the m e m b e r s h i p grades.
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Ronald R. Yager
It is interesting to note that the above normalization p r o c e d u r e does not affect the specificity of the fuzzy subset. Consider first the subnormal fuzzy subset F. W i t h o u t loss of generality we shall assume F ( x i) > F ( x j ) for i < j. T h e n if C a r d ( X ) = q, q
S p ( F ) = F ( x l)
Y'~ w j F ( x j ) . j=2
Consider now E where E(x)
= F(x)
+ ol,
ol = 1 - F ( x l ) . T h e n again E ( x i) >_ E ( x j ) for i < j ; hence q
Sp(E~
=
F(x,) + ~ -
~ w j [ F ( x j ) + o~] j=2 q
= F ( x 1) + a -
q
o~ ~_, wj + ~ , w j F ( x j ) . j=2
j=2
Since Ejwj = 1, we get q
S p ( E ) = F ( x 1) + c~ - o~ + Y', w j F ( x j ) j-2
= Sp(F). W e note that this type of normalization, when viewed from the perspective of a D e m p s t e r structure, essentially corresponds to a change in the relation R relating the spaces Y and X. Namely, we transform R into R', where R and R' are the same except for the row corresponding to the null set. If Yn corresponds to the o u t c o m e in the space Y which generates the null set, then we change R into R' by letting R'(y~,x)
= 1
for all x, whereas
R ( y . , x) = 0 for all x. T h e extension of this normalization process to pseudo belief structures can be easily accomplished. A s s u m e m* is a p s e u d o belief structure whose focal elements are the fuzzy subsets F 1. . . . . Fq, where H e i g h t ( ~ ) = 1 - ~i. T h e n we can f o r m the normalized belief structure m with focal elements
Normalization of Fuzzy Belief Structures
151
E 1. . . . . Eq and where E i ( x ) = Fi(x) + ~i-
The weights associated with the E i are m(Ei) = m*(Fi).
In this section we have investigated a normalization procedure different in spirit from the one suggested by Dempster. In this method, instead of proportionally allocating the weight associated with nonnormal focal elements, we convert the nonnormal focal elements into normal focal elements by increasing their membership grades by adding to each focal element the fixed amount needed to raise the maximum membership to one.
9. C O N C L U S I O N In this work we have considered the problem of normalization of belief structures which have fuzzy focal elements. We first noted that normalization is required in this environment when we have nonnormal focal elements. A normalization procedure, called the smooth normalization procedure, was introduced. This procedure was shown to be an extension of the D e m p s t e r normalization as well as an extension of the process used for normalizing fuzzy subsets. An interpretation of this procedure as a conditioning operation was provided. We looked at the effect of this normalization on the specificity, belief, and plausibility measures. We showed that the process of normalization has no effect on the defuzzification process. This has important implications for the use of fuzzy belief structures in fuzzy system modeling as discussed in [24]. An alternative normalization procedure involving the addition of membership weight to nonnormal focal elements was also studied.
References
1. Shafer, G., A Mathematical Theory of Euidence, Princeton U.P., Princeton, N.J., 1976. 2. Smets, P., Belief functions, in Non-standard Logics for Automated Reasoning (P. Smets, E. H. Mamdani, D. Dubois, and H. Prade, Eds.), Academic, London, 253-277, 1988. 3. Yager, R. R., Kacprzyk, J., and Fedrizzi, M., Advances in the Dempster-Shafer Theory of Euidence, Wiley, New York, 1994.
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4. Zadeh, L. A., On the validity of Dempster's rule of combination of evidence, Memo U C B / E R L , M79/32, Univ. of California, Berkeley, 1979. 5. Zadeh, L. A., A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination, AI Mag., Summer, 85-90, 1986. 6. Smets, P., and Kennes, R., The transferable belief model, Artif. Intell. 66, 191-234, 1994. 7. Kohlas, J., and Monney, P. A., Theory of evidence: A survey of its mathematical foundations, applications and computations, ZOR Math. Method Oper. Res. 39, 35-68, 1994. 8. Smets, P., "Nonstandard probabilistic and nonprobabilistic representations of uncertainty," in Advances in Intelligent Computing--IPMU'94 (B. BouchonMeunier, R. R. Yager, and L. A. Zadeh, Eds.), Springer-Verlag, Berlin, 13-38, 1995. 9. Dubois, D., Prade, H., "On several representations of an uncertain body of evidence," in Fuzzy Information and Decision Processes (M. M. Gupta and E. Sanchez, Eds.), North-Holland, Amsterdam, 309-322, 1982. 10. Dempster, A. P., Upper and lower probabilities induced by a multi-valued mapping, Ann. Math. Statist. 38, 325-339, 1967. 11. Dempster, A. P., A generalization of Bayesian inference, J. Roy. Statist. Soc., 205-247, 1968. 12. Yager, R. R., On the Dempster-Shafer framework and new combination rules, Inform. Sci. 41, 93-137, 1987. 13. Yen, J., Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Trans. Systems Man Cybernet. 20, 559-570, 1990. 14. Zadeh, L. A., Fuzzy sets and information granularity, in Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, and R. R. Yager, Eds.), North-Holland, Amsterdam, 3-18, 1979. 15. Goodman, I. R., and Nguyen, H. T., Uncertainty Models for Knowledge-Based Systems, North-Holland, Amsterdam, 1985. 16. Negoita, C. V., and Ralescu, D., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975. 17. Dubois, D., and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic, New York, 1980. 18. Yager, R. R., "Arithmetic and other operations on Dempster-Shafer structures," Internat. J. Man-Machine Stud. 25, 357-366, 1986. 19. Yager, R. R., "Measuring tranquility and anxiety in decision making: An application of fuzzy sets," Internat. J. Gen. Systems 8, 139-146, 1982. 20. Yager, R. R., "On the specificity of a possibility distribution," Fuzzy Sets and Systems 50, 279-292, 1992.
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21. Yager, R. R., "Entropy and specificity in a mathematical theory of evidence," Internat. J. Gen. Systems 9, 249-260, 1983. 22. Yager, R. R., and Filev, D. P., On the issue of defuzzification and selection based on a fuzzy set, Fuzzy Sets and Systems 55, 255-272, 1993. 23. Filev, D., and Yager, R. R., A generalized defuzzification method under BAD distributions, Internat. J. Intell. Systems 6, 687-697, 1991. 24. Yager, R. R., and Filev, D. P., Including probabilistic uncertainty in fuzzy logic controller modeling using Dempster-Shafer theory, IEEE Trans. Systems Man Cybernet. 25, 1221-1230, 1995.