Interval Type-3 Fuzzy Systems: Theory and Design (Studies in Fuzziness and Soft Computing, 418) 3030965147, 9783030965143

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Studies in Fuzziness and Soft Computing

Oscar Castillo Juan R. Castro Patricia Melin

Interval Type-3 Fuzzy Systems: Theory and Design

Studies in Fuzziness and Soft Computing Volume 418

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at https://link.springer.com/bookseries/2941

Oscar Castillo · Juan R. Castro · Patricia Melin

Interval Type-3 Fuzzy Systems: Theory and Design

Oscar Castillo Division of Graduate Studies Tijuana Institute of Technology Tijuana, Baja California, Mexico

Juan R. Castro School of Engineering UABC University Tijuana, Baja California, Mexico

Patricia Melin Division of Graduate Studies Tijuana Institute of Technology Tijuana, Baja California, Mexico

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-96514-3 ISBN 978-3-030-96515-0 (eBook) https://doi.org/10.1007/978-3-030-96515-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The use of type-2 fuzzy systems has become widespread in the leading economy sectors, especially in industrial and application areas, such as services, health, defense, and so on. Recent studies and research have focused on type-2 fuzzy systems that enable handling uncertainty in many applications, such as intelligent control, robotics, pattern recognition, time series prediction, and medical diagnosis. However, more recently, the use of interval type-3 fuzzy systems has been receiving increasing attention, and some successful applications have been developed in the last year. These issues were taken into consideration when defining the scope of this book, as we did realize that there was a need to offer the main theoretical concepts of type-3 fuzzy logic theory, as well as methods to design, develop, and implement the type-3 fuzzy systems. A review of basic concepts and their use in the design and implementation of interval type-3 fuzzy systems, which are relatively new models of uncertainty and imprecision, are considered in this book. The main focus of this work is based on the basic reasons of the need for interval type-3 fuzzy systems in different areas of application. Recently, type-2 fuzzy systems have emerged as powerful approach for solving complex problems, in a wide range of application areas. In this book, we briefly review the basic concepts of type-2 fuzzy systems and then describe the proposed definitions for interval type-3 fuzzy sets and relations, also interval type3 inference and systems. In addition, we describe methods for designing interval type-3 fuzzy systems and illustrate this with some examples and simulations. We also provide a comparison of the type-3 fuzzy approach with respect to type-2 fuzzy systems. This book is intended to be an important reference for scientists and engineers interested in applying interval type-3 fuzzy logic techniques for solving problems in diverse areas of application. This book can also be used as a reference for graduate courses like the following: soft computing, fuzzy logic, hybrid intelligent systems, and others. We consider that this book can also be used to obtain interesting original ideas for new lines of research, or to continue the lines of research proposed by the authors of this work.

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Preface

The structure of the book is as follows: in Chap. 1, we offer an introduction to the area of interval type-3 fuzzy systems; in Chap. 2, we put forward a review of type-2 fuzzy logic concepts; in Chap. 3, we present the definitions and methods of Interval type-3 fuzzy sets, their membership functions, and operations. Also, in Chap. 4, we describe the concepts and methods regarding interval type-3 fuzzy logic systems. Finally, in Chap. 5, the conclusions about the book and possible future works are presented. We end this preface of the book by giving thanks to all the people who have helped or encouraged us to write this book that opens a new area of research in fuzzy logic, namely type-3 fuzzy logic. First of all, we would like to thank our colleague and friend Prof. Janusz Kacprzyk for always supporting and motivating our work. We would also like to thank our colleagues working in soft computing, which are too many to mention each by their name. Of course, we need to thank our supporting agencies, CONACYT and TecNM, in our country for their help during this project. We have to thank our institutions, Tijuana Institute of Technology and UABC University, for always supporting our projects. Finally, we thank our respective families for their continuous support during the time during the process of writing this book. Tijuana, Mexico October 2021

Oscar Castillo Juan R. Castro Patricia Melin

Contents

1 Introduction to Interval Type-3 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2 Type-2 Fuzzy Logic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fuzzifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Type Reducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Defuzzifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 7 7 8 9 10 10 10

3 Interval Type-3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions of Interval Type-3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . 3.2 Formulation and Parameterization of IT3 MFs . . . . . . . . . . . . . . . . . . 3.3 Mathematical Representation of IT3 FSs . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Representation by the Vertical Cut (Vertical-Slice) . . . . . . . . 3.3.2 Representation by the Horizontal Cut (Horizontal-Slice) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Theoretical Operations of IT3FSs Calculated by the Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Theoretical Operations of IT3 FSs Calculated Using Horizontal Cuts (Horizontal-Slices) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Definition of an Interval Type-3 Fuzzy Relation . . . . . . . . . . . . . . . . . 3.6.1 Theoretical Operations with Interval Type-3 Fuzzy Relations Using the Extension Principle . . . . . . . . . . . . . . . . . 3.6.2 Interval Type-3 Fuzzy Relations and Their Composition in Different Spaces U × V and V × W . . . . . . 3.7 Cartesian Product of Two IT3 FSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 20 28 29 29 30 37 39 39 40 42 42 43

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4 Interval Type-3 Fuzzy Logic Systems (IT3FLS) . . . . . . . . . . . . . . . . . . . 4.1 Non-singleton Interval Type-3 Mamdani Fuzzy Logic Systems (NSIT3 MAMIT3FLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Approach to Evaluate a Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS) by Using α-planes . . . . . . . . 4.5 Approach to Evaluate a Singleton TSK Interval Type-2 Fuzzy Logic System Using α-planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) Using the Principle of Justifiable Granularity . . . . . . . . . . . . . . . . . . . 4.6.1 Methodology for the Design of Singleton Mamdani Interval Type-3 Fuzzy Logic Systems (SMAMIT3FLS) Using the Principle of Justifiable Granularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Methodology for the Design of Singleton TSK Interval Type-3 Fuzzy Logic Systems (STSKIT3FLS) Using the Principle of Justifiable Granularity . . . . . . . . . . . . . 4.7 Comparison of Results of Type-3 Versus Type-2 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

92 93 98

5 Conclusions of Type-3 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

45 57 66 75 79 82

84

88

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Chapter 1

Introduction to Interval Type-3 Fuzzy Systems

A review of the basic concepts and their use in the design of interval type-3 fuzzy systems, which are relatively new models of uncertainty and imprecision, is considered in this book. The fundamental focus of the work is based on the basic reasons of the need for interval type-3 fuzzy systems for different areas of application. Recently, type-2 fuzzy systems have emerged as powerful approach for solving complex problems, in a wide range of application areas. In this book, we review the basic concepts of type-2 fuzzy systems and then describe the proposed definitions for interval type-3 fuzzy sets and relations, also interval type-3 inference and systems. In addition, we describe methods for designing interval type-3 fuzzy systems and illustrate these methods with some examples and simulations. We also provide a comparison of results between the type-3 approach with respect to type-2 fuzzy systems. Uncertainty affects decision-making and emerges in a number of different forms. The concept of information is inherently associated with the concept of uncertainty [1, 2]. The most fundamental aspect of this connection is that the uncertainty involved in any problem-solving situation is a result of some information deficiency, which may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. Uncertainty is an attribute of information [3]. The general framework of fuzzy reasoning allows handling much of this uncertainty. In the case that fuzzy systems employ type-1 fuzzy sets, these sets try to represent uncertainty by using numbers in the range [0, 1]. When an entity is uncertain, like a measurement, it is difficult to specify its exact value, and of course a type-1 fuzzy set makes more sense than a traditional set [3, 4]. However, it is not reasonable to use an accurate membership function for something uncertain, so in this case what we need is another type of fuzzy sets, those which are able to handle these uncertainties, and the so-called type-2 fuzzy sets [5, 6] were put forward for achieving this purpose. The amount of uncertainty in a system

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0_1

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1 Introduction to Interval Type-3 Fuzzy Systems

can be reduced by using type-2 fuzzy logic because this logic offers better capabilities to handle linguistic uncertainties by modeling vagueness and unreliability of information [7, 8]. Recently, there has been some works dealing with interval type-3 fuzzy models, as they can provide even better capabilities for handling uncertainty in control applications [9] and other areas, and this is the main motivation of this book. Type-2 fuzzy models have emerged as an interesting generalization of fuzzy models based upon type-1 fuzzy sets [5, 10]. There have been a number of claims put forward as to the relevance of type-2 fuzzy sets being regarded as generic building constructs of fuzzy models [11–13]. Likewise, there is a record of some experimental evidence showing some improvements in terms of accuracy of fuzzy models of type-2 over their type-1 counterparts [14–18]. Unfortunately, no systematic and general design framework has been provided and while improvements over type-1 fuzzy models have been evidenced, it is not clear whether this effect could always be expected. Furthermore, it is not demonstrated whether the improvement is substantial enough and fully legitimized given the heavy optimization overhead associated with the design of this category of models. There have been a lot of applications of type-2 in intelligent control [19–26], pattern recognition [27–31], intelligent manufacturing [8, 16, 32], time series prediction [14, 33], and others [8]. Recently, there has been some evidence [9] that type-3 fuzzy systems can improve results with respect to type-2 in some cases, and for this reason the importance of this book that provides the basic concepts to develop these type-3 fuzzy systems. In general, the methods for designing a type-2 fuzzy model based on experimental data can be classified into two categories The first category of methods assumes that an optimal (with respect to some criteria) type-1 fuzzy model has already been designed and later a type-2 fuzzy model is constructed through some sound augmentation and generalization of the existing model. The second class of design methods is concerned with the construction of the type2 fuzzy model directly from experimental data. In both cases, an optimization method can help in obtaining the optimal type-2 fuzzy model for the particular application. Now for the situation of type-3 fuzzy models, we have a similar scenario. We can start from a well-constructed type-2 fuzzy model to extend it to become a type-3 fuzzy model, or we can build directly the type-3 fuzzy model from data, which can be more difficult, but it is also possible. In this sense, the main contribution of this book is presenting a way to extend the concepts and knowledge of how to design and implement fuzzy systems from type-2 to type-3, and illustrating this with examples and benchmark datasets.

References

3

References 1. Melin, P., Castillo, O.: Modelling, Simulation and Control of Non-Linear Dynamical Systems. Taylor and Francis, London, Great Britain (2002) 2. Mendel, J.M.: Uncertainty, fuzzy logic, and signal processing. Sig. Process. J. 80, 913–933 (2000) 3. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8, 43–80 (1975) 4. Jang, J.R., Sun, C.T., Mizutani, E.: Neuro-Fuzzy and Soft Computing. Prentice Hall, Upper Saddle River, NJ, USA (1997) 5. Castillo, O., Melin, P.: Type-2 Fuzzy Logic: Theory and Applications. Springer-Verlag, Heidelberg, Germany (2008) 6. Karnik, N.N., Mendel, J.M.: An introduction to type-2 fuzzy logic systems. Technical Report, University of Southern California (1998) 7. Wagenknecht, M., Hartmann, K.: Application of fuzzy sets of type 2 to the solution of fuzzy equations systems. Fuzzy Sets Syst. 25, 183–190 (1988) 8. Zarandi, M.H.F., Turksen, I.B., Kasbi, O.T.: Type-2 fuzzy modelling for desulphurization of steel process. Expert Syst. Appl. 32, 157–171 (2007) 9. Mohammadzadeh, A., Castillo, O., Band, S.S., et al.: A novel fractional-order multiple-model type-3 fuzzy control for nonlinear systems with unmodeled dynamics. Int. J. Fuzzy Syst. 23, 1633–1651 (2021) 10. Hagras, H.: Hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots. IEEE Trans. Fuzzy Syst. 12, 524–539 (2004) 11. Coupland, S., John, R.: New geometric inference techniques for type-2 fuzzy sets. Int. J. Approx. Reason. 49, 198–211 (2008) 12. Starczewski, J.T.: Efficient triangular type-2 fuzzy logic systems. Int. J. Approx. Reason. 50, 799–811 (2009) 13. Walker, C., Walker, E.: Sets with type-2 operations. Int. J. Approx. Reason. 50, 63–71 (2009) 14. Bajestani, N.S., Zare, A.: Application of optimized type-2 fuzzy time series to forecast Taiwan stock index. In: 2nd International Conference on Computer. Control and Communication, pp. 275–280 (2009) 15. Castro, J.R., Castillo, O., Melin, P., Rodriguez-Diaz, A.: A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks. Inf. Sci. 179, 2175–2193 (2009) 16. Dereli, T., Baykasoglu, A., Altun, K., Durmusoglu, A., Turksen, I.B.: Industrial applications of type-2 fuzzy sets and systems: A concise review. Comput. Ind. 62, 125–137 (2011) 17. Leal-Ramirez, C., Castillo, O., Melin, P., Rodriguez-Diaz, A.: Simulation of the bird agestructured population growth based on an interval type-2 fuzzy cellular structure. Inf. Sci. 181, 519–535 (2011) 18. Martinez, R., Castillo, O., Aguilar, L.T.: Optimization of interval type-2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot using genetic algorithms. Inf. Sci. 179(13), 2158–2174 (2009) 19. Castillo, O., Melin, P.: Soft Computing for Control of Non-Linear Dynamical Systems. Springer, Heidelberg, Germany (2001) 20. Castillo, O., Aguilar, L.T., Cazarez-Castro, N.R., Cardenas, S.: Systematic design of a stable type-2 fuzzy logic controller. Appl. Soft Comput. J. 8, 1274–1279 (2008) 21. Hsiao, M., Li, T.H.S., Lee, J.Z., Chao, C.H., Tsai, S.H.: Design of interval type-2 fuzzy slidingmode controller. Inf. Sci. 178(6), 1686–1716 (2008) 22. Melin, P., Castillo, O.: A new method for adaptive model-based control of non-linear dynamic plants using a neuro-fuzzy-fractal approach. J. Soft. Comput. 5, 171–177 (2001) 23. Melin, P., Castillo, O.: A new method for adaptive model-based control of nonlinear plants using type-2 fuzzy logic and neural networks. In: Proceedings of IEEE FUZZ Conference, pp. 420–425 (2003)

