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

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EXAMPLE 8.11. Consider a tuple T =·T

, T

Ò of trapezoidal probability

granules

whose a-cuts are

This tuple, which is shown in Figure 8.9a, is proper since b

+ b

= 0.9 < 1

and c

+ c

= 1.4 > 1. However, it is not reachable since, for example, the

inequality (8.39) is violated for i = 1 (b

+ b

+ c

= 1.1 ⱕ 1) and i = 3 (b

+ b

+ c

= 1.1 ⱕ 1). To use the probability granules for further processing, for

example, as prior probabilities in Bayesian inference, we need to convert T to

its reachable counterpart F. By using Eqs. (8.41) and (8.42) for each i = 1, 2, 3

and a Œ [0, 1], we obtain the following a-cuts of the probability granules (not

necessarily trapezoidal) in F:

004 02

01 02 05 01

04 02 09 01

[]

=+ -

[]

=+ -

[]

,. . ,

..,..,

..,...

00 00 02 04

01 03 04 05

04 06 08 09

., ., ., .

342 8. FUZZIFICATION OF UNCERTAINTY THEORIES

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Figure 8.8. A reachable tuple of probability granules (Example 8.10).

Graphs of F

, F

are shown in Figure 8.9b. We can see that F

π T

and

π T

. Moreover, F

is not of the trapezoidal shape.

Once it is veriﬁed that a given tuple F of probability granules on set X is

reachable, it can be used for calculating probability granules for any subset of

X. These calculations are facilitated by applying Eqs. (5.71) and (5.72) to the

a-cuts of F.

For each A ŒP(X) and each a Œ [0, 1], let

denote the a-cut of the probability granule, F

, of set A.Then, using Eqs. (5.71)

and (5.72), we have

(8.43)

(8.44)

EXAMPLE 8.12. Consider the reachable tuple of four probability granules

on X = {x

, x

} that is discussed in Example 8.10. Graphs of the granules

are shown in Figure 8.8, and their a-cut representations are:

The a-cuts of probability granules

for all sets A ŒP(X), calculated by Eqs.

(8.43) and (8.44), are given in Table 8.1. For convenience, sets A are identiﬁed

01 02 01

01 01 025 005

02 01 04 01

03 01 06 02

[]

=+ -

[]

=+ -

[]

=+ -

[]

.,. . ,

..,. .,

..,..,

..,...

uul

aaa

()

Œœ

ÂÂ

min , .1

llu

aaa

()

Œœ

ÂÂ

max , ,1

aaFl u

AA A

() ()

[]

()

{}

()

=- -

{}

-Œ

[

)

-Œ

[]

()

=+ -

1 0 02 04 0

04 02 05 04

04 02 0 05

05 1

20102030

:max,..,

min . . , . .

.. ,.

., ,

:max..,.

aaa

when

0.5 0.4 when

.. . . ,

min..,.. ..,

:max..,....,

min . . , . .

30102

05 01 06 02 05 01

3 0402 0103 0402

09 01 09 02

{}

()

=- -

{}

()

=++

{}

()

=- -

{}

aaaa

aaa

u ==-09 02...a

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 343

by the index k deﬁned in the table. For the singletons, clearly, k = i.For

example, for set A = {x

, x

}, identiﬁed by k = 5, we have

llluu

uuull

51234

01 02 03

01 02

045 015 05 03

aaaaa

()

(){}

{}

()

(){}

=- -

{}

max ,

max . . , .

..,

min ,

min . . , . .

0045 015..;- a

344 8. FUZZIFICATION OF UNCERTAINTY THEORIES

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

(a)

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

(b)

Figure 8.9. Illustration to Example 8.11: (a) A proper tuple of probability granules that is not

reachable; (b) the reachable tuple of probability granules derived from the one in part (a).

for the set A = {x

, x

}, identiﬁed by k = 13, we have

and, similarly, all the remaining values of l

(a) and u

(a) in Table 8.1 are cal-

culated. Observe the following:

(a) For all A ŒP(X) and all a Œ [0, 1],

(b) For all A ŒP(X), the probability granules F

have triangular shapes;

are additive for a = 1 (l

(1)

= u

(1) for all A ŒP(X)).

()

llllu

uuuul

13 1 3 4 2

05 03 075 005

075 005

12 04 09 0

aaaaa

()

(){}

=+ +

{}

()

(){}

=- -

max ,

max . . , . .

