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

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Table 7.1. Probabilities in (a) Example 7.10; (b) Exercise 7.29

t 71 72 73 74 75 76 77 78 79 80 81 82 83

p(t) 0.01 0.03 0.11 0.15 0.21 0.16 0.14 0.11 0.04 0.02 0.01 0.006 0.004

(a)

r 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

p(r) 0.005 0.005 0.01 0.02 0.05 0.05 0.04 0.05 0.05 0.05 0.1 0.3 0.2 0.04 0.02

(b)

292

where R is a fuzzy relation on X ¥ Y. It is assumed here that R is determined

for each x ŒX and each y Œ Y by the formula

where the symbol J stands for a binary operation on [0, 1] that represents an

appropriate fuzzy implication in the given application context. Clearly,

for all ·x, yÒŒX ¥ Y. Moreover, if a truth qualiﬁcation or a probability qual-

iﬁcation is employed, R must be composed with the respective qualiﬁers to

obtain for each ·x, yÒŒX ¥ Y the degree of truth of the conditional and qual-

iﬁed proposition.

As is well known, operations that qualify as fuzzy implications form a class

of binary operations on [0, 1], similarly as fuzzy intersections and fuzzy unions.

An important class of fuzzy implications, referred to as Lukasiewicz implica-

tions, is deﬁned for each a Œ [0, 1] and each b Œ [0, 1] by the formula

(7.39)

where l > 0 is a parameter by which individual implications are distinguished

from one another.

Fuzzy propositions of any of the introduced types may also be quantiﬁed.

In general, fuzzy quantiﬁers are fuzzy intervals. This subject is beyond the

scope of this overview.

7.6.2. Approximate Reasoning

Reasoning based on fuzzy propositions of the various types is usually referred

to as approximate reasoning. The most fundamental components of approxi-

mate reasoning are conditional fuzzy propositions, which may also be truth

qualiﬁed, probability qualiﬁed, quantiﬁed, or any combination of these.Special

procedures are needed for each of these types of fuzzy propositions.This large

variety of fuzzy propositions makes approximate reasoning methodologically

rather intricate. This reﬂects the richness of natural language and the many

intricacies of common-sense reasoning, which approximate reasoning based

upon fuzzy set theory attempts to emulate.

To illustrate the essence of approximate reasoning, let us characterize the

fuzzy-logic generalization of one of the most common inference rules of clas-

sical logic: modus ponens. The generalized modus ponens is expressed by the

following schema:

J ab a b,min, ,

()

=-+

[]

fxyRxy

()

Rx y Ax Bx,,,

()

() ()()

:, ,XY

()

7.6. FUZZY LOGIC 293

Clearly, in this schema A and F are fuzzy sets deﬁned on X, while B and C are

fuzzy sets deﬁned on Y. Assuming that the fuzzy rule is already converted to

the alternative form

where R represents the fuzzy implication employed, the fuzzy conclusion C is

obtained by composing F with R. That is

or, more speciﬁcally,

(7.40)

for all y ŒY. This way of obtaining the conclusion according to the general-

ized modus ponens schema is called a compositional rule of inference.

To use the compositional rule of inference, we need to choose a ﬁtting fuzzy

implication in each application context and express it in terms of a fuzzy rela-

tion R. There are several ways in which this can be done. One way is to derive

from the application context (by observing or by expert’s judgements) pairs

F, C of fuzzy sets that are supposed to be inferentially connected (facts and

conclusions). Relation R, which represents a fuzzy implication, is then deter-

mined by solving the inverse problem of fuzzy relation equations. This and

other issues regarding fuzzy implications in approximate reasoning are dis-

cussed fairly thoroughly in the literature (see Note 7.9).

7.7. FUZZY SYSTEMS

In general, each classical system is ultimately a set of variables together with

a relation among states (or values) of the variables. When the states of vari-

ables are fuzzy sets, the system is called a fuzzy system. In most typical fuzzy

systems, the states are fuzzy intervals that represent linguistic terms such as

very small, small, medium, large, very large, as interpreted in the context

of each particular application. If they do, the variables are called linguistic

variables.