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24. Ozen, T., Garibaldi, J.M.: Investigating adaptation in type-2 fuzzy logic systems applied to umbilical acid-base assessment. In: European Symposium on Intelligent Technologies, Hybrid Systems and their implementation on Smart Adaptive Systems (EUNITE 2003), Oulu, Finland (2003) 25. Sepulveda, R., Castillo, O., Melin, P., Montiel, O.: An efficient computational method to implement type-2 fuzzy logic in control applications. Adv. Soft Comput. 41, 45–52 (2007) 26. Sepulveda, R., Castillo, O., Melin, P., Rodriguez-Diaz, A., Montiel, O.: Experimental study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic. Inf. Sci. 177(10), 2023–2048 (2007) 27. Melin, P., Castillo, O.: Hybrid Intelligent Systems for Pattern Recognition. Springer, Heidelberg, Germany (2005) 28. Mendoza, O., Melin, P., Castillo, O., Licea, G.: Type-2 fuzzy logic for improving training data and response integration in modular neural networks for image recognition. Lect. Notes Artif. Intell. 4529, 604–612 (2007) 29. Mendoza, O., Melin, P., Castillo, O.: Interval type-2 fuzzy logic and modular neural networks for face recognition applications. Appl. Soft Comput. J. 9, 1377–1387 (2009) 30. Mendoza, O., Melin, P., Licea, G.: Interval type-2 fuzzy logic for edges detection in digital images. Int. J. Intell. Syst. 24, 1115–1133 (2009) 31. Urias, J., Hidalgo, D., Melin, P., Castillo, O.: A method for response integration in modular neural networks with type-2 fuzzy logic for biometric systems. Adv. Soft Comput. 41, 5–15 (2007) 32. Melin, P., Castillo, O.: An intelligent hybrid approach for industrial quality control combining neural networks, fuzzy logic and fractal theory. Inf. Sci. 177, 1543–1557 (2007) 33. Castillo, O., Melin, P.: Hybrid intelligent systems for time series prediction using neural networks, fuzzy logic and fractal theory. IEEE Trans. Neural Netw. 13, 1395–1408 (2002)

Chapter 2

Type-2 Fuzzy Logic Systems

In this chapter, a brief overview of type-2 fuzzy systems is presented. This overview is intended to provide the basic concepts needed to understand the methods and algorithms presented later in this book [1–3]. If for a type-1 membership function, as in Fig. 2.1, we can blur it to the left and to the right, as illustrated in Fig. 2.2, then a type-2 membership function is produced. In this case, for a specific value x  , the membership function (u  ), takes on different values, which do not have not the same weights, so we can assign membership grades to all of those points. By doing this for all x ∈ X , we form a three-dimensional membership function –a type-2 membership function—that characterizes a type-2 fuzzy set [2, 3]. A type-2 ˜ is represented by the membership function: fuzzy set A, A˜ =

   (x, u), μ A˜ (x, u) |∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]

(2.1)

in which 0 ≤ μ A˜ (x, u) ≤ 1. In fact, Jx ⊆ [0, 1] represents the primary membership of x, and μ A˜ (x, u) is a type-1 fuzzy set known as the secondary set. Hence, a type-2 membership grade can be any subset in [0, 1], the primary membership, and corresponding to each primary membership, there is a secondary membership (which can also be in [0, 1]) that defines the possibilities for the primary membership. Uncertainty is represented by a region, which is named the footprint of uncertainty (FOU). When μ A˜ (x, u) = 1, ∀ u ∈ Jx ⊆ [0, 1] we have an interval type-2 membership function, as shown in Fig. 2.3. The uniform shading for the FOU represents the complete interval type-2 fuzzy set and it can be described in terms of an upper membership functionμ A˜ (x) and a lower membership function μ A˜ (x). A fuzzy logic system (FLS) described using at least one type-2 fuzzy set is called a type-2 FLS. Type-1 FLSs are unable to directly handle rule uncertainties, because they use type-1 fuzzy sets that are certain (viz, fully described by single numeric © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0_2

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2 Type-2 Fuzzy Logic Systems

Fig. 2.1 An example of a type-1 membership function

Fig. 2.2 Blurred type-1 membership function

values). On the other hand, type-2 FLSs, are useful in circumstances where it is difficult to determine an exact numeric membership function, and there are measurement uncertainties [3]. A type-2 FLS is characterized by IF–THEN rules, where their antecedent or consequent sets are now type-2 fuzzy sets. Type-2 FLSs, can be used when the situation is too uncertain to determine exact membership grades such as when the training data is corrupted by noise. Similarly, to the type-1 FLS, a type-2 FLS includes a fuzzifier, a rule base, fuzzy inference engine, and an output processor, as we can find in Fig. 2.4. The output processor includes type-reducer and defuzzifier; it generates a type-1 fuzzy set output (from the type-reducer) or a number (from the defuzzifier) [2]. Now we describe each of the blocks illustrated in Fig. 2.4.

2.1 Fuzzifier

7

Fig. 2.3 Interval type-2 membership function

Fig. 2.4 Structure of the Type-2 Fuzzy Logic System

2.1 Fuzzifier The fuzzifier maps a numeric vector x = (x1 , …, xp )T ∈ X 1 xX 2 x…xX p ≡ X into a type-2 fuzzy set A˜ x in X [3], an interval type-2 fuzzy set in this case. We use a type-2 singleton fuzzifier for the system in Fig. 2.4. In singleton fuzzification, the input fuzzy set has only a single point on nonzero membership. A˜ x is a type-2 fuzzy singleton if μ A˜ x (x) = 1/1 for x = x and μ A˜ x (x) = 1/0 for all other x = x.

2.2 Rules The structure of the rules in a type-1 FLS and a type-2 FLS is basically the same but in the latter the antecedents and the consequents are represented by type-2 fuzzy sets. So for a type-2 FLS with p inputs x 1 ∈ X 1 , …, x p ∈ X p and one output y ∈ Y, In

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2 Type-2 Fuzzy Logic Systems

the case of the Multiple Input Single Output (MISO) systems, we assume there are M rules, the lth rule in the type-2 FLS can be expressed as follows (where the F’s and G are fuzzy sets for each rule): R l : IF x1 is F˜1l and · · · and xp is F˜ pl , THEN y is G˜ l

l = 1, . . . , M

(2.2)

2.3 Inference In the type-2 FLS, the inference engine combines the outputs of the rules and provides a mapping from the input type-2 fuzzy sets to output type-2 fuzzy sets. It is necessary to compute the join , (unions) and the meet  (intersections), as well as the extended sup-star compositions of type-2 relations. If F˜1l × · · · × F˜ pl = A˜ l , then (2.2) can be re-written as follows R l : F˜1l × · · · × F˜ pl → G˜ l = A˜ l → G˜ l l = 1, . . . , M

(2.3)

Rl is described by the membership function μ Rl (x, y) = μ Rl (x1 , ..., x p , y), where μ Rl (x, y) = μ A˜ l →G˜ l (x, y)

(2.4)

μ Rl (x, y) = μ A˜ l →G˜ l (x, y) = μ F˜1l (x1 ) · · ·   p   μ F˜il (xi )  μG˜ l (y) μ F˜ pl (x p )  μG˜ l (y) =

(2.5)

can be written as:

i=1

In general, the p-dimensional input to Rl is given by the type-2 fuzzy set A˜ x whose membership function becomes μ A˜ x (x) = μx˜1 (x1 )  · · ·  μx˜ p (x p ) =

p i=1

μ F˜il (xi )

(2.6)

where X˜ i (i = 1, ..., p) are the labels of the fuzzy sets describing the inputs. Each rule Rl determines a type-2 fuzzy set B˜ l = A˜ x ◦ R l such that: μ B˜ l (y) = μ A˜ x ◦Rl =



μ A˜ x (x) μ Rl (x, y)] x∈X

y ∈ Y l = 1, . . . , M

(2.7)

2.3 Inference

9

This dependency is the input/output relation shown in Fig. 2.3, which holds between the type-2 fuzzy set that activates a certain rule in the inference engine and the type-2 fuzzy set at the output of that engine [3]. In the FLS, we used interval type-2 fuzzy sets and intersection under product t-norm, so the result of the input and antecedent operations, which are contained in  p the firing set  i=1 μ F˜ii (xi ≡ F l (x ), is an interval type-1 set   l l F l (x ) = f l (x ), f (x ) ≡ f l , f

(2.8)

where 



f l (x ) = μ F˜ l (x1 ) ∗ · · · ∗ μ F˜ l (x p )

(2.9)

p

1

and 

l



f (x ) = μ F˜1l (x1 ) ∗ · · · ∗ μ F˜ pl (x p )

(2.10)

where * stands for the product operation.

2.4 Type Reducer The type reducer produces a type-1 fuzzy set output which is then converted in a numeric output through applying the defuzzifier. This type-1 fuzzy set is also an interval set, for the case of our FLS we used the center of sets (cos) type reduction, Y cos , which is expressed as [3]

Ycos (x) = [yl , yr ] =

y 1 ∈[yl1 ,yr1 ]

M

···

···

f

M ∈[

f ,f M

M

1/ i=1 M ] i=1

y M ∈[ylM ,yrM ]

1

f 1 ∈[ f 1 , f ]

f i yi fi

(2.11)

This interval set is defined by its two end points, yl and yr , which corresponds to the centroid of the type-2 consequent set G˜ i ,

C G˜ i =

θ1 ∈Jy1 · · ·

N i=1 yi θi 1/ = [yli , yri ] N θ N ∈Jy N θ i=1 i

(2.12)

before the calculation of Y cos (x), we must evaluate Eq. (2.12), and its two end points, yl and yr . If the values of f i and yi that are associated with yl are denoted f l i and yl i ,

10

2 Type-2 Fuzzy Logic Systems

respectively, and the values of f i and yi that are associated with yr are denoted f r i and yr i , respectively, from Eq. (2.13), we have [3] M yl = i=1 M

fli yli fli

i=1

M

yr = i=1 M

fri yri

i=1

fri

(2.13)

(2.14)

The values of yl and yr define the output interval of the type-2 fuzzy system, which can be utilized to verify if training or testing data are contained in the output of the fuzzy system. This measure of data coverage is considered as one of the design criteria in finding an optimal interval type-2 FS design. The other optimization criteria, is that the size of this output interval should be as small as possible.

2.5 Defuzzifier From the type reducer, we obtain an interval set Y cos , to defuzzify it we use the average of yl and yr , so the defuzzified output of an interval singleton type-2 FLS is (cf. [3]). y(x) =

yl + yr 2

(2.15)

2.6 Summary In this chapter we have provided a summary of the basic concepts of type-2 fuzzy as the basis for the proposed concepts of type-3 fuzzy systems that will be outlined in the following chapters. For more details on type-2 fuzzy systems the reader can follow reference books on type-2 fuzzy sets and systems such as [2, 4].

References 1. Castillo, O., Melin, P.: Soft Computing and Fractal Theory for Intelligent Manufacturing. Springer, Heidelberg, Germany (2003) 2. Castillo, O., Melin, P.: Type-2 Fuzzy Logic: Theory and Applications. Springer, Heidelberg, Germany (2008)

References

11

3. Karnik, N.N., Mendel, J.M.: An introduction to type-2 fuzzy logic systems. Technical Report. University of Southern California (1998) 4. Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper-Saddle River, NJ (2001)

Chapter 3

Interval Type-3 Fuzzy Sets

In this chapter, an provide an overview of type-3 fuzzy sets and their operations. This overview is intended to offer the basic concepts required for understanding the methods and algorithms presented later in this book. We start by defining interval type-3 fuzzy sets and related concepts.

