20

CHAPTER-1

INTRODUCTION

----------Lotfi Zadeh

about its behaviour diminishes until a threshold is reached beyond which precision and significance

become mutually exclusive characteristics.

1.1 Introduction to Fuzzy Sets

In mathematics a set or crisp set is a collection of things that belong to some

definition. Any item either belongs to that set or does not belong to that set. Fuzzy sets

were first proposed by Lofti A. Zadeh (1965) in his paper entitled, Fuzzy . A fuzzy

set A in X is characterized by a membership function xA which associates with each

point in X a real number in the interval [0, 1], with the values of xA at x representing

the "grade of membership" of x in A . Thus, nearer the value of xA to unity, the

higher the grade of membership of x in A .

the crisp sets can be taken as special cases of fuzzy sets. Let A be a crisp set defined

over the universe X orU . Then, for any element x in X either x is a member of A or

not. In fuzzy set theory, this property is generalized. Therefore, in a fuzzy set, it is not

necessary that x is a full member of the set or not a member. It can be a partial member

of the set.

The generalization is performed as follows: For any crisp set A , it is possible to

define a characteristic function X i.e. the characteristic function takes either

21

of the values 0 or 1 in the classical set. Figure 1.1 shows the membership function

characterizing the crisp set 26and20betweenliesxxA .

Figure 1.1.1 Membership Function Characterizing the Crisp Set

For a fuzzy set, the characteristic function can take any value between zero and

one. The membership function xA of a fuzzy set A is a function ]1,0[xA . So every

element x in X has membership degree: ]1,0[xA . A is completely determined by

the set of tuple: Xxx,xA A .

Thus, the elements may belong in the fuzzy set to a greater or lesser degree as

indicated by a larger or smaller membership grade. The membership function may be

described as follows:

A or x A whether x ambiguity maximum is thereif 0.5ambiguity, no is thereandA x if 1ambiguity, no is thereandA x if 0

xA

The key difference between a crisp set and a fuzzy set is their membership function.

A crisp set has unique membership function, whereas a fuzzy set can have an infinite

number of membership functions to represent it. For example, one can define a possible

membership function for the set of real numbers close to 0 as follows:

.x;x

xA 21011

22

Here the number 3 is assigned a grade of 0.01, the number 1 is assigned a grade of

0.09 and the number 0 is assigned a grade 1. For fuzzy sets uniqueness is sacrificed, but

flexibility is gained because the membership function can be adjusted to maximize the

utility/ sensitivity for a particular application. It may be noted that elements in a fuzzy

set, because the membership need not be complete, can also be member of other fuzzy

sets on the same universe.

Membership functions might formally take any arbitrary form as they express only

an element- wise membership condition. However, they usually exhibit smooth and

monotonic shapes. This is due to the fact that membership functions are generally used to

represent linguistic units described in the context of a coherent universe of discourse i.e.,

the closer the elements, the more similar the characteristics they represent, as in the case

for variables with physical meaning.

The operation that assigns a membership value x to a given value Ux is

called fuzzification, e.g., Figure 1.2 shows the membership function of the fuzzy set

26and20betweenalmostisxxA i.e., the fuzzy set representing

approximately the same concept as that of the crisp set of Figure 1.1.1

Figure 1.1.2 The Membership Function of the Fuzzy Set

Operations on Fuzzy Sets

The operations on fuzzy sets are extension of the most commonly used crisp

operations. This extension imposes a prime condition that all the fuzzy operations which

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are extensions of crisp concepts must reduce to their usual meaning when the fuzzy sets

reduce themselves to crisp sets i.e., when they have only 1 and 0 as membership values.

For the definitions of the following operations, we assume A and B are two fuzzy subsets

of U with the membership functions xA and xB ; x denotes an arbitrary element

of U:

(i) Intersection/ AND operation is defined as

x,xminx BABA (ii) Union/OR operation is defined as

xxx BABA ,max

(iii) Complement or Negation of any fuzzy set A is defined as

xx AA 1 .

