Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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Several ideas of how to measure the fuzziness of fuzzy sets (i.e., standard

fuzzy sets) have been pursued in the literature. One of them, which is currently

predominant in the literature and is followed in this section, is to measure the

fuzziness of a fuzzy set by the lack of distinction between the fuzzy set and its

complement.This is a sound idea. It is precisely the lack of distinction between

sets and their complements that distinguishes fuzzy sets from crisp sets. The

less a set differs from its complement, the fuzzier it is.

Measuring fuzziness in terms of distinctions between sets and their com-

plements is dependent on the deﬁnition of the complementation operation.

The simplest way of expressing the local distinctions (one for each x ŒX) of

a given set A and its complement, c(A), is to calculate the difference

The maximum of this difference, obtained for crisp sets, is 1. The lack of dis-

tinction between A and its complement is thus expressed for each x Œ X by

the value

When X is ﬁnite, a measure of fuzziness, f(A), of the set A is then obtained by

adding these values for all x in the support of A, S(A). That is,

(8.6)

or, alternatively,

(8.7)

Equations (8.6) and (8.7) are justiﬁed by the fact that 1 -|A(x) - c[A(x)]|=0

for all x œ S(A). Clearly,

where f(A) = 0 if and only if A is a crisp set, and f(A) =|X| if and only if

for all x Œ X, which means that the set is equal to its complement. When f(A)

is divided by |X|, we obtain a normalized value, f

(A), of fuzziness. Clearly,

When X = [x

, x

] and S(A) = [a, b] 債 X, the summation in Eq. (8.6) must be

replaced with integration. That is,

01£

()

.fA

Ax c Ax

()

[]

0 £

()

£fA X,

fA SA Ax cAx

xSA

()

[]

()

fA Ax cAx

xSA

()

[]

()

1 -

()

[]

Ax c Ax .

Ax c Ax

()

[]

322 8. FUZZIFICATION OF UNCERTAINTY THEORIES

(8.8)

or alternatively,

(8.9)

Clearly,

and for the associated normalized version, f

(A) = f(A)/(x

- x

–

When c is the standard fuzzy complement, Eq. (8.7) becomes

(8.10)

and Eq. (8.9) becomes

(8.11)

EXAMPLE 8.3. To calculate the amount of fuzziness of the fuzzy relation R

in Example 8.1 by Eq. (8.10) (i.e., assuming the use of standard operation of

complementation), it is convenient to ﬁrst calculate the following matrix of

the differences |2R(x) - 1| for all pairs x =·x

, x

ÒŒS(R):

The sum of all these differences is 14.2, |X|=49, and |S(R)|=25. Hence, f(R)

= 25 - 14.2 = 10.8 and f

(R) = 10.8/49 = 0.22.

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1 234567

—

——

fA b a Ax dx

()

=--

()

21.

fA SA Ax

xSA

()

21,

01£

()

.fA

0 £

()

£-fA x x

fA b a Ax cAx dx

()

=--

()

[]

fA Ax cAx dx

()

[]

()

8.2. MEASURES OF FUZZINESS 323

EXAMPLE 8.4. To illustrate the use of Eq. (8.11), let

for all x Œ X = [0, 5]. It is obvious that the support of A is the interval [0, 2].

Hence,

The expression 4x - 2x

- 1 is positive within the interval [x

, x

] and negative

within the intervals [0, x

] and [x

, 2], where x

= (2

+ Hence,

and f

(A) = 0.78/5 = 0.156.

Two alternative functionals for measuring the fuzziness of fuzzy sets are

well known in the literature. Both of them are based on the idea that fuzzi-

ness is manifested by the lack of distinction between a fuzzy set and its com-

plement. One of them, say functional f ¢, expresses this lack of distinction by

the standard intersection of the set and its complement. That is,

(8.12)

when X is ﬁnite, or

(8.13)

when S(A) = [a, b]. The minimum, f ¢(A) = 0, is clearly obtained only if A is a

crisp set. The maximum is obtained when

for all x ŒX. The value for which this equation is satisﬁed is called an equi-

librium of complement c. For example, the equilibrium of the standard com-

plement is 0.5. Denoting the equilibrium of complement c by

c, it is obvious

that

Ax c Ax

()

[]

()

() ()

[]

{}

fA AxcAx dx

min ,

()

() ()

[]

{}

()

fA AxcAx

xSA

min ,

f A x x dx x x dx

xxdx

()