3.1 Definitions of Interval Type-3 Fuzzy Sets Definition 3.1 A type-3 fuzzy sets (T3 FS) [1, 2], denoted by A(3) , is represented by the plot of a trivariate function, called a membership function (MF) of A(3) , in the Cartesian product X × [0, 1] × [0, 1] in [0, 1], where X is the universe of the primary variable of A(3) , x. The MF of μ A(3) is denoted by μ A(3) (x, u, v) (or μ A(3) to abbreviate) and it is called a type-3 membership function (T3 MF) of the T3 FS. In other words, more formally,

μ A(3) : X × [0, 1] × [0, 1]→[0, 1] A(3) = {(x, u(x), v(x, u), μ A(3) (x, u, v))|x ∈ X, u ∈ U ⊆ [0, 1], v ∈ V ⊆[0, 1]} (3.1) where U is the universe for the secondary variable u and V is the universe for tertiary variable v. A T3FS, A(3) can also be expressed in continuous notation as follows: A(3) =





 μ A(3) (x, u, v)/(x, u, v)

(3.2)

x∈X u∈[0,1] v∈[0,1]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0_3

13

14

3 Interval Type-3 Fuzzy Sets

A(3) =

⎡



⎣

x∈X

⎡



⎤



⎣

⎤

μ A(3) (x, u, v)/v ⎦/u ⎦/x

(3.3)

v∈[0,1]

u∈[0,1]

˝ where denotes the union over all the admissible x, u, v values. Equation (3.3) is represented as a mapping of the T3 FS membership functions specified by the following equations: 

A(3) =

μ A(3) (u, v)/x x x∈X

 μ A(3) (v)/u

μ A(3) (u, v) = x

(x,u)

u∈[0,1]

 μ A(3) (v) =

μ A(3) (x, u, v)/v

(x,u)

v∈[0,1]

where μ A(3) (u, v) is the primary membership function, μ A(3) (u, v) is the secondary x x membership function and μ A(3) (v) is the tertiary membership function of the T3 FS. (x,u)

If μ A(3) (x, u, v) = 1, the T3 FS, A(3) , is reduced to an interval type-3 fuzzy set (IT3 FS) denoted A, defined by the following equation 

 A= x∈X



  v∈ μA (x,u),μA (x,u)

u∈[0,1]



1/v /u /x

(3.4)

where  μA(x,u) (v) =

1/v v∈[μA (x,u),μA (x,u)

⎡



⎢ ⎣

μA(x) (u, v) = u∈[0,1]

 A= x∈X



⎤ ⎥ 1/v ⎦/u

v∈[μA (x,u),μA (x,u)

μA(x) (u, v)/x

 Assuming that v ∈ μA (x, u), μA (x, u) and the lower and upper membership functions μA (x, u), μA (x, u) are general type-2 membership functions (T2 MF) over

3.1 Definitions of Interval Type-3 Fuzzy Sets

15

the plane (x, u), Eq. (4) can be simplified as a bivariate isosurface  simplification with an interval type-3 membership function (IT3 MF), μ˜ A (x, u) ∈ μA (x, u), μA (x, u) , defined by Eq. (3.5), that is 

 μ˜ A (x, u)/(x, u)

A=

(3.5)

x∈X u∈[0,1]

where the lower T2 MF μA (x, u), is contained in the upper T2 MF μA (x, u), this is, μA (x, u) ⊆ μA (x, u), then μA (x, u) ≤ μA (x, u), and as a consequence, an IT3 FS is represented by two T2 FSs, one inferior A with T2 MF μA (x, u) and another superior A, with T2 MF μA (x, u) defined by Eqs. (3.6) and (3.7) (see Fig. 3.1), that is ⎡ ⎤     ⎣ μA (x, u)/(x, u) = f x (u)/u ⎦/x A= (3.6) x∈X u∈[0,1]

x∈X

u∈[0,1]

Fig. 3.1 IT3 FS with IT3MF μ(x, ˜ u) where μ(x, u) is the LMF and μ(x, u) is the UMF. The embedded secondary T1 MFs in x  of A and A are f x  (u) and f x  (u) respectively

16

3 Interval Type-3 Fuzzy Sets





A=

 μA (x, u)/(x, u) =

x∈X u∈[0,1]

x∈X

⎡ ⎣



⎤ f x (u)/u ⎦/x

(3.7)

u∈[0,1]

where, the secondary MFs of A and A are T1 MFs of T1FS given by the Eqs. (3.8) and (3.9), that is  μ A(x) (u) =

f x (u)/u

(3.8)

f x (u)/u

(3.9)

u∈Jx

 μ A(x) (u) = u∈Jx

Another simple way to represent the IT3 MF from IT3 FS, A, is like a bivariate isosurface, μ˜ A (x, u), because of the union of vertical-slices in x, where each vertical (IT2 cut in x = x  is an embedded secondary interval type-2 membership function

  MF) [1, 3], f˜x  (u), in other words, f˜x  (u) ∈ 1/ μA x  , u , μA x  , u or f˜x  (u) ∈  f x  (u), f x  (u) . As a consequence, Eq. (3.5) can be simplified to Eq. (3.10) (see Fig. 3.2), that is

 Fig.  3.2 IT3 FS, A, with IT3 MF, μ˜ A (x, u) and an embedded vertical cut, μA(x ) (u) ∈ f x  (u), f x  (u) with the FOU in green color

3.1 Definitions of Interval Type-3 Fuzzy Sets



17







A=

 ˜ f x (u)/u /x

μ˜ A (x, u)/(x, u) = x∈X

x∈X u∈[0,1]

(3.10)

u∈[0,1]

 where μ˜ A (x, u) ∈ μA (x, u), μA (x, u) . Equation (3.10) can be expressed as Eqs. (3.11) and (3.12): 





 ˜ f x (u)/u /x

μA(x) (u)/x =

A= x∈X

x∈X

(3.11)

u∈[0,1]

where μA(x) (u) is an Interval type-2 fuzzy set (IT2 FS), denoted by Eq. (3.12), and  μA(x) (u) = u∈Jx



f˜x (u)/u =

1/u

(3.12)

u∈[ f x (u), f x (u)

Definition 3.2 An Interval type-3 fuzzy set (IT3 FS), denoted by A, is an isosurface with a bivariate function, called MF of A (see Fig. 3.4) over the Cartesian product X × [0, 1] in [0, 1], where X is the universe for the primary variable of A, x. The membership function, MF, of A is denoted by μ˜ A (x, u), (or μ˜ A˜ for simplicity) and it is called an Interval type-3 membership function MF (IT3 MF), that is

A = {(x, u, μ˜ A (x, u))|x ∈ X, u ∈ U ≡ [0, 1]} where μ˜ A (x, u) ⊆ [0, 1]; U is the universe of discourse for the secondary variable u, and in this work is always assumed that U is [0, 1]. An IT3 FS, A, can also be expressed in the notation of fuzzy sets as in Eqs. (3.9) and (3.10) given below. 



 A= 

x∈X

u∈[0,1]



= x∈X

μ˜ A (x, u)/(x, u) =  ˜ f x (u)/u /x

μ A(x) (u)/x x∈X

u∈[0,1]

where μA(x) (u) is an Interval type-2 fuzzy set (IT2 MF).  μA(x) (u) = u∈[0,1]

f˜x (u)/u =

 1/u u∈[ f x (u), f x (u)

18

3 Interval Type-3 Fuzzy Sets

˜ and denotes the union over all the admisible x and u. The 3D plot of the IT3MF is an isosurface with volume in between the layers of the surface formed by all the secondary IT2MFs μA(x) (u) in green color in Fig. 3.3, which forms the domain of uncertainty (DOU) of IT3 FS. Definition 3.3 A secondary membership function, μA(x) (u), is a restricted function μA : X × [0, 1] → [0, 1] for x ∈ X , in other words, μA(x) : [0, 1] → [0, 1], or in the notation of fuzzy sets: 

 μ˜ A (x, u)/u =

μA(x) (u) = u∈[0,1]

f˜x (u)/u =

u∈[0,1]

 1/u u∈[ f x (u), f x (u)

Fig. 3.3 Isosurface of the membership function of the IT3 FS

Fig. 3.4 The isosurface of the

left plot shows

 the IT3 MF of an IT3 FS and the right plot visualizes the F OU (A), where F OU A ⊆ F OU A

3.1 Definitions of Interval Type-3 Fuzzy Sets

19

The support of A consists of all the (x, u) ∈ X × [0, 1] such that μA (x, u) > 0 and μA (x, u) > 0, which is also called a domain of uncertainty of A, D OU (A), or in other words,  J˜x DOU(A) = {(x, u) ∈ X × [0, 1]|μ˜ A (x, u)0} = x∈X

 where μ˜ A (x, u) ∈ μA (x, u), μA (x, u) .

The primary membership of the Interval type-2 fuzzy set in x, J˜x , is expressed as a subset of {x} × I˜x , where I˜x is the support of the secondary IT2 MF, μA(x) (u)—in other words J˜x = {x} × I˜x . Then

 I˜x = {u ∈ [0, 1]|μ˜ A (x, u)0} = u˜ A (x), u˜ A (x) where  u˜ A (x) = u A (x), u A (x) , where u˜ A (x) is the support of the lower T2 MF, A in x: u A (x) = sup { u|u ∈ [0, 1], μA (x, u) > 0} u A (x) = inf { u|u ∈ [0, 1], μA (x, u) > 0}

  That is, u˜ A (x) = u A (x), u A (x) , where u˜ A (x) is the support of the upper T2 MF, A in x, that is u A (x) = sup { u|u ∈ [0, 1], μA (x, u) > 0} u A (x) = inf { u|u ∈ [0, 1], μA (x, u) > 0}  footprint

of uncertainty of A, is a strong α-cut, A0+ , where F OU (A) =  Then,

the F OU A , F OU A , or in other words,  

  F OU A = (x, u)|x ∈ Xandu ∈ u A (x), u A (x) 

   F OU A = (x, u)|x ∈ Xandu ∈ u A (x), u A (x)

 The footprint of uncertainty of the IT3 FS, F OU (A), is formed by the F OU A

 and F OU A of the lower and upper T2 FSs of IT3 FS A, respectivelly, where



 F OU A ⊆ F OU A (see Fig. 3.4).

20

3 Interval Type-3 Fuzzy Sets

Fig. 3.5 The left plot is an isosurface of an IT3 FS when the lower and upper T2MFs are μA (x, u) = μA (x, u) and the right plot shows the reduction of the FOU of an IT3 FS to a T2 FS

The FOU of an IT3 FS tends to the FOU of a T2 FS when the FOUs of the secondary IT2 MFs tends to zero; in other words, μA (x, u) = μA (x, u), then the IT3 FS reduces to a T2 FS (see Fig. 3.5).

3.2 Formulation and Parameterization of IT3 MFs There exists several approaches for formulating and parameterizing the Interval type2 membership functions (IT2 MF) [4, 5] and the general type-2 (T2 MF) [5]. The same thing can be said about the footprint of uncertainty (FOU) for Interval type-2 fuzzy sets (IT2 FS) and general type-2 fuzzy sets (T2 FS). In the first approach, the IT2 MFs are formulated based on the parameters of a type-1 membership function, called parameters of the upper membership function (UMF), U pper Parameter s, and the parameters of the lower membership function (LMF) are evaluated with the parameter of core scaling of the UMF (Lower Scale) and the vector Lower Lag, which represents a relative lag in the values of the LMF on both sides, in comparison with the upper membership function, in other words, is a relative delay of the parameters that form the support of the UMF. The Lower Scale and Lower Lag parameters form the footprint of uncertainty of IT2 FS. The same approach can be used for formulating and parameterizing the type-n membership functions. The constraints for the parameters are the following: An interval type-2 membership function (IT2MF) is represented with the upper and lower membership functions. The values for UMF are always greater or equal to the values of LMF. The difference between the values of UMF and LMF introduces a source of uncertainty which is called the footprint of uncertainty (FOU) of an IT2MF [5, 6].

3.2 Formulation and Parameterization of IT3 MFs

21

UpperParameters: This is numeric vector that specifies the parameter values of an upper membership function (UMF). LowerScale: This is an scaling factor with numeric scalar values for the lower membership function (LMF), the values are between 0 and 1 (inclusive), this is used to scale the values of LMF. The predefined value is 1. LowerLag: This is a scalar or vector that specifies delay values for LMF. LowerLag, is a numeric scalar or a vector that has values between 0 and 1, inclusive. This represents a relative delay in the values of the LMF on both side in comparison with the upper membership function. LowerLag is used to create the footprint of uncertainty of an Interval type-2 membership function by introducing a delay in the lower values of the membership function. For the formulation and parameterization of the IT3 MFs of the IT3 FS, we can use approach number 1 to build the DOU with primary IT2 MFs and the FOU of the secondary IT2 MF, μA(x) (u) to build the FOU of the IT3 FS. We show below the formulation and parameterization of several IT3 MFs: The Interval type-3 Gaussian membership function, ScaleGaussScaleGaussIT3MF, with Gaussian F OU (A), is characterized by two parameters [σ, m] (UpperParameters) for the UMF and for the LMF the parameters s (LowerScale),  (LowerLag) to form the DOU. The vertical cuts μA(x) (u) are IT2 FSs with Gaussian IT2 MFs, μA(x) (u), and the parameters [σu , m u ] for the UMF and s (LowerScale),  (LowerLag) for the LMF (see Fig. 3.6). The following equations define the IT3 MF, μ˜ A (x, u) μ˜ A (x, u) : ScaleGauss ScaleGauss I T 3M F(x, u, [σ, m], s, )

Fig. 3.6 Plot of the isosurface of the IT3 MF, ScaleGauss ScaleGauss I T 3M F(x, u, [σ, m], s, ) of an IT3 FS

22

3 Interval Type-3 Fuzzy Sets



  1 x −m 2 u(x) = ex p − 2 σ    1 x −m 2 u(x) = s · ex p − 2 σ∗  ln() where σ ∗ = σ ln(ε) and ε = 0.01. If  = 0, then σ ∗ = σ . Then u(x) and u(x) are the upper and lower limits of the DOU. δ(u) = u(x) − u(x) δ(u) σu = √ + ε 2 3 where δ(u) is the range, σu is the radius of the DOU and ε is the epsilon machine number to avoid that σu is zero. 