Consider a fuzzy set of tall men described by

tall men (say)190

1,5.187

75.0,185

5.0,5.182

25.0,180

0,175

0,165

0 A ;

average tall men (say)190

0,5.187

0,185

0,5.182

25.0,180

5.0,175

1,165

0 B ; then

1900,

5.1870,

1850,

5.18225.0,

1800,

1750,

1650BA

1901,

5.18775.0,

1855.0,

5.18225.0,

1805.0,

1751,

1650BA

Complement of tall men 190

0,5.187

25.0,185

5.0,5.182

75.0,180

1,175

1,165

1A

Fundamental Properties of the Fuzzy Sets

(i) Idempotent:

AAAAAAAA ;;

(ii) Commutative:

.; ABBAABBA

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(iii) Associative:

.; CBACBACBACBA

(iv) Distribution:

.; CBCACBACBCACBA

(v) Absorption:

.; ABAAABAA

(vi) Zero Law:

.; UUAA

(vii) Identity Law:

.; AUAAA

.; BABABABA

1.2. Basic Concepts Some important concepts related to fuzzy set theory are defined as given below: Equality of two Fuzzy Sets

Two fuzzy sets are said to be equal if and only

if Xxxx iiBiA );()( .

Standard Fuzzy Sets

Fuzzy sets are said to be standard if

Xxx iiA ;5.0)( .

Support of a Fuzzy Set

Given a fuzzy set A which is a subset of the universal set U, the support of A denoted by

supp(A), is an ordinary set defined as the set of elements whose degree of membership in

A is greater than 0 i.e.,

supp (A) = .0iAi xUx

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Fuzzy Number and Fuzzy Interval

A fuzzy number is a quantity whose value is imprecise rather than exact as is the

-valued) numbers. Mathematically, a fuzzy number is a

convex and normalized fuzzy set whose membership function is at least segmentally

continuous having bounded support and has the functional value 1xA at precisely one

element which is called modal value of fuzzy number. For instance: A symmetric

triangular fuzzy number (STFN) is defined by the membership function

otherwise0,

when1 jjjjj

j

Arccrc,

rcc

xj

,

where jc is known as the middle value for which 0r and1 jjj cA is the spread of jAIt can be represented by fig.1.1.1

Figure 1.2.1: Symmetrical Triangular Fuzzy Number

Membership Functions

A membership function is a curve that defines how each point in the input space is

mapped to a membership value between 0 and 1. The input space is sometimes referred to

as the universe of discourse. The only condition a membership function must really

satisfy is that it must vary between 0 and 1. The function itself can be an arbitrary curve

whose shape can be defined as a function that suits the point of view of simplicity,

convenience, speed and efficiency.The most commonly used membership functions are

as follows:

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(i) Triangular membership function is specified by three parameters, defined by

trim 0,bcxc,

abaxminmaxc,b,a;xf and is illustrated in fig.1.2.2 (a).

(ii) Trapezoidal membership function is specified by four parameters, defined by

trapm 01 ,cdxd,,

abaxminmaxd,c,b,a;xf and is illustrated in fig.1.2.2 (b).

(iii) General bell shaped function is specified by three parameters, defined by

gbellm b

acx

cbaxf 2

1

1,,; and is illustrated in fig.1.2.2 (c).

(iv) Gaussian membership is specified by two parameters and is defined as

Gaussian2

2mxexpm,;x , where mand denote the width and the

centre of the function respectively. We control the shape of the function by adjusting the

parameter . A small will generate a thin membership function, increasing the value

of will flatten the membership function.

Figure 1.2.2 Membership Functions

The Important points related to membership function

(i) Fuzzy sets describe vague concepts.

(ii) Fuzzy sets admit the possibility of partial membership in it.

(iii) The degree an object belongs to a fuzzy set is denoted by a membership value [0, 1]

(iv) A membership function associated with a given fuzzy set maps an input value to its

appropriate membership value.

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1.3. Fuzzy Uncertainty and Probability

Uncertainty and fuzziness are the basic nature of human thinking and of many real

world objectives. Fuzziness is found in our decision, in our language and in the way we

process information. The main use of information is to remove uncertainty and fuzziness.

