Generalized Belief Function, Plausibility
Function, and Dempster’s Combinational
Rule to Fuzzy Sets
Miin-Shen Yang,* Tsang-Chih Chen, Kuo-Lung Wu†
Department of Applied Mathematics, Chung Yuan Christian University,
Chung-Li, Taiwan 32023, R.O.C.
Uncertainty always exists in nature and real systems. It is known that probability has been used
traditionally in modeling uncertainty. Since a belief function was proposed as an another type of
measuring uncertainty, Dempster-Shafer theory (DST) has been widely studied and applied in
diverse areas. Because of the advent of computer technology, the representation of human
knowledge can be processed by a computer in complex systems. The analysis of fuzzy data
becomes increasingly important. Up to date, there are several generalizations of DST to fuzzy
sets proposed in the literature. In this article, we propose another generalization of belief
function, plausibility function, and Dempster’s combinational rule to fuzzy sets. We then make
the comparisons of the proposed extension with some existing generalizations and show its
effectiveness. © 2003 Wiley Periodicals, Inc.
1.
INTRODUCTION
In 1967, Dempster1 proposed upper and lower probabilities using a multivalued mapping. Let (⍀, G, P) be a probability space and let T be a multivalued
mapping from ⍀ to S, which assigns a subset T( ␻ ) 傺 S to every ␻ 僆 ⍀; i.e., T
is a set-valued function from ⍀ to the power set P S of S. For any A 僆 P S , define
A * ⫽ 兵 ␻ 僆 ⍀兩T共 ␻ 兲 傺 A, T共 ␻ 兲 ⫽ A其
(1)
A* ⫽ 兵 ␻ 僆 ⍀兩T共 ␻ 兲 艚 A ⫽ A其
(2)
and
*Author to whom all correspondence should be addressed: e-mail: msyang@
math.cycu.edu.tw.
†
e-mail: [email protected].
INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 18, 925–937 (2003)
© 2003 Wiley Periodicals, Inc. Published online in Wiley InterScience
(www.interscience.wiley.com). • DOI 10.1002/int.10126
926
YANG, CHEN, AND WU
In particular, S* ⫽ S * ⫽ ⍀. Let H be a class of subsets A of S so that A * and A*
belong to G, Dempster1 defined the lower probability P and upper probability P*
*
of A in H as
P * 共A兲 ⫽ P共A * 兲/P共S*兲
and P*共A兲 ⫽ P共A*兲/P共S*兲
(3)
where P * ( A) and P*( A) are defined only if P(S*) ⫽ 0. It can be shown that
P * ( A) ⫽ 1 ⫺ P*( A c ), P ( A) ⫹ P ( A c ) 聿 1, P*( A) ⫹ P*( A c ) 肁 1, and
*
*
P * ( A) 聿 P*( A). Note that if T is a (single-valued) function, then T becomes a
random variable and P * ( A) ⫽ P*( A). If T is a multivalued mapping, then an
outcome ␻ may be mapped to more than one value. This is similar to one case
having more than two interpretations or points of view. Dempster2 then used it to
infer hypotheses and a generalized Bayesian inference.
Suppose that S is a finite set. Shafer3–5 defined a set function m : PS 3 [0,
1] with the conditions (1) m(A) ⫽ 0 and (2) ¥A僆PS m( A) ⫽ 1. The set function
m is called a basic probability assignment (BPA). An element A in PS with
m( A) ⫽ 0 is called a focal element. Note that if m is defined as m(B) ⫽ ¥T(␻)⫽B
P({␻}) with m(A) ⫽ 0 for any B 僆 PS and the mapping T is a function, then m
becomes an induced probability measure. Shafer called P* a belief (Bel)
function and P* a plausibility (Pl) function and showed that for any A 僆 PS,
Bel共A兲 ⫽ P*共A兲 ⫽
Pl共A兲 ⫽ P*共A兲 ⫽
冘 m共B兲
冘 m共B兲
B傺A
B艚A⫽A
Bel共A兲 ⫽ 1 ⫺ Pl共Ac 兲
In addition, it should be mentioned that Zadeh16 proposed a possibility theory
on the basis of fuzzy sets. Eventually, possibility becomes one type of
plausibility.
All of these (nonadditivity) measures may be regarded as measures for the
information content of an evidence, and the theory usually is called the DempsterShafer theory (DST; see Refs. 7–9). Since the advent of computer technology, the
representation of human knowledge can be processed by a computer in complex
systems. The analysis of fuzzy-valued data becomes increasingly important (see
Ref. 10). Yang et al.11–13 had treated fuzzy-valued data in clustering, possibility,
and fuzzy regression analysis. Zadeh14 first generalized the DST to fuzzy sets.
