Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

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that contains 14 position papers covering various aspects of the role and future prospects of fuzzy sets [Dubois, 2005]. The mathematical basis for formal fuzzy logic can be found in infinite-valued logics, first studied by the Polish logician Jan Lukasiewicz in the 1920s (see [Borkowski, 1970]). Lukasiewicz constructed a series of multi-valued logical systems, generalizing from small finite numbers of truth-values to those containing infinite sets of truth values. His work and calculation formulae are ingrained in modern fuzzy set theory and fuzzy logic, the genesis of which is credited to Zadeh in his seminal three part treatise on the theory and applications of linguistic variables [Zadeh, 1975a; Zadeh 1975b; Zadeh 1976]. Perhaps the biggest boost to the visibility and perceived utility of fuzzy set theory came from the application of rule-based fuzzy systems to problems in control [Mamdani and Assilian, 1975; Mamdani, 1977; Takagi and Sugeno, 1985; Sugeno, 1985, Verbruggen and Babuska, 1999, Passino and Yurkovich, 1998]. In what has become commonplace now, sets of linguistically described rules were created and inserted into a variety of non-linear control systems. The ease of design and the smoothness of the control surface from only a handful of rules made fuzzy controllers very popular in a variety of products from the automotive industry, consumer electronics markets, etc. Fuzzy controllers are well suited for low-cost embedded systems. While the big economic impact of fuzzy set theory and fuzzy logic centers on control, particularly in consumer electronics, there has been, and continues to be, much research and application of these technologies in pattern recognition, information fusion, data mining, and automated decision making [Bezdek et al., 1999, Keller et al., 1996]. There are national, multi-national, and international fuzzy systems professional societies around the globe whose purposes are to foster research, development and application of fuzzy set theory and fuzzy logic. Fuzzy systems are one of the core pillars of the IEEE Computational Intelligence Society. An introduction to the key components of fuzzy set theory and fuzzy logic is now given with the view toward computational methods of use in bioinformatics. After a discussion of the general principles of fuzzy set theory, membership functions and fuzzy connective operators, we focus on those areas for which we present specific applications within bioinformatics: fuzzy logic rule based systems, fuzzy clustering, fuzzy classifiers, particularly, the Fuzzy K-Nearest Neighbor algorithm, fuzzy measures and the fuzzy integral. The reader is referred to [Klir and Yuan, 1995; Bezdek et al., 1999] for more extensive development of the theory and selected applications.
X.4 Fuzzy membership Functions and Operators X.4.1 Membership functions Traditional set theory is based on binary, or two-valued, logic. Given a “universe” set X, a subset A of X can be defined in several ways. Suppose that X is the set of integers. The subset of prime numbers less than 10 can be specified by listing its members: A = { 2,3,5,7} or by providing defining properties { x ∈X x is a positive integer less than 10 and x has only 2 distinct divisors :1and x } A = . Alternately, we define a subset A by its characteristic function, which is also denoted by the set name, A : X → {0,1} from X into the binary set {0,1} given by A (x) ⎧1 = ⎨ ⎩0 if x ∈A if x ∉A. Zadeh [1965] simply defined a fuzzy subset of X as a function A : X → [0,1] , i.e., a characteristic function from X into the interval [0,1]. The value A(x) is called the membership of the point x in the fuzzy set A or the degree to which the point x belongs to the set A. For example, the fuzzy subset of “big positive integers” could be defined by A (x) ⎧ 1 ⎪1 − = ⎨ x ⎪ ⎩ 0 if x > 0 else. An example closer to the topic of this book is a membership function, A, that describes the similarity of a protein sequence S to that of a specific sequence called S’ in a protein database. A typical measure for sequence similarity is the expectation value, Eval, returned from BLAST [Altschul et al., 1997]. Assume the score for the alignment between S and S’ under a scoring scheme is R. Eval indicates the number of different alignments with scores equivalent to or better than R that are expected to occur in a database search by chance. Although expectation value is a good descriptor for sequence search from a database, it does not reflect all the information of the protein similarity. For example, when two sequences are identical, the expectation value varies accordingly to the length, although biologically the two proteins are the same. Two different long sequences can have better expectation value than two identical short sequences. To address this issue, we can describe the similarity between any protein sequence and one
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