Fuzzy measure theory

The second broad framework for dealing with uncertainty—fuzzy measure theory—was founded by Sugeno in 1974, even though some basic ideas of fuzzy measures had already been recognized by Choquet in 1953. Fuzzy measure theory is an outgrowth of classical measure theory, which is obtained by replacing the additivity requirement of classical measures with the weaker requirements of monotonicity (with respect to set inclusion) and continuity (or semicontinuity) of fuzzy measures. 

Z. Wang and G. J. Klir, Fuzzy Measure Theory. Plenum Press, New York, 1992. 

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... 

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. 

Molecular Similarity Measures and Chirality Measures Based on Resolution and Fuzzy Set Theory... 

In 1960, Lotfi Zadeh proposed what he called Fuzzy Logic to describe the way people think and speak in their natural language. Fuzzy set theory creates classes or groupings of data with boundaries not sharply defined. Fuzzy techniques are used to solve real-world problems where we must deal with imprecision in the variables and parameters that are measured... 

These ideas, of course, still require development. Just as we have theorems of probability theory, decision theory, reliability theory, it will be possible to develop fuzzy probability theory, fuzzy decision theory and fuzzy reliability theory perhaps based on the measures presented here. 

It is very clear from the complexity of the situations described in the case studies of the last two chapters, that simple factors of safety, load factors, partial factors or even notional probabilities of failure can cover only a small part of a total description of the safety of a structure. In this chapter we will try to draw some general conclusions from the incidents described as well as others not discussed in any detail in this book. The conclusions will be based upon the general classification of types of failure presented in Section 7.2. Subjective assessments of the truth and importance of the checklist of parameter statements within that classification are analysed using a simple numerical scale and also using fuzzy set theory. This leads us on to a tentative method for the analysis of the safety of a structure yet to be built. The method,however, has several disadvantages which can be overcome by the use of a model based on fuzzy logic. At the end of the chapte(, the discussion of the various possible measures of uncertainty is completed. 

Kosterev, V.V., Bolyatko, V.V., et al. (1998) Methods of fuzzy set theory applied to railway transportation risk assessment. International conference on soft computing and measurements (SCM 98), St. Petersburg, June. 

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. 

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]. 

According to fuzzy set theory of Zadeh [27, 31] there are two kinds of measure one is probability measure-Po, the other is necessity measure-Aec. The probability of fuzzy event is the biggest probability in all values which can make the event true while the necessity is defined as impossible which is the opposite of the event. Let be a fuzzy variable with membership function u(x). Let r be a real number. Then Pos > r and Nec > r show respectively the possibility and necessity of every fuzzy event > r. Then credibility measure [29] is... 

Fuzzy Set Theory (FST) was formalised by Prof. Lofti Zadeh at the University of California in 1965. The significance of fuzzy variables is that they facilitate gradual transition between states and consequently, possess a natural capability to express and deal with observation and measurement uncertainties. 

The validity discriminant discussed in this section is the descendant of an earlier cluster validity measure used by Gunderson ( ) to assess the quality of cluster configurations obtained in an application of the Fuzzy ISODATA algorithms. It is closely related to a method suggested by Sneath ( ) for testing the distinctness, i.e. separation, of two clusters, and also borrows from the ideas of Fisher s linear discriminant theory (see chapt. 4, Duda and Hart,(2 0). The validity discriminant attempts to measure the separation between the classes of a cluster configuration usually, but not necessarily, obtained by application of the FCV algorithms. A brief description follows ... 

From these basic properties of possibility measures, the full calculus of possibility theory, analogous to the calculus of probability theory, has been developed. Its primary role is to deal with incomplete information expressed in terms of fuzzy propositions. Due to limited space, it is not possible to cover here details of this calculus. 

Alternative symmetry deficiency measures of fuzzy sets are defined following the treatment of symmetry deficiency of ordinary subsets of finite n-dimensional Euclidean spaces, introduced earlier. To this end, we shall use certain concepts derived as generalizations of concepts in crisp set theory. 

Pictures of high resolution appear crisp, whereas pictures of low resolution appear fuzzy. A decrease of resolution is accompanied by an increase of fuzziness. Consequently, similarity measures based on the minimum level of resolution required to distinguish objects can be formulated in terms of the maximum level of fuzziness at which the objects are distinguishable. Similarity can be regarded as fuzzy equivalence. This principle provides an alternative mathematical basis for using the methods of topological resolution [262] in similarity analysis the theory of fuzzy sets [382-385]. 

Fuzzy set operations are derived from classical set theory. In addition, there exist theories for calculating with fuzzy numbers, functions, relations, measures, or integrals. 

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