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

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8.2. Show that the standard operations of intersection and union of fuzzy sets

are cutworthy.

8.3. Verify that the binary fuzzy relation R deﬁned by the matrix

is a fuzzy equivalence relation that is cutworthy and determine its par-

tition tree.

8.4. Repeat Example 8.2 for function y = 9 + 1/x, where x Œ [1, 10], and two

triangular fuzzy members by which the value x is approximately

assessed: A

=·1.5, 2, 2, 2.5Ò and A

=·2.5, 3, 3, 3.5Ò.

8.5. Assuming the standard complement and using the functional f deﬁned

by Eq. (8.10), calculate the amount of fuzziness and its normalized value

for the following fuzzy sets:

(a) A = 0.5/x

+ 0.7/x

+ 1/x

+ 0.9/x

+ 0.6/x

+ 0.4/x

;

(b) Fuzzy relation R in Exercise 8.3.

8.6. Repeat Example 8.4 for the following fuzzy numbers:

(a) Triangular fuzzy number A =·0, 1, 1, 2Ò;

(b) A fuzzy number B whose a-cut representation is ;

-|5x-10|

for all x Œ [1, 3] and C(x) = 0 for x œ [1, 3].

8.7. Repeat the calculations in Examples 8.3 and 8.4 and Exercises 8.5 and

8.6 for some nonstandard complements introduced in Section 7.3.1.

8.8. Let f and f ¢ be deﬁned by Eqs. (8.7) and (8.12), respectively, and let c be

the standard complement of fuzzy sets. Show that for any fuzzy set A

deﬁned on a ﬁnite set X:

(a) f(A) = 2f ¢(A)

(b) f

(A) = f

¢(A)

8.9. Repeat Exercise 8.8 for f and f ¢ deﬁned by Eqs. (8.9) and (8.13), respec-

tively, and X = ⺢.

8.10. Investigate the relationship between the functional f deﬁned by Eq.

(8.10) and the functional f ≤ deﬁned by Eq. (8.14).

Bx x=-

[]

xx xxxx

123456

R =

352 8. FUZZIFICATION OF UNCERTAINTY THEORIES

8.11. Using the functional f≤ deﬁned by Eq. (8.15), calculate the amount of

fuzziness and its normalized value for the following fuzzy intervals:

(a) A trapezoidal fuzzy interval A =·1, 3, 4, 7Ò;

(b) A trapezoidal fuzzy interval B =·0, 0, 1, 5Ò;

-|5x-10|

for all x Œ [1, 3] and C(x) = 0 for x œ [1, 3].

8.12. Verify the values of GH(m

), S

(Nec

), S(Nec

), f(R), and f(R

) in

Example 8.7.

8.13. Verify the values of f(R), and f(R

) in Example 8.8 by using Eq. (8.11),

as well as by geometrical reasoning in Figure 8.5.

8.14. Consider a discrete variable X whose values are in the set X = ⺞

200

. Given

information that “X is F,” where F is a fuzzy set deﬁned by

calculate the following:

(a) The nonspeciﬁcity of the possibilistic representation of the given

information;

(b) The degrees of possibility and necessity in the sets A = {55, 56,...,

62}, B = {53, 54, 55}, and C = {54, 55, 56}.

8.15. Consider a variable X whose values are in the interval X = [0, 100].

Assume that the value of the variable is assessed in a given context in

terms of a fuzzy propositions “X is F,” in which F is a fuzzy interval

deﬁned for all x Œ X as follows:

Calculate:

(a) The amount of nonspeciﬁcity and its normalized version for the pos-

sibilistic representation of the given information;

(b) The degrees of possibility and necessity in the fuzzy sets whose a-

cut representations are

=+ -

[]

=+ -

[]

=+ -

[]

72 10

65 85

.,. .

()

[]

()

[]

14 65 65 7

07 7 8

810

.. .,,

.,,

when

3.5 10 when

otherwise.

F =+++++++01 52 02 53 04 54 06 55 08 56 05 57 03 58 0150........,

EXERCISES 353

8.16. Show that fuzzy partitions are not cutworthy regardless of which oper-

ations are used for intersections and unions of fuzzy sets.

