Fuzzy sets

We are continuously having to recognize people, objects, handwriting, voice, images, and other patterns, using distorted or unfamiliar, incomplete, oc-cluded, fuzzy, and inconclusive data, where a pattern should be allowed to have membership or belongingness to more than one class. This is also very significant in (say) medical diagnosis, where a patient afflicted with a certain set of symptoms can be simultaneously suffering from more than one disease.

Again, the symptoms need not necessarily be strictly numerical. It would be in natural terms, defined as linguistic and/or set variables such as very high, more or less low, between 50°(7 and 55°C. This is how the concept offuzziness comes into the picture.

Let us explain the concept of membership with an example. You ask a friend to meet you at 10 a.m. tomorrow. It is highly likely that your friend will arrive any time around 10 a.m., say, from 9.55 a.m. to 10.05 a.m. This defines the concept of a membership function along the time axis, with a peak (membership of 1) at 10 a.m. sharp having a bandwidth of 10 min. As you move away either side from the peak, the membership approaches the value 0. The bandwidth, again, is problem- and context-dependent. Hence if the person is serious, the bandwidth would be less, whereas otherwise the band-width would usually be more. Thus we see that although 10 a.m. is a crisp concept with {0,1} hard characterizing function, in reality it becomes fuzzy with [0,1] graded membership function. One may note that the membership value reflects the degree of compatibility or similarity of an event with an im-precise concept representing a fuzzy set, whereas the probability of an event is related to the number of times it occurs (i.e., its frequency).

Fuzzy sets were introduced in 1965 by Zadeh [10] as a new way of repre-senting vagueness in everyday life. This theory provides an approximate and yet effective means for describing the characteristics of a system that is too complex or ill-defined to admit precise mathematical analysis [11, 12]. The fuzzy approach is based on the premise that the key elements in human think-ing are not just numbers but can be approximated to tables of fuzzy sets, or,

40 SOFT COMPUTING

in other words, classes of objects in which the transition from membership to nonmembership is gradual rather than abrupt. Much of the logic behind human reasoning is not the traditional two-valued or even multivalued logic, but logic with fuzzy truths, fuzzy connectives, and fuzzy rules of inference.

Fuzzy set theory is reputed to handle, to a reasonable extent, uncertainties (arising from deficiencies of information) in various applications particularly in decision-making models under different kinds of risks, subjective judgment, vagueness, and ambiguity. The deficiencies may result from various reasons, namely, incomplete, imprecise, not fully reliable, vague, or contradictory in-formation depending on the problem. Since this theory is a generalization of the classical set theory, it has greater flexibility to capture various aspects of incompleteness or imperfection in information about a situation.

The use of linguistic variables may be viewed as a form of data compression, which can be termed granulation [I]. The same effect can also be achieved by conventional quantization. However, in the case of quantization the values are intervals, whereas in the case of granulation the values are overlapping fuzzy sets. The advantages of granulation over quantization are that

• It is more general.

• It mimics the way in which humans interpret linguistic values.

• The transition from one linguistic value to a contiguous linguistic value is gradual rather than abrupt, resulting in continuity and robustness.

Again, the uncertainty in classification or clustering of patterns may arise from the overlapping nature of the various classes. This overlapping may result from fuzziness or randomness. In the conventional classification technique, it is usually assumed that a pattern belongs to only one class. This is not necessarily realistic physically, and certainly not mathematically. A pattern can and should be allowed to have degrees of membership in more than one class. It is therefore necessary to convey this information while classifying a pattern or clustering a dataset.

Let us now consider the problem of processing and recognizing a gray tone image pattern. In a conventional vision system, each operation in low level, middle level, and high level involves crisp decisions to make regions, features, primitives, relations, and interpretations crisp. Since the regions in an im-age are not always crisply defined, uncertainty can arise at every phase of recognition tasks. Therefore it becomes convenient and natural and may be appropriate to avoid committing ourselves to specific (hard) decision by al-lowing the segments or contours to be fuzzy subsets of the image; the subsets are characterized by the possibility (degree) of a pixel belonging to them.

A fuzzy set A in a space of points R ~ {r} is a class of events with a continuum of grades of membership, and it is characterized by a membership function /M(r) that associates with each element in R a real number in the interval [0,1] with the value of HA (r) at r representing the grade of member-ship of r in A. Formally, a fuzzy set A with its finite number of supports

WHAT IS SOFT COMPUTING? 41

r2, . . . , rn is defined as a collection of ordered pairs A = {(^(ri),ri),i = 1,2, ...,n

where the support of A is an ordinary subset of R and is defined as S(A) = {r\r E R and n^A(r) > 0}.