=- + -

()

---

()

-+ -

()

ÚÚ

2124 421

12 4 078

)

222=-

()

and

fA x x dx

()

=- - -

242 1

Ax x x

()

{}

max ,02

324 8. FUZZIFICATION OF UNCERTAINTY THEORIES

when X is ﬁnite, or

when X = (x, x

). For the standard fuzzy complement, it is easy to show that

f(A) = 2f ¢(A) for any fuzzy set A, regardless whether it is deﬁned on a ﬁnite

set X or on ⺢. In this case, f and f ¢ differ only in the measurement unit, which

does not affect their normalized versions. Hence, f

(A) = f

¢(A) for all fuzzy

sets.

Another functional, f ≤, extensively covered in the literature, is applicable

only to the standard fuzzy complement. It is deﬁned by the formula

(8.14)

when X is ﬁnite, or by the formula

(8.15)

when X = [x, x

]. This functional is based on the recognition that for each x Œ

X, the values of A(x) and the standard fuzzy complement, 1 - A(x), add to 1.

Hence, for each x Œ X, the pair ·A(x), 1 - A(X)Ò may be viewed as a proba-

bility distribution on two elements. The functional utilizes, for each x Œ X, the

Shannon entropy of this elementary distribution, S(A(x), 1 - A(x)), as a con-

venient measure of the lack of distinction between A(x) and 1 - A(x).

EXAMPLE 8.5. To compare functionals f and f ≤, let us repeat Example 8.3

for f ≤. First, for each entry of the matrix we calculate the Shannon entropy

S(R(x), 1 - R(x)):

After adding all entries in this matrix, we obtain f ≤(R) = 13.2 and f

≤(R) =

13.2/49 = 0.27.

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1234567

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072

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000

072

000

097

000

047

100

097

000

000.

0000

000

047

100

000

047

000

047

000

100

000

100

000

100

000

¢¢

()

() ()

()()

[]

f A Ax Ax Ax Ax dxlog log

⺢

¢¢

()

() ()

()()

[]

f A Ax Ax Ax Ax

log log

0 £ ¢

()

£-

()

◊fA x x c

0 £ ¢

()

£◊fA X c

8.2. MEASURES OF FUZZINESS 325

EXAMPLE 8.6. To illustrate the use of Eq. (8.15), let us repeat Example 8.4

for f ≤. We have,

Moreover, f

≤(A) = 1.18/5 = 0.236.

8.3. FUZZY-SET INTERPRETATION OF POSSIBILITY THEORY

The term “possibility theory” is used in this section for the theory of graded

possibilities introduced in Section 5.2. As a formal mathematical system, pos-

sibility theory has various interpretations, some of which are mentioned in

Section 5.2.6. Perhaps the most visible and useful interpretation of possibility

theory, which is the subject of this section, is its fuzzy-set interpretation. In this

interpretation, possibility proﬁles are derived from information expressed in

terms of fuzzy sets.

In order to explain this interpretation of possibility theory, let X denote a

variable that takes values on a universal set X, and assume that information

about the actual value of the variable is expressed by a proposition “X is F,”

where F is, in general, a fuzzy set on X.This clearly also covers the special case

when F is a crisp set or even a singleton. To express information captured

by this fuzzy proposition in measure–theoretic terms, it is natural, to interpret

the membership degree F(x) for each x ŒX as the degree of possibility that X

= x. This interpretation induces a unique possibility proﬁle r

on X that is

deﬁned for all x ŒX by the equation

(8.16)

Given this possibility proﬁle, the corresponding possibility measure, Pos

,is

then determined for all A ŒP(X) by the formula

(8.17)

where c

denotes the characteristic function of A. When A is a fuzzy set, the

characteristic function in Eq. (8.17) is replaced with the membership function

of A, which results in the more general formula

Pos A x r x

()

() (){}{}

sup min , ,c

rx Fx

()

¢¢

()

=- -

()

+- +

()

[]

=+ -

()

---

()

fA xx xx xx xx dx

xxxxxx

xxx x

2 2 12 12

33 12

log log

log log log

log ++

()

118..

326 8. FUZZIFICATION OF UNCERTAINTY THEORIES

(8.18)

The use of Eqs. (8.17) and (8.18) is illustrated in Figure 8.3a and 8.3b,

respectively.