8.17. Calculate probabilities of crisp and fuzzy events deﬁned in Figure 8.7a

and 8.7b, respectively, for the following probability density functions q

on ⺢:

(a)

(b)

8.18. Suggest some reasonable ways of approximating the probability granule

in Figure 8.9b by a trapezoidal granule (see also Note 8.11).

8.19. Determine for each of the following tuples T =·T

, T

Ò of trapezoidal

probability granules on X = {x

, x

} whether it is proper and

reachable:

(a) T

=·0, 0, 0, 0.1Ò, T

=·0.2, 0.3, 0.4, 0.5Ò, T

=·0.4, 0.6, 0.8, 0.9Ò

(b) T

=·0, 0, 0.3, 0.5Ò, T

=·0, 0.2, 0.5, 0.6Ò, T

=·0.2, 0.4, 0.4, 0.7Ò

=·0.2, 0.3, 0.3, 0.4Ò, T

=·0, 0.2, 0.2, 0.5Ò, T

=·0.3, 0.5, 0.5, 0.8Ò

Convert each of the tuples that is proper but not reachable to its reach-

able counterpart.

8.20. In analogy with Example 8.12, determine for each of the reachable tuples

(or their reachable counterparts) in Exercise 8.19 the following functions

of a for all A ŒP(X):

(a) l

(a)

(b) u

(a)

(c)

(A)

8.21. Using the functions determined in Exercise 8.20, calculate:

(a) GH(

m) and GH

(b) S

(a)) and S

8.22. For Example 8.13, determine the following:

(a) l(a)

(b) The a-cuts of the lower probabilities;

()

◊◊ Œ

[]

4 6875 10 0 40

.,when

otherwise

()

-Œ

[

)

-Œ

[]

0 0016 0 032 20 45

0 112 0 0016 45 70

.. ,

when

otherwise

354 8. FUZZIFICATION OF UNCERTAINTY THEORIES

METHODOLOGICAL ISSUES

355

There is nothing better than to know that you don’t know.

Not knowing, yet thinking your know—

This is sickness.

Only when you are sick of being sick

Can you be cured.

—Lao Tsu

9.1. AN OVERVIEW

From the methodological point of view, two complementary features of gener-

alized information theory (GIT) are signiﬁcant. One of them is the great diver-

sity of prospective uncertainty theories that are subsumed under GIT. This

diversity increases whenever new types of formalized languages or monotone

measures are recognized. At any time, of course, only some of the prospective

uncertainty theories are properly developed.

Complementary to the diversity of uncertainty theories subsumed under

GIT is their unity, which is manifested by common properties that the theo-

ries share. From the methodological point of view, particularly notable are the

common forms of functionals that generalize the Hartley and Shannon mea-

sures of uncertainty in the various uncertainty theories, and the invariance of

equations and inequalities that express the relationship among joint, marginal,

and conditional measures of uncertainty.

The diversity of GIT offers an extensive inventory of distinct uncertainty

theories, each characterized by speciﬁc assumptions embedded in its axioms.

Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir

This allows us to choose, in any given application context, a theory that is com-

patible with the application of concern. On the other hand, the unity of GIT

allows us to work within GIT as a whole. That is, it allows us to move from

one theory to another, as needed.

The primary aim of this chapter is to examine the following four method-

ological principles of uncertainty:

1. Principle of minimum uncertainty

2. Principle of maximum uncertainty

3. Principle of requisite generalization

4. Principle of uncertainty invariance

In general, these principles are epistemologically based prescriptive proce-

dures that address methodological issues involving uncertainty that cannot be

resolved solely by using calculi of the individual uncertainty theories. Due to

the connection between uncertainty and uncertainty-based information, these

principles also can be interpreted as principles of information.

The principle of minimum uncertainty is basically an arbitration principle.

It facilitates the selection of meaningful alternatives from solution sets

obtained by solving problems in which some amount of the initial information

is inevitably lost.According to this principle, we should accept only those solu-

tions for which the loss of the information is as small as possible. This means,

in turn, that we should accept only solutions with minimum uncertainty.

