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

Подождите немного. Документ загружается.

The nested body of evidence associated with this 2-dimensional possibility

proﬁle is shown in Figure 8.4, where the pairs of integers denote the subscripts

of pairs ·x

, y

Ò. Calculating the various components of uncertainty associated

with this possibilistic body of evidence, we obtain: GH(m

) = 3.12, S

(Nec

) =

4.29, and S(Nec

) = 0. Viewing r

as the normalized version, R

, of the given

relation R, we readily obtain f(R

) = 11.2 and f

) = 0.56, while for the given

fuzzy relation R, we get f(R) = 9.6 and f

(R) = 0.48.

Consider now a real-valued variable X that takes values in X = [x, x

]. Infor-

mation about the actual value of the variable is expressed again by the propo-

sition “X is F,” where F is in this case a fuzzy interval deﬁned on X. Differences

in Eq. (8.20) are now differentials. Denoting the focal elements correspond-

ing to the a-cuts

F by

, the necessity function for all focal elements is

expressed by the simple formula

(8.25)

For other crisp sets

Nec r d

()

yyyyy

rxy

Ri j

12345

()

[]

332 8. FUZZIFICATION OF UNCERTAINTY THEORIES

, y

): 1 0.9 0.8 0.7 0.6 0.5 0.3 ¨ Eq. (8.19)

: A

): 0.1 0.1 0.1 0.1 0.1 0.2 0.3 ¨ Eq. (8.20)

Nec

): 0.1 0.2 0.3 0.4 0.5 0.7 1.0 ¨ Eq. (8.21)

Pos

): 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ¨ Eq. (8.22)

Figure 8.4. Possibilistic body of evidence in Example 8.7.

(8.26)

which can be easily generalized to fuzzy set A via Eq. (8.18).

Equations (8.25) and (8.26) are counterparts of Eq. (8.21) for any convex

and bounded universal set X Ã ⺢. A generalization to ⺢

(n ≥ 2) is straight-

forward, but it is not pursued here.

EXAMPLE 8.8. Let evidence regarding the value of a real-valued variable

X, whose values are in the interval X = [0, 4], be expressed by the proposition

“X is F,” where F is a triangular fuzzy number shown in Figure 8.5 and deﬁned

for each x ŒX by the formula

F is a subnormal fuzzy set; h

= 0.5. According to Eq. (8.19), the associated

possibility proﬁle r

is deﬁned for each x ŒX by

and its graph is shown in Figure 8.5. For each a Œ(0, 1], the focal elements of

are

rx Fx

()

+ 05.,

()

[

)

()

[]

12 12

223

when

3 when

otherwise.

Nec A r x

()

1 sup

8.3. FUZZY-SET INTERPRETATION OF POSSIBILITY THEORY 333

(4–x)/2

x/2

4-2a

Figure 8.5. Illustration to Example 8.8.

and their Lebesgue measures are

The generalized Hartley-like measure of nonspeciﬁcity, GHL, is then calcu-

lated by the following integral:

Using Eq. (8.11), the degree of fuzziness of F and its normalized counterpart,

= r

, are f(F) = 1 and f(F

) = 3, and their normalized versions are f

(F) = 0.25

and f

) = 0.75. The numbers can be obtained in this case by simple geomet-

rical considerations in Fig. 8.5.

8.4. PROBABILITIES OF FUZZY EVENTS

A simple fuzziﬁcation of classical probability theory is obtained by extending

the classical concept of an event from classical sets to fuzzy sets. Notwith-

standing its simplicity, this fuzziﬁcation makes probability theory more expres-

sive and, consequently, enlarges the domain of its applicability.The use of fuzzy

events adds new capabilities to probability theory. Among them is the capa-

bility to capture the meanings of linguistic expressions of natural language,

and the capability to express borderline uncertainties in measurement.

