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2013 | Buch

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Advances in Type-2 Fuzzy Sets and Systems

Theory and Applications

herausgegeben von: Alireza Sadeghian, Jerry M. Mendel, Hooman Tahayori

Verlag: Springer New York

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book explores recent developments in the theoretical foundations and novel applications of general and interval type-2 fuzzy sets and systems, including: algebraic properties of type-2 fuzzy sets, geometric-based definition of type-2 fuzzy set operators, generalizations of the continuous KM algorithm, adaptiveness and novelty of interval type-2 fuzzy logic controllers, relations between conceptual spaces and type-2 fuzzy sets, type-2 fuzzy logic systems versus perceptual computers; modeling human perception of real world concepts with type-2 fuzzy sets, different methods for generating membership functions of interval and general type-2 fuzzy sets, and applications of interval type-2 fuzzy sets to control, machine tooling, image processing and diet. The applications demonstrate the appropriateness of using type-2 fuzzy sets and systems in real world problems that are characterized by different degrees of uncertainty.

Inhaltsverzeichnis

Frontmatter

Theoretical Foundations

Frontmatter

Interval Type-2 Fuzzy Logic Systems and Perceptual Computers: Their Similarities and Differences

Abstract

In this chapter, we compare the interval type-2 fuzzy logic system and perceptual computer, so as to eliminate confusion among researchers about whether or not there really are differences between them. We show that there are many more differences than similarities between them by focusing on the following six issues: inputs and membership functions, fuzzifier versus encoder, rules versus computing with words (CWW) engines, inference versus output of CWW engine, output processing versus decoder, and outputs versus recommendation plus data.

Jerry M. Mendel

A Survey of Continuous Karnik–Mendel Algorithms and Their Generalizations

Abstract

Karnik–Mendel (KM) algorithms are important tools for type-2 fuzzy logic. This survey chapter summarizes some extensions of continuous Karnik–Mendel algorithms. It is shown that the solution of KM algorithms can be transformed into the solution of root-finding problems, and that the iteration formula in KM algorithms is equivalent to the Newton-Raphson root-finding method in numerical analysis. New iteration formulas are summarized that accelerate the convergence speed and it is shown that numerical integration methods can be used to improve computation accuracy. This chapter demonstrates that properties and structures of KM algorithms can be understood and improved with the techniques from numerical analysis.

Xinwang Liu

Two Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers: Adaptiveness and Novelty

Abstract

Interval type-2 fuzzy logic controllers (IT2 FLCs) have been attracting great research interests recently. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and hence better make use of them. This chapter explains two fundamental differences between IT2 and T1 FLCs: (1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and, (2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase.

Dongrui Wu

Interval Type-2 Fuzzy Markov Chains

Abstract

Uncertainties in fuzzy Markov chains can be treated in different ways. The use of interval type-2 fuzzy sets (IT2FS) allows describing the distributional behavior of an uncertain discrete-time Markov process through infinite type-1 fuzzy sets embedded in its Footprint of Uncertainty. In this way, a finite state fuzzy Markov chain process is defined in an interval type-2 fuzzy environment. To do so, its limiting properties and its type-reduced behavior are defined and applied to two explanatory examples.

Juan Carlos Figueroa-García

zSlices Based General Type-2 Fuzzy Sets and Systems

Abstract

This chapter provides a concise introduction to zSlices based general type-2 fuzzy sets and their associated set-theoretic operations. zSlices based general type-2 fuzzy sets allow the representation of and computation with general type-2 fuzzy sets by modeling each fuzzy set as a series of zSlices, i.e., modified interval type-2 fuzzy sets, thus greatly reducing computational as well as design and implementation complexity. The chapter proceeds to illustrate the role and application of zSlices based general type-2 fuzzy sets as part of general type-2 fuzzy systems and reviews their utility as part of both traditional, control style, as well as more recent applications such as fuzzy set based agreement modeling.