The second principle, the principle of maximum uncertainty, is essential for

any problem that involves ampliative reasoning. This is reasoning in which

conclusions are not entailed in the given premises. Using common sense, the

principle may be expressed as follows: in any ampliative inference, use all

information supported by available evidence, but make sure that no additional

information (unsupported by the given evidence) is unwittingly added.

Employing the connection between information and uncertainty, this deﬁni-

tion can be reformulated in terms of uncertainty: any conclusion resulting from

ampliative inference should maximize the relevant uncertainty within con-

straints representing given premises. This principle guarantees that we fully

recognize our ignorance when we attempt to make inferences that are beyond

the information domain deﬁned by the given premises and, at the same time,

that we utilize all information contained in the premises. In other words, the

principle guarantees that our inferences are maximally noncommittal with

respect to information that is not contained in the premises.

The principles of minimum and maximum uncertainty are well developed

within classical, probability-based information theory. They are referred to

as principles of minimum and maximum entropy. These classical principles of

uncertainty are extensively covered in the literature. A survey of this litera-

ture is presented in Notes 9.1–9.3.A few examples in Sections 9.2 and 9.3 illus-

trate the important role of these principles in classical information theory.

356 9. METHODOLOGICAL ISSUES

Optimization problems that emerge from the minimum and maximum

uncertainty principles outside classical information theory have yet to be prop-

erly investigated and tested in praxis. One complication is that two types of

uncertainty coexist in all the nonclassical uncertainty theories. Material pre-

sented in this chapter is thus by and large exploratory, oriented primarily to

examining the issues involved and stimulating further research.

Principles of minimum and maximum uncertainty are fundamentally dif-

ferent from the remaining principles: the principle of requisite generalization

and the principle of uncertainty invariance. While the former are applicable

within each particular uncertainty theory, the latter facilitate transitions from

one theory to another. Clearly, the latter principles have no counterparts in

classical information theory.

The principle of requisite generalization is based on the assumption that we

work within GIT as a whole.According to this principle, we should not a priori

commit to any particular uncertainty theory. Our choice should be determined

by the nature of the problem we deal with. The chosen theory should be

sufﬁciently general to allow us to capture fully our ignorance. Moreover, when

the chosen theory becomes incapable of expressing uncertainty resulting from

deﬁcient information at some problem-solving stage, we should move to a more

general theory that has the capability of expressing the given uncertainty.

The last principle, the principle of uncertainty invariance (also called the

principle of information preservation), was introduced in GIT to facilitate

meaningful transformations between various uncertainty theories. According

to this principle, the amount of uncertainty (and the associated uncertainty-

based information) should be preserved in each transformation from one

uncertainty theory to another. The primary use of this principle is to approxi-

mate in a meaningful way uncertainty formalized in a given theory by a for-

malization in a theory that is less general.

The roles of each of the four principles of uncertainty are illustrated in

Figure 9.1, where uncertainty theory T

is assumed to be more general than

uncertainty theory T

. While the principles of minimum and maximum uncer-

tainty (1 and 2) are applicable within a single theory (theory T

in the ﬁgure),

the principles of requisite generalization (3) and uncertainty invariance (4)

deal with transitions from one theory to another.

9.2. PRINCIPLE OF MINIMUM UNCERTAINTY

The principle of minimum uncertainty is applicable to all problems that are

prone to lose information.The principle may be viewed as a safeguard against

losing more information than necessary to solve a given problem. It is thus

reasonable to also refer to it as the principle of minimum information loss. In

this section, the principle is illustrated by two classes of problems: simpliﬁca-

tion problems and conﬂict-resolution problems.

