X

Pls(A) ¼ m(B) (5:7)

A>B

The importance of the Dempster“ Shafer method is that it furnishes a method of

combining beliefs based on different evidence. Let Bel1 and Bel2 denote to belief

functions. To these belief functions there will correspond two basic probability

assignments, m1 and m2. We now wish to compute a new basic probability assign-

ment m(A) ¼ m1 È m2(A) and a new belief Bel(A) ¼ Bel1(A) È Bel2(A) based on

the combined evidence. Dempster™s rule is

!

X X

m(A) ¼ m1 È m2 ¼ m1 (X)m2 (Y) 1 À m1 (X)m2 (Y) (5:8)

x>y¼A X>Y¼0

Bel(A) ¼ Bel1 È Bel2 can now be computed by (5.8).

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95

5.5 THE DEMPSTER“SHAFER METHOD

We might also wish to combine evidence from two different sources. [Our treat-

ment of this topic is taken from Klir and Yuan (1995), pp 183 ff.] We need basic

probability assignments m1 and m2 for the set of all hypotheses and for all its

subsets, the power set of the set of hypotheses Q. There is no unique way of com-

bining the evidence, but a standard way is given by

X

m1,2 (A) ¼ m1 (B) Á m2 (C)=(1 À K) (5:9)

B>C¼A

where

X

K¼ m1 (B) Á m2 (C) (5:10)

B>C¼0

Klir and Yuan (1996) also give a simple example of Bayes™ method to a problem

of the origin of a painting. They have three hypotheses: the painting is by Raphael

(hypothesis R); by a disciple of Raphael (hypothesis D); or a counterfeit (hypothesis

C). Two experts examine the painting, and provide basic probability assignments m1

and m2, respectively, for the origin of the painting; R, D, C, R < D, R < C, D < C,

and R < D < C. Table 5.5 shows the basic assignments m1 and m2, the correspond-

ing measures of belief Bel1 and Bel2, and the combined evidence m1,2 and belief

Bel1,2 using the Dempster “Shafer formulas.

An advantage of Dempster “ Shafer over Bayesian methods is that Dempster “

Shafer does not require prior probabilities; it combines current evidence.

However, a great deal of current evidence is required for a sizeable set of hypoth-

eses, and if this is available the method is computationally expensive. For fuzzy

expert systems, there is an important failure of Dempster “ Shafer; the requirement

that the hypotheses be mutually exclusive. Since the members of fuzzy sets

are inherently not mutually exclusive, this raises doubts as to the applicability of

Dempster “ Shafer in fuzzy systems. Nevertheless, Baldwin™s FRIL system

(Baldwin et al., 1995) uses a generalization of Dempster“ Shafer to good effect.

TABLE 5.5 Example of Dempster “ Shafer Method

Combined

Expert 1 Expert 2 Evidence

Focal

Elements m1 Bel1 m2 Bel2 m1,2 Bel1,2

R 0.05 0.05 0.15 0.15 0.21 0.21

D 0 0 0 0 0.01 0.01

C 0.05 0.05 0.05 0.05 0.09 0.09

R<D 0.15 0.2 0.05 0.2 0.12 0.34

R<C 0.1 0.2 0.2 0.4 0.2 0.5

D<C 0.05 0.1 0.05 0.1 0.06 0.16

R<D<C 0.6 1 0.5 1 0.31 1

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96 COMBINING UNCERTAINTIES

5.6 SUMMARY

A problem faced by users of multivalued logics is to select which of a wide variety

of the logical operators AND and OR to use in evaluating a complex fuzzy logical

proposition such as those found in the antecedent of fuzzy rules. [Almost everyone

uses the same operator for NOT: NOT A ¼ 1 2 truth(A).]

Almost all de¬nitions of the AND and OR logical operators fail to obey the clas-

sical laws of Excluded Middle (P and NOT P ¼ 0) and Non-Contradiction (P OR

NOT P ¼ 1), which some ¬nd disconcerting. The Zadehian max “min logic has

the advantage that it is idempotent (A AND A ¼ A, A OR A ¼ A) and there is

an enormous amount of experience with it. In an attempt to rescue the classical

laws for fuzzy logic, we devised a family of operators for AND and OR that pre-

serves the classical laws. The family has one parameter, the correlation coef¬cient

between the truth values of the operands obtained either from past experience or

from the structure of the logical expression being evaluated, if the expression con-

tains both A and NOT A, where A is a logical proposition.

If the expression being evaluated does not include both a proposition and its

negation, and if there is insuf¬cient historical data to establish a reliable correlation

coef¬cient between elements of the complex proposition, the user has a free choice

of any operator pair for AND and OR, without violating either excluded middle or

non-contradiction. We suggest that the Zadehian max “min operator pair is a

desirable default. The Zadeh operators have the nicest mathematical properties;

there is a great deal of experience with them; and they do not restrict the complexity

of rule antecedents.

The most important need for fuzzy logic is in evaluating the antecedent of a rule.

We list ¬ve types of antecedent clauses and discuss the evaluation of their truth

values: test of truth value of discrete fuzzy set member; test of attribute value

against a literal; test of attribute value against previously de¬ned variable; test of

attribute™s truth value against a literal; and test of attribute™s truth value against a

previously de¬ned variable. For an antecedent clause of the type

A (comparison operator) B

the truth value of equal to the truth value of A; the truth value of the comparison; and

the truth value of B. In many cases, such as literal values for B, the truth value of B

or C will be one by default.

Fuzzy numbers may be combined in a similar fashion, except that the truth values

are no longer scalars, but are functions of numbers from the real line. Suppose that

fuzzy number A and B are de¬ned as a(x) and b(x), where a(x) and b(x) are the

grades of membership of x in A and B, respectively. Then the fuzzy numbers

C ¼ A AND B, and D ¼ A OR B are de¬ned by

C = A AND B, c(x) = a(x) AND b(x) for all x on real line

D = A OR B, d(x) = a(x) OR b(x) for all x on real line

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97

5.7 QUESTIONS

In effect, we calculate the union (or intersection) of two fuzzy numbers point by

point.

ANDing or ORing fuzzy numbers using the Zadehian max “min logic can give

rather peculiar results. Of particular interest is the combining of fuzzy numbers

such as in “less than OR equal to”, useful in approximate numerical comparisons.

While there is more theoretical work to be done here, the use of the concepts in

the parameterized family of logics in Section 5.1.1 can produce more sensible

results, as shown in Figures 5.1 “5.3, since the laws of Excluded Middle and

Non-Contradiction are preserved.

5.7 QUESTIONS

What properties do the classical logic operators posses that are not shared by

5.1

fuzzy logic operators?

When combining the fuzzy numbers A and NOT A, what fuzzy logical oper-

5.2

ators should be used?