Equation (8.18) is also applicable when X is a multidimensional variable

and F is then a fuzzy relation, generally n-dimensional (n ≥ 2); the induced

possibility distribution function r

is also n-dimensional in this case. When the

given fuzzy proposition is truth qualiﬁed, probability qualiﬁed, or modiﬁed in

some other way, Eq. (8.18) is still applicable, provided that the fuzzy set F in

Eq. (8.16) represents a relevant composition of functions involved in the mod-

iﬁcation, as is explained in Section 7.6.1.

The fuzzy-set interpretation of possibility theory emerges quite naturally

from the similarity between the mathematical structures of possibility mea-

sures (or, alternatively, necessity measures) and fuzzy sets. In both cases, the

Pos A A x r x

()

() (){}{}

sup min , .

8.3. FUZZY-SET INTERPRETATION OF POSSIBILITY THEORY 327

A (x)

F(x)=r (x)

F(x)=r

c (x)

Pos (A)

r (a)=Pos ({a})

Pos (A)

(a)

(b)

(x)

Figure 8.3. Illustration of (a) Eq. (8.17); (b) Eq. (8.18).

underlying mathematical structures are families of nested sets. In possibility

theory, these families consist of focal elements; in fuzzy sets, they consist of

a-cuts.

Equation (8.16) is usually referred to as the standard fuzzy-set interpreta-

tion of possibility theory. This interpretation is simple and natural, but it is

applicable only to fuzzy sets that are normal. Recall that a fuzzy set F is normal

when its height, h

, is equal to 1. When a subnormal fuzzy set F is involved in

Eq. (8.16), for which h

< 1 by deﬁnition, then

and, by Eq. (8.18),

This violates one of the axioms of possibility theory, and hence, the theory

looses its coherence.To illustrate this problem, let S

and S

denote the support

of F and its complement, respectively. When h

< 1 for the fuzzy set F in Eq.

(8.16), we obtain

by Eq. (8.18). Then, by the duality between Pos and Nec, expressed by Eq.

(5.1), we have

Hence,

which violates one of the key properties of possibility theory: whatever is

necessary must be possible at least to the same degree, as expressed by the

inequality in Eq. (5.12).

To assume that F in Eq. (8.16) is always normal is overly restrictive. For

example, if the given fuzzy proposition “X is F” represents the conjunction of

several fuzzy propositions that express information about X obtained from

disparate sources, it is quite likely that F is subnormal. In this case, the value

of 1 - h(F ) indicates the degree of inconsistency among the individual infor-

mation sources. It is thus important that a sound fuzzy-set interpretation of

possibility theory be coherent for all fuzzy sets, regardless of whether they are

normal or not.

In order to formulate a genuine fuzzy-set interpretation of possibility

theory, we need to address the following question: How should we interpret

Nec S Pos S

FF FF

()

Nec S

()

= 1.

Pos S h Pos S

FF F FF

()

=and 0

Pos X h

()

=<1.

sup

rx h

(){}

=<1

328 8. FUZZIFICATION OF UNCERTAINTY THEORIES

information given in the form “X is F,” where F is an arbitrary fuzzy subset of

X, in terms of a possibility proﬁle? An associated, but more speciﬁc, question

is: How should we assign values of r

to the values of F(x) for all x ŒX?

The position taken here is that the deﬁnition of the possibility proﬁle r

terms of a given fuzzy set F should be such that it does not change the evi-

dence conveyed by F and, at the same time, preserves the required possibilis-

tic normalization, which is expressed in general by the equation sup

xŒX

(x)}

= 1. Since the standard deﬁnition of r

given by Eq. (8.16) does not satisfy the

normalization for subnormal fuzzy sets, it must be appropriately modiﬁed.

To satisfy the possibilistic normalization, we must deﬁne r

(x) = 1 for at least

one x ŒX. Since there is no reason to treat distinct elements of X differently,

the only sensible way to achieve the required normalization is to increase the

values of r

equally for all x ŒX by the amount of 1 - h

. This means that the

revised fuzzy-set interpretation of possibility theory is expressed for all x ŒX

by the equation

(8.19)

This is a generalized counterpart of the standard interpretation, Eq. (8.16); it

is applicable to all fuzzy sets, regardless whether they are normal or not. For

normal fuzzy sets, clearly, Eq. (8.19) collapses to Eq. (8.16).