9.2. PRINCIPLE OF MINIMUM UNCERTAINTY 357

9.2.1. Simpliﬁcation Problems

A major class of problems for which the principle of minimum uncertainty is

applicable consists of simpliﬁcation problems. When a system is simpliﬁed, it

is usually unavoidable to lose some information contained in the system. The

amount of information that is lost in this process results in the increase of an

equal amount of relevant uncertainty. Examples of relevant uncertainties are

predictive, retrodictive, or prescriptive uncertainty. A sound simpliﬁcation of

a given system should minimize the loss of relevant information (or the

increase in relevant uncertainty) while achieving the required reduction of

complexity. That is, we should accept only such simpliﬁcations of a given

system at any desirable level of complexity for which the loss of relevant infor-

mation (or the increase in relevant uncertainty) is minimal. When properly

applied, the principle of minimum uncertainty guarantees that no information

is wasted in the process of simpliﬁcation.

There are many simpliﬁcation strategies, all of which can perhaps be clas-

siﬁed into three main classes:

•

Simpliﬁcations made by eliminating some entities from the system (vari-

ables, subsystems, etc.).

•

Simpliﬁcations made by aggregating some entities of the system (vari-

ables, states, etc.).

•

Simpliﬁcations made by breaking overall systems into appropriate

subsystems.

Regardless of the strategy employed, the principle of minimum uncertainty is

utilized in the same way. It is an arbiter that decides which simpliﬁcations to

358 9. METHODOLOGICAL ISSUES

Uncertainty

theory

Uncertainty

theory

Deﬁcient

information

Figure 9.1. Principles of uncertainty: an overview. Assumption: Theory T

is more general than

theory T

. ➀ Principle of minimum uncertainty; ➁ principle of maximum uncertainty; ➂ principle

of requisite generalization; ➃ principle of uncertainty invariance.

choose at any given level of complexity or, alternatively, which simpliﬁcations

to choose at any given level of acceptable uncertainty. Let us describe this

important role of the principle of minimum uncertainty in simpliﬁcation prob-

lems more formally.

Let Z denote a system of some type and let Q

denote the set of all sim-

pliﬁcations of Z that are considered admissible in a given context. For example,

may be the set of simpliﬁcations of Z that are obtained by a particular sim-

pliﬁcation method. Let £

and £

denote preference orderings on Q

that are

based upon complexity and uncertainty, respectively. In general, systems with

smaller complexity and smaller uncertainty are preferred. The two preference

orderings can be combined in a joint preference ordering, £, deﬁned as follows:

for any pair of systems Z

, Z

ŒQ

, Z

£ Z

if and only if Z

and Z

The joint preference ordering is usually a partial ordering, even though it may

be only a weak ordering in some simpliﬁcation problems (reﬂexive and tran-

sitive relation on Q

). The use of the uncertainty preference ordering, which,

in this case, exempliﬁes the principle of minimum uncertainty, enables us to

reduce all admissible simpliﬁcations to a small set of preferred simpliﬁcations.

The latter forms a solution set, SOL

, of the simpliﬁcation problem, which con-

sists of those admissible simpliﬁcations in Q

that are either equivalent or

incomparable in terms of the joint preference ordering. Formally,

Observe that the solution set SOL

in this formulation, which may be called

an unconstrained simpliﬁcation problem, contains simpliﬁcations at various

levels of complexity and with various degrees of uncertainty.The problem can

be constrained, for example, by considering simpliﬁcations admissible only

when their complexities are at some designated level or when they are below

a speciﬁed level, when their uncertainties do not exceed a certain maximum

acceptable level, and the like. The formulation of these various constrained

simpliﬁcation problems differs from the preceding formulation only in the def-

inition of the set of admissible simpliﬁcations Q

EXAMPLE 9.1. The aim of this example is to illustrate the role of the prin-

ciple of minimum uncertainty in determining the solution set of preferable

simpliﬁcations of a given system. The example deals with a simple diagnostic

system, Z, with one input variable, x, and one output variable, y, whose states

are in sets X = {0, 1, 2, 3} and Y = {0, 1}, respectively. It is assumed that the

states are ordered in the natural way: 0 £ 1 £ 2 £ 3. The relationship between

the variables is expressed by the joint probability distribution function on X

¥ Y that is given in Table 9.1a. The diagnostic uncertainty of the system is

SY X SXY S X

()

,()

2 571 1 904

0 667

SOL Q Q

ijjiijZZ Z

ZZZZZZ=Œ Œ £ £

{}

for all implies ,.