Afterward, more generalizations were made by several different researchers (see
Refs. 15–20). In this article, an advanced generalization of Bel, Pl, and Dempster’s
combinational rule to fuzzy sets is proposed. In Section 2, we propose the
generalized Bel and Pl functions in the DST by extending Bel and Pl functions to
fuzzy sets. We then make the comparisons with some other existing generalizations. Section 3 presents an extension of Dempster’s combination rule and conclusions are made in Section 4.
PLAUSIBILITY FUNCTION AND DEMPSTER’S COMBINATIONAL RULE 927
2.
A GENERALIZATION OF Bel AND Pl FUNCTIONS
TO FUZZY SETS
After Zadeh14 proposed information granularity and extended Shafer’s Bel
function to fuzzy sets, Ishizuka et al.,15 Ogawa et al.,17 and Yager19 made similar
extensions according to a different defined measure I(Ã 傺̃ B̃) of fuzzy inclusion
for which
Bel共B̃兲 ⫽
冘̂ I共Ã 傺̃ B̃兲m共Ã兲
(4)
A
They defined the measure of fuzzy inclusion as follows:
Ishizuka et al.15:
I I共Ã 傺̃ B̃兲 ⫽
minx 兵1, 1 ⫹ 共␮B̃ 共x兲 ⫺ ␮Ã 共x兲其
maxx ␮Ã 共x兲
(5)
¥x min兵␮Ã 共x兲, ␮B̃ 共x兲其
¥x ␮B̃ 共x兲
(6)
Ogawa et al.17:
I O共Ã 傺̃ B̃兲 ⫽
Yager19:
I Y共Ã 傺̃ B̃兲 ⫽ min兵max ␮Ã 共x兲, ␮B̃ 共x兲其
(7)
x
Afterward, Yen20 formulated linear programming problems for computing the Bel
and Pl functions of fuzzy sets. He showed that the optimal solutions of linear
programming problems are the minimum and maximum probability masses that
can be allocated to a fuzzy set B̃ from a (crisp) set A as follows:
m * 共B̃ : A兲 ⫽ m共A兲 ⫻ inf ␮B̃ 共x兲
x僆A
m*共B̃ : A兲 ⫽ m共A兲 ⫻ sup ␮B̃ 共x兲
(8)
x僆A
To deal with a fuzzy focal element Ã, Yen20 used the resolution form Ã
⫽ 艛␣ ␣Ã␣ and defined the probability assignment as follow:
m共Ã ␣i兲 ⫽ 共 ␣ i ⫺ ␣ i⫺1兲 ⫻ m共Ã兲
(9)
where ␣ 0 ⫽ 0, ␣ n ⫽ 1, ␣ i⫺1 ⬍ ␣ i , i ⫽ 1, . . . , n. He defined the probability
assignment that a fuzzy focal à contributes to a fuzzy set B̃ as
冘 m*共B̃ : Ã 兲
m*共B̃ : Ã兲 ⫽ 冘 m*共B̃ : Ã 兲
m * 共B̃ : Ã兲 ⫽
␣
␣
␣
␣
(10)
YANG, CHEN, AND WU
928
Finally, the formulas for computing the Bel and Pl functions of fuzzy sets are
obtained:
冘̃ m*共B̃ : Ã兲 ⫽ 冘̃ m共Ã兲 冘 共␣ ⫺ ␣
Pl共B̃兲 ⫽ 冘̃ m*共B̃ : Ã兲 ⫽ 冘̃ m共Ã兲 冘 共␣ ⫺ ␣
Bel共B̃兲 ⫽
i
A
A
␣i
A
A
␣i
i⫺1
i
i⫺1
兲 inf ␮B̃ 共x兲
x僆Ã ␣ i
兲sup ␮B̃ 共x兲
(11)
x僆Ã ␣ i
Note that if the conditional factor C(B̃ : Ã) ⫽ ¥ ␣ i ( ␣ i ⫺ ␣ i⫺1 )infx僆Ã ␣ ␮ B̃ ( x), of