Each linguistic variable is deﬁned in terms of a base variable, whose values

are usually real numbers within a speciﬁc range. A base variable is a variable

in the classical sense, as exempliﬁed by any physical variable (temperature,

pressure, tidal range, grain size, etc.). Linguistic terms involved in a linguistic

By Fx Rx y

()

() ( )

[]

{}

max min , ,

BFR= o

XY,,

()

is R

Fuzzy rule If is then is

Fuzzy fact is

Fuzzy conclusion is

294 7. FUZZY SET THEORY

variable are used for approximating the actual values of the associated base

variable. Their meanings are captured, in the context of each particular appli-

cation, by appropriate fuzzy intervals. That is, each linguistic variable consists

of:

•

A name, which should reﬂect the meaning of the base variable involved.

•

A base variable with its range of values (usually a closed interval of real

numbers).

•

A set of linguistic terms that refers to values of the base variable.

•

A set of semantic rules, which assign to each linguistic term its meaning

in terms of an appropriate fuzzy interval (or some other fuzzy set) deﬁned

on the range of the base variable.

An example of a linguistic variable is shown in Figure 7.9. Its name “interest

rate” captures the meaning of the base variable—a real variable whose range

is [0, 20]. Five linguistic states are distinguished by the linguistic terms very

small, small, medium large, large, and very large. Each of these terms is repre-

sented by a trapezoidal-shaped fuzzy interval, as shown in the ﬁgure.

Consider a linguistic variable whose base variable has states (values) in set

X. Fuzzy sets representing a ﬁnite set of linguistic states of the linguistic vari-

able are often deﬁned in such a way that the sum of membership degrees in

these fuzzy sets is equal to 1 for each x Œ X. Such a family of fuzzy subsets

of X is called a fuzzy partition of X. Formally a ﬁnite family {A

Œ F(X),

i Œ ⺞

, n ≥ 1} of n fuzzy subsets of X is a fuzzy partition if and only if A

π ⭋

for all i Œ ⺞

and

An example of a fuzzy partition of X = [0, 20] is the family of fuzzy intervals

in Figure 7.9.

7.7.1. Granulation

Representing states of variables by fuzzy sets (usually fuzzy numbers or inter-

vals) is called a granulation. It is a fuzzy counterpart of classical quantization,

which is any meaningful grouping of states of variables into quanta (or aggre-

gates).An example of quantization of a real variable whose range is [0, 1] into

eleven quanta (semiopen or closed intervals of real numbers) is shown in

Figure 7.10a. One of many possible fuzzy counterparts of this quantization, a

particular granulation of the variable by triangular fuzzy numbers, is shown in

Figure 7.10b.

While viewing physical variables, such as temperature, pressure, and

electric current, as real variables is mathematically convenient, this view

introduces a fundamental inconsistency between the inﬁnite precision

Ax x X

()

=Œ

1 for each .

⺞

7.7. FUZZY SYSTEMS 295

296 7. FUZZY SET THEORY

0 2.5 5 7.5 10 12.5 15 17.5 20

Very small Small

Medium

Large Very large

Interest rate

Linguistic

variable

Semantic

rule

Fuzzy

intervals

Base variable

v (interest rate in %)

Linguistic

values (states)

Figure 7.9. An example of a linguistic variable.

(a)

(b)

Around Around

Around Around Around

Around Around

0 0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10.950.85

0.75

0.65

0.55

0.45

0.35

0.25

0.15

0.05

Figure 7.10. Quantization versus granulation.

required to distinguish real numbers and the ﬁnite precision of any measur-

ing instrument. Appropriate quantization, whose coarseness reﬂects the pre-

cision of a given measuring instrument, is thus inevitable to resolve this

inconsistency. Consider, for example, a real variable that represents electric

current whose values range from 0 to 1 ampere. Assume, for the sake of sim-

plicity, that measurements of the variable can be made to an accuracy of 0.1

ampere. Then, according to the usual quantization, the interval [0, 1] is parti-

tioned into 10 semiopen intervals and one closed interval (quanta, aggregates),

[0, 0.5), [0.5, 1.5), [1.5, 2.5),...,[0.85, 0.95), [0.95, 1], which are labeled by their

ideal representatives 0, 0.1, 0.2,...,0.9, 1,respectively.This is exactly the quan-

tization shown in Figure 7.10a.