Given a probability distribution function p on X (when X is ﬁnite) or a

probability density function q on X (when X = ⺢

for some n ≥ 1), the classi-

cal probability measure, Pro(A), can be expressed for each set A in a given

s-algebra by the formulas

Pro A x p x

Pro A x q x dx

()

()()

()

()()

⺢

GHL r r d

()

[]

=+-

()

[]

()

ÚÚ

log

log log

554

54 54

2 077

aaa

()

[]

-Œ

(

]

4005

44 051

when

., .

[]

(

]

[]

(

]

04 005

242 051

,,.

,.,,

when

334 8. FUZZIFICATION OF UNCERTAINTY THEORIES

respectively, where X

denotes the characteristic function of set A. These

formulas are readily generalized to fuzzy sets (events) A by replacing the

characteristic function X

with the membership function A:

(8.27)

(8.28)

Observe that the classical case of crisp events is captured by Eqs. (8.27) and

(8.28) as well. Observe also that probabilities of fuzzy events preserve some

properties of classical (additive) probabilities. For example, using the standard

operations on fuzzy sets and assuming that X is ﬁnite, we obtain for any A, B

ŒF(X)

This equation conforms to the calculus of classical probability theory, even

though A and B are fuzzy events. However, some other properties of the cal-

culus are not preserved for fuzzy events. For example, Pro(A » A

) £ 1 and

Pro(A « A

) ≥ 0 when A is a fuzzy event; the equalities are obtained only in

the special case when A becomes a crisp event.

EXAMPLE 8.9. Consider a random variable whose values are positive real

numbers and whose probability distribution function p (shown in Figure 8.6)

is characterized by the probability density function

This function is also shown in Figure 8.6, together with the membership

function

which represents (in a given application context) a fuzzy event “around 80.”

Using Eq. (8.28), the probability that x is around 80, Pro(A), is calculated as

follows:

()

[

)

()

[]

76 4 76 80

86 4 80 84

when

otherwise.

()

[

)

()

[]

76 40 76 84

86 10 84 86

when

otherwise.

Pro A B A x B x

Ax Bx Ax Bx

Pro A Pro B Pro A B

xX xX xX

()

() (){}

()

() (){}

()

-«

()

ŒŒŒ

ÂÂÂ

max ,

min ,

Pro A A x q x dx

()

()()

⺢

Pro A A x p x

()

()()

8.4. PROBABILITIES OF FUZZY EVENTS 335

A random variable whose values are real numbers in an interval X = [x, x

]

is often approximated by partitioning X into a ﬁnite set of disjoint intervals.

This approximation is usually referred to as quantization. Assuming that the

partition consists of n intervals (quanta) of equal length, D

= (x - x

)/n, the

partition, p

(X), is deﬁned in general terms as

Observe that all intervals except the last one in this set are semiclosed inter-

vals; the last interval is closed to ensure that the union of all the intervals is

equal to X (as required for a partition). For convenience, let

Recall that these intervals may be represented by their characteristic functions

An example of partition p

(X), where X = [0, 100] and n = 5, is shown in

Figure 8.7a. The ﬁve intervals in this partition may be characterized as classes

of values of the variable that are small, medium, large, and so on, as is indi-

cated in the ﬁgure. A more expressive representation of these classes is

obtained when the intervals I

Œ p

(X) are replaced with appropriate fuzzy

xi xi i

xi xi in

nn n

()

[]

()

[]

when

⺞

nnn

Xxi xii xn xn

()

=+-

()

[]

{}

()

[]

{}

DD D D,,,,.⺞

Pro A

()

◊

ÚÚ

336 8. FUZZIFICATION OF UNCERTAINTY THEORIES

78 80 82 84 86

0.2

0.4

0.6

0.8

Figure 8.6. Probability distribution p, probability density function q, and fuzzy event A in

Example 8.9.

intervals A

, as shown in Figure 8.7b. The main advantage of fuzzy intervals is

their capability to make transitions between adjacent classes gradual rather

than abrupt.