  1 x −m 2 u(x) = ex p − 2 ρ where ρ =

σ +σ ∗ 2

and u(x) is the main embedded T1 MF, and   1 u − u(x) 2 μA(x) (u) = ex p − 2 σu    1 x − u(x) 2 μA(x) (u) = s · ex p − 2 σu∗ 

 ln() where σu∗ = σu ln(ε) and ε = 0.01. If  = 0, then σu∗ = σu . Then, μA(x) (u) and μA(x) (u) are the UMF and LMF of the IT2 FS of the vertical cuts of the secondary IT2MF of the IT3 FS. The second approach that we propose to formulate and parameterize an IT2 MF is based on the parameters with a source of uncertainty (represented as Interval real numbersIR ) of an IT2 MF, to form the FOU of the IT2 FS. For example, we show below the formulation for some IT2 MFs, in which all the parameters have distinct values of uncertainty:

3.2 Formulation and Parameterization of IT3 MFs

23

triIntervalType2MF:  μ(x) ˜ = μ(x), μ(x) = tri I nter valT ype2M F(x, params)  ˜ c˜ and t r i I nt erval T ype2M F is an IT2MF with where params = a, ˜ b, ˜ c˜ ∈ IR , a˜ ∈ [a1 , a2 ]∀a1 ≤ a2 , b˜ ∈ [b1 , b2 ]∀b1 ≤ b2 and parameter s a, ˜ b, c˜ ∈ [c1 , c2 ]∀c1 ≤ c2 , and ⎧ 0 x ≤ a1 ⎪ ⎪ ⎪ x−a1 ⎪ ⎪ ⎨ b1 −a1 a1 < x < b1 μ(x) = 1 b1 ≤ x ≤ b2 ⎪ c2 −x ⎪ ⎪ b2 < x < c2 ⎪ ⎪ ⎩ c2 −b2 0 x ≥ c2 ⎧ ⎪ 0 x ≤ a2 ⎪ ⎪ ⎨ s x−a2 a < x ≤ x c xc −a2 2 μ(x) = c1 −x ⎪ s x < x < c c 1 ⎪ c −x ⎪ ⎩ 1 c 0 x ≥ c1 xc =

a2 (b1 − c1 ) + c1 (a2 − b2 ) (a2 − c1 ) − (b2 − b1 ) s=

x c − a2 b2 − a2

where s, is the scale factor (LowerScale) for the LMF, μ(x), and xc is the value in which LMF achieves its máximum value (supremum of LMF). The ranges for the ˜ c˜ define the vector LowerLag with a máximum value of 1. parameters a, ˜ b, gaussIntervalType2MF:  μ(x) ˜ = μ(x), μ(x) = gauss I nter valT ype2M F(x, params)   where params = σ˜ , m˜ and gauss I nt erval T ype2M F is an IT2MF with parameters σ˜ , m˜ ∈ IR , σ˜ ∈ [σ1 , σ2 ] and m˜ ∈ [m 1 , m 2 ], and 

  1 x − m˜ 2 μ(x) ˜ = ex p − 2 σ˜

24

3 Interval Type-3 Fuzzy Sets

  ⎧ ⎪ 1 ⎪ ex p − 21 x−m ⎪ σ2 ⎪ ⎨ μ(x) = 1   ⎪ ⎪ ⎪ 2 ⎪ ⎩ ex p − 21 x−m σ2 xc =

2

2

 x < m1  m1 ≤ x ≤ m2 x > m2

m1 + m2 2



  1 xc − m 2 2 s = ex p − 2 σ1 ⎧    2 ⎪ 1 x−m 2 ⎪ x < xc ⎨ ex p − 2 σ1    μ(x) = 2 ⎪ 1 ⎪ x ≥ xc ⎩ ex p − 21 x−m σ1 where s, is the scale factor (LowerScale) for the LMF, μ(x), and xc is the value where the LMF schieves its máximum value (supremum of LMF). The ranges for the parameters σ˜ , m˜ ∈ IR define the vector LowerLag with máximum value of 1. tsGaussIntervalType2MF:  μ(x) ˜ = μ(x), μ(x) = tsGauss I nter valT ype2M F(x, params)   where params = σ˜ l , m, ˜ σ˜ r and t sGauss I nt erval T ype2M  l l F is a Gaussian IT2 , m, ˜ σ ˜ ∈ IR , σ ˜ ∈ σ1 , σ2 , m˜ ∈ [m 1 , m 2 ] and MF on both sides, with parameters σ ˜ l r l   σ˜ r ∈ σ1r , σ2r , and   ⎧ ⎪ 1 ⎪ ex p − 21 x−m ⎪ ⎪ σ2l ⎨ μ(x) = 1   ⎪ ⎪ ⎪ 1 x−m 2 ⎪ ⎩ ex p − 2 σ r 2

2

2

 x < m1  m1 ≤ x ≤ m2 x > m2

σ1r m 2 + σ1l m 1 σ1r + σ1l    1 xc − m 2 2 s = ex p − 2 σ1 xc =

3.2 Formulation and Parameterization of IT3 MFs

25

⎧   ⎪ 2 ⎪ ⎨ ex p − 21 x−m σl   1 μ(x) = ⎪ 1 ⎪ ⎩ ex p − 21 x−m σr 1

2



2



x < xc x ≥ xc

where s, is the factor of scale (LowerScale) for the LMF, μ(x), and xc is the value where the LMF achieves its máximum value (supremum of the LMF). The ranges ˜ σ˜ r ∈ IR define the vector LowerLag with máximum value for the parameters σ˜ l , m, of 1. For he formulation and parameterization of the IT3 MFs of the IT3 FSs, we can use the approach number 2 to evaluate the DOU with primary IT2 MF and with approach number 1 we define the FOU of the secondary IT2 MF, μA(x) (u) to form the FOU del IT3 FS. In the following we outline the forml;ation and parameterization of several IT3 MF. a.

The Interval type-3 Gaussian membership function, GaussScaleGaussIT3MF with Gaussian F OU (A), is characterized with parameters σ˜ , m˜ to form the DOU. The vertical cuts μA(x) (u) are IT2 FSs with Gaussian IT2 MFs and parameters [σu , m u ] for the UMF and LMF s (LowerScale),  (LowerLag) (see Fig. 3.7). The following equations define the IT3 MF, μ˜ A (x, u):

μ˜ A (x, u) : Gauss ScaleGauss I T 3M F(x, u, params, s, )

Fig. 3.7 Plot of the isosurface of the IT3 MF, Gauss ScaleGauss I T 3M F(x, u, params, s, ) of an IT3 FS

26

3 Interval Type-3 Fuzzy Sets

  where params = σ˜ , m˜ and GaussScal eGauss I T 3M F is an IT3MF with parameters σ˜ , m˜ ∈ IR , σ˜ ∈ [σ1 , σ2 ] and m˜ ∈ [m 1 , m 2 ], and   u(x), u(x) = gauss I nter valT ype2M F(x, params) δ(u) δ(u) = u(x) − u(x) y σu = √ + ε 2 3 where δ(u) is the range, σu is the radius of the DOU and ε is the machine epsilon number to avoid that σu becomes zero, and 

  1 x −m 2 u(x) = exp − 2 σ where σ =

σ1 +σ2 , 2

m=

m 1 +m 2 2

and u(x) is the main embedded T1MF, and

  1 u − u(x) 2 μA(x) (u) = ex p − y 2 σu    1 x − u(x) 2 μA(x) (u) = s · ex p − 2 σu∗ 

 ln() where σu∗ = σu ln(ε) and ε = 0.01. If  = 0, then σu∗ = σu . Then, μA(x) (u) and μA(x) (u) are the UMF and LMF of the IT2 FS of the vertical cuts of the secondary IT2MF of IT3 FS. b.

The Interval type-3 Gaussian membership function, tsGaussScaleGaussIT3MF with Gaussian F OU (A) on both sides, is characterized with the parameters ˜ σ˜ r ∈IR to form the DOU. The vertical cuts μA(x) (u) are IT2 FSs with σ˜ l , m, Gaussian IT2 MFs and parameters [σu , m u ] for the UMF and LMF with s (LowerScale),  (LowerLag) (see Fig. 3.8). The following equations define the IT3 MF, μ˜ A (x, u):

μ˜ A (x, u) : tsGauss ScaleGauss I T 3M F(x, u, params, s, )   eGauss I T 3M F is an IT3MF with where params = σ˜ l , m, ˜ σ˜ r andt sGaussScal   ˜ σ˜ r ∈ IR , σ˜ l ∈ σ1l , σ2l , m˜ ∈ [m 1 , m 2 ] and σ˜ r ∈ σ1r , σ2r , and parameters σ˜ l , m,     ˜ σ˜ r u(x), u(x) = tsGauss I nter valT ype2M F(x, σ˜ l , m,

3.2 Formulation and Parameterization of IT3 MFs

27

Fig. 3.8 Plot of the isosurface of the IT3 MF, tsGauss ScaleGauss I T 3M F(x, u, params, s, ) of an IT3 FS

δ(u) = u(x) − u(x) δ(u) σu = √ + ε 2 3 where δ(u) is the range, σu is the radius of the DOU and ε is the epsilon machine number to avoid that σu becomes zero, and u(x) = tsgausssm f t ype1(x, [σl mσr ]) σ l +σ l

where σl = 1 2 2 , m = double sided Gaussian.

m 1 +m 2 , 2

σr =

σ1r +σ2r 2



and u(x) is the main embedded T1MF,

  1 u − u(x) 2 μA(x) (u) = ex p − 2 σu    1 x − u(x) 2 μA(x) (u) = s · ex p − 2 σu∗

28

3 Interval Type-3 Fuzzy Sets

 ln() where σu∗ = σu ln(ε) and ε = 0.01. If  = 0, then σu∗ = σu . Then, μA(x) (u) and μA(x) (u) are the UMF and LMF of the IT2 FS of the vertical cuts from the secondary IT2MF of IT3 FS.

3.3 Mathematical Representation of IT3 FSs In the literature there are four methods to mathematically represent a T2 FS [5, 7, 8]: (a) (b) (c) (d)

collection of points; union of vertical slices (over x ∈ X ), where each vertical slice is a T1 FS (is a secondary MF); union of wavy slices, where each wavy slice is an embedded T2 FS; and, fuzzy union of horizontal cuts (over α ∈ [0, 1]), where each horizontal cut is like an IT2 FS elevated to level α.

Utilizing the previous methods, an IT3 FS, A, is represented by the union of the vertical cuts(cf. [9]), of the lower T2MF, μA (x, u) and upper, μA (x, u), where each vertical cut is a lower and upper T1 FS (see Fig. 3.3). In the same way, the union of the vertical cuts of, A, where each vertical cut is an IT2 FS (see Fig. 3.9). Another alternative is representing an IT3 FS by the fuzzy union of the horizontal cuts [10, 11] (over α ∈ [0, 1]) for the T2 FSs with lower T2MF, μA (x, u) and upper, μA (x, u), where each horizontal cut is an IT2 FS elevated to a level α (see Fig. 3.10).