The signiﬁcance of the possibility proﬁle r

deﬁned by Eq. (8.19) is that it

is the only one that does not change the evidence conveyed by F.An easy way

to show this uniqueness is to use the Möbius representation of this possibility

proﬁle (see Section 5.2.1).As in Section 5.2.1, assume that elements of the uni-

versal set X = {x

, x

,...,x

} are ordered in such a way that

for all i Œ⺞

n-1

, and let r

i+1

) = 0 by convention. Moreover, let A

= {x

, x

...,x

}. Then, the Möbius representation, m

, of the possibility proﬁle is given

by the formula

(8.20)

Now substituting for r

from Eq. (8.19), we obtain

We can see that the evidential support, m

(A), for the various sets A Ã X is

based on the original evidence expressed by the values F(x

) for all i Œ⺞

.The

Fx Fx A A i

Fx h A X A

AA i

ii i n

iF n

()

=Œ

()

+- = =

()

πŒ

when for some

when

when for all

⺞

inf

rx rx A A i

AA i

Fi Fi i n

()

=Œ

πŒ

when for some

when for all

⺞

⺞ .

rx rx

Fi Fi

()

≥

()

rx Fx h

()

+-1.

8.3. FUZZY-SET INTERPRETATION OF POSSIBILITY THEORY 329

only change is in the value of m

(X), which expresses the degree of our

ignorance. Since inconsistency in evidence, which is expressed by the value

1 - h

, is a form of ignorance, it is natural to place it in m

(X).

The necessity function, Nec

, which is based on the evidence expressed by

F, is uniquely determined by m

via the usual formula

(8.21)

This function, again, does not modify in any way the evidence expressed by F.

Moreover,

(8.22)

which is always the case for all possibilistic bodies of evidence.

It is easy to see that deﬁning r

in any way different from Eq. (8.19) would

either violate the possibilistic normalization or the evidence conveyed by F.

For example, it has often been suggested in the literature to obtain a normal-

ized possibility proﬁle, r¢

, for a subnormal fuzzy set by the formula

(8.23)

for all x ŒX. However, this possibility proﬁle does not preserve the evidence

conveyed by F. Clearly,

for all i Œ⺞

, where the symbols x

, x

i+1

, and A

have the meaning introduced

earlier in this section. We can see that the evidential support for sets A

(focal

elements), expressed by the values of m ¢

), is inﬂated by the factor of 1/h

Under the generalized fuzzy-set interpretation of possibility theory, Eq.

(8.18) can be written more explicitly as

(8.24)

where A and F are arbitrary fuzzy sets. Thus, for example,

Pos S S x F x h

Fx h

()

() ()

{}{}

()

{}

sup min ,

sup

Pos A A x F x h

()

() ()

{}{}

sup min , ,1

()

=¢

()

-¢

()

[]

mA rr rr

Fx Fx

Fi Fi i

ii11

()

Pos A m A i

Fi k n

()

=Œ

«π∆

1 for all ⺞ ,

Nec A

mA r x A A i

kFi

()

=Œ

()

when for some

otherwise.

⺞ .

max

330 8. FUZZIFICATION OF UNCERTAINTY THEORIES

Now consider the two extreme cases when F = X and F = ⭋. In the former

case, the proposition “X is X” does not carry any information; in the latter

case, the proposition “X is ⭋” represents evidence that is totally conﬂicting,

and hence, it does not carry any information either. Applying Eq. (8.19), we

obtain

for all x ŒX. This means that both propositions are represented by the same

possibility and necessity measures:

These results are exactly the same as we would expect on intuitive grounds.

EXAMPLE 8.7. To illustrate the generalized fuzzy-set interpretation of pos-

sibility theory, let evidence regarding the relation between two discrete

variables X and Y with states in sets X = {x

|a Œ⺞

} and Y = {y

|b Œ⺞

}, respec-

tively, be expressed in terms of a fuzzy relation R deﬁned by the following

matrix:

The possibility proﬁle, r

, based on this evidence is determined by Eq. (8.19):

yyyy y

1234 5

R =

Pos A Pos A

Nec A Nec A

()

π∆

=∆

()

∆

when

rx rx

()

∆

Pos S S x F x h

Nec S Pos S

hSX

Nec S Pos S

FF FF

()

() ()

{}{}

-π∆

=∆

()

sup min ,

when

8.3. FUZZY-SET INTERPRETATION OF POSSIBILITY THEORY 331