In fact, we measure information supplied by the amount of probabilistic uncertainty

removed in an experiment and the measure of uncertainty removed is also called as a

measure of information while measure of fuzziness is the measure of vagueness and

ambiguity of uncertainties.

Fuzzy and probability are different ways of expressing uncertainty. While both

fuzzy and probability theory can be used to represent subjective belief, fuzzy set theory

uses the concept of fuzzy set membership (i.e. how much a variable is in a set),

probability theory uses the concept of subjective probability (i.e. how probable a variable

is in a set). While this distinction is mostly philosophical, the fuzzy-logic-derived

possibility measure is inherently different from the probability measure; hence they are

not directly equivalent.

Suppose a die is thrown and it is asked to guess the top face. The uncertainty about

the outcome is attributed to randomness. The best way to approach this question might be

to describe the status of the die in terms of a probability distribution on the six faces.

Uncertainty that arises due to chance is called Probabilistic Uncertainty.

Next, suppose an editor of a magazine sends an article to n reviewers for their

opinions. Each reviewer is asked to grade the article at some point in the scale 0 to 1. The

grade 0 means that the article is completely useless and the grade 1 means the article is

completely useful and important and there is no uncertainty in the mind of reviewer in

both the cases. Grade 0.1 means that the article is almost useless, but there is little content

in the article which create some uncertainty in the mind of the reviewer. Similarly, grade

0.9 means the article is almost useful but there are some undesirable content which create

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uncertainty in the mind of reviewer about its utility. This type of uncertainty is called

fuzzy uncertainty which is different from probabilistic uncertainty.

Since fuzzy set theory makes statements about one concrete object, therefore, it

helps in modelling local vagueness. On the other hand, probability theory makes

statements about a collection of objects from which one is selected; therefore it helps in

modelling global uncertainty.

A very good example that demonstrates the conceptual difference between

probability and fuzzy classification is: A person who is dying of thirst in the desert is

given two bottles of fluid. One bottle label says that it has a 0.9 membership in the class

a 90% probability of being pure drinking water and a 10% probability of being poison.

Which bottle would the person choose?

In this example the probability bottle contains poison; this is quite probable since

there was 1 in 10 chance of it being poisonous. The fuzzy bottle contains swamp water,

this makes sense since swamp water would have a 0.9 membership in the class of non-

poisonous fluids, and the point is that probability involves crisp set.

Fuzzy uncertainty differs from probabilistic uncertainty because it deals with the

situations where the boundaries are not sharply defined. Probabilistic uncertainties are not

due to ambiguity about set- boundaries, but rather about the belongingness of elements or

events to crisp sets.

However, many statisticians are of the opinion that only one kind of mathematical

uncertainty is needed and thus fuzzy logic is unnecessary. Lofti Zadeh (1965) argues that

fuzzy logic is different in character from probability, and is not a replacement for it. He

fuzzified probability to fuzzy probability and also generalized it to what is called

possibility theory. Note, however, that fuzzy logic is not controversial to probability but

rather complementary.

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1.4. Probabilistic and Fuzzy Information Measure Let X is a discrete random variable with probability distribution

npppP ..,........., 21

uncertain degree of the randomness in a probability distribution. The information

contained in this experiment is given by

i

n

ii ppPH

1log (1.4.1)

which is well known Shannon entropy.

The concept of entropy has been widely used in different areas, e.g.,

communication theory, statistical mechanics, finance, pattern recognition, and neural

network etc. Fuzzy set theory developed by Lofti A. Zadeh (1965) has found wide

applications in many areas of science and technology, e.g., clustering, image processing,

decision making etc. because of its capability to model non-statistical imprecision or

vague concepts.

Zadeh (1965) developed the concept of fuzzy set and defined the entropy of a

concept is needed in order to define it. It may be noted that fuzzy entropy deals with

vagueness and ambiguous uncertainties, while Shannon (1948) entropy deals with

randomness (probabilistic) of uncertainties. Fuzzy entropy is a measure of fuzziness of a

set which arises from the intrinsic ambiguity or vagueness carried by the fuzzy set.