Fig. 3.9 IT3 FS, A, represented by the vertical cuts IT2 FSs with embedded IT2MFs, f˜x (u)

3.3 Mathematical Representation of IT3 FSs

29

Fig. 3.10 IT3 FS, A, represented by horizontal cuts of embedded IT2 FSs, elevated to level α. Cuts of the lower IT2MF, μA (x, u) (blue color) and upper, μA (x, u) (red color)

3.3.1 Representation by the Vertical Cut (Vertical-Slice) Definition 3.4 The representation of the vertical cut of an IT3 FS is centered in each value of the primary variable x, and expressed as (3.10) with the union of all the secondary IT2 FS, in other words.  μr omanA(x) (u)/x

A= x∈X

where,  μA(x) (u) =



f˜x (u)/u =

1/u  u∈ f x (u), f x (u)

u∈Jx

3.3.2 Representation by the Horizontal Cut (Horizontal-Slice) Definition 3.5 An α-plane for an IT3 FS, denoted by Aα , is the union of all the primary memberships of (Eq. (3.6)) and (Eq. (3.7)) with the secondary degrees higher or equal to α ∈ [0, 1], that is.

x∈X

 

sup α/Aα (x) /x =

μA(x) (u)/x =

A=



 



x∈X

α∈[0,1]

x∈X

   sup α/ Aα (x), Aα (x) /x

α∈[0,1]

30

3 Interval Type-3 Fuzzy Sets

where;   Aα (x) = a α (x), bα (x) a α (x) = inf { u|u ∈ [0, 1], μA (x, u) ≥ α} bα (x) = sup { u|u ∈ [0, 1], μA (x, u) ≥ α}   Aα (x) = a α (x), bα (x) a α (x) = inf { u|u ∈ [0, 1], μA (x, u) ≥ α} bα (x) = sup { u|u ∈ [0, 1], μA (x, u) ≥ α} In the case of an IT3 FS the secondary membership function is an IT2 FS, μA(x) (u), then the α-cut of an interval type-2 fuzzyset (see Fig. 3.11), depends on the height of the lower membership function, 0 < h f x (u) ≤ 1, in the case that the height of 

the upper membership is h f x (u) = 1, OR, in other words, is normalized !  μAα (x) (u) =

u Aα (x) , u Aα (x) ], [u Aα (x) , u Aα (x)    ∅, u Aα (x) , u Aα (x)



 α ≤ h f˜x (u) other wise

   where h f˜x (u) = h f x (u) , 1 .

3.4 Theoretical Operations of IT3FSs Calculated by the Extension Principle If we consider two IT3 FSs, A and B that are expressed using the representation of vertical cuts to evaluate the union A ∪ B, intersection A ∩ B and complement A operations (see Fig. 3.12):  A=

 μA(x) (v)/x =

x∈X

x∈X

⎡ ⎣



v∈[0,1]

⎤ f˜x (v)/v ⎦/x

3.4 Theoretical Operations of IT3FSs Calculated …

31

Fig. 3.11 The upper plot shows the intervals of the α-cut of a secondary IT3 MF, μAα (x) (u) of the IT3 FS. The lower figure illustrates the intervals in the plane (X, U)

 B=

 μB(x) (w)/x =

x∈X

x∈X

⎡



⎣

⎤ g˜ x (w)/w ⎦/x

w∈[0,1]

Union of IT3 FSs The union of two IT3FSs, A ∪ B is calculated using THE vertical cuts as: 

 μ(A∪B)x (u)/x =

A∪B= x∈X

μA(x) (v) x∈X

where; u ≡v∨w μ(A∪B)x (u) ≡ μA(x) (v)

" μB(x)

(w)

"

μB(x) (w)/x

32

3 Interval Type-3 Fuzzy Sets

Fig. 3.12 Operations of Interval type-3 fuzzy sets





=

f˜x (v)

#

g˜ x (w)/(v ∨ w)

v∈[0,1] w∈[0,1]

f˜x (v)



# g˜ x

(w) =

1/u  u∈ f x (v)g x (w), f x (v)g x (w)

$

% and denote the join and meet operations, respectively. In the same way,  is the T-Norm operator that indicates the minimum or the product, and ∨ indicates the maximum. Intersection of IT3 FS The intersection of two IT3 FSs, A ∩ B, is calculated using the vertical cuts as: 

 μ(A∩B)x (u)/x =

A∩B= x∈X

μA(x) (v) x∈X

where: u ≡v∧w

# μB(x)

(w)/x

3.4 Theoretical Operations of IT3FSs Calculated …

μ(A∩B)x (u) ≡ μA(x) (v) 

33

#



=

μB(x)

(w)

f˜x (v)

#

g˜ x (w)/(v ∧ w)

v∈[0,1] w∈[0,1]

f˜x (v)



# g˜ x

(w) =

1/u  u∈ f x (v)g x (w), f x (v)g x (w)

% where: denotes the meet operation,  is the T-Norm operator that indicates the minimum or product, and ∧ indicates the minimum. Complement of IT3 FSs The complement of an IT3 FS, A , is calculated using the vertical cuts as:  A=

 μ(A) (v)/x = x

x∈X

¬μA(x) (v)/x x∈X

where: ¬, denotes the complement operation or negation.  μ(A)x (v) =

v∈[0,1]

f˜x (v)/(1 − v)

Example 3.1 Suposse we have the vertical cut, μA(x) and μB(x) , of the IT3 FSs f A and B for a paticular element x:

μA(x) = [0.4, 0.6]/0.0 + [0.6, 0.8]/0.1 + [0.2, 0.4]/0.2 μB(x) = [0.8, 1.0]/0.0 + [0.5, 0.7]/0.1 + [0.1, 0.3]/0.2 We calculate: μA(x)

"

μB(x) = [0.4, 0.6]∧[0.8, 1.0]/0.0 ∨ 0.0 + [0.4, 0.6]∧[0.5, 0.7]/0.0 ∨ 0.1 + [0.4, 0.6]∧[0.1, 0.3]/0.0 ∨ 0.2 + [0.6, 0.8]∧[0.8, 1.0]/0.1 ∨ 0.0 + [0.6, 0.8]∧[0.5, 0.7]/0.1 ∨ 0.1

34

3 Interval Type-3 Fuzzy Sets

+ [0.6, 0.8]∧[0.1, 0.3]/0.1 ∨ 0.2 + [0.2, 0.4]∧[0.8, 1.0]/0.2 ∨ 0.0 + [0.2, 0.4]∧[0.5, 0.7]/0.2 ∨ 0.1 + [0.2, 0.4]∧[0.1, 0.3]/0.2 ∨ 0.2 μA(x)

"

μB(x) = [0.4, 0.6]/0.0 + [0.4, 0.6]/0.1 + [0.1, 0.3]/0.2 + [0.6, 0.8]/0.1 + [0.5, 0.7]/0.1 + [0.1, 0.3]/0.2 + [0.2, 0.4]/0.2 + [0.2, 0.4]/0.2 + [0.1, 0.3]/0.2

μA(x)

"

μB(x) = [0.4, 0.6]/0.0 + {[0.4, 0.6]∨[0.6, 0.8]∨[0.5, 0.7]}/0.1 + {[0.1, 0.3] ∨ [0.1, 0.3] ∨[0.2, 0.4]∨[0.2, 0.4]∨[0.1, 0.3]}/0.2

μA(x)

"

μB(x) = [0.4, 0.6]/0.0 + [0.6, 0.8]/0.1 + [0.2, 0.4]/0.2

μA(x)

#

μB(x) = [0.4, 0.6]∧[0.8, 1.0]/0.0 ∧ 0.0 + [0.4, 0.6]∧[0.5, 0.7]/0.0 ∧ 0.1 + [0.4, 0.6]∧[0.1, 0.3]/0.0 ∧ 0.2 + [0.4, 0.6]∧[0.8, 1.0]/0.0 ∧ 0.0 + [0.4, 0.6]∧[0.5, 0.7]/0.0 ∧ 0.1 + [0.4, 0.6]∧[0.1, 0.3]/0.0 ∧ 0.2 + [0.6, 0.8]∧[0.8, 1.0]/0.1 ∧ 0.0 + [0.6, 0.8]∧[0.5, 0.7]/0.1 ∧ 0.1 + [0.6, 0.8]∧[0.1, 0.3]/0.1 ∧ 0.2 + [0.2, 0.4]∧[0.8, 1.0]/0.2 ∧ 0.0 + [0.2, 0.4]∧[0.5, 0.7]/0.2 ∧ 0.1 + [0.2, 0.4]∧[0.1, 0.3]/0.2 ∧ 0.2

μA(x)

#

μB(x) = [0.4, 0.6]/0.0 + [0.4, 0.6]/0.0 + [0.1, 0.3]/0.0 + [0.6, 0.8]/0.0 + [0.5, 0.7]/0.1 + [0.1, 0.3]/0.1 + [0.2, 0.4]/0.0 + [0.2, 0.4]/0.1 + [0.1, 0.3]/0.2

3.4 Theoretical Operations of IT3FSs Calculated …

μA(x)

#

35

μB(x) = {[0.4, 0.6] ∨[0.4, 0.6]∨ [0.1, 0.3]∨ [0.6, 0.8]∨ [0.2, 0.4]} /0.0 + {[0.5, 0.7]∨[0.1, 0.3]∨[0.2, 0.4]}/0.1 + [0.1, 0.3]/0.2

μ(A) ≡ ¬μA(x) = [0.4, 0.6]/1.0 + [0.6, 0.8]/0.9 + [0.2, 0.4]/0.8 x Example 3.2 Suppose that the vertical cut μA(x) and μB(x) of the IT3 FSs of A and B for a particular element x is:

μA(x) = [0.5, 0.7]/0.2 + [0.7, 0.9]/0.3 + [0.1, 0.3]/0.4 μB(x) = [0.6, 0.8]/0.2 + [0.2, 0.4]/0.5 + [0.0, 0.2]/0.8 The: μA(x)

"

μB(x) = [0.5, 0.7]∧[0.6, 0.8]/0.2 ∨ 0.2 + [0.5, 0.7]∧[0.2, 0.4]/0.2 ∨ 0.5 + [0.5, 0.7]∧[0.0, 0.2]/0.2 ∨ 0.8 + [0.7, 0.9]∧[0.6, 0.8]/0.3 ∨ 0.2 + [0.7, 0.9]∧[0.2, 0.4]/0.3 ∨ 0.5 + [0.7, 0.9]∧[0.0, 0.2]/0.3 ∨ 0.8 + [0.1, 0.3]∧[0.6, 0.8]/0.4 ∨ 0.2 + [0.1, 0.3]∧[0.2, 0.4]/0.4 ∨ 0.5 + [0.1, 0.3]∧[0.0, 0.2]/0.4 ∨ 0.8

μA(x)

"

μB(x) = [0.5, 0.7]/0.2+[0.2, 0.4]/0.5 + [0.0, 0.2]/0.8 + [0.6, 0.8]/0.3 + [0.2, 0.4]/0.5 + [0.0, 0.2]/0.8 + [0.1, 0.3]/0.4 + [0.1, 0.3]/0.5 + [0.0, 0.2]/0.8

μ A(x)

"

μB(x) = [0.5, 0.7]/0.2 + [0.6, 0.8]/0.3 + [0.1, 0.3]/0.4 + {[0.2, 0.4] ∨ [0.2, 0.4] ∨ [0.1, 0.3]}/0.5 + {[0.0, 0.2] ∨ [0.0, 0.2] ∨ [0.0, 0.2]}/0.8

36

3 Interval Type-3 Fuzzy Sets

μ A(x)

"

μB(x) = [0.5, 0.7]/0.2 + [0.6, 0.8]/0.3 + [0.1, 0.3]/0.4 + [0.2, 0.4]/0.5 + [0.0, 0.2]/0.8

μA(x)

#

μB(x) = [0.5, 0.7] ∧ [0.6, 0.8]/0.2 ∧ 0.2 + [0.5, 0.7] ∧ [0.2, 0.4]/0.2 ∧ 0.5 + [0.5, 0.7] ∧ [0.0, 0.2]/0.2 ∧ 0.8 + [0.7, 0.9] ∧ [0.6, 0.8]/0.3 ∧ 0.2 + [0.7, 0.9] ∧ [0.2, 0.4]/0.3 ∧ 0.5 + [0.7, 0.9] ∧ [0.0, 0.2]/0.3 ∧ 0.8 + [0.1, 0.3] ∧ [0.6, 0.8]/0.4 ∧ 0.2 + [0.1, 0.3] ∧ [0.2, 0.4]/0.4 ∧ 0.5 + [0.1, 0.3] ∧ [0.0, 0.2]/0.4 ∧ 0.8

μA(x)

#

μB(x) = [0.5, 0.7]/0.2+[0.2, 0.4]/0.2 + [0.0, 0.2]/0.2 + [0.6, 0.8]/0.2 + [0.2, 0.4]/0.3 + [0.0, 0.2]/0.3 + [0.1, 0.3]/0.2 + [0.1, 0.3]/0.4 + [0.0, 0.2]/0.4

μA(x)

#

μB(x) = {[0.5, 0.7] ∨ [0.2, 0.4] ∨[0.0, 0.2]∨[0.6, 0.8]∨[0.1, 0.3]}/0.2 + {[0.2, 0.4]∨[0.0, 0.2]}/0.3 + {[0.1, 0.3]∨[0.0, 0.2]}/0.4

μA(x)

#

μB(x) = [0.6, 0.8]/0.2 + [0.2, 0.4]/0.3 + [0.1, 0.3]/0.4

μ(A) ≡ ¬μA(x) = [0.5, 0.7]/0.8 + [0.7, 0.9]/0.7 + [0.1, 0.3]/0.6 x

3.5 Theoretical Operations of IT3 FSs Calculated Using Horizontal …

37

3.5 Theoretical Operations of IT3 FSs Calculated Using Horizontal Cuts (Horizontal-Slices) Consider two IT3 FSs, A and B, that are expressed, by usuing the representation of the horizontal cuts, as:  