Although the usual quantization of real variables is capable of capturing

the limited resolutions of measuring instruments employed, it completely

ignores the issue of measurement errors. While representing the inﬁnite

number of values by a ﬁnite number of quanta (disjoint intervals of real

numbers and their ideal representatives) the unavoidable measurement errors

make the sharp boundaries between the quanta highly unrealistic. The repre-

sentation can be made more realistic by a ﬁnite number of granules (fuzzy

numbers or intervals), as illustrated for our example in Figure 7.10b. The fun-

damental difference is that transitions from each granule to its adjacent gran-

ules are smooth rather than abrupt. Moreover, any available knowledge

regarding measurement errors in each particular application context can be

utilized in molding the granules.

7.7.2. Types of Fuzzy Systems

In principle, fuzzy systems can be knowledge-based, model-based, or hybrid.

In knowledge-based fuzzy systems, relationships between variables are

described by collections of fuzzy inference rules (conditional fuzzy proposi-

tional forms). These rules attempt to capture the knowledge of a human

expert, expressed often in natural language. Model-based fuzzy systems are

based on traditional systems modeling, but they employ appropriate areas of

fuzzy mathematics (fuzzy analysis, fuzzy differential equations, etc.). These

mathematical areas, based on the notion of fuzzy numbers or intervals, allow

us to approximate classical mathematical systems of various types via appro-

priate granulation to achieve tractability, robustness, and low computational

cost. Hybrid fuzzy systems are combinations of knowledge-based and model-

based fuzzy systems. At this time, knowledge-based fuzzy systems are more

developed than model-based or hybrid fuzzy systems.

In knowledge-based fuzzy systems, the relation between input and output

linguistic variables is expressed in terms of a set of fuzzy inference rules (con-

ditional propositional forms). From these rules and any information describ-

ing actual states of input variables, the actual states of output variables are

derived by an appropriate compositional rule of inference. Assuming that the

input variables are X

, X

,...,and the output variables are Y

, Y

,...,we

7.7. FUZZY SYSTEMS 297

have the following general scheme of inference to represent the input–output

relation of the system:

This general formulation of knowledge-based fuzzy systems involves many

complex issues regarding the construction of the inference rules as well as pro-

cedures for using them to make inferences.These issues have been extensively

addressed in the literature, but they are beyond the scope of this overview.

Although fuzzy systems that approximate relationships among numerical

base variables have special signiﬁcance, due to their extensive applicability,

they are not the only fuzzy sets. Any classical (crisp) system whose variables

are not numerical (e.g., ordinal-scale or nominal-scale variables) can be fuzzi-

ﬁed as well.

7.7.3. Defuzziﬁcation

The result of each fuzzy inference based on a system of input and output lin-

guistic variables that approximate numerical base variables is, in general, a set

of fuzzy intervals, one for each output variable. Some applications (such as

control or decision making) require that each of these fuzzy intervals be con-

verted to a single real number that, in the context of a given application, best

represent the fuzzy interval. This conversion of a given fuzzy interval A, to its

real-number representation d(A), is called a defuzziﬁcation of A.

The most common defuzziﬁcation method, which is called a centroid

method, is deﬁned by the formula

(7.41)

or, when A is deﬁned on a ﬁnite universal set of numbers X, by the formula

(7.42)

xAx

()

◊

()

xAxdx

Ax dx

()

◊

()

⺢

Rule If is and is and then is and is

and

Rule If is and is and then is and is

and

111 2 21 111 2

112 2 22 112 2

...

........................................

XX Y Y

AA B

.......................................

...

Rule If is and is and then is and is

and

Fact is and is and

is and is and

nAA B

CXC

Conclusion D D

nn n

XX Y Y

11 2 2 11 2

11 2 2

11 22

298 7. FUZZY SET THEORY

This method may be viewed as unbiased since it treats all values A(x) equally.

However, it has been observed that other methods of defuzziﬁcation are

preferable in some applications. In these methods, differences in values A(x)

are either upgraded or downgraded to various degrees. Such methods are

members of the general class of defuzziﬁcation methods deﬁned either by the

formula

(7.43)

when A is deﬁned on ⺢, or by the formula

(7.44)

when A is deﬁned on a ﬁnite universal set X. Individual methods in this class

are distinguished by the value of parameter l Œ (0, •). When l = 1, we obtain

the centroid method. When l > 1, differences in values A(x) are upgraded;

when l < 1, they are downgraded.

