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Dempster – Shafer methods (Dempster, 1967) use dual truth values: a lower level, calledbelief, representing the extent to which the evidence supports a hypothesis; and an upper level, calledplausibility, representing the extent to which the evidence fails to refute the hypothesis. These are closely analogous to the dual measures

necessity and possibilityin fuzzy systems theory. The method is concerned with

combining evidence regarding the truth of a hypothesis from different sources. Our presentation here is paraphrased from that of Jackson (1999), Chapter 21. We seek to establish belief and plausibility of some set of hypotheses from evidence. The representation and manipulation of possibility and necessity in rule-based systems will be taken up in Chapter 8.

Ahypothesis spacein Dempster–Shafer theory is represented byQ, a space that

holds all the individual hypotheses hi. All hypotheses are assumed to be mutually exclu- sive, and the set of hypothesesQis assumed to be exhaustive. We assume that it is poss- ible to obtain evidence that each single subset ofQ, A1, A2,. . ., is true. (A subset Aimay be a single hypothesis, or may be the entire hypothesis setQ.) The hypotheses in each subset may overlap those in other subsets. We also have pieces of evidence yi, included in a setC. Each piece of evidence will point to a subset AiofQthat holds all the hypoth- eses that are supported by yj; the subset Aito which yjpoints is called afocal element. Since the hypotheses are exhaustive, that is, that there is at least one hypothesis consistent with every evidence, no evidence will point to a null set.

Key to the Dempster – Shafer method is the idea of aprobability assignment.A

basic probability assignment(bpa) is defined as a function m(Ai) that maps each

subset Ai of the hypotheses to a value included in [0, 1]. The sum of all m(Ai) over all subsets ofQis 1. The belief Bel in any focal element A is the sum of all the basic probability assignments for all subsets of A:

Bel(A)¼X

B.A

m(B) (5:6)

The plausibility Pls of A represents the evidence is consistent with A: Pls(A)¼X

A>B

m(B) (5:7)

The importance of the Dempster – Shafer method is that it furnishes a method of combining beliefs based on different evidence. Let Bel1and Bel2denote to belief functions. To these belief functions there will correspond two basic probability assignments, m1and m2. We now wish to compute a new basic probability assign- ment m(A)¼m1m2(A) and a new belief Bel(A)¼Bel1(A)Bel2(A) based on the combined evidence. Dempster’s rule is

m(A)¼m1m2¼ X x>y¼A m1(X)m2(Y) 1 X X>Y¼0 m1(X)m2(Y) ! (5:8)

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 hypothesesQ. There is no unique way of com- bining the evidence, but a standard way is given by

m1,2(A)¼ X B>C¼A m1(B)m2(C)=(1K) (5:9) where K¼ X B>C¼0 m1(B)m2(C) (5:10)

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 m1and m2, the correspond- ing measures of belief Bel1and Bel2, and the combined evidence m1,2and belief Bel1,2using 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 inherentlynotmutually 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

Focal Elements

Expert 1 Expert 2

Combined Evidence

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

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 A¼12truth(A).]

Almost all definitions of the AND and OR logical operators fail to obey the clas- sical laws of Excluded Middle (P and NOT P¼0) and Non-Contradiction (P OR NOT P¼1), which some find disconcerting. The Zadehian max – min logic has the advantage that it is idempotent (A AND A¼A, A OR A¼A) 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 coefficient 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 insufficient historical data to establish a reliable correlation coefficient 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 five 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 defined variable; test of attribute’s truth value against a literal; and test of attribute’s truth value against a previously defined 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 defined 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 C¼A AND B, and D¼A OR B are defined 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

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

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

5.2 When combining the fuzzy numbers A and NOT A, what fuzzy logical oper- ators should be used?

5.3 When combining the fuzzy quantities A and A, what fuzzy logical operators should be used?

5.4 Should the bounded sum and difference operators be used when combining semantically inconsistent membership functions in

a. Fuzzy control applications?

b. In general-purpose fuzzy reasoning applications?

5.5 Temperature is a scalar whose value is 78 and whose truth value is 0.6. What are the truth values of the following antecedent clauses?

a. “Temperature¼75”

b. “Temperature is 78”

c. “Temperature is,X.”

d. “size is Large”. (The grade of membership of Large in fuzzy set size is 0.356.)

e. “Temperature.cf.0.5”?

5.6 We have two fuzzy numbers, A and B, shown below in Figure Question 5.6.

a. What is the truth value of the proposition “A¼B”?

b. Of the proposition “A,B”?

c. Of the proposition “A.¼B”?

5.7 Using the fuzzy numbers A and B in Question 5.6, we wish to construct the fuzzy numbers A AND B and A OR B, with the min – max logic as our default. Should we use min – max or the bounded operators in combining these two fuzzy numbers?

5.8 Assume that the fuzzy numbers A and B in Question 5.5 are in fact member- ship functions used to describe the same numeric quantity. Should we use the default min – max or the bounded operators in combining these two fuzzy numbers?

5.9 What is the main problem with the use of Bayesian methods?

5.10 What is the relation between Dempster – Shafer methods and fuzzy logic?

5.11 In the lack of any knowledge about a hypothesis, what is its possibility? Its necessity?