9.2. PRINCIPLE OF MINIMUM UNCERTAINTY 359

360 9. METHODOLOGICAL ISSUES

The amount of diagnostic information contained in the system, I(Y | X), is the

difference between the maximum uncertainty allowed within the experimen-

tal frame, S

max

(Y | X) = log

8 - log

4 = 1, and the actual uncertainty: I(Y | X) =

1 - 0.667 = 0.333. Now assume that we want to simplify the system by quan-

tizing states of the input variable. Since the states of X are ordered, there are

six meaningful quantizations (with at least two states), which are expressed by

the following partitions on X:

Table 9.1. Preferred Simpliﬁcations Determined by the Principle of Minimum

Uncertainty (Example 9.1)

Z: p(x, y) YXp

(x)

X 0 0.10 0.30 0 0.40 S(X, Y) = 2.571

1 0.00 0.15 1 0.15 S(X) = 1.904

2 0.15 0.05 2 0.20 S(Y | X) = 0.667

3 0.05 0.20 3 0.25 I(Y | X) = 0.333

(a)

: p

(x, y) YX

(x)

{0,1} 0.10 0.45 {0,1} 0.55 S(X

,Y) = 2.158

{2} 0.15 0.05 {2} 0.20 S(X

) = 1.439

{3} 0.05 0.20 {3} 0.25 S(Y | X

) = 0.719

I(Y | X

) = 0.281

: p

(x, y) YX

(x)

{0} 0.10 0.30 {0} 0.40 S(X

,Y) = 2.409

{1,2} 0.15 0.20 {1,2} 0.35 S(X

) = 1.559

{3} 0.05 0.20 {3} 0.25 S(Y| X

) = 0.850

I(Y | X

) = 0.150

: p

(x, y) YX

(x)

{0} 0.10 0.30 {0} 0.40 S(X

,Y) = 2.228

{1} 0.00 0.15 {1} 0.15 S(X

) = 1.458

{2,3} 0.20 0.25 {2,3} 0.45 S(Y | X

) = 0.770

I(Y | X

) = 0.230

(b)

Simpliﬁcations Z

, Z

, which are based on partitions X

, X

, respectively,

are shown in Table 9.1b.These simpliﬁcations have the same complexity, which

is expressed in this example by the number of states of the input variable.

Among them, simpliﬁcation Z

has the smallest diagnostic uncertainty,

S(Y | X

) = 0.719, and the smallest loss of information with respect to the given

system Z:

Simpliﬁcations Z

, Z

, which are based on partitions X

, X

, respectively,

are shown in Table 9.1c. They have the same complexity and, clearly, they are

IY X IY X

()

0 052..

01 2 3

0123

01 23

012 3

0123

{}{}{}{}

{}{ }{}{}

{}{}{ }{}

{}{}{}

{}{ }{}

,, , ,

,,, ,

,,,,

,,, ,

,, , ,

,,, .

9.2. PRINCIPLE OF MINIMUM UNCERTAINTY 361

: p

(x, y) YX

(x)

{0,1} 0.10 0.45 {0,1} 0.55 S(X

,Y) = 1.815

{2,3} 0.20 0.25 {2,3} 0.45 S(X

) = 0.993

S(Y| X

) = 0.822

I(Y | X

) = 0.178

: p

(x, y) YX

(x)

{0,1,2} 0.25 0.50 {0,1,2} 0.75 S(X

,Y) = 1.680

{3} 0.05 0.20 {3} 0.25 S(X

) = 0.811

S(Y | X

) = 0.869

I(Y | X

) = 0.131

: p

(x, y) YX

(x)

{0} 0.10 0.30 {0} 0.40 S(X

,Y) = 1.846

{1,2,3} 0.20 0.40 {1,2,3} 0.60 S(X

) = 0.971

S(Y | X

) = 0.875

I(Y | X

) = 0.125

(c)

Table 9.1. Continued