Let nxxx ,...,, 21 be the members of the universe of discourse, then all ),( 1xA

),( 2xA )( nA x lie between 0 and 1, but these are not probabilities because their sum

is not unity. However,

,)(

)()(

1

n

iiA

iAiA

x

xx .,...,2,1 ni (1.4.2)

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is a probability distribution. On considering (1.4.2) Kaufman (1980) defined entropy of a

fuzzy set A having n support points nxxx ,...,, 21 by

)(AHlogn

1 n

1iiAiA ).x(log)x( (1.4.3)

Uncertainty and fuzziness are the basic nature of human thinking and of many

real world objectives. Fuzziness is found in our decision, in our language and in the way

we process information. The main use of information is to remove uncertainty and

fuzziness. In fact, we measure information supplied by the amount of probabilistic

uncertainty removed in an experiment and the measure of uncertainty removed is also

called as a measure of information while measure of fuzziness is the measure of

vagueness and ambiguity of uncertainties. In Fuzzy set theory, the entropy is a measure

of fuzziness which means the amount of average ambiguity or difficulty in making a

decision whether an element belongs to a set or not.

A measure of fuzziness )(AH of a fuzzy set A should satisfy at least the following

four properties (P-1) to (P-4):

(P-1) )(AH is minimum if and only if A is a crisp set, i.e.

.,...2,1: allfor 1or 0)( nixx iiA

(P-2) )(AH is maximum if and only if A is most fuzzy set, i.e.

nixx iiA ,...2,1: allfor 5.0)( .

(P-3) )(AH )(AH , where A is sharpened version of A.

(P-4) )(AH )(AH , where A is the complement of A.

Since )( iA x and 1 )( iA x

fuzziness, therefore,

distribution Deluca and Termini (1971) defined the following measure of fuzzy entropy:

(1.4.4) .)(1log)(1)(log)()(

1

n

iiAiAiAiA xxxxAH

31

It may be seen that (1.4.4) satisfies all the four properties (P-1) to (P-4) and hence it is a

valid measure of fuzzy entropy. The fuzzy information measures the uncertainty due to

vagueness and ambiguity.

However, we have other fuzzy information measures, but (1.4.4) can be

considered as the first correct measure of ambiguity of a fuzzy set. In addition, Yager

(1979) also defined an information measure of a fuzzy set based on the distance from the

set to its complement set. Similarly, Kosko (1986) introduced other kind of fuzzy

information measure by considering the distance from a set to its nearest non fuzzy set

and the distance from the set to its farthest non fuzzy set.

The fuzzy information measure with an exponential function was introduced by

Pal and Pal (1989). Later on, they introduced the concept of higher r th order entropy of a

fuzzy set in their paper Pal and Pal (1992). Further, Bhandari and Pal (1993) made a

survey on information measures on fuzzy sets and gave some new measures of fuzzy

information. Kapur (1997), Prakash (1998, 2001) and Hooda (2004) also suggested and

studied new measures of fuzzy information. Next, we enumerate the similarities and

dissimilarities between the two types of measures - fuzzy information measure and

probabilistic entropy.

Similarities

(i) For all probability distributions 10 ip for each i and for every fuzzy set

10 iA x for each i.

(ii) The probabilistic entropy measures the closeness of the probability distribution P

nppp ..,........., 21 with uniform distribution nnn1.....,.........1,1 and fuzzy information

measure, measures the closeness of fuzzy distribution with the most fuzzy vector

distribution21.....,.........

21,

21 .

(iii) Probabilistic and fuzzy information measures are concave functions of

nppp ..,........., 21 and nAAA xxx ..,........., 21 respectively. Starting with any

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values of nAAA xxx ..,........., 21 and approaching the vector 21.....,.........

21,

21 the

fuzzy information measure will increase. Also, starting with any probability vector

nppp ..,........., 21 and approaching the vectornnn1.....,.........1,1 , the probabilistic

entropy will increase.

(iv) Probabilistic directed divergence measures and fuzzy directed divergence measures

are convex functions with minimum value zero. Further, analogous to each measure of

probabilistic entropy and directed divergence, we have measure of fuzzy information and

fuzzy directed divergence.