Equation 11 is replaced by C(B̃ : Ã) ⫽ min{ x兩 ␮ Ã( x)⬎0} ␮ B̃ ( x), then the Bel
function of fuzzy sets becomes
i
Bel共B̃兲 ⫽
冘̃ m共Ã兲
min
兵x兩 ␮Ã 共x兲⬎0其
A
␮B̃ 共x兲
(12)
which is defined by Smets.18
Now, we extend the Bel and Pl functions to fuzzy sets along with Yen’s
approach.20 Let ␪ ␣ ⫽ { x兩 ␮ Ã ( x) ⫽ ␣ }, ␣ 僆 [0, 1]. Define
m * 共B̃ : Ã ␣兲 ⫽ m⬘共Ã ␣兲 ⫻ inf ␮B̃ 共x兲
(13)
x僆Ã ␣
m⬘共Ã ␣兲 ⫽ m共Ã兲 ⫻
兩 ␪ ␣兩
兩Ã兩
(14)
where 兩Ã兩 ⫽ ¥ x ␮ à ( x) and 兩 ␪ ␣ 兩 ⫽ ¥ x僆 ␪ ␣ ␮ à ( x) are the cardinalities of à and ␪˜ ␣
respectively. The Bel function of a fuzzy set B̃ is formulated as
冘̃ m*共B̃ : Ã兲
⫽ 冘̃ 冘 m 共B̃ : à 兲 共according to a resolution form à ⫽ 艛 ␣à 兲
*
⫽ 冘̃ 冘 m⬘共Ã 兲 ⫻ inf ␮ 共x兲
Bel共B̃兲 ⫽
A
␣
A
␣
A
␣
␣
␣
⫽
B̃
x僆Ã ␣
冘̃ m共Ã兲 冘 兩兩Ã兩␪ 兩 ⫻ inf ␮ 共x兲
␣
(15)
B̃
A
␣
␣
x僆Ã ␣
Similarly, the plausibility function of a fuzzy set B̃ is given as
Pl共B̃兲 ⫽
冘̃ m共Ã兲 冘 兩兩Ã兩␪ 兩 ⫻ sup ␮ 共x兲
␣
B̃
A
␣
(16)
x僆Ã ␣
The contribution of a fuzzy focal element à is decomposed to the contribution of
each crisp set à ␣ under level ␣. The contribution proportion of à ␣ to overall
contribution of à is considered as the membership proportion of elements with
membership ␣ to overall memberships. This approach can catch more information
PLAUSIBILITY FUNCTION AND DEMPSTER’S COMBINATIONAL RULE 929
to the change of fuzzy focal elements according to its generalization structure. We
now discuss some properties of our generalized Bel and Pl measures.
PROPERTIES.
(1) If B̃ 1 傺 B̃ 2 , then Bel(B̃ 1 ) 聿 Bel(B̃ 2 )
(2) Bel(B̃) ⫽ 1 ⫺ Pl(B̃ c )
(3) Bel(B̃) ⫹ Bel(B̃ c ) 聿 1 and Pl(B̃) ⫹ Pl(B̃ c ) 肁 1
where B̃1 傺 B̃2 if ␮B̃1(x) 聿 ␮B̃2(x) for all x and ␮B̃c(x) ⫽ 1 ⫺ ␮B̃(x).
Proof.
(1) If B̃ 1 傺 B̃ 2 then infx僆Ã ␣ ␮ B̃ 1( x) 聿 infx僆Ã ␣ ␮ B̃ 2( x) for all ␣, this implies that
Bel(B̃ 1 ) 聿 Bel(B̃ 2 ).
(2) 1 ⫺ Pl(B̃ c ) ⫽ 1 ⫺ ¥ Ã m(Ã) ¥ ␣ 兩 ␪ ␣ 兩/兩Ã兩 ⫻ supx僆Ã ␣ ␮ B̃ c( x) ⫽ ¥ Ã m(Ã) ¥ ␣ 兩 ␪ ␣ 兩/兩Ã兩
⫻ (1 ⫺ supx僆Ã ␣ ␮ B̃ c( x)) ⫽ Bel(B̃)
(3) Since 0 聿 Bel(B̃) 聿 1, 0 聿 Pl(B̃) 聿 1, and Bel(B̃) 聿 Pl(B̃) this implies ⫺1 聿 Bel(B̃) ⫺
Pl(B̃) 聿 0 and hence Bel(B̃) ⫹ Bel(B̃ c ) ⫽ Bel(B̃) ⫹ 1 ⫺ Pl(B̃) 聿 1. Similarly,
Pl(B̃) ⫹ Pl(B̃ c ) 肁 1
■
Here, the cardinality of 兩Ã兩 is considered finite. To compare the proposed Bel
function with the other Bel functions, we use an example similar to Yen’s20 to
illustrate.