 μA(x) (u)/x =

A= x∈X

      sup α/Aα (x) /x = sup α/ Aα (x), Aα (x) /x

α∈[0,1]

x∈X

x∈X

α∈[0,1]

where:   Aα (x) = a α (x), bα (x) a α (x) = inf { u|u ∈ [0, 1], μA (x, u) ≥ α} bα (x) = sup { u|u ∈ [0, 1], μA (x, u) ≥ α}   Aα (x) = a α (x), bα (x) a α (x) = inf { u|u ∈ [0, 1], μA (x, u) ≥ α} bα (x) = sup { u|u ∈ [0, 1], μA (x, u) ≥ α} 





B=

μB(x) (u)/x = 

x∈X

= x∈X



x∈X

 sup α/Bα (x) /x

α∈[0,1]

   sup α/ Bα (x), Bα (x) /x

α∈[0,1]

where:   B α (x) = cα (x), d α (x) cα (x) = inf { u|u ∈ [0, 1], μB (x, u) ≥ α} d α (x) = sup { u|u ∈ [0, 1], μB (x, u) ≥ α}   B α (x) = cα (x), d α (x)

38

3 Interval Type-3 Fuzzy Sets

cα (x) = inf { u|u ∈ [0, 1], μB (x, u) ≥ α} d α (x) = sup { u|u ∈ [0, 1], μB (x, u) ≥ α} Union of IT3 FSs The union of two IT3FSs, A ∪ B, is calculated using the horizontal cuts as:  μ(A∪B)x (u)/x

A∪B= x∈X

   sup α/(Aα ∪ Bα ) /x = α∈[0,1]

x∈X





= x∈X

   sup α/ Aα (x) ∪ Bα (x), Aα (x) ∪ Bα (x)

α∈[0,1]

where:   Aα (x) ∪ B α (x) = a α (x) ∨ cα (x), bα (x) ∨ d α (x) y   Aα (x) ∪ B α (x) = a α (x) ∨ cα (x), bα (x) ∨ d α (x) Intersection of IT3 FSs The intersection of two IT3FSs, A ∩ B, is calculated using the horizontal cuts as: A

&

 B=

  = x∈X

μ(A x∈X

'

    & sup α / Aα B /x B)x (u)/x = α x∈X

α∈[0,1]

  & & sup α/ Aα (x) B α (x), Aα (x) B α (x) /x

α∈[0,1]

where:   Aα (x) ∩ Bα (x) = a α (x) ∧ cα (x), bα (x) ∧ d α (x)   Aα (x) ∩ Bα (x) = a α (x) ∧ cα (x), bα (x) ∧ d α (x) Complement of IT3 FSs The complement of an IT3 FS, A, is calculated using the horizontal cuts as:

3.5 Theoretical Operations of IT3 FSs Calculated Using Horizontal …

 A= x∈X

39

        sup α /¬μAα (x) /x = sup α / ¬ Aα (x), ¬ Aα (x) /x μ(A) (v)/x = x α∈[0,1] α∈[0,1] x∈X

x∈X

  ¬ Aα (x) = 1 − bα (x), 1 − a α (x)   ¬ Aα (x) = 1 − bα (x), 1 − a α (x)

3.6 Definition of an Interval Type-3 Fuzzy Relation Consider two universes U and V . Let R(U, V ) be an Interval type-3 fuzzy relation that can be defined in the product space U × V . The secondary membership function of R(U, V ) is an IT2 MF, denoted μ˜ r(u,v) . The Interval type-3 fuzzy relation can be expressed as:  R(U, V ) =

 μR (u, v)/(u, v) =

U ×V

U ×V

⎡ ⎣



⎤ μ˜ r(u,v) (ξr )/ξr ⎦/(u, v)

ξr ∈[0,1]

3.6.1 Theoretical Operations with Interval Type-3 Fuzzy Relations Using the Extension Principle Interval type-3 fuzzy relations in the same space U × V Consider two universes U and V . Let R(U, V ) and S(U, V ) be two interval type-3 fuzzy relations that are defined in the same product space U × V . The elements of R(U, V ) and S(U, V ) are the IT2 FSs (in other words, the secondary IT2MFs), denoted μ˜ r(u,v) and μ˜ s(u,v) , respectively. The relations R(U, V ) and S(U, V ) can be expressed as follows:  R(U, V ) =

 μR (u, v)/(u, v) =

U ×V

U ×V





S(U, V ) =

μS (u, v)/(u, v) = U ×V

U ×V

⎡ ⎣



ξr ∈[0,1]

⎡ ⎣



ξs ∈[0,1]

⎤ μ˜ r(u,v) (ξr )/ξr ⎦/(u, v) ⎤ μ˜ s(u,v) (ξs )/ξs ⎦/(u, v)

40

3 Interval Type-3 Fuzzy Sets

Union of relations over IT3 FSs

( The union of two relations on the IT3 FSs, R S is calculated using the vertical cuts as follows:    " R S= μR∪S (u, v)/(u, v) = μR (u, v) μS (u, v)/(u, v) U ×V

μR∪S (u, v) = μR (u, v)  =

"

U ×V

μS (u, v) μ˜ r(μ,ν) (ξr )

ξr ∈[0,1]ξs ∈[0,1]

#

μ˜ s(u,ν) (ξs )/(ξr ∨ ξs )

Intersection of relations over IT3 FSs The intersection of two relations on the IT3 FSs, R∩S,is calculated using the vertical cuts as follows:  R∩S= μR∩S (u, v)/(u, v) U ×V # = μR (u, v) μS (u, v)/(u, v) U ×V

μR∩S (u, v) = μR (u, v)  =

#

μS (u, v)  # μ˜ r(μ,y) (ξr ) μ˜ s(μ,y) (ξs )/(ξr ∧ ξs )

ξ,∈[0,1]ξs ∈[0,1]

Complement of a relation on IT3 FSs The complement of a relation on the IT3 FSs, R , is calculated using the vertical cuts as follows: ⎡ ⎤    ⎣ R(U, V ) = μR (u, v)/(u, v) = μ˜ r(u,v) (ξr )/(1 − ξr )⎦/(u, v) U ×V

U ×V

ξr ∈[0,1]

3.6.2 Interval Type-3 Fuzzy Relations and Their Composition in Different Spaces U × V and V × W If R and S are two Interval type-3 fuzzy relations on U × V and V × W , respectively, and μR (u, v) and μS (v, w) are normal IT2 FSs, then the membership for any pair (u, w), u ∈ U and w ∈ W , is different than zero if and only if it exists at least a

3.6 Definition of an Interval Type-3 Fuzzy Relation

41

v ∈ V such that μR (u, v) = 1/0 and μS (v, w) = 1/0. The elements of R(U, V ) and S(V, W ) are IT2 FSs (in other words, secondary IT2MFs), called μ˜ r(u,v) and μ˜ s(v,w) , respectively. The relations R(U, V ) and S(U, V ) can be expressed as follows:  R(U, V ) =

⎡



⎣

μR (u, v)/(u, v) = U ×V

U ×V

 S(V, W ) =



V ×W

μ˜ r(u,v) (ξr )/ξr ⎦/(u, v)

ξr ∈[0,1]

⎡ ⎣

μS (v, w)/(v, w) =

⎤



⎤



μ˜ s(v,w) (ξs )/ξs ⎦/(v, w)

ξs ∈[0,1]

V ×W

This is equivalent to the following composition operation, an extended supremum(T-Norm) (extended sup-star composition) (u ∈ U, w ∈ W ):  R◦S=

μR◦S (u, w)/(u, w) U ×W

where: μR◦ S (u, w) =

"

μR (u, v)

v∈V

μR (u, v)

#

#

 # μS (v, w) = sup μR (u, v) μS (v, w)



μS (v, w) =

v∈V



ξr ∈[0,1]

ξs ∈[0,1]

μ˜ r(u,v) (ξr )

#

μ˜ s(v,w) (ξs )/(ξr ∧ ξs )

Composition of an IT3 FS with an Interval type-3 fuzzy relation Consider the case in which one of the relations involved in the extended composition is only an IT3 FS. The composition of an IT3 FS, R(U ), and an Interval type-3 fuzzy relation, S(U, V ) is given by, v ∈ V :  R(U ) =

 μR (u)/u =

U

U



⎡ ⎣ 

U ×V

μ˜ r(u) (ξr )/ξr ⎦/u

ξr ∈[0,1]

μS (u, v)/(u, v) =

S(U, V ) =

⎤



U ×V

⎡ ⎣



ξs ∈[0,1]

 R◦S=

μR◦S (v)/v V

⎤ μ˜ s(u,v) (ξs )/ξs ⎦/(u, v)

42

3 Interval Type-3 Fuzzy Sets

μR·S (v) =

 " # # μR (u) μS (u, v) = sup μR (u) μS (u, v) u∈U

u∈U

μR (u)

#





μS (u, v) =

ξr ∈[0,1]

ss ∈[0,1]

μ˜ r(x) (ξr )

#

μ˜ s(u,u) (ξs )/(ξr ∧ ξs )

in which μR(u) is a secondary IT2 MF of R. This equation plays an important role in the inference mechanism for a fuzzy rule in which the antecedents (or consequents) are IT3 FSs and it is the fundamental inference mechanism for rules in an Interval type-3 fuzzy logic system (IT3FLS).

3.7 Cartesian Product of Two IT3 FSs Consider two IT3 FSs, A and B, in domains X and Y , respectively. Using the vertical cut representation, we obtain:  A=

⎡



⎣

μA(x) (v)/x = x∈X

x∈X



 μB(y) (w)/y =

B= y∈Y

y∈Y

⎡

⎤



f˜x (v)/v ⎦/x

v∈[0,1]

⎣



⎤ g˜ y (w)/w ⎦/y

w∈[0,1]

 μA×B(x,y) (v, w)/(x, y)

A×B= X ×Y

 μA×B(x,y) (v, w) =

v∈[0,1]

 w∈[0,1]

f˜x (v)

#

g˜ y (w)/v ∧ w

3.8 Summary We have provided in this chapter the basic definitions for the Interval type-3 fuzzy sets, relations and their operations. These theoretical concepts form a basis for the methods and concepts that will be presented in Chap. 4 of this book in which the interval type-3 fuzzy systems are developed for particular cases and datasets.

References

43

References 1. Mohammadzadeh, A., Sabzalian, M.H., Zhang, W.: An interval type-3 fuzzy system and a new online fractional-order learning algorithm: theory and practice. IEEE Trans. Fuzzy Syst. 28(9), 1940–1950 (2020) 2. Rickard, J.T., Aisbett, J., Gibbon, G.: Fuzzy subsethood for fuzzy sets of type-2 and generalized type-n. IEEE Trans. Fuzzy Syst. 17(1), 50–60 (2009) 3. Liu, Z., Mohammadzadeh, A., Turabieh, H., Mafarja, M., Band, S.S., Mosavi, A.: A new online learned interval type-3 fuzzy control system for solar energy management systems. IEEE Access 9, 10498–10508 (2021) 4. Liang, Q., Mendel, J.M.: Interval type-2 fuzzy logic systems: theory and design. IEEE Trans. Fuzzy Syst. 8, 535–550 (2000) 5. Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper-Saddle River, NJ (2001) 6. Mendel, J.M., Hagras, H., Tan, W.-W., Melek, W.W., Ying, H.: Introduction to Type-2 Fuzzy Logic Control. Wiley and IEEE Press, Hoboken, NJ (2014) 7. Karnik, N.N., Mendel, J.M.: Operations on type-2 fuzzy sets. Fuzzy Sets Syst. 122, 327–348 (2001) 8. Sakalli, A., Kumbasar, T., Mendel, J.M.: Towards systematic design of general type-2 fuzzy logic controllers: analysis, interpretation, and tuning. IEEE Trans. Fuzzy Syst. 29(2), 226–239 (2021) 9. Cao, Y., Raise, A., Mohammadzadeh, A., et al.: Deep learned recurrent type-3 fuzzy system: application for renewable energy modeling. Prediction. Energy Reports (2021) 10. Moreno, J.E., et al.: Design of an interval type-2 fuzzy model with justifiable uncertainty. Inf. Sci. 513, 206–221 (2020) 11. Qasem, S.N., Ahmadian, A., Mohammadzadeh, A., Rathinasamy, S., Pahlevanzadeh, B.: A type-3 logic fuzzy system: optimized by a correntropy based Kalman filter with adaptive fuzzy kernel size Inform. Sci. 572, 424–443 (2021)

Chapter 4

Interval Type-3 Fuzzy Logic Systems (IT3FLS)

In this chapter we present the basic concepts and methods for building the Interval type-3 fuzzy systems.