..,

min ,

min . . , . .11

09 01

{}

=-..;

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 345

Table 8.1. Lower and Upper Probability Granulas for the Reachable Tuple of

Probability Granules Discussed in Example 8.10 and Shown in Figure 8.8 (illustration

of Example 8.12)

m(a)

(a) u

(a)

A:0000 0 0 0 0

1000 1 0.1a 0.2 - 0.1a 0.1a

0 1 0 0 2 0.1 + 0.1a 0.25 - 0.05a 0.1 + 0.1a

0 0 1 0 3 0.2 + 0.1a 0.4 - 0.1a 0.2 + 0.1a

0 0 0 1 4 0.3 + 0.1a 0.6 - 0.2a 0.3 + 0.1a

1 1 0 0 5 0.1 + 0.2a 0.45 - 0.15a 0

1 0 1 0 6 0.2 + 0.2a 0.6 - 0.2a 0

1 0 0 1 7 0.35 + 0.15a 0.7 - 0.2a 0.05 - 0.05a

0 1 1 0 8 0.3 + 0.2a 0.65 - 1.5a 0

0 1 0 1 9 0.4 + 0.2a 0.8 - 0.2a 0

001110 0.55 + 0.15a 0.9 - 0.2a 0.05 - 0.05a

111011 0.4+ 0.2a 0.7 - 0.1a 0.1 - 0.1a

110112 0.6+ 0.1a 0.8 - 0.1a 0.15 - 0.15a

101113 0.75 + 0.05a 0.9 - 0.1a 0.15 - 0.15a

011114 0.8+ 0.1a 1.0 - 0.1a 0.15 - 0.15a

111115 1 1 -0.25 + 0.25a

Once the a-cuts of probability granules F

are determined for all A ŒP(X),

the Möbius representation of these a-cuts,

m, can be calculated by the usual

formula

(8.45)

for all A ŒP(X). For the lower probability in Example 8.12, the Möbius rep-

resentation of is shown in Table 8.1. For example,

Observe that

for each a Œ [0, 1].

Using the Möbius representation, we can compute the average nonspeci-

ﬁcity embedded in the associated tuple of probability granules. First, we cal-

culate the generalized Hartley measure as a function of a,

(8.46)

which is followed by computing the average value, GH

, via the integral

(8.47)

Applying Eq. (8.46) to

m in Example 8.12 (given in Table 8.1), we obtain

Clearly, GH(

m) = 0.47 and GH(

m) = 0. Applying now Eq. (8.47), we obtain

= 0.235.

Computing S

is more difﬁcult.Algorithm 6.1 can be employed, but it is now

applicable to the ratios l

(a)/|A|, which are functions of a. First, we need to

express the dependence of S

on a, S

(a)).Then, the average value, S

, is com-

puted by the integral

(8.48)

SSl d

()()

aa.

GH m

()

=-047 047...

GH GH m d

()

GH m m A A

()

log ,

()

aa a a

aaaa

mxx x

123

04 02 01 02 02 02 03 02

01 01 01 02 01 01 01

,, .. .. .. ..

........

{}()

=+ -+

()

+++++=-

amA l

BB A

()

() ()

346 8. FUZZIFICATION OF UNCERTAINTY THEORIES

Applying Algorithm 6.1 to the a-cuts l

(a) in Table 8.1, we obtain the fol-

lowing probabilities of S

(a)) in the given order, one at each iteration of the

algorithm:

Now, we have

The plot of S

(a)) for a Œ [0, 1] is shown in Figure 8.10. The extreme values

are S

(0)) = 1.985 and S

(1)) = 1.846. The average value, obtained by Eq.

(8.48), is S

= 1.93.

When tuples of probability granules are employed in any computation

involving arithmetic operations (such as computing expected values or poste-

rior probabilities in Bayesian inference), it is essential that constrained fuzzy

arithmetic be used. Computing with fuzzy probabilities requires that not only

the requisite equality constraints be observed, but also the probabilistic con-

straints, as is explained in Section 7.4.2.

aaa

()()

=- +

()

03 01 03 01

025 005 025 05

025 005 025 005

02 01 02 01

..log..

..log...

03 01

025 005

02 01

()

..,

...

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 347

0 0.2 0.4 0.6 0.8 1

1.86

1.88

1.9

1.92

1.94

1.96

1.98

Figure 8.10. Dependence of S

on a in Example 8.12.

8.6. OTHER FUZZIFICATION EFFORTS

Efforts to fuzzify uncertainty theories have not been restricted to the three

fuzziﬁcations described in Section 8.3–8.5. However, among the various efforts

to fuzzify uncertainty, which have been discussed in the literature (see Note

8.12), the three fuzziﬁcations described in this chapter have some visible

advantages.They are well developed,conceptually and computationally sound,

and they have features that are suitable for applications.

To illustrate the point that some uncertainty theories are less practical

to fuzzify than others, let us examine the following very simple example

of fuzziﬁcation within the theory of uncertainty based on the Sugeno

l-measures.

