Fuzzy sets theory

2 Fuzzy logic control systems 10.2.1 Fuzzy set theory 

Fuzzy logie was first proposed by Zadeh (1965) and is based on the eoneept of fuzzy sets. Fuzzy set theory provides a means for representing uneertainty. In general, probability theory is the primary tool for analysing uneertainty, and assumes that the 

Fuzzy sets represented by symmetrical triangles are commonly used because they give good results and computation is simple. Other arrangements include non-symmetrical triangles, trapezoids, Gaussian and bell shaped curves. 

Let the fuzzy set medium temperature be called fuzzy set M. If an element u of the universe of discourse U lies within fuzzy set M, it will have a value of between 0 and 1. This is expressed mathematically as 

When the universe of discourse is discrete and finite, fuzzy set M may be expressed as 

Uncertainty theory is also referred to as probability theory, credibility theory, or reliability theory and includes fuzzy random theory, random fuzzy theory, double stochastic theory, double fiizzy theory, the dual rough theory, fiizzy rough theory, random rough theory, and rough stochastic theory. This section focuses on the probability theory and fiizzy set theory, including probability spaces, random variables, probability spaces, credibility measurement, fuzzy variable and its expected value operator, and so on. 

Definition 2.1 Let Q be a nonempty set. A is a class of sets which consists of subsets of Q. If the following conditions are true, 

Definition 2.3 A random variable is a measurable function from a probability space ( 2, A, Pr) to the set of real numbers (H, and n-dimensional random vector is a measurable function from a probability space (12, A, Pr) to the space of n-dimen-sional real vectors 3 [26]. 

Theorem 2.2 Let be an n-dimensional random vector defined on the space of probability (Q, A, Pr), and f 3(i 3 a measurable function. Then f if) is a random variable (proof see Literature [26]). 

Fuzzy set theory was firstly proposed by Zadeh [27] via membership function in 1965. In 1970, Bellman and Zadeh [28] published a paper Decision-Making under Fuzzy Environment in Management Science which is a pioneer in the domain. From then on, many researchers have devoted to this study and consequentiy gready promoted its evolution. Liu and Liu [29] presented the concept of credihUily measure in 2002 and then Li X. and Liu B. refined the concept of credihUily measure later [30]. Based on his research, Liu B. further proposed uncertain theory [22]. 

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Fuzzy logic and fuzzy set theory are applied to various problems in chemistry. The applications range from component identification and spectral Hbrary search to fuzzy pattern recognition or calibrations of analytical methods. 

An overview over different applications of fuzzy set theory and fuzzy logic is given in [15] (see also Chapter IX, Section 1.5 in the Handbook). 

If a spectrum lacks certain Lines or contains extra lines from additional unknown components, or if the true line positions are blurred, fuzzy set theory can improve the matching. 

APPLICATION OF FUZZY SETS THEORY TO SOLVING TASKS OF MULTICOMPONENT QUALITATIVE ANALYSIS... 

Data collected by modern analytical instalments are usually presented by the multidimensional arrays. To perform the detection/identification of the supposed component or to verify the authenticity of a product, it is necessary to estimate the similarity of the analyte to the reference. The similarity is commonly estimated with the use of the distance between the multidimensional arrays corresponding to the compared objects. To exclude within the limits of the possible the influence of the random errors and the nonreproductivity of the experimental conditions and to make the comparison of samples more robust, it is possible to handle the arrays with the use of the fuzzy set theory apparatus. 

The central concept of fuzzy set theory is that the membership function /i, like probability theory, can have a value of between 0 and 1. In Figure 10.3, the membership function /i has a linear relationship with the x-axis, called the universe of discourse U. This produces a triangular shaped fuzzy set. 

Bayesian probability theory and methods that are based on fuzzy-set theory. The principles of both theories are explained in Chapter 16 and Chapter 19, respectively. Both approaches have advantages and disadvantages for the use in expert systems and it must be emphasized that none of the methods, developed up to now are satisfactory [7,11]. 

Zadeh [1975] extended the classical set theory to the so-called fuzzy set theory, introducing membership functions that can take on any value between 0 and 1. As illustrated by the intersection of the (hard) reference data set (A) and the fuzzed test data set (C), the intersection (E) shows an agreement of about 80%. Details on application of fuzzy set theory in analytical chemistry can be found in Blaffert [1984], Otto and Bandemer [ 1986a,b] and Otto et al. [1992],... 

Cornelissen AMG, van den Berg J, Koops WJ, Udo HMJ (2001) Assessment of the contribution of sustainability indicators to sustainable development a novel approach using fuzzy set theory. Agric Ecosys Environ 86 173-185... 

Distances in these spaces should be based upon an Zj or city-block metric (see Eq. 2.18) and not the Z2 or Euclidean metric typically used in many applications. The reasons for this are the same as those discussed in Subheading 2.2.1. for binary vectors. Set-based similarity measures can be adapted from those based on bit vectors using an ansatz borrowed from fuzzy set theory (41,42). For example, the Tanimoto similarity coefficient becomes... 

Fuzzy control, which is based on fuzzy sets theory proposed by Zadeh [9], can easily utilize empirical knowledge gained from skilled operators by employing... 