Dissimilarities

(i) While 11

n

iip for all probability distributions,

n

iiA x

1

need not be equal to one

and it need not even be the same for all fuzzy sets i.e., the probabilities of 1n outcomes

will determine the probability of the nth outcome, but the knowledge of fuzziness of 1n

elements gives no information about the fuzziness of the nth element.

(ii) The probabilities ip and ip1 make different contributions to probabilistic entropy.

However, iA x gives the same degree of fuzziness as iA x1 because both are

equidistant from 21 and the crisp set values 0 and 1.

(iii) Most of the measures of probabilistic entropy are of the form n

iipf

1while most

measures of fuzzy information measure are of the form n

iiA

n

iiA xfxf

111

(iv) Similarly, for probability distributions npppP ..,........., 21 and

nqqqQ ..,........., 21 , most of the probabilistic directed divergence measures are of

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the form n

iii qpf

1, while for fuzzy sets A and B, the fuzzy directed divergence

measures are of the form n

iiBiA

n

iiBiA xxfxxf

111,1, .

1.5. Fuzzy Directed Divergence Measure Let npppP ....,,........., 21 and nq....,,.........q,qQ 21 be two probability

distributions of a discrete random variable. The measure of directed divergence of P

from Q is defined as a function QPD : satisfying the following conditions:

0: QPD

0: QPD if and only if QP

PHUHUPD : ; where U is the uniform probability distribution and PH is

the measure of probabilistic entropy

Kullback and Leibler (1951) defined the measure of directed divergence of

probability distribution P from the probability distribution Q as

i

in

ii q

ppQPD log:1

(1.5.1)

which is called the distance measure as it measures how far the probability distribution

P is from the probability distribution Q .

Further, Kullback (1959) defined the measure of symmetric divergence as follows:

i

in

iii q

pqpPQDQPDQPJ log:::1

. (1.5.2)

Analogously, the fuzzy measures of directed divergence is defined as

0, BAI

0, BAI if and only if BA

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AHAHAAI FF, where FA is the most fuzzy set i.e. .;21 ixiAF

Let nAAA xxxA ,........,, 21 and nBBB xxxB ,........., 21 be

two fuzzy sets, then the simplest measures of fuzzy directed divergence and fuzzy

symmetric divergence introduced by Bhandari and Pal (1993) are

n

i iB

iAiA

iB

iAiA x

xxxxxBAI

1 11log1log,

(1.5.3)

n

i iAiB

iBiAiBiA xx

xxlogxxA,BIB,AIB,AJ1 1

1 (1.5.4)

This measure of fuzzy symmetric divergence can discriminate between two fuzzy

sets. It may be noted that BAJ , is symmetric with respect to A and B . It also

satisfies the following properties:

0, BAJ

0, BAJ if and only if BA

ABJBAJ ,, .

and BAJ , does not satisfy the triangular inequality property of a metric.

Therefore, BAJ , can be called pseudo metric. It may be noted that if we take

FAB i.e. ,;21 ixiB then we have

n

iiAiAiAiAF xxxxnAAI

11log1log2log,

(1.5.5)

or AHAHAAI FF, it gives the measures of non-fuzziness in the set A. Some

generalizations have been studied in chapter-3. The measures (1.5.3) and (1.5.4) can be

generalized parametrically in so many ways.

The uncertainty is the state of being uncertain (i.e. not certain to occur) which gives

rise to fuzziness and ambiguity. Ambiguity can be viewed in non specificity (i.e.

indistinguishable alternatives) and conflict (i.e. distinguishable alternatives) while

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fuzziness can be viewed a slack of distinction between a set and its complement and

vagueness is non- specific knowledge about lack of distinction.

Harvda and Charvat (1967) defined the directed divergence measure of the

probability distribution P and Q as

,,;qpQ:PDn

iii

1011

11

1

(1.5.6)

Corresponding to (1.5.6) Hooda (2004) suggested the following measure of fuzzy

directed divergence and symmetric divergence: n

iiiii xxxxB,AI

BABA1

11 1111

1

(1.5.7)

and 10,;A,BIB,AIB,AJ respectively.