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy Logic Systems (NSIT3 MAMIT3FLS) The first case is the non-singleton Mamdani interval type-3 fuzzy systems. The structure of the k-th generic IF–THEN Zadeh fuzzy rule for a Mamdani fuzzy system is the following: R kZ : I F x1 is Fk1 and . . . and xi is Fik and . . . and xn is Fkn T H E N y1 is Gk1 , . . . , y j is Gkj , . . . , ym is Gkm where i = 1,…, n (number of inputs), j = 1,…, m (number of outputs) and k = 1,…, r (number of rules). To begin the approach based on the Zadeh rules and a MAMIT3FLS with the Mamdani type reasoning, we should represent the antecedents of the rules as a fuzzy relation Ak , using the Cartesian product with the Interval type-3 fuzzy sets (IT3 FS), Fik , and the implication with the consequent of the j-the output, Gkj . Then, the fuzzy relation of the rule, Rkj , can be expressed as: Ak = Fk1 × . . . × Fkn Rkj = Ak → Gkj   If Rkj , is described as a membership function of the rules, μRkj x, y j , then © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0_4

45

46

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

    μRkj x, y j = μAk →Gkj x, y j As a consequence, when the Mamdani implication is used, Ak → Gkj , with multiple antecedents, Ak , and consequents, Gkj , and these are connected by the meet () operator, then       μAk →Gkj x, y j = μFk1 ×...×Fkn →Gkj x, y j = μFk1 ×...×Fkn (x)μGkj y j        ... μFkn (xn ) μGkj y j μAk →Gkj x, y j = μFk1 (x1 )  n   μFik (xi ) μGkj y j = i=1

The n-dimensional input is given by the fuzzy relation A X  , with the MF expressed as n μXi (xi |xi ) A X  (x) = μX1 (x1 |x1 ) . . . μXn (xn |xn ) = i=1

Each fuzzy relation of Rkj determines a fuzzy set of the consequent of the rule Bkj = A◦X  Rkj in Y such that     μBkj y j |x  = μA◦  RkZ y j |x  X    = sup A X  (x)μRk →Gkj x, y j , y ∈ Y 

x∈X

This equation is an input–output relation between the IT3 FS that activates the inference of one rule and the IT3 FS of the output. The composition (◦ ) is a highly    non-linear mapping from the input vector x to an IT3 FS μBkj y j |x ( y ∈ Y ) as output vector. The reasoning is a fuzzy inference mechanism, that can be interpreted as a system that maps an IT3 FS into an IT3 FS by using the composition, that is      μBkj y j |x  = sup A X  (x)μRk →Gkj x, y j 

x∈X

     n n μXi (xi |xi ) Ti=1 μFik (xi ) μGkj y j = sup i=1 

x∈X

     n μXi (xi |xi )μFik (xi ) μGkj y j μBkj y j |x = sup i=1 



x∈X ⎧⎡ ⎤ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎥ ⎢   ⎢ ⎥ ⎥  . . .  sup μBk y j |x = ⎢ μ |x )μ μ |x )μ sup (x (x ) (x (x ) ⎣ ⎦ k k 1 ⎦ i X1 1 1 Xn n n ⎣ F F ⎪ ⎪ 



1 j i ⎪ ⎪ ⎭ ⎩ x1 ∈X 1 xn ∈X n     μBk y j |x = μGk y j j

j

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy …

47

⎧⎡ ⎤ ⎡ ⎤⎫ ⎪ ⎪ ⎨ ⎬   ⎢ ⎥ ⎢ ⎥ k ˜  x = ⎣ sup μX1 (x1 |x1 )μFk1 (x1 )⎦ . . . ⎣ sup μXn (xn |xn )μFik (xi )⎦ 

⎪ ⎪ ⎩ 

⎭ x1 ∈X 1



xn ∈X n



max |xi  = sup μXi (xi |xi )μFik (xi ) ≡ sup μQik (xi |xi ) μQik xk,i 



xi ∈X i

xi ∈X i

where    max ≡ arg max sup μQik xi |xi xk,i   

xi

xi ∈X i

μQik (xi |xi ) = μXi (xi |xi )μFik (xi )   ˜ k x  , is the IT2 MF value resulting from the  operation The firing strength   max   applied to all the supremum membership values μQik xk,i |xi from the intersection   of each input μXi xi |xi with its antecedent μFik (xi ) that contributes to the level of activation of the rule, that is    max  n ˜ k x  = i=1  μQik xk,i |xi  ;   The level of activation of the rule is the membership value μBkj y j |x  resulting   ˜ k x  and the membership value of from the operation  of the strength   firing  the consequent of rule μGkj y j . This is, the composition operation (◦ ) between the facts and the rules of the knowledge base that describes the relational function, Bkj = A◦X  Rkj , that is       ˜ k x  μGk y j μBkj y j |x  =  j The results of the rules are combined by using the fuzzy  union (as an aggregation operation), that is, in other words, we use the join ( ) operator to calculate the   aggregation of the values μBkj y j |x  , and obtain B j = B1j ∪ . . . ∪ Bkj ∪ . . . ∪ Brj = Brk=1 Bkj               μB j y j  x = μB1j y j  x  ... μBkj y j  x  ... μBrj y j  x     r    μBkj y j  x  μB j y j  x  = k=1     r  k       ˜ x ∪ μGk y j μB j y j x =  j k=1

48

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

The type-reduction is expressed as follows     yˆ j = t ype Reduction y j , μB j y j  x  Case study: Evaluation of a non-singleton Mamdani Interval type-3 fuzzy system (NSMAMIT3FLS) from zero We assume that the NSMAMIT3FLS has 2 inputs and one output with two fuzzy rules. The objective is to illustrate the inference process according to the interval type-3 fuzzy sets theory. Definition of the Rules: pr emise 1(r ule1)

: I F x1 is F11 and x2 is F12 T H E N y is G1

pr emise 2(r ule2)

: I F x1 is F21 and x2 is F22 T H E N y is G2

pr emise 3( f act)

: x1 is X1 and x2 is X2

consequence (conclusion)

: y is B

Process of inference: Interval type-3 fuzzy relations are: R1 = F11 ∩ F12 → G1 = (F11 → G1 ) ∩ (F12 → G1 ) = (F11 × G1 ) ∩ (F12 × G1 ) R2 = F21 ∩ F22 → G2 = (F21 → G2 ) ∩ (F22 → G2 ) = (F21 × G2 ) ∩ (F22 × G2 ) H = X1 ∩ X2 = X1 × X2 The output fuzzy sets of the activated rules in the inference process are:   B1 = H◦ R1 = X◦1 (F11 × G1 ) ∩ X◦2 (F12 × G1 )   B2 = H◦ R2 = X◦1 (F21 × G2 ) ∩ X◦2 (F22 × G2 ) Aggregation This process consists in the combination of the output fuzzy sets of the activated rules, that is B = B1 ∪ B2 where ×, ◦ , ∩ y ∪ are the fuzzy sets operators, Cartesian product, composition, intersection and union, respectively.

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy …

49

Type reduction The Interval type-3 fuzzy set (IT3 FS), B, is reduced to an Interval type-2 fuzzy set (IT2 FS), B, and this is then reduced to a real interval (IR). Now, we show a step by step implementation of this inference process in the Matlab® programming language: % exa00_nsmamit3fls212.m % study case: Evaluating a Non-Singleton Mamdani IT3 FLS from scratch clearvars; x1 = linspace(0,10,100); x2 = linspace(0,10,100); y = linspace(0,10,100); u = linspace(0,1,100); %

% Rules % % R1: IF x1 is F11 and x2 is F12 THEN y is G1 % R2: IF x1 is F21 and x2 is F22 THEN y is G2 % H : IF x1 is X1 and x2 is X2 % ----------------------------------------------------------% % Fuzzy inference engine %

y is B

50

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

% Fuzzy relations % R1 = (F11->G1) I (F12->G1) = (F11xG1) I (F12xG1) % R2 = (F21->G2) I (F22->G2) = (F21xG2) I (F22xG2) % H = X1 x X2 % % Fired-rule output fuzzy set % B1 = H o R1 = [X1 o (F11xG1)] I [X2 o (F12xG1)] % B2 = H o R2 = [X1 o (F21xG2)] I [X2 o (F22xG2)] % where I is intersection operator % % Combining fired-rule output fuzzy sets (Agregation) % B = B1 U B2 % % Antecedent interval type-3 membership functions F11 = ScaleGaussScaleGaussIT3MF(x1,u,{{[1.0 4.0]},0.8,0.2}); F21 = ScaleGaussScaleGaussIT3MF(x1,u,{{[1.0 7.0]},0.6,0.2}); X1 = ScaleGaussScaleGaussIT3MF(x1,u,{{[0.8 6.0]},0.8,0.2}); F12 = ScaleGaussScaleGaussIT3MF(x2,u,{{[1.0 5.0]},0.8,0.3}); F22 = ScaleGaussScaleGaussIT3MF(x2,u,{{[1.0 4.5]},0.6,0.3}); X2 = ScaleGaussScaleGaussIT3MF(x2,u,{{[0.8 3.0]},0.9,0.1});

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy …

% Consequent interval type-3 membership functions G1 = ScaleGaussScaleGaussIT3MF(y,u,{{[1.5 2.0]},0.75,0.2}); G2 = ScaleGaussScaleGaussIT3MF(y,u,{{[2.0 5.5]},0.75,0.2}); % % Rule View % Rule 1 figure; subplot(2,3,1); plot3dMeshIsocapsIT3MF(F11); text(4.0,1.05,1.10,'F11') view(-5,75);

subplot(2,3,2); plot3dMeshIsocapsIT3MF(F12); text(5.0,1.05,1.10,'F12') view(-5,75);

51

52

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

subplot(2,3,3); plot3dMeshIsocapsIT3MF(G1); text(2.0,1.05,1.10,'G1') view(-5,75);

% Rule 2 subplot(2,3,4); plot3dMeshIsocapsIT3MF(F21); text(7.0,1.05,1.10,'F21') view(-5,75); subplot(2,3,5); plot3dMeshIsocapsIT3MF(F22); text(4.5,1.05,1.10,'F22') view(-5,75); subplot(2,3,6); plot3dMeshIsocapsIT3MF(G2); text(5.5,1.05,1.10,'G2') view(-5,75); % % Fuzzy inference engine % Inference Rule 1 F11xG1 = prodCartIntervalType3(F11,G1,'min'); F12xG1 = prodCartIntervalType3(F12,G1,'min');

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy …

X1oF11xG1 = maxStarIntervalType3(X1,F11xG1,'min'); X2oF12xG1 = maxStarIntervalType3(X2,F12xG1,'min'); % Fired-rule output set, B1 B1 =intersectionIntervalType3(X1oF11xG1,X2oF12xG1,'min'); % Inference Rule 2 F21xG2 = prodCartIntervalType3(F21,G2,'min'); F22xG2 = prodCartIntervalType3(F22,G2,'min'); X1oF21xG2 = maxStarIntervalType3(X1,F21xG2,'min'); X2oF22xG2 = maxStarIntervalType3(X2,F22xG2,'min'); % Fired-rule output set, B2 B2 =intersectionIntervalType3(X1oF21xG2,X2oF22xG2,'min'); % % Combining Fired-Rule Output Sets (Agregation), B B=unionIntervalType3(B1,B2,'min'); % % Fuzzy inference engine Rule View % Rule 1 figure; subplot(3,3,1); plot3dMeshIsocapsIT3MF(F11); hold on plot3dIsocapsIT3MF(X1); hold on text(4.0,1.05,1.10,'F11'); text(6.0,1.05,1.10,'X1');

53

54

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

view(-5,75);

subplot(3,3,2); plot3dMeshIsocapsIT3MF(F12); hold on plot3dIsocapsIT3MF(X2); hold on text(5.0,1.05,1.10,'F12'); text(3.0,1.05,1.10,'X2'); view(-5,75);

subplot(3,3,3); % plot3dMeshIsocapsIT3MF(G1); hold on plot3dIsocapsIT3MF(B1); text(2.0,1.05,1.10,'B1'); view(-5,75);

% Rule 2 subplot(3,3,4); plot3dMeshIsocapsIT3MF(F21); hold on plot3dIsocapsIT3MF(X1); hold on text(7.0,1.05,1.10,'F21'); text(6.0,1.05,1.10,'X1'); view(-5,75);

4.1 Non-singleton Interval Type-3 Mamdani Fuzzy …

subplot(3,3,5); plot3dMeshIsocapsIT3MF(F22); hold on plot3dIsocapsIT3MF(X2); hold on text(4.5,1.05,1.10,'F22'); text(3.0,1.05,1.10,'X2'); view(-5,75);

subplot(3,3,6); % plot3dMeshIsocapsIT3MF(G2); plot3dIsocapsIT3MF(B2); text(5.0,1.05,1.10,'B2'); view(-5,75);

subplot(3,3,9); plot3dIsocapsIT3MF(B); hold on text(5.0,1.05,1.10,'B') view(-5,75); % % Type Reduction to IT2 FS (IT3 FS to IT2 FS) it2fs = typeReductionIT3MF(B); % Type Reduction to Real Interval (IT2 FS to IR) Yh = kmTypeReduction(it2fs.X{1}',it2fs.LU{1}(:,2)',it2fs.LU{1}(:,1)'); % Defuzzification

55

56

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

Yc = (Yh(1)+Yh(2))/2; % figure; plot2dtype2(it2fs); hold on line([Yc Yc],[0 1],'Color','g','LineWidth',2); line([Yh(1) Yh(1)],[0 1],'Color','b','LineStyle',':'); line([Yh(2) Yh(2)],[0 1],'Color','b','LineStyle',':'); line([Yh(1) Yh(2)],[0 0],'Color','g','LineStyle','-','LineWidth',2); xlabel('y'); ylabel('\mu_{B}(y)'); axis([-inf +inf 0 1]) hold off

We show in Fig. 4.1 the fuzzy rules of the Interval type-3 fuzzy system. We illustrate in Fig. 4.2 the inference process for the non-singleton Mamdani Interval type-3 fuzzy system. Finally, we illustrate in Fig. 4.3 the type reduction process.