Example 1. Let S ⫽ {1, 2, . . . , 10}. Let à and C̃ be fuzzy sets in S̃ with
à ⫽ 兵0.25/1, 0.5/2, 0.75/3, 1/4, 1/5, 0.75/6, 0.5/7, 0.25/8其
C̃ ⫽ 兵0.5/5, 1/6, 0.8/7, 0.4/8其
where each member of the list is in the form of ␮ Ã ( x i )/x i . Let B̃ be a fuzzy set in
P̃ S with
B̃ ⫽ 兵0.5/2, 1/3, 1/4, 1/5, 0.9/6, 0.6/7, 0.3/8其
The decomposition of the fuzzy focal à consists of four nonfuzzy focal elements:
à 0.25 ⫽ 兵1, 2, . . . , 8其
à 0.5 ⫽ 兵2, 3, . . . , 7其
à 0.75 ⫽ 兵3, 4, 5, 6其
à 1.0 ⫽ 兵4, 5其
with mass m共Ã0.25 兲 ⫽ 0.1 ⫻ m共Ã兲
with mass 0.2 ⫻ m共Ã兲
with mass 0.3 ⫻ m共Ã兲
with mass 0.4 ⫻ m共Ã兲
and the decomposition of the fuzzy focal C̃ consists of four nonfuzzy focal
elements:
YANG, CHEN, AND WU
930
C̃ 0.4 ⫽ 兵5, 6, 7, 8其
C̃ 0.5 ⫽ 兵5, 6, 7其
C̃ 0.8 ⫽ 兵6, 7其
C̃ 1.0 ⫽ 兵6其
with mass 0.148 ⫻ m共C̃兲,
with mass 0.186 ⫻ m共C̃兲,
with mass 0.296 ⫻ m共C̃兲,
with mass 0.370 ⫻ m共C̃兲.
Then,
m * 共B̃ : Ã兲 ⫽
冘̃ m共Ã兲 冘 兩兩Ã兩␪ 兩 ⫻ inf ␮ 共x兲
␣
B̃
A
␣
x僆Ã ␣
⫽ m共Ã兲共0.1 ⫻ 0 ⫹ 0.2 ⫻ 0.5 ⫹ 0.3 ⫻ 0.9 ⫹ 0.4 ⫻ 1兲
⫽ 0.77 ⫻ m共Ã兲
m * 共B̃ : C̃兲 ⫽ m共C̃兲共0.148 ⫻ 0.3 ⫹ 0.186 ⫻ 0.6 ⫹ 0.296
⫻0.6 ⫹ 0.370 ⫻ 0.9) ⫽ 0.6666 ⫻ m共C̃兲
Thus, we obtain
Bel共B̃兲 ⫽ 0.77 ⫻ m共Ã兲 ⫹ 0.666 ⫻ m共C̃兲
Similarly, we have
Pl共B̃兲 ⫽ m共Ã兲 ⫹ 0.9334m共C̃兲
The degrees of Bel in the fuzzy set B̃ computed using the other methods are listed
as follows:
Ishizuka et al.15: Bel(B̃) ⫽ 0.75m(Ã) ⫹ 0.8m(C̃)
Ogawa et al.17: Bel(B̃) ⫽ 0.8962m(Ã) ⫹ 0.434m(C̃)
Yager19: Bel(B̃) ⫽ 0.5m(Ã) ⫹ 0.6m(C̃)
Yen20: Bel(B̃) ⫽ 0.6m(Ã) ⫹ 0.54m(C̃)
We compare how these results are changed in response to a change in a fuzzy focal
element, such as Yen’s work,20 by setting
Ã⬘ ⫽ 兵0.166/1, 0.5/2, 0.833/3, 1/4, 1/5, 0.75/6, 0.5/7, 0.25/8其
Ã⬙ ⫽ 兵0.25/1, 0.75/2, 1/3, 1/4, 1/5, 0.75/6, 0.5/7, 0.25/8其
Ã⵮ ⫽ 兵0/1, 0.5/2, 0.75/3, 1/4, 1/5, 0.75/6, 0.5/7, 0.25/8其
Similar to Yen,20 Table I lists the portion of each modified focal mass that
contributes to B̃’s Bel measure and Table II shows how Bel(B̃) computed by
different methods changes as the focal element à changes in three ways. According
to the results of Table II, the comparison indicates that our method is similar to
Yen’s method20 in that it is more responsive to any change in a focal element’s
PLAUSIBILITY FUNCTION AND DEMPSTER’S COMBINATIONAL RULE 931
Table I. Contribution to Bel(B̃) from the focal element à and its variations.