,,;qplogQ:PDn

iii

101

11

1

(1.5.8)

Bajaj and Hooda (2010) define the following measure of fuzzy directed divergence and

symmetric divergence:

;xxxxlogB,AIn

iiBiAiBiA

1

11 111

1 (1.5.9)

and 10,;A,BIB,AIB,AJ respectively.

1.6. Fuzzy Information Improvement and Total Ambiguity The probabilistic measure of information improvement, suggested by Theil (1967), is

given by

i

in

ii q

rpRPDQPD log::1

(1.6.1)

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where P and Q are observed and predicted probability distributions respectively of a

random variable and nrrrR ....,,........., 21 is the revised probability distribution of Q.

Similarly, suppose the correct fuzzy set is A and originally our estimate for it was the

fuzzy set B that was revised to C, the original ambiguity was BAI , and final ambiguity

is CAI , .

Analogously the reduction in ambiguity is given by

n

i iB

iCiA

iB

iCiA x

xxxxxCAIBAI

1 11

log1log,,

(1.6.2)

which may be called fuzzy information improvement measure.

The total ambiguity of the fuzzy set A about fuzzy set B is the sum of two

components:

(i) Fuzzy entropy of the fuzzy set A.

(ii) Fuzzy directed divergence of fuzzy set A from fuzzy set B i.e.,

BAIAHTA ,

Kapur (1997) suggested the measure of total fuzzy ambiguity which can be

obtained by taking the sum of measure of fuzzy directed divergence and corresponding

measure of fuzzy entropy. From (1.4.4) and (1.5.3) we have n

iiBiA

n

iiBiA xlogxxlogxTA

1111 (1.6.3)

1.7. Fuzzy Matrix and Fuzzy Binary Relation

A matrix is a table of numbers with a finite number of rows and finite number of

columns. The following

A =

324985103

(1.7.1)

is an example of a matrix with three rows and three columns.

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Fuzzy Matrix

If A enjoys an additional property, such that all the entries in A are from the unit

interval [0, 1] i.e., from the fuzzy set [0, 1] then we can say that A is a fuzzy matrix and

denoted as X. For example

X =

3.02.04.09.08.05.0

103.0 (1.7.2)

is a fuzzy matrix i.e. X = )( n x mijx , where ]1,0[ijx refer to Vasantha Kandasamy, et.al.(2007) The basic concept of the fuzzy matrix theory is very simple and can be applied to

social and natural situations. A branch of fuzzy matrix theory uses algorithms and algebra

to analyse data. It is used by social scientists to analyse interactions between actors and

can be used to complement analyses carried out using game theory or other analytical

tools.

A standard fuzzy matrix is the fuzzy matrix of the following form where all the

entries are less than or equal to 0.5:

X =

3.02.04.05.02.05.01.001.0

(1.7.3)

Fuzzy Binary Relation A fuzzy relation is characterized in the same manner as a fuzzy set. It consists of the

set of ordered pairs containing the ordered relations and their membership grades by

George and Yuan. (2006). For example

nnnn yxyxyxyx ,,............,,,,,,,, 133112211111 .

The elements of the fuzzy relations are defined as ordered pairs

,,,, 2111 yxyx nn yx ,..... . These elements are again grouped with their

38

membership grades nn.....,, 131211 , whose values range from 0 to 1, including 0

and 1.

A fuzzy relation R is a mapping from the Cartesian space X Y to the interval [0,

1], where the strength of the mapping is expressed by the membership function of the

relation ., yxR where ]1,0[: BAR such that

y torelatednotisor xy torelated isthat x ambiguity maximum0.5 y torelatedisxthatambiguityno1 y torelatednotisxthatambiguityno0

, yxR

The crisp relation R represents the presence or absence of association, interaction or

interconnection between the elements of these two sets. While fuzzy relations can be

generalized to allow for various degrees or strength of association or interaction between

elements. Degree of association can be represented by membership grades in a fuzzy

relation in the same way as degree of the set membership are represented in a fuzzy set.