I. Pattern generation using fuzzy set theory. J. Comput. Chem. 23(12) 1176—1187... 

TABLE I Some of the Key Historical Developments in Our Understanding of Uncertainty That Led to the Emergence of Fuzzy Set Theory and Fuzzy Logic... 

We shall not discuss the properties or mathematics of fuzzy sets here, as this topic constitutes the subject matter of Chapter 2. What we will mention, however, is that fuzzy set theory, unlike most other attempts to establish a new approach to uncertainty, gradually gained in acceptance and eventually went on to become a stellar performer. 

The focus in this chapter is on the mathematics pertaining to fuzzy set theory and its role in science. However, other approaches to uncertainty are also briefly introduced. 

As is well known, fuzzy set theory was introduced by Zadeh in 1965. Its objects—fuzzy sets—are sets whose boundaries are not required to be precise that is, membership in a fuzzy set is not necessarily a matter of affirmation or denial, but it is, in general, a matter of degree. The degrees of membership of elements of a designated universal set in a given set are usually characterized by numbers in the unit interval [0,1], but various other characterizations are also employed. 

In each particular application of classical set theory as well as fuzzy set theory, all the sets of concern (classical or fuzzy) are subsets of a fixed set, which consists of all objects relevant to the applications. This set is called a uniuersal set and it is always denoted in this chapter by X. To distinguish classical (nonfuzzy) sets from fuzzy sets, the former are referred to as crisp sets. 

Contrary to the symbolic role of numbers 1 and 0 in characteristic functions of crisp sets, numbers assigned to relevant objects by membership functions of fuzzy sets have a numerical significance. This significance is preserved when crisp sets are viewed (from the standpoint of fuzzy set theory) as special fuzzy sets. 

As an example, several possible membership functions that are reasonable for defining the set of real numbers that are close to 3 are shown in Fig. 1. Which of these functions captures best the concept close to 3 depends on the context within which the concept is applied. It turns out, however, that most current applications of fuzzy set theory are not overly sensitive to changes in shapes of the membership functions employed. Since triangular shapes (function A in Fig. 1) and trapezoidal shapes (Fig. 

The problem of constructing membership functions in the contexts of various applications is not a problem of fuzzy set theory per se. It is a problem of knowledge acquisition, which is a subject of a relatively new field referred to as knowledge engineering. 

Any property or operation extended from classical set theory into the domain of fuzzy set theory that is preserved in all a-cuts is called a cutworthy property or operation-, if it is preserved in all strong a-cuts, it is called a strong cutworthy property or operation. It is important to realize that only some properties and operations involving fuzzy sets are cutworthy or strong cutworthy. They are of special significance since they bridge fuzzy set theory with classical set theory. They are like reference points from which other fuzzy properties or operations deviate to various degrees. 

Fuzzy sets that are defined on the set K of real numbers (i.e., A = J ) have special significance in fuzzy set theory. They can be interpreted as fuzzy numbers provided they satisfy the following requirements ... 

Propositions may also contain uncertainties of both types. To deal with information contained in such propositions, a measure-theoretic counterpart of fuzzy set theory was introduced by Zadeh under the name possibility theory.The following are basic notions of the theory. 

My purpose in this section is to explain why the various novel theories of uncertainty are important for science and why it is reasonable to view their use in science as a new scientific paradigm. 1 discuss several roles of these uncertainty theories in science, in particular the roles played by fuzzy set theory and fuzzy logic. 

We can see in these debates once again that most opponents of fuzzy set theory from the area of probability theory attempt to compare probabilities with degrees of truth (or degrees of membership). However, these are not comparable. As explained in Section III in the context of possibility theory, degrees of truth result from linguistic uncertainty, while probabilities result from information deficiency. These two types of uncertainty may be combined, but their comparison is meaningless. 

It is significant that some probabilists have been supportive of fuzzy set theory and recognized its complementarity to probability theory. One of them is Kapur, a well-known contributor to classical (probability-based) information theory. The following excerpt from a published interview expresses his views regarding fuzzy set theory ... 

In mathematics, earlier, algebra and topology were fighting for the soul of mathematics. Ultimately both are co-existing and are enriching each other. Similarly today there is a struggle between probability theory and fuzzy set theory to capture the soul of uncertainty. I am sure ultimately both will co-exist and enrich each other. Already the debate has led to a deeper understanding of what we mean... 

Another probabilist endorsing fuzzy set theory is Viertl, who has been active in developing reasoning methods that combine probability theory with fuzzy logic. ... 

Responses of opponents of fuzzy set theory to these examples of very successful applications of fuzzy set theory have been of two kinds. In the first kind of response, the examples are accepted as legitimate applications of fuzzy set theory, but it is maintained that some traditional methodology (classical control theory, Bayesian methodology, classical logic, etc.) would solve the problems even better. An example of this kind of response is the following excerpt from a personal letter I received from Anthony Garrett, one of my professional acquaintances and a devoted Bayesian, after I informed him about the fuzzy helicopter control ... 

While this kind of response does not challenge fuzzy set theory in any way, it challenges the Bayesian methodology to substantiate the expressed claims. 

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