Focal element
Yager19
Ishizuka et al.15
Ogawa et al.17
Yen20
Ours
Ã
Ã⬘
Ã⬙
Ã⵮
0.5
0.5
0.5
0.5
0.75
0.834
0.75
1
0.8962
0.9119
0.9434
0.8962
0.6
0.6252
0.5
0.675
0.77
0.8466
0.727
0.8263
membership function than the other methods. More explanation can be obtained by
referring to Yen.20
Example 2. In this example, we compare our Bel function with Smets’18 and
Yen’s20 Bel functions. Continuing with Example 1, recall that
B̃ ⫽ 兵0.5/2, 1/3, 1/4, 1/5, 0.9/6, 0.6/7, 0.3/8其
Let D̃ be a fuzzy set in S̃ with
D̃ ⫽ 兵0.75/2, 0.9/3, 1/4, 1/5, 0.5/6, 0.25/7, 0.1/8其
Now, compare how the results from Smets’,18 Yen’s,20 and our Bel functions are
changed in response to a change in a fuzzy focal element. Let D 1 ⬃ D 6 be fuzzy
sets constituting six changes in D̃ with
D̃ 1 ⫽ 兵1/4, 1/5其
D̃ 2 ⫽ 兵0.9/3, 1/4, 1/5其
D̃ 3 ⫽ 兵0.75/2, 0.9/3, 1/4, 1/5其
D̃ 4 ⫽ 兵0.75/2, 0.9/3, 1/4, 1/5, 0.5/6其
D̃ 5 ⫽ 兵0.75/2, 0.9/3, 1/4, 1/5, 0.5/6, 0.25/7其
D̃ 6 ⫽ 兵0.75/2, 0.9/3, 1/4, 1/5, 0.5/6, 0.25/7, 0.1/8其 ⫽ D̃
As in Example 1, we can analyze the impact on the Bel functions by comparing the
contributions of the focal element D̃ ⫽ D̃ 6 and its variations D̃ 1 ⫽ D̃ 5 to the
degree of Bel in B̃. Table III lists the portion of each modified focal’s mass that
contributes to B̃’s Bel measure [i.e., the ratio m * (B̃ : D̃)/m(D̃) for each method].
Table II. Changes to Bel(B̃) caused by changes in the focal element Ã.
Changes of focal
element of Ã
Yager19
Ishizuka et al.15
Ogawa et al.17
Yen20
Ours
à 3 Ã⬘
à 3 Ã⬙
à 3 Ã⵮
U
U
U
I
U
I
I
I
U
I
D
I
I
D
I
U, I, and D denote unchanged, increased, and decreased, respectively.
YANG, CHEN, AND WU
932
Table III. Contribution to Bel(B̃) from focal element and its variations.
Focal element of
Smets18
Yen20
Ours
D̃ 1
D̃ 2
D̃ 3
D̃ 4
D̃ 5
D̃ 6
1
1
0.5
0.5
0.5
0.3
1
1
0.625
0.625
0.625
0.605
1
1
0.8973
0.8494
0.8295
0.8178
Hence, the comparisons in Table III indicate that our generalization of the Bel
measure applied to fuzzy-valued data actually give a decreasing contribution to
Bel(B̃) from the changes D̃ 1 ⬃ D̃ 6 in the focal element in increasing fuzziness and
also responsive to every change in the focal element. However, those of Smets18
and Yen20 are not always responsive to a change in the focal element.
In general, Yen’s Bel function20 could not measure the changing information
to a focal element’s change unless it results in a change in the “critical point,” a
point in which its membership value is the minimal value on its decomposition of
the focal element. For example the focal element D̃ 2 can be decomposed into two
crisp sets D̃ 2,1.0 ⫽ {4, 5} and D̃ 2,0.9 ⫽ {3, 4, 5}. These two decomposed crisp
sets have the critical point with regard to fuzzy set B̃ with any of 3, 4, and 5
[because ␮ B̃ (3) ⫽ ␮ B̃ (4) ⫽ ␮ B̃ (5) ⫽ 1.0]. Thus, the contributions of D̃ 1 and D̃ 2
to B̃ are both the same because they do not result in any change in a critical point
in its minimal membership value. The focal elements D̃ 3 , D̃ 4 , and D̃ 5 have the
same critical point with 2 (its membership value ⫽ 0.5) for decomposed crisp sets
D̃ 3,0.75 , D̃ 4,0.75 , D̃ 4,0.5 , D̃ 5, 0.75 , and D̃ 5,0.25 . Thus, the contributions of D̃ 3 , D̃ 4 ,
and D̃ 5 to B̃ do not make any change (all with the contribution 0.25 ⫻ 1 ⫹ 0.75 ⫻
0.5 ⫽ 0.625). Until D̃ 6 , the critical point for D̃ 6,0.1 changes to 8 with a membership
value of 0.3. The contribution of D̃ 6 to B̃ changes to 0.25 ⫻ 1 ⫹ 0.65 ⫻ 0.5 ⫹
0.1 ⫻ 0.3 ⫽ 0.605.