1.8. Fuzzy Logic

The concept of a fuzzy logic is one that it is very easy for the ill-informed to

dismiss as trivial and/or insignificant. It refers not to a fuzziness of logic but instead to

logic of fuzziness, or more specifically to the logic of fuzzy sets. Those that examined

Lofti A. Zadeh's (1965) concept more closely found it to be useful for dealing with real

world phenomena. From a strictly mathematical point of view the concept of a fuzzy set

is a brilliant generalization of the classical notion of a set.

Now the concept of a fuzzy set is well established as an important and practical

construct for modelling. Moreover, Zadeh's (1965) formulation makes one realize how

artificial the classical black-white formulation of classical logic is. In a world of shades

of gray a black-white dichotomy involves an unnecessary arbitrariness, an artificiality

imposed upon that world. In this way an attempt is made to apply a more human-like way

of thinking in the programming of computers.

39

Gradual transition from precision to imprecision/specificity to non-specificity/ sharpness

to vagueness has made fuzzy logic a profitable tool for the controlling of subway-systems

and complex industrial processes. It is also a profitable tool for household-electronics,

entertainment-electronics, diagnosis systems and other expert systems.

Fuzzy Logic was initiated by Lofti A. Zadeh (1965), professor of computer

science at the University of California in Berkeley. However, the idea of an extended

multivalued logic had been considered by physicists early in the 20th century, but had not

become a standard part of science (there was the concept of vague sets). Fuzzy system is

an alternative to traditional concepts of set membership and logic that has its origins in

ancient Greek philosophy. Fuzzy logic has ability to capture (mathematically) reasoning

about the notions with inherited fuzziness, such as being tall, young, fat, hot etc. Similar

to probabilistic logic, we have the real valued truth in fuzzy logic also i.e., the truth of

certain statement can be any real number in the interval [0, 1].

Fuzziness is found in our decision, in our thinking, in the way we process

information and particularly in our language. A sunny day may contain some clouds

see you later a little more I don't feel very

well" are fuzzy expressions. However, for most of the problems that we face, Zadeh

(1968, 1973, 1975, 1984, 1995 and 2002) suggests that we can do a better job in

accepting some level of imprecision. For any field S and any theory T can be fuzzified by

replacing the concept of a crisp set in S and T by that of a fuzzy set.

Fuzzification leads basic field such as arithmetic to fuzzy arithmetic, topology to

fuzzy topology, graph theory to fuzzy graph theory, probability theory to fuzzy

probability theory. Similarly, in application to applied fields such as neural network

theory, stability theory, pattern recognition and mathematical programming, fuzzification

leads to fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and

fuzzy mathematical programming. Fuzzification gives greater generality, higher

expressive power, and an enhanced ability to model real world problems. Most

importantly, it gives a methodology for exploiting the tolerance for imprecision i.e., a

methodology which serves to achieve tractability, robustness and lower solution cost.

40

(1965) work had a profound influence on the thinking about uncertainty

because it challenged not only probability theory as the sole representation for

uncertainty, but the very foundations upon which probability theory was based classical

binary (two valued) logic with reference to George and Yuan (1995).

Fuzzy logic is a powerful problem-solving methodology with a myriad of

remarkably simple way, in a sense fuzzy logic resembles human decision making with its

ability to work from approximate data and find precise solutions. Unlike classical logic

which requires a deep understanding of a system, exact way of thinking, which allows

modelling complex systems using a higher level of abstraction originating from our

knowledge and experience. Fuzzy Logic allows expressing this knowledge with

subjective concepts such as very hot, bright red, and a long time which are mapped into

exact numeric ranges.

Fuzzy Logic has been gaining increasing acceptance during the past few years.

There are over two thousand commercially available products using Fuzzy Logic, ranging

from washing machines to high speed trains. Nearly every application can potentially

realize some of the benefits of fuzzy logic, such as performance, simplicity, lower cost,

and productivity.

Fuzzy Logic has been found to be very suitable for embedded control

applications. Several manufacturers in the automotive industry are using fuzzy

technology to improve quality and reduce development time. In aerospace, fuzzy logic

enables very complex real time problems to be tackled using a simple approach. In

consumer electronics, fuzzy improves time to market and helps reduce costs. In

manufacturing, fuzzy is proven to be invaluable in increasing equipment efficiency and

diagnosing malfunctions.