Christian Wagner, Hani Hagras

Geometric Type-2 Fuzzy Sets

Abstract

This chapter gives a review and technical overview of the geometric representation of a type-2 fuzzy set and explores logical operators used to manipulate this representation. Geometric fuzzy logic provides a distinct way of understanding a fuzzy system, where fuzzy sets and fuzzy logic operators are seen purely as geometric objects which are manipulated only using knowledge of geometry. This approach is simple and intuitive, ideal for those who are not well versed in discrete mathematics. For researchers working with fuzzy systems regularly, this approach can raise some interesting questions about how fuzzy sets and systems are constructed.

Simon Coupland, Robert John

Type-2 Fuzzy Sets and Bichains

Abstract

This chapter is a continuation of the study of the variety generated by the truth value algebra of type-2 fuzzy sets. That variety and some of its reducts were shown to be generated by finite algebras, and in particular to be locally finite. A basic question remaining is whether or not these algebras have finite equational bases, and that is our principal concern in this chapter. The variety generated by the truth value algebra of type-2 fuzzy sets with only its two semilattice operations in its type is generated by a four element algebra that is a bichain. Our initial goal is to understand the equational properties of this particular bichain, and in particular whether or not the variety generated by it has a finite equational basis.

John Harding, Carol L. Walker, Elbert Walker

Type-2 Fuzzy Sets and Conceptual Spaces

Abstract

Conceptual spaces provide a rich interpretation for computing with words, offering additional structure to that provided by fuzzy set models alone. In fuzzy conceptual spaces, properties are type-2 fuzzy sets on domains, concepts are type-2 fuzzy sets on pairs of properties and an observation is a family of fuzzy sets on domains relevant to a context. These type-2 fuzzy set structures are derived and manipulated using subsethood. This chapter relates such a theory of conceptual spaces to conventional multivariate classification and computing with words (CWW), and illustrates its application to land use assessment tasks.

Janet Aisbett, John T. Rickard

Type-2 Fuzzy Set Membership Function Generation

Frontmatter

Modeling Complex Concepts with Type-2 Fuzzy Sets: The Case of User Satisfaction of Online Services

Abstract

Specific characteristics of human perception, like context-dependency, imprecision, and diversity, demand capable formal frameworks for modeling the human mind. This chapter discusses a two-phase method for deriving type-2 fuzzy sets that model human perceptions of the linguistic terms used in describing online satisfaction. In the first phase, we describe the identification of the determinants of user satisfaction of online tourism services. We will demonstrate how the decomposition of the satisfaction concept into a set of simpler, albeit covering subconcepts, would be used to calculate a type-1 fuzzy set model of an individual’s perception. In the second phase, type-2 fuzzy sets modeling online user satisfaction are derived based on the obtained type-1 fuzzy sets. The construction of type-2 fuzzy sets is based on the exploitation of the fuzzy approach to represent uncertainty and by stacking the α-planes calculated at different levels of confidence around the estimated mean values of the type-1 fuzzy set.

Masoomeh Moharrer, Hooman Tahayori, Alireza Sadeghian

Construction of Interval Type-2 Fuzzy Sets From Fuzzy Sets: Methods and Applications

Abstract

In this chapter, we present some methods to construct interval type-2 membership functions from fuzzy membership functions and their applications in image processing, classification, and decision making. First, we review some basic concepts of interval type-2 fuzzy sets (IT2FSs). Next, we analyze three different approaches to construct IT2FSs starting from fuzzy sets and their applications in different fields.

Miguel Pagola, Edurne Barrenechea, Javier Fernández, Aranzazu Jurio, Mikel Galar, Jose Antonio Sanz, Daniel Paternain, Carlos Lopez-Molina, Juan Cerrón, Humberto Bustince

Interval Type-2 Fuzzy Membership Function Generation Methods for Representing Sample Data

Abstract

Type-2 fuzzy sets (T2 FSs) have been shown to manage uncertainty more effectively than type-1 fuzzy sets (T1 FSs) in several areas of engineering. However, computing with T2 FSs can require an undesirably large amount of computations since it involves numerous embedded T2 FSs. To reduce the complexity, interval type-2 fuzzy sets (IT2 FSs) can be used, since the secondary memberships are all equal to one. In this chapter, three novel interval type-2 fuzzy membership function (IT2 FMF) generation methods are proposed. The methods are based on heuristics, histograms, and interval type-2 fuzzy C-means (IT2 FCM). For each method, the footprint of uncertainty (FOU) is only required to be obtained, since the FOU can completely describe an IT2 FMF. The performance of the methods is evaluated by applying them to back-propagation neural networks (BPNNs). Experimental results for several data sets are given to show the effectiveness of the proposed membership assignments.

