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**Contents:**

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Info for. Tools Directory Info Desk usi. News and events. Intelligent Systems. Michael Bronstein Prof. More information. Ilia Horenko Prof. Type-2 fuzzy sets [53—55] are special kinds of fuzzy sets, the membership grades of which are themselves fuzzy i. The idea of type-2 fuzzy sets emerged from a paper by Zadeh [53], where he tried to address a typical problem with type-1 fuzzy sets that the membership function of a type-1 fuzzy set has no uncertainty associated with it.

Thus this finding sometimes contradicts the word fuzzy, since that word has the connotation of lots of uncertainty.

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Now, in type-2 fuzzy sets, there is no longer a single value for the membership function for any input measurement or x value, but there are a few. This fact has been illustrated in Figure 1. The A denotes a type-1 fuzzy set, A 1. A cross section of one slice of the third dimension is shown.

This cross section, as well as all others, sits on the FOU. We can either assign the same weighting or a variable weighting to the interval of membership function values [a, b]. When the former is done, the resulting type-2 fuzzy set is called either an interval type-2 fuzzy set or an interval valued fuzzy set although different names may be used, they are the same fuzzy set.

When the latter is done, the resulting type-2 fuzzy set is called a general type-2 fuzzy set to distinguish it from the special interval type-2 fuzzy set. It is illustrated in Figure 1. For an interval type-2 fuzzy set that 3D value is the same e. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set with its useful third dimension is sometimes referred to as a second-order uncertainty fuzzy set model.

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It has recently emerged as a major mathematical tool for managing uncertainty that arises from granularity in the domain of discourse i. R: Equivalence relation on U , called indiscernibility relation. A definable set in A is any finite union of elementary sets in A. Many different problems can be addressed by RST. During the last few years, this formalism has been approached as a tool used in connection with many different areas of research.

There have been investigations of the relations between RST and the Dempster—Shafer theory and between rough sets and fuzzy sets. It has also been used for, among many others, knowledge representation; data mining; dealing with imperfect data; reducing knowledge representation, and for analyzing attribute dependencies.

The notions of rough relations and functions are based on RST and can be applied as a theoretical basis for rough controllers, among others. It, in fact, has a major influence on an emerging field of study known as granular computing GrC [58—60].

The theory of rough sets and the theory of granularity offer artificial intelligence perspectives on granular computing. Specifically, granular computing can be viewed as a study of human-inspired problem solving and information processing. Granular computing concerns the processing of complex information entities called information granules, which arise in the process of data abstraction and derivation of knowledge from information. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like.

Currently, granular computing is more a theoretical perspective than a coherent set of methods or principles. As a theoretical perspective, it encourages an approach to data that recognizes and exploits the knowledge present in data at various levels of resolution or scales. In this sense, it encompasses all methods that provide flexibility and adaptability in the resolution at which knowledge or information is extracted and represented.

Like other biologically inspired techniques, it tries to extract ideas from a natural system, in particular the vertebrate immune system, in order to develop computational tools for solving engineering problems. The artificial immune system also plays a great role to maintain its own system against dynamically changing environment. The immune system thus aims at providing a new methodology suitable for dynamics problems dealing with unknown—hostile environment.

In recent years, much attention has been focused on behavior-based AI for its proven robustness and flexibility in a dynamically changing environment. Artificial immune systems are one such behavior-based reactive system that aims at developing a decentralized consensus making mechanism, following the behavioral characteristics of biological immune system. The human blood circulatory system contains roughly distinct types of B-lymphocytes, each of which has a distinct molecular structure and produces Y-shaped [63] antibodies from its surface.

Antibodies can recognize foreign substances, called antigens, that invade a living creature. Virus, cancer cells, and so on, are typical examples of antigens. To cope with a continuously changing environment, a living system possesses an enormous repertoire of antibodies in advance. The T-lymphocytes, on the other hand, are the cells maturing in the thymus, and are used to kill infected cells and regulate the generation of antibodies from B-lymphocytes as outside circuits of B-lymphocyte networks. It is interesting to note that an antibody recognizes an antigen by part of its structure called epitope.

The portion of the antibody that has the recognizing capability of an antigen is called paratope. Usually, epitope is the key portion of the antigen, and paratope is the keyhole portion of the antibody. Recent study in immunology reveals that each type of antibody has its specific antigen determinant, called idiotope. Jerne [63—65] proposed the idiotypic network hypothesis to explain the biological communication among different species of antibodies. According to the hypothesis, antibodies—lyphocytes are not isolated, but they communicate to each other among their variant species.

The third term denotes the stimulation from the antigen, and the fourth term represents the natural decay of the ith antibody. Equation 1. As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions with no random elements involved.

This behavior is known as deterministic chaos, or simply chaos. Chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a peculiar phenomenon known as chaos. One important characteristic of the chaotic systems is their sensitivity to initial conditions popularly referred to as the butterfly effect. Because of this sensitivity, the behavior of these systems appears to be random, even though the dynamics is deterministic in the sense that it is well defined and contains no random parameters.

Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, and population growth.

Currently, fuzzy logic and chaos theory form two of the most intriguing and promising areas of mathematical research. Recently, fuzzy logic and chaos theory have merged to form a new discipline of knowledge, called fuzzy chaos theory [68,69].

The detailed implications of fuzzy chaotic models are beyond the scope of this chapter. It has reportedly outperformed a few evolutionary algorithms EAs and other search heuristics like the PSO when tested over both benchmark and real-world problems. Differential evolution is a population-based global optimization algorithm that uses a floating-point real-coded representation. It is the method of creating this donor vector that distinguishes the various DE schemes.

Computational Intelligence and Pattern Analysis in Biology Informatics (Wiley Series in Bioinformatics): Medicine & Health Science Books. This book synthesizes current research in the integration of computational intelligence and pattern analysis techniques, either individually or in a hybridized .

To save space here, we briefly describe the binomial crossover, which is also employed by the modified DE algorithm.

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