A nonempty family of fuzzy intervals A

, i Œ⺞

, for some n ≥ 1 (or, more

generally, fuzzy sets) deﬁned on X is called a fuzzy partition of X if and only

if the family does not contain the empty set and

It is left to the reader to show that fuzzy partitions are not cutworthy.The ﬁve

trapezoidal-shaped fuzzy intervals in Figure 8.7b, whose deﬁnitions are

form a fuzzy partition on [0, 100].

0 0 15 25

15 25 35 45

35 45 55 65

55 65 75 85

75 85 100 100

,, ,

,,,

,, , ,

Ax x X

()

=Œ

1 for each .

⺞

8.4. PROBABILITIES OF FUZZY EVENTS 337

40 60 80 100

(a)

Very small

Small

Medium Large

Very large

(b)

40 60

A : Very

large

A : Very

small

A : Small

A : Medium

A : Large

100

Figure 8.7. Examples of partitions of the interval [0, 100]: (a) crisp partition; (b) fuzzy partition.

Now assume that we have a set of m observations of the variable

Using these data, probabilities of the individual intervals, Pro(I

), are calcu-

lated in the usual way. For each i Œ⺞

(8.29)

where

(8.30)

Counterparts of Eqs. (8.29) and (8.30) for fuzzy intervals A

are, respectively

(8.31)

where

If the family {A

|i Œ⺞

} forms a fuzzy partition of X, then

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED

PROBABILITY DISTRIBUTIONS

A natural fuzziﬁcation of the theory of reachable interval-valued probability

distributions (introduced in Section 5.5) is obtained by extending relevant

probability intervals to fuzzy intervals via the a-cut representation. To discuss

basic issues of this fuzziﬁed uncertainty theory, let the following notation be

used.

Assume, as in Section 5.5, that we deal with a ﬁnite set X = {x

|i Œ⺞

} of

considered alternatives.These alternatives may, in general, be viewed as states

of variable X. Assume further that each alternative x

Œ X is associated with

an imprecise probability expressed by a fuzzy interval, F

, deﬁned on [0, 1]. It

is assumed that F

(p) is deﬁned for all p Œ [0, 1] in terms of the canonical form

am ProA

()

ŒŒ

ÂÂ

and 1.

⺞⺞

aAx

iik

()

Pro A

()

= ,

()

Pro I

()

= ,

xXk

ŒŒ

{}

⺞ .

338 8. FUZZIFICATION OF UNCERTAINTY THEORIES

expressed by Eq. (7.25). As always, the fuzzy interval is uniquely represented

by the family of its a-cuts,

for all a Œ [0, 1]. In each particular application, evidence in this fuzziﬁed uncer-

tainty theory is expressed by an n-tuple, F, of fuzzy intervals F

. That is,

(8.32)

Again, each F is uniquely represented by the family of its a-cuts,

for all a Œ [0, 1]. For convenience, let fuzzy intervals F

in Eq. (8.32) be called

probability granules, and let each tuple F deﬁned by Eq. (8.32) be called a tuple

of probability granules.

The various properties of tuples of probability intervals, which are intro-

duced in Section 5.5, are extended to tuples of probability granules via the a-

cut representations of the latter. Thus, for example, a tuple of probability

granules is called proper iff all its a-cuts are proper in the classical sense; it is

called reachable iff all its a-cuts are reachable in the classical sense; and so

forth.

To make computation with probability granules as efﬁcient as possible, it is

desirable to represent them by trapezoidal membership functions. That is, it is

desirable to deal only with special tuples of probability granules,

(8.33)

where each probability granule T

is a trapezoidal-shaped fuzzy interval with

support [a

, d

] and core [b

, c

,]. However, when the operations of multiplica-

tion or division of fuzzy arithmetic are applied to trapezoidal granules, the

resulting granules are not trapezoidal. This is unfortunate since we often need

to use the resulting granules as inputs for further processing. For efﬁcient com-

putation, it is thus desirable to approximate the membership functions of

resulting probability granules at each stage of computation by appropriate

trapezoidal granules.