On the other hand, Smets’ Bel function18 could not measure the changing
information to the change in a focal element either unless it results in a change in
the critical point, a point in which its membership value is the minimal value over
the focal element. Note that Smets18 did not consider the decomposition of a focal
element. For example, the critical point of D̃ 2 is any of 3, 4, and 5 with the
membership value of 1.0. The critical point of D̃ 3 , D̃ 4 , and D̃ 5 is 2 with the
membership value of 0.5 and that of D̃ 6 is 8 with the membership value of 0.3.
Thus, the contributions of D̃ 1 ⬃ D̃ 6 are 1.0, 1.0, 0.5, 0.5, 0.5, and 0.3.
It is seen that our method for computing Bel measures is on the basis of
weighting on the cardinality of the decomposition of the focal element. Our method
actually could catch the changing information to the change of focal elements. In
summary, the foregoing comparisons indicate that our method is more effective
and responsive to any change in fuzziness and membership value of a focal element
than the other methods.
The foregoing extension of Bel function to fuzzy sets is considered only for
兩Ã兩 finite and ␮ Ã ( x) discrete types. If 兩Ã兩 ⫽ ⬁, then we define
PLAUSIBILITY FUNCTION AND DEMPSTER’S COMBINATIONAL RULE 933
m * 共B̃ : Ã兲 ⫽ m共Ã兲 ⫻
Bel共B̃兲 ⫽
inf
兵x兩 ␮Ã 共x兲⬎0其
␮B̃ 共x兲
冘̃ m*共B̃ : Ã兲 ⫽ 冘̃ m共Ã兲 ⫻
A
A
inf
兵x兩 ␮Ã 共x兲⬎0其
␮B̃ 共x兲
(17)
This is similar to Smets’ method.18
3.
AN EXTENSION TO DEMPSTER’S COMBINATION RULE
Let Bel1 and Bel2 be two Bel functions of crisp sets over the same frame of
discernment according to two independent evidential sources. If m 1 and m 2 are
BPAs of Bel1 and Bel2, respectively, then the combined BPA for a crisp set C is
computed by Dempster’s rule of combination as
m 1 丣 m 2共C兲 ⫽
¥ A艚B⫽C m 1共A兲m 2共B兲
1 ⫺ ¥ A艚B⫽A m 1共A兲m 2共B兲
(18)
Ishizuka et al.15 extended Dempster’s rule of combination to fuzzy sets by taking
into account the degree J(Ã, B̃) of fuzzy intersection of two fuzzy sets as
m 1 丣 m 2共C̃兲 ⫽
¥Ã艚B̃⫽C̃ J共Ã, B̃兲m 1共A兲m 2共B兲
1 ⫺ ¥Ã,B̃ 共1 ⫺ J共Ã, B̃兲兲m 1共Ã兲m 2共B̃兲
(19)
maxx ␮Ã艚B̃ 共x兲
min兵maxx ␮Ã 共x兲, maxx ␮B̃ 共x兲其
(20)
where
J共Ã, B̃兲 ⫽
Yen20 extended Dempster’s rule of combination to fuzzy sets by using two
operators with a cross-product operation and a normalization process as follows:
(1) Cross-product
m⬘共C̃兲 ⫽ m 1 丢 m 2共C̃兲 ⫽
冘
m 1共Ã兲m 2共B̃兲
Ã艚B̃⫽C̃
(2) Normalization
N关m⬘兴共D̃兲 ⫽
៮ D̃ maxx ␮C̃ 共x兲m⬘共C̃兲
¥C⫽
1 ⫺ ¥C̃ 共1 ⫺ maxx ␮C̃ 共x兲兲m⬘共C̃兲
where C៮ is the normalization of C̃. The normalization of fuzzy focal elements is
needed for Yen’s approach.20 Suppose that à ⫽ {0.5/1, 0.8/ 2}, the decomposition of this fuzzy focal element for computing the Bel function are à 0.5 ⫽ {1,
2} with mass 0.5m(Ã), à 0.8 ⫽ {2} with mass 0.3m(Ã), and à 1.0 ⫽ A with mass
0.2m(Ã). The empty set A will give contributions and the focal elements are
normalized to avoid this problem. In the special case that there are only two fuzzy
BPAs to be combined and all fuzzy focal elements are normalized, the combined
BPA of a subnormal fuzzy set can be
YANG, CHEN, AND WU
934
Table IV. BPAs m 1 and m 2 of two experts for à 1 ⬃ à 5 .