EXAMPLE 8.13. Consider the set X = {x

, x

} of alternatives and assume

that we want to deal with uncertainty regarding these alternatives in terms of

l-measures (Section 5.3). This requires that we are able to assess either the

lower probabilities or the upper probabilities on singletons. From these assess-

ments, we compute the value of parameter l by Eq. (5.30), which makes it pos-

sible, in turn, to compute lower and upper probabilities for all subsets of X.

Once uncertainty regarding the alternatives is formalized in this way, we can

apply the rules of the calculus of l-measures to manipulate the uncertainty in

any desirable way within this theory.

Now assume that we are able to make only fuzzy assessments of lower (or

upper) probabilities on singletons, and that these assessments are represented

by appropriate fuzzy intervals or numbers. As a speciﬁc example, let the

triangular-shape fuzzy numbers

represent a given assessment of lower probabilities. These fuzzy numbers are

reasonable representations of linguistic assessments such as “the lower prob-

ability of x

is p

or a little less,” where p

= 0.2, p

= 0.3, and p

= 0.4. The a-

cuts of the numbers are

Applying Eq. (5.30) to these a-cuts results in the equation

01 01 02

02 01 03

03 01 04

{}()

[]

{}()

[]

{}()

[]

..,.,

..,..

01 02 02 02

02 03 03 03

03 04 04 04

{}()

., ., . . ,

348 8. FUZZIFICATION OF UNCERTAINTY THEORIES

whose positive root deﬁnes the value l(a), for each a Œ [0, 1].The plot of l(a)

is shown in Figure 8.11a. The analytic expression for l(a) is too complex to

be practical, even in this very simple example. It is more practical to calculate

l(a) for selected values of a, such as those shown in Figure 8.11b.

Once values of l(a) are determined, the a-cuts of lower probabilities,

m(A), can be determined via Eq. (5.28) for all subsets A of X, as is shown in

Table 8.2. Considering the fact that l(a) is determined by a rather complex

expression, we can see that some expressions in this table would be quite for-

midable when completed. Clearly, expressions for the Möbius representation

will be even more complex.

0 001 0 006 0 11 0 006 0 12 0 11 0 3 0 4 0

32 2

.... ....,aaalala+++

()

+-=

8.6. OTHER FUZZIFICATION EFFORTS 349

Table 8.2. a-Cuts of Lower Probabilities,

(A), in Example 8.13

(A)

A:0000

1 0 0 [0.1 + 0.1a, 0.2]

0 1 0 [0.2 + 0.1a, 0.3]

0 0 1 [0.3 + 0.1a, 0.4]

1 1 0 [0.3 + 0.2a + l(a) (0.1 + 0.1a) (0.2 + 0.1a), 0.5 + l(a)·0.06]

1 0 1 [0.4 + 0.2a + l(a) (0.1 + 0.1a) (0.3 + 0.1a), 0.6 + l(a)·0.08]

0 1 1 [0.5 + 0.2a + l(a) (0.2 + 0.1a) (0.3 + 0.1a), 0.7 + l(a)·1.2]

1111

l(a)

0.0

3.1091

0.1 2.62262

0.2 2.20969

0.3 1.85613

0.4 1.55108

0.5 1.28614

0.6 1.05468

0.7 0.851409

0.8 0.672067

0.9 0.51317

1.0 0.371852

(b)

0.2 0.4 0.6 0.8 1

0.5

1.5

2.5

3.5

(a)

Figure 8.11. Plot of l(a) and some numerical values of l(a) in Example 8.13.

It is sensible to conclude from this strikingly simple example that the uncer-

tainty theory based on l-measures is conceptually and computationally difﬁ-

cult to fuzzify. This contrasts with the theory based on reachable

interval-valued probability distributions, whose fuzziﬁcation (discussed in

Section 8.5) is much more transparent and computationally tractable. By com-

paring the fuzziﬁcations of these two theories, it is reasonable to expect that

only the latter will survive as useful for various applications. It is likely that

among the many possible fuzziﬁcations of uncertainty theories some will be

found useful and some will be discarded.

NOTES

8.1. The concept of a-cuts of fuzzy sets was already introduced in the seminal paper

by Zadeh [1965]. In another paper [Zadeh, 1971], he introduced the a-cut rep-

resentation of fuzzy sets and showed how equivalence, compatibility, and order-

ing relations can be fuzziﬁed via this representation. However, the term

“cutworthy” was coined much later by Bandler and Kohout [1993]. A highly

general and comprehensive investigation of cutworthy properties was made in

terms fuzzy predicate logic by Be˘lohlávek [2003].