Frank Chung-Hoon Rhee, Byung-In Choi

Applications

Frontmatter

Type-2 Fuzzy Logic in Image Analysis and Pattern Recognition

Abstract

Interval type-2 fuzzy systems can be of great help in image analysis and pattern recognition applications. In particular, edge detection is a process usually applied to image sets before the training phase in recognition systems. This preprocessing step helps to extract the most important shapes in an image, ignoring the homogeneous regions and remarking the real objective to classify or recognize. Many traditional and fuzzy edge detectors can be used, but it is difficult to demonstrate which ones are better before the recognition results are obtained. In this work we show experimental results, where several edge detectors were used to preprocess the same image sets. Each resulting image set was used as training data for a neural network recognition system, and the recognition rates were compared. The goal of these experiments is to find the better edge detector that can be used to improve the training data of a neural network for an image recognition system.

Patricia Melin, Oscar Castillo

Reliable Tool Life Estimation with Multiple Acoustic Emission Signal Feature Selection and Integration Based on Type-2 Fuzzy Logic

Abstract

Reliable tool life estimation of cutting tool in micromilling is essential for planning machining operations for maximum productivity and quality. This chapter presents type-2 fuzzy tool life estimation system. In this system, type-2 fuzzy analysis is used as not only a powerful tool to model acoustic emission signal features, but also a great estimator for the ambiguities and uncertainties associated with them. Depending on the estimation of root-mean-square-error and variations in modeling results of all signal features, reliable ones are selected and integrated to cutting tool life estimation.

Qun Ren, Luc Baron, Marek Balazinski, Krzysztof Jemielniak

A Review of Cluster Validation with an Example of Type-2 Fuzzy Application in R

Abstract

Interval valued type-2 fuzziness can be represented by means of membership functions obtained with upper and lower values of the level of fuzziness. These upper and lower values for the level of fuzziness in FCM algorithm were obtained in our previous studies. A particular application of Interval valued type-2 fuzziness is shown for cluster validity analysis in this chapter. For this purpose, we introduce a brief taxonomy for cluster validity indices to clarify the contribution of our novel approach. To provide reproducibility of our technique, the source code is written in freely available language ‘R’ and can be found on our web site.

Ibrahim Ozkan, I. Burhan Türkşen

Type-2 Fuzzy Set and Fuzzy Ontology for Diet Application

Abstract

Nowadays, most people can get enough energy to maintain one-day activity, while few people know whether they eat healthily or not. It is quite important to analyze nutritional facts of foods eaten for those who are losing weight or suffering chronic diseases such as diabetes. However, diet is a problem with a high uncertainty, and it is widely pointed out that classical ontology is not sufficient to deal with imprecise and vague knowledge for some real-world applications like diet. On the other hand, a fuzzy ontology can effectively help handle and process uncertain data and knowledge. This chapter proposes a type-2 fuzzy set and fuzzy ontology for diet application and uses the type-2 fuzzy markup language (T2FML) to describe the knowledge base and rule base of the diet, including ingredients and the contained servings of six food categories of some common foods in Taiwan. The experimental results show that type-2 fuzzy logic system (FLS) performs better than type-1 FLS, proving that type-2 FLS can provide a powerful paradigm to handle the high level of uncertainties present in diet.

Chang -Shing Lee, Mei -Hui Wang, Chin -Yuan Hsu, Zhi -Wei Chen

Backmatter

Titel: Advances in Type-2 Fuzzy Sets and Systems
herausgegeben von: Alireza Sadeghian
Jerry M. Mendel
Hooman Tahayori
Verlag: Springer New York
Electronic ISBN: 978-1-4614-6666-6
Print ISBN: 978-1-4614-6665-9
DOI: https://doi.org/10.1007/978-1-4614-6666-6