1.9. Fuzzy Clustering Techniques

Nobel Laureate Herbert Simon

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Dynamic Pattern Recognition

Traditionally, methods of fuzzy pattern recognition consider objects at a fixed

moment in time, but sometimes for better recognition of an object, it is not only sufficient

to consider the properties of object at a certain moment in time but also to analyse

properties characterizing the temporal development. The fuzzy pattern recognition

techniques in which dynamic viewpoint are desirable is known as dynamic fuzzy pattern

recognition.

Objects are called dynamic if they represent measurements or observations of a

dynamic system and contain a history of their dynamic development. In other words,

each dynamic object is a time dependent sequence of observations and is described by a

discrete function of time. This time-dependent function is called a trajectory of an object.

Thus, in contrast to static objects, dynamic objects are represented not by points but by

multidimensional trajectories in the feature space extended by an additional dimension

nting a dynamic object are not real

numbers but vectors describing the development of the corresponding feature over time.

The spectral analysis in the spirit of traditional Fourier transform does not preserve the

time dependence of the patterns when a signal is non stationary. Wavelet analysis, on the

other hand, has emerged as a remarkable tool for decomposition of functions. General

procedures of wavelet-based regression estimators assume the time invariant coefficients.

To accommodate the stochastic and dynamic properties that are typical of many

applications, Zuohong Pan and Xiaodi Wang (1996) incorporate the state-space model in

the wavelet estimator. The coefficients of the wavelet estimators are formulated as

dynamic (random) processes so that the Kalman filtering approach can be applied. The

resulting estimator is a stochastic nonlinear wavelet-based estimator.

Pattern recognition techniques for time-series forecasting are beginning to be

realized as an important tool for predicting chaotic behaviour of dynamic systems. Singh

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and Paul (1998) develop the concept of a Pattern Modelling and Recognition System

which is used for predicting future behaviour of time-series using local approximation

and also studied the effect of noise filtering on the Performance of the proposed system.

Fourier analysis is used for noise-filtering the time-series. The results show that Fourier

analysis is an important tool for improving the performance of the proposed forecasting

system. The results are discussed on three benchmark series and the real US, S&P

financial index.

To deal with document clustering, a new Fuzzy Hierarchical clustering method for

clustering documents based on dynamic document cluster centres is presented by Shyi-

Ming Chen and Liang-Yu Chen (2007). We use the terms in documents to construct

dynamic cluster centres and the cluster centres will change when different documents are

merged. The degrees of similarities between document clusters are calculated based on

these dynamic document cluster centres.

A new Adaptive Fuzzy Clustering algorithm for remotely sensed images combines

the capability of Fuzzy Mathematics and Adaptation in image segmentation is proposed

by Chih-Cheng Hung, Wenping Liu and Bor-Chen Kuo (2008). The proposed adaptive

Fuzzy c- means clustering algorithm is to improve the Fuzzy c- means which needs prior

information about the number of classes. This new clustering algorithm combines the

capability of Fuzzy Mathematics and Adaptation. The Adaptive capability is achieved by

using the mechanism of splitting and merging.

Unlike most of the Fuzzy clustering algorithms which require a prior knowledge

about the data set this new algorithm can learn the number of classes dynamically. First

the membership and cluster centre are updating using the FCM Fuzzy sets. Secondly the

FCM employs the splitting and merging methods for forming the cluster. It determines

the splitting by merging the cluster membership that weighs the cohesion degree and

decides the merging by the number of samples in both clusters that weighs independency

degree. This clustering algorithm arbitrary chooses initial cluster centres. The AFCM

provides a statistical method based on the sample space density to initialize the number of

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clusters and cluster centres. So that, the adaptive capability of the algorithm can be

enhanced.

Bajaj, Shrivastav and Hooda (2009) have studied applications of fuzzy clustering

and fuzzy linear regression models. Here we develop and study fuzzy clustering

techniques for dynamic pattern recognition and image compression in chapter 6 and 7.