An alternative kind of defuzziﬁcation is to replace fuzzy intervals with rep-

resentative crisp intervals rather than with individual real numbers. This kind

of defuzziﬁcation can be formalized rationally in information–theoretic terms,

as is shown in Chapter 9.Another problem, somewhat connected with defuzzi-

ﬁcation, is the problem of converting a given fuzzy set to a linguistic expres-

sion. This problem, which is usually referred to as linguistic approximation,

can also be dealt with in information-theoretic terms.

7.8. NONSTANDARD FUZZY SETS

In addition to standard fuzzy sets, several other types of fuzzy sets have been

introduced in the literature. Each of them is a nucleus of a particular formal-

ized language, which may be viewed as a branch of the overall fuzzy set theory.

Distinct types of fuzzy sets are distinguished from one another by the domain

and range of their membership functions.

The following are deﬁnitions of the most visible of nonstandard fuzzy sets.

In each of them, symbols X and A denote, respectively, the universal set of

concern and the fuzzy set deﬁned.

Interval-Valued Fuzzy Sets. Membership functions have the form

AX CI:,,Æ

[]

()

xAx

()

◊

()

xAxdx

()

◊

()

⺢

7.8. NONSTANDARD FUZZY SETS 299

where CI([0, 1]) denotes the set of all closed intervals contained in [0, 1]. For

each x ŒX, A(x) is a closed interval of real numbers in [0, 1]. An alternative

formulation:

where A

and A

are standard fuzzy sets such that A(x) £ A

(x) for all x Œ X.

Fuzzy sets deﬁned in this way are sometimes called gray fuzzy sets. Clearly,

for each x ŒX

An example of an interval-valued fuzzy set is shown in Figure 7.11.

Fuzzy Sets of Type 2. Membership functions have the form

where FI([0, 1]) denotes the set of all fuzzy intervals deﬁned on [0, 1]. For each

x Œ X, A(x) = I

, where I

is a fuzzy interval deﬁned on [0, 1]. This fuzzy inter-

val deﬁnes (imprecisely) the membership degree of x in A. An example is

shown in Figure 7.12, where the fuzzy intervals are assumed to be of a trape-

zoidal shape. For each x Œ X, the closed interval deﬁned by the shaded area

is the core of I

, and the closed interval deﬁned by the uppermost and lower-

most curves is the support of I

AX FI:,,Æ

[]

()

Ax Ax Ax CI

()

() ()

[]

()

,,.01

AAA= ,,

300 7. FUZZY SET THEORY

A(x)

0.5

Figure 7.11. Example of an interval-valued fuzzy set (or a gray set).

Fuzzy Sets of Type t (t > 2). For each t > 2, membership functions are deﬁned

recursively by the form

where FI

t-1

([0, 1]) denote the set of all fuzzy intervals of type t - 1.These types

of fuzzy sets were introduced in the literature as theoretically possible gener-

alizations of fuzzy sets of type 2. However, they have not been seriously inves-

tigated so far, and their practical utility remains to be seen.

Fuzzy Sets of Level 2. Membership functions have the form

where F(X) denotes a family of fuzzy sets deﬁned on X. That is, a fuzzy set of

level 2 is deﬁned on a family of fuzzy sets, each of which is deﬁned, in turn, on

a given universal set X. This mathematical structure allows us to represent

a higher-level concept by lower-level concepts, all expressed in imprecise

linguistic terms of natural language.

Fuzzy Sets of Level l (l > 2). For each l > 2, membership functions are deﬁned

recursively by the form

where F

(l-1)

(X) denotes a family of fuzzy sets of level l - 1. Sets of these types

are natural generalizations of fuzzy sets of level 2.They are sufﬁciently expres-

sive to facilitate representation of high-level concepts embedded in natural

:,,F

()

[]

AX:,,F

()

[]

AX FI

:,,Æ

[]

()

-1

7.8. NONSTANDARD FUZZY SETS 301

A(x)

(y) I

(y)

Figure 7.12. Example of a fuzzy set of type 2.