A simple way of approximating an arbitrary granule, F, by a trapezoidal

one, T, is to keep the values a, b, c, d of the canonical form of F (see Eq. (7.25))

unchanged and replace its nonlinear functions in the intervals [a, b] and [c, d]

with their linear counterparts

and ,

T == ŒTabcdi

iiiii n

,,, ,⺞

F =ŒFi

⺞

F =ŒFi

⺞ .

aaFl u

ii i

() ()

[]

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 339

respectively. The advantage of this approximation method is its simplicity.

However, when multiplication or division operations are used repeatedly, the

accumulated error can become sufﬁciently large to produce misleading results.

Other approximation methods have been proposed to reduce the accumulated

error. Although some recent results seem to indicate that this approximation

problem can be adequately solved, further research is needed to obtain more

conclusive results (see Note 8.11).

Assume, for the sake of simplicity, that we deal in this section only with

tuples of trapezoidal probability granules and that we specify them either in

the form of Eq. (8.33) or via the associated a-cut representation

(8.34)

To be proper, a tuple of trapezoidal probability granules must satisfy the

inequalities

for all a Œ [0, 1]. Clearly, if the inequalities are satisﬁed for a = 1, then they

are satisﬁed for all a Œ [0, 1]. It is thus sufﬁcient to check the inequalities

Moreover, this convenient feature holds for all tuples of probability granules,

not only for the trapezoidal ones.

To be reachable, a tuple of trapezoidal probability granules must satisfy the

inequalities

(8.35)

(8.36)

for all i Œ⺞

and all a Œ [0, 1]. Since the left-hand sides of these inequalities

are linear functions of a, it is sufﬁcient to check the inequalities only for a =

0 and a = 1. That is, it is sufﬁcient to check the inequalities

(8.37)

(8.38)

+≥

+£

ddc aba

jjj

iii

()

[]

++ -

()

≥

aa1

aba ddc

jjj

iii

()

[]

+- -

()

aa1,

£≥

ŒŒ

ÂÂ

11and .

⺞⺞

aba

ddc

iii

()

[]

()

[]

≥

⺞

aaT ==+-

()

[]

ŒTaba ddc i

iiii iii n

,.⺞

340 8. FUZZIFICATION OF UNCERTAINTY THEORIES

(8.39)

(8.40)

for all i Œ ⺞

. Clearly, this convenient feature does not hold for tuples of prob-

ability granules that are not trapezoidal.

EXAMPLE 8.10. Consider a tuple with four trapezoidal probability granules,

T =·T

| i Œ ⺞

Ò, where

This tuple, which is shown in Figure 8.8, is proper since

It is also reachable since the inequalities (8.37)–(8.40) are satisﬁed for all

i Œ⺞

, as can be easily veriﬁed.

When a given tuple of trapezoidal fuzzy granules,

is proper but not reachable, it can be converted to its reachable counterpart,

for all a Œ [0, 1] and all i Œ⺞

via the formulas

(8.41)

(8.42)

where [l

(a), u

(a)] =

. Observe that, due to the max and min operations in

these formulas, granules F

of the resulting tuple F may not have trapezoidal

shapes.

uddcaba

iiiijjj

aa a

()

=--

()

-+-

()

[

)

min , 1

labaddc

iiiijjj

aaa

()

=+-

()

---

()

[

)

max , 1

F =ŒFi

,⺞

T =ŒTi

,⺞

iŒŒ

ÂÂ

⺞⺞

0010102

01 02 02 025

02 03 03 04

03 04 04 06

,.,.,.,

., ., ., . ,

., ., ., . .

+≥

+£

8.5. FUZZIFICATION OF REACHABLE INTERVAL-VALUED PROBABILITY DISTRIBUTIONS 341