Here /Zj, the grade of membership of r^ in A, denotes the degree to which an event TJ may be a member of A or belong to A. Note that /Zj = 1 indicates the strict containment of the event TJ in A. If, on the other hand, T-J does not belong to A, then /Zj = 0.

If the support of a fuzzy set is only a single point r\ 6 R, then

is called a fuzzy singleton. Thus A = (1/ri), for p,\ = 1, would obviously denote a nonfuzzy singleton.

In terms of the constituent singletons the fuzzy set A with its finite number of supports ri, r2, . . . , rn can also be expressed in union form as

A - *f + g + "- + ^

= E*?1. * = l , 2 , . . . , n (2.1)

= (J*£. » = 1,2,. ...n, where the + sign denotes the union.

Fuzzy logic is based on the theory of fuzzy sets and, unlike classical logic, aims at modeling the imprecise (or inexact) modes of reasoning and thought processes (with linguistic variables) that play an essential role in the remark-able human ability to make rational decisions in an environment of uncertainty and imprecision. This ability depends, in turn, on our ability to infer an ap-proximate answer to a question based on a store of knowledge that is inexact, incomplete, or not totally reliable. In fuzzy logic, everything, including truth, is a matter of degree [13]. Zadeh has developed a theory of approximate rea-soning based on fuzzy set theory. By approximate rearea-soning we refer to a type of reasoning that is neither very exact nor very inexact. This theory aims at modeling the human reasoning and thinking process with linguistic variables [11] in order to handle both soft and hard data, as well as various types of uncertainty. Many aspects of the underlying concept have been incorporated in designing decision-making systems [14, 15].

Assignment of membership functions of a fuzzy subset is subjective in na-ture and reflects the context in which the problem is viewed. It cannot be assigned arbitrarily. In many cases, it is convenient to express the membership function of a fuzzy subset in terms of standard S and IT functions. Note that fuzzy membership function and probability density function are conceptually different.

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Fig. 2.1 Standard S function.

2.2.2.1 Membership functions It is frequently convenient to employ stan-dardized functions with adjustable parameters (e.g., the S and TT functions) which are denned in the following equations (see also Fig. 2.1):

5(r; a,(3,c) = 0

= 1

for r < a for a < r < ft for /3 < r < c for r > c.

(2.2)

(2.3) 7r(r; c, A) = 5(r;c — A,c — ^,c) for r < c

= 1 — 5(r;c, c + f,c-f A) for r > c.

In S(r; a, /5, c), the parameter /?,/?= (a-f c)/2, is the crossover point, that is, the value of r at which S takes the value 0.5. In ?r(r; c, A), A is the bandwidth, that is, the distance between the crossover points of TT, while c is the central point at which TT is unity.

Let us consider the linguistic variable age (x). Here the linguistic values young and old play the role of primary fuzzy sets which have a specified meaning, for example,

Vyoung = 1-5(20,30,40),

Void = 5(50,60,70),

(2.4) (2.5) where the S and TT functions are defined by Eqs. (2.2) and (2.3), and and n0id denote the membership functions of young and old, respectively.

In pattern recognition problems we often need to represent a class with fuzzy boundary in terms of a TT function. A representation for such a TT

1 function, with range [0,1] and r € JRⁿ, may be given as [2]

7r(r; c, A) = 0,

for |< ||r-c||<A for 0 < ||r - c|| < | otherwise,

(2.6)

WHAT IS SOFT COMPUTING? 43

Fig. 2.2 TT function when r € R2

-where A > 0 is the radius of the -n function with c as the central point and ||.||

denotes the Euclidean norm. This is shown in Fig. 2.2 for r e JR2. Note that when the pattern r lies at the central point c of a class, then ||r — c|| = 0 and its membership value is maximum, that is, TT(C;C, A) = 1. The membership value of a point decreases as its distance from the central point c (i.e., ||r — c||) increases. When ||r — c|| = A/2, the membership value of r is 0.5, and this is called a crossover point.

2.2.2.2 Basic operations Basic operations related to fuzzy subsets A and B of R having membership values /MO") and HB(T}, r e R respectively, are summarized here [15].

• A is equal to B (i.e., A = B) => HA(T) = VB(r), for all r 6 R.

^ r = --r = I —

• A is a complement of B (i.e., A = B) =$• //

all r 6 R.

• A is contained in B (A C B) =$• HA^T) <

• The union of A and B (A U B) => P.A\JB(T} ==

r 6 R, where V denotes maximum.

• The intersection of A and B (A n B) all r 6 R, where A denotes minimum.

for all r € R.

(r) for

for all r for We also have the modifiers not, very, and more or less. These are explained, in terms of the linguistic value young, as follows:

fJ"not young — •*• (2.7)

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P-very young ~ (Pyoung) i

A*not very young = 1 ~ (f^young) >

A^more or less young = {fAyoung) •