8.2. The ﬁrst formulation of the extension principle was introduced by Zadeh [1965],

even though it was described under the heading “fuzzy sets induced by map-

pings.”The term “extension principle” was introduced in [Zadeh, 1975–76], where

the principle and its utility are thoroughly examined.

8.3. Fuzziﬁcation via fuzzy morphisms and category theory was pioneered by Goguen

[1967, 1968–69, 1974]. Among other notable references are [Arbib and Manes,

1975], [Rodabaugh et al., 1992], [Rodabaugh and Klement, 2003], [Höhle and

Klement, 1995], [Höhle and Rodabaugh, 1999], and Walker [2003].

8.4. The idea that the degree of fuzziness of a fuzzy set can be most naturally

expressed in terms of the lack of distinction between the set and its complement

was proposed by Yager [1979, 1980b]. A general formulation based on this idea,

which is applicable to all possible fuzzy complements, was developed by Higashi

and Klir [1982]. They proved that every measure of fuzziness of this type can be

expressed in terms of a metric distance that is based on an appropriate aggregate

of the absolute values of the individual differences between membership grades

of the given fuzzy set and its complement (of a type chosen in a given applica-

tion) for all elements of the universal set.

8.5. Observe that the summation term in Eq. (8.7) and the integral in Eq. (8.9) express

the Hamming distance between A and c(A). It is of course possible to use other

distance functions for this purpose. The whole range of measures of fuzziness

based on the Minkowski class of distance functions is examined by Higashi and

Klir [1982].

8.6. The measure of fuzziness based on local Shannon entropies (deﬁned by Eq. (8.14)

or Eq. (8.15)) was introduced by De Luca and Termini [1972, 1974].This measure,

which is usually called an entropy of fuzzy sets, has been investigated in the lit-

erature fairly extensively by numerous authors.

350

8. FUZZIFICATION OF UNCERTAINTY THEORIES

8.7. Broader classes of measures of fuzziness were characterized on axiomatic

grounds by Knopfmacher [1975] and Loo [1977]. A good overview of the litera-

ture dealing with measures of fuzziness was prepared by Pal and Bezdek [1994].

8.8. The standard fuzzy set interpretation of possibility theory was introduced by

Zadeh [1978a] as a natural tool for dealing with uncertainty associated with infor-

mation expressed in natural language and represented by fuzzy propositions

[Zadeh, 1978b, 1981]. The generalized fuzzy-set interpretation of possibility

theory deﬁned by Eq. (8.19) was introduced in a paper by Klir [1999]. The

paper also surveys other, unsuccessful attempts to revise the standard fuzzy-set

interpretation.

8.9. Probabilities of fuzzy events were introduced by Zadeh [1968], just three years

after he introduced the concept of a fuzzy set [Zadeh, 1965].

8.10. Fuzziﬁcation of the uncertainty theory based on reachable interval-valued prob-

ability distributions was investigated by Pan [1997a] and Pan and Yuan [1997].

They also developed a fuzziﬁed Bayesian inference within this uncertainty

theory, which is based on methods of linear programming.

8.11. Some methods for approximating arbitrary probability granules by trapezoidal

ones were examined by Giachetti and Young [1997] from the standpoint of the

accumulated error for repeated use of multiplication or division. A method pro-

posed in the paper is shown to lead to acceptable results in most practical cases.

An interesting approximation method was proposed by Pan [1997a,b] in the

context of Bayesian inference with probability granules. It is based on keeping

the core unchanged and calculating new support in terms of the weighted least-

square-error method in which the weights are monotone increasing with values

of a according to some rule. This method seems promising, but no error analysis

has been made for it as yet.

8.12. Perhaps the most fundamental approach to fuzzifying uncertainty theories, at

least from the theoretical point of view, is to fuzzify monotone measures. This is

explored in a paper by Qiao [1990], which is also reprinted in [Wang and Klir,

1992]. Interesting approaches to fuzzifying the Dempster–Shafer theory were

proposed by Yen [1990] and Yang et al. [2003]. The concept of a fuzzy random

variable is covered in the literature quite extensively. A broad introduction is in

[Zwick and Wallsten, 1989] and [Negoita and Ralescu, 1987]; a Special Issue

edited by Gil [2001] covers more recent developments, as well as valuable re-

ferences to previous work. The following are some useful references that deal

with fuzzy probabilities, even though they do not speciﬁcally address the issue of

fuzzifying uncertainty theories: [Buckley, 2003; Cai, 1996; Fellin et al., 2005;

Manton et al., 1994; Möller and Beer, 2004; Ross et al., 2002; Viertl, 1996]. The

book by Mordeson and Nair [1998] is a good overview of other fuzziﬁcations in

mathematics.

EXERCISES

8.1. Show that the standard complement of a fuzzy set is not a cutworthy

operation.

EXERCISES 351