Assignment of experts
à 1
à 2
à 3
à 4
Experts 1 (m 1 )
Experts 2 (m 2 )
0.6
0
0.2
0.4
0.2
0.3
0
0.1
m 1 丣 m 2共C̃兲 ⫽
à 5
0
0.2
¥Ã艚B̃⫽C̃ maxx ␮Ã艚B̃ 共x兲m1 共Ã兲m2 共B̃兲
1 ⫺ ¥Ã,B̃ 共1 ⫺ maxx ␮Ã艚B̃ 共x兲兲m1 共Ã兲m2 共B̃兲
which will be equivalent to Ishizuka’s generalization.15
We propose another extension of Dempster’s rule of combination to fuzzy sets
by constructing a weighted variable W(C̃, Ã), which expresses the weight of
contribution to the fuzzy set C̃ from a focal element Ã. It is defined as
W共C̃, Ã兲 ⫽
兩C̃兩
兩Ã兩
(21)
where 兩B̃兩 ⫽ ¥ x ␮ B̃ ( x) is the cardinality of B̃. The combined fuzzy BPA m 1 Q m 2
of two fuzzy BPAs m 1 and m 2 is defined as
m 1 丣 m 2共C̃兲 ⫽
1 ⫺ ¥Ã,B̃
¥Ã艚B̃ ⫽C̃ W共C̃, Ã兲m 1共Ã兲W共C̃, B̃兲m 2共B̃兲
共1 ⫺ W共Ã 艚 B̃, Ã兲W共Ã 艚 B̃, B̃兲兲m 1共Ã兲m 2共B̃兲
(22)
Example 3. Let S ⫽ {2, 3, 4, 5, 6} and let all focal elements be as follows:
à 1 ⫽ 兵0.75/2, 0.5/3, 0.75/4, 1/5其
à 2 ⫽ 兵0.5/3, 1/4, 0.5/5其
à 3 ⫽ 兵0.25/2, 1/3, 0.75/4其
à 4 ⫽ 兵0.5/5, 1/6其
à 5 ⫽ 兵0.25/2, 1/4, 0.75/6其
Assume that there are two experts assigning the two BPAs m 1 and m 2 listed in
Table IV. The three different methods are used to compute the combined fuzzy
BPA m 1 Q m 2 with the results listed in Table V. To make comparisons of different
methods, how the the results are changed in response to a change in the fuzzy focal
element à 1 can be seen. We change à 1 to Ã⬘1 , à ⬙1 , and Ã⵮1 so that the combined
fuzzy focal elements C̃ 1 to C̃ 8 are not changed where Ã⬘1 , Ã ⬙1 , and Ã⵮1 are listed
as
Ã⬘1 ⫽ 兵1/2, 0.5/3, 0.75/4, 1/5其
à ⬙1 ⫽ 兵0.5/2, 0.5/3, 0.75/4, 1/5其
Ã⵮1 ⫽ 兵0.75/2, 0.5/3, 0.75/4, 0.5/5其
PLAUSIBILITY FUNCTION AND DEMPSTER’S COMBINATIONAL RULE 935
Table V. Combined focal elements and BPAs for different methods.
Combined focal element
C̃ 1
C̃ 2
C̃ 13
C̃ 14
C̃ 5
C̃ 16
C̃ 23
C̃ 7
C̃ 26
C̃ 8
C̃ 24
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
à 1
à 1
à 1
à 1
à 2
à 2
à 2
à 2
à 3
à 3
à 3
艚
艚
艚
艚
艚
艚
艚
艚
艚
艚
艚
à 2
à 3
à 4
à 5
à 2
à 3
à 4
à 5
à 2
à 3
à 5
Ishizula et al.15
Yen20
Ours
0.2416
0.1812
0.0403
0.1208
0.1074
0.0604
0.0134
0.0537
0.0805
0.0604
0.0403
0.2416
0.1812
0.0403
0.1208
0.1074
0.0604
0.0134
0.0537
0.0805
0.0604
0.0403
0.2851
0.1571
0.0078
0.0465
0.1862
0.0545
0.0039
0.0233
0.0727
0.1396
0.0233
C̃ 1 ⫽ {0.5/3, 0.75/4, 0.5/5}, C̃ 2 ⫽ {0.25/ 2, 0.5/3, 0.75/4}, C̃ 13 ⫽
C̃ 23 ⫽ {0.5/5}, C̃ 14 ⫽ C̃ 24 ⫽ {0.25/ 2, 0.75/4}, C̃ 5 ⫽ {0.5/3, 1/4, 0.5/
5}, C̃ 16 ⫽ C̃ 26 ⫽ {0.5/3, 0.75/4}, C̃ 7 ⫽ {1/4}, and C̃ 8 ⫽ {0.25/ 2,
1/3, 0.75/4}.
The results in Table VI show how the combined BPA calculated by different
methods changes as the fuzzy focal element à 1 changes with regard to Ã⬘1 , à ⬙1 ,
and Ã⵮1 .
The comparisons in Table VI indicate that our generalized Dempster’s rule of
combination applied to fuzzy sets could catch more changing informations than the
others. Generally, Ishizuka’s methods15 could not measure the changing information to changes in a focal element unless it results in a change in the degree J(Ã,
B̃) of fuzzy intersection. Yen’s method20 could not measure the changing information to the change in a focal element either because it directly extends Dempster’s rule of combination and then uses the normalization process.
It is mentioned that Isizuka’s,15 Yen’s,20 and our extensions of Dempster’s
rule of combination are not associative, i.e.,
Table VI. Changes to combined BPA caused by changes in fuzzy focal element.
à 1 3 à ⬙1
à 1 3 Ã⬘1
à 1 3 Ã⵮1
Combined
focal element
Ishizuka15
Yen20
Ours
Ishizuka15
Yen20
Ours
Ishizuka15
Yen20
Ours
C̃ 1
C̃ 2
C̃ 13
C̃ 14
C̃ 5
C̃ 16
C̃ 23
C̃ 7
C̃ 26
C̃ 8
C̃ 24
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
D
D
D
D
I
I
I
I
I
I
I
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
U
I
I
I
I
D
D
D
D
D
D
D
I
I
I
I
D
D
D
D
D
D
D
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
⫻
I
I
I
I
D
D
D
D
D
D
D
U, I, D, and ⫻ denote unchanged, increased, decreased, and cannot calculate, respectively.
YANG, CHEN, AND WU
936
共m 1 丣 m 2兲 丣 m 3共C̃兲 ⫽ m 1 丣 共m 2 丣 m 3兲共C̃兲
It is known that association is important for any evidence combination rule. For
solving this problem, our generalization is divided into two steps that are analogous
to Yen’s process.20 Hence, we first calculate fuzzy BPAs by applying Dempster’s
rule of combination directly to fuzzy focal elements and then continue to combine
all fuzzy BPAs without normalization. Finally, the normalization process using a
weighted variable is applied at the end so that the summation of fuzzy BPAs is
equal to one. The two steps in the combination rule are formulated as follows:
Step 1.
m 1 丣 · · · 丣 m n共C̃兲 ⫽
冘
m 1共Ã 1兲 · · · m n共Ã n兲 ⫽ m 1· · ·n共C̃兲
Ã1艚· · ·艚Ãn⫽C̃
Step 2.
N关m1 丣 · · · 丣 mn兴共B̃兲 ⫽
¥C̃⫽B̃ W共B̃ : Ã1兲 · · · W共B̃ : Ãn兲m1· · ·n共C̃兲
1 ⫺ ¥Ã1,...,Ãn 共1 ⫺ W共B̃ : Ã1兲 · · · W共B̃ : Ãn兲兲m1· · ·n共C̃兲
4.
CONCLUSIONS
In this article, the generalized DST to fuzzy sets was considered. We especially focused on generalized Bel, Pl functions and Dempster’s combination rule to
fuzzy sets. We extended the Bel and Pl functions for processing fuzzy data and
showed that our generalization could catch more changing information to the
change of fuzzy focal elements than Yen’s20 and others. We also gave good
properties of the generalized Bel and Pl functions. In the generalization of Dempster’s combination rule, the degree of fuzzy intersection of two fuzzy sets was
divided into two weighted variables. The proposed combination rule is more
effective to the change of fuzzy focal elements than Ishizuka’s15 and Yen’s20
generalization.
Acknowledgment
This work was supported in part by the National Science Council of Taiwan, R.O.C., under
the Grant NSC-87-2118-M-033-001.
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