Fig. 4.1 Fuzzy rules of NSMAMIT3FLS

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS)

57

Fig. 4.2 Inference process for NSMAMIT3FLS

Fig. 4.3 Type reduction process

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS) The structure of the IF–THEN fuzzy rule of the generic Zadeh form for the Mamdani fuzzy system is

58

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

R kZ : I F x1 is Fk1 and . . . and xi is Fik and . . . and xn is Fkn T H E N y1 is Gk1 , . . . , y j is Gkj , . . . , ym is Gkm where i = 1,…, n (number of inputs), j = 1,…, m (number of outputs) and k = 1,…, r (number of rules).   ˜ k x  , is the value of the IT2 MF of the  operation over The firing strength  all the membership values of the antecedents μFik (xi ) that contribute to the level of activation that contributes to the level of activation of the rule, that is n    k ˜ μFik (xi )  x = i=1

  The level of activation of the rule, is the membership value, μBkj y j |x  , resulting   ˜k  from the operation  of the firing   strength  x and the membership value of the consequent of the rule μGkj y j . In other words, the composition operation (◦ ) between the facts and the rules of the knowledge base that describes the relational function, Bkj = A◦X  Rkj , where A X  is a fuzzy singleton, that is       ˜ k x  μGk y j μBkj y j |x  =  j Combining the rules using fuzzy union as an aggregation operation, in otherwords, using the join ( ) operator for calculating the agregation of values of μBkj y j |x  yields B j = B1j ∪ . . . ∪ Bkj ∪ . . . ∪ Brj = ∪rk=1 Bkj           μB j y j |x  = μB1j y j |x  ... μBkj y j |x  ... μBrj y j |x  r      μB j y j |x  = μBkj y j |x  k=1

  μB j y j |x  =

r  

    ˜ k x  μGk y j  j

k=1

Type-Reduction    yˆ j = t ype Reduction y j , μB j y j |x 

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS)

59

Case study: Singleton Mamdani Interval type-3 fuzzy logic system (SMAMIT3FLS) for the basic tip problem We consider the basic tip problem to illustrate the use of a SMAMIT3FLS model with two inputs and one output and three fuzzy rules. The objective is to illustrate the design process of a SMAMIT3FLS using Matlab®. 1.

Generating the IT3 FIS in line commands for the basic tip problem

% tipperSingletonMamit3fls.m % % Date : 20-June-2021 % % Basic Tipping Problem. Given a number from 0 through 10 that % represents the quality of service at a restaurant (where 10 is % excellent), what should the tip be? % This problem is based on tipping as it is typically practiced in the % United States. An average tip for a meal in the US is 15%, though % the actual amount can vary depending on the quality of the service % provided. %

60

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

clearvars; % Create IT3FIS Model % Sintaxis: % fis=newfisIntervalType3(fisName,fisType,inputsFuzzifier,andMethod,... %

orMethod,impMethod,aggMethod,defuzMethod,

%

typeReductionMethod)

% % defuzMethod: centroid, csums,height,mdheight,cos % typeReductionMethod: km, ekm, iasc, eiasc, eods, wmub, %

nt, bmm

% fis = newfisIntervalType3('tipper','mamdani','singleton','min','max',... 'min','max','centroid','km'); % % Input 1 fis=addvarIntervalType3(fis,'input','service',[0 10]); fis=addmfIntervalType3(fis,'input',1,'poor','ScaleGaussScaleGaussIT3MF',{ {[1.5 0.0]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',1,'good','ScaleGaussScaleGaussIT3MF', {{[1.5 5.0]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',1,'excellent','ScaleGaussScaleGaussIT3 MF',{{[1.5 10.0]},0.9,0.2}); % Input 2 fis=addvarIntervalType3(fis,'input','food',[0 10]); fis=addmfIntervalType3(fis,'input',2,'rancid','ScaleTrapScaleGaussIT3MF',{ {[0 0 1 3]},0.9,[0.0 0.2]});

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS)

61

fis=addruleIntervalType3(fis,ruleList); % % write it3fis % writefisIntervalType3(fis,'tipper213');

Now we show the tipper213.fis file that contains sequence of command lines and the Matlab® functions for the Interval type-3 fuzzy inference system (SMAMIT3FLS) for the basic tip problem: [System] Name='tipper213' Type='mamdani' InputsFuzzifier='singleton' Version=1.0 NumInputs=2 NumOutputs=1 NumRules=3 AndMethod='min' OrMethod='max' ImpMethod='min' AggMethod='max' DefuzMethod='centroid' TypeReductionMethod='km'

62

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

[Input1] Name='service' Range=[0 10] NumMFs=3 MF1='poor':'ScaleGaussScaleGaussIT3MF',{{[1.5 0]},0.9,[0.2]} MF2='good':'ScaleGaussScaleGaussIT3MF',{{[1.5 5]},0.9,[0.2]} MF3='excellent':'ScaleGaussScaleGaussIT3MF',{{[1.5 10]},0.9,[0.2]} [Input2] Name='food' Range=[0 10] NumMFs=2 MF1='rancid':'ScaleTrapScaleGaussIT3MF',{{[0 0 1 3]},0.9,[0 0.2]} MF2='delicious':'ScaleTrapScaleGaussIT3MF',{{[7 9 10 12]},0.9,[0.2 0]} [Output1] Name='tip' Range=[0 30] NumMFs=3 MF1='cheap':'ScaleTriScaleGaussIT3MF',{{[0 5 10]},0.8,[0.3 0.3]} MF2='average':'ScaleTriScaleGaussIT3MF',{{[10 15 20]},0.8,[0.3 0.3]} MF3='generous':'ScaleTriScaleGaussIT3MF',{{[20 25 30]},0.8,[0.3 0.3]}

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS)

[Rules] 1 1, 1 (1) : 2 2 0, 2 (1) : 1 3 2, 3 (1) : 2

2.

Evaluating the SMAMIT3FLS system for the basic tip problema

% demo01_singletonMamit3flsTipper.m clearvars;

fis = readfisIntervalType3('singletonMamit3fls213Tipper');

inputs = [2 5; 6 9];

% Evaluate fuzzy inference system tipper = evaluateSingletonMamdaniIT3FIS(inputs,fis);

% Linguistic variables and linguistic values figure; subplot(2,2,1); plotIntervalType3MF(fis,'input',1);

63

64

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

xlabel('service'); view(0,90);

subplot(2,2,2); plotIntervalType3MF(fis,'input',2); xlabel('food'); view(0,90);

subplot(2,2,3); plotIntervalType3MF(fis,'output',1); xlabel('tip'); view(0,90); % Overall input-output surface figure; gensurfIntervalType3(fis,[1 2],1,15); grid on; box on

3.

Results of evaluating the SMAMIT3FLS system for the basic tip problem. Figure 4.4 shows the membership functions of the system. Figure 4.5 shows the nonlinear surface of the model.

4.2 Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS)

Fig. 4.4 Linguistic variables of the SMAMIT3FLS model for the basic tip problem

Fig. 4.5 Solution of the SMAMIT3FLS model for the basic tip problema

65

66

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

inputs = 2

5

6

9

pper = 6.9050 20.1518

4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS) The structure of the k-th IF–THEN generic rule of the TSK fuzzy system is RTk S K : I F x1 is Fk1 and . . . and xi is Fik and . . . and xn is Fkn T H E N y1 is g˜˜ 1k , . . . , y j is g˜ kj , . . . , ym is g˜ mk where i = 1,…, n (number of inputs), j = 1,…, m (number of outputs) and k = 1,…, r (number of rules). The functions of the rule consequents are linear interval equations ˙˙ . . . + C xi + · · · + C xn + C = g˜ kj (x) = Ck,1 x1 + k,i k,n k,0 j

j

j

j

n !

j

j

Ck,i xi + Ck,0

i=1

  j j j j j where the coefficients Ck,i ∈ ck,i − sk,i , ck,i + sk,i are the Interval type-1 fuzzy numbers (intervals), the IT3 TSK fuzzy system that uses these rules is called (A3– C1), which means that the antecedents are the IT3 FSs while the consequents are j j the T1 FSs. The coefficient ck,i is the center and sk,i is the radius of uncertainty of j the interval Ck,i . Simplifying g˜ kj (x), we obtain the linear interval functions of the   −k k Takagi—Sugeno—Kang (TSK) rules, g˜ kj (x) = g− j (x), g j (x) , where g kj (x) =

n ! i=1

j

j

ck,i xi + ck,0 −

n ! i=1

j

j

sk,i |xi | − sk,0

4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS)

g kj (x) =

n !

j

j

ck,i xi + ck,0 +

i=1

n !

j

67

j

sk,i |xi | + sk,0

i=1

  ˜ k x  , is the value of the IT2 MF of the  operation of all The firing strength  membership values of the antecedents μFik (xi ) that contribute to the level of activation of the rule, that is   n ˜ k x  = i=1  μFik (xi )         ˜ k x  = 1/ φlk x  , φrk x  F OU  The evaluation of the mathematical functionsrepresenting the consequents of the        rules, g˜ kj x  = g kj x  , g kj x  , gives. n n   !   !  j j j   j g kj x  = ck,i xi + ck,0 − sk,i xi  − sk,0

  g kj x  =

i=1 n !

j



j

ck,i xi + ck,0 +

i=1

i=1 n !

  j   j sk,i xi  + sk,0

i=1

Type-reduction The Normalized Weighted Average (wta) yields          y lj x  = le f t point g kj x  , φlk x  , φrk x     k    "r   k   "L k k   k=1 φr x g j x + k=L+1 φl x g j x l yj x =    "r   "L k k k=1 φr x + k=L+1 φl x          y rj x  = right point g kj x  , φlk x  , φrk x     k    "r   k   "R k k   k=1 φl x g j x + k=R+1 φr x g j x r yj x =    "r   "R k k k=1 φl x + k=R+1 φr x Defuzzifier yˆ j =

    y lj x  + y rj x  2

68

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

Case study: Evaluation of a Singleton TSK interval type-3 fuzzy logic system (STSKIT3FLS) Assume that we have a STSKIT3FLS with two inputs and two outputs with three rules. The objective is here to illustrate the design process of a STSKIT3FLS using the Matlab® programming language. 1.

Generate the IT3 FIS using line commands.

% SingletonTSKit3fls223.m % % Date : 20-June-2021 % clearvars; % Create IT3FIS Model % Sintaxis:

% fis=newfisIntervalType3(fisName,fisType,inputsFuzzifier,andMethod,... %

orMethod,impMethod,aggMethod,defuzMethod,

%

typeReductionMethod)

% % defuzMethod: centroid, csums,height,mdheight,cos,wtaver % typeReductionMethod: km, ekm, iasc, eiasc, eods, wmub, % %

nt, bmm

4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS)

69

fis = newfisIntervalType3('singletonTSKit3fls223','sugeno','singleton','prod','prob or',... 'prod','sum','wtaver','km'); % % Input 1 fis=addvarIntervalType3(fis,'input','x1',[0 10]); fis=addmfIntervalType3(fis,'input',1,'F11','ScaleGaussScaleGaussIT3MF',{{ [1.3633 5.4931]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',1,'F21','ScaleGaussScaleGaussIT3MF',{{ [1.0384 4.5038]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',1,'F31','ScaleGaussScaleGaussIT3MF',{{ [0.9766 6.2121]},0.9,0.2}); % Input 2 fis=addvarIntervalType3(fis,'input','x2',[0 10]); fis=addmfIntervalType3(fis,'input',2,'F12','ScaleTrapScaleGaussIT3MF',{{[1 .1706 2.9989]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',2,'F22','ScaleTrapScaleGaussIT3MF',{{[1 .1185 5.9512]},0.9,0.2}); fis=addmfIntervalType3(fis,'input',2,'F32','ScaleTrapScaleGaussIT3MF',{{[1 .4871 5.1395]},0.9,0.2}); % Output 1 fis=addvarIntervalType3(fis,'output','y1',[0 10]); fis=addmfIntervalType3(fis,'output',1,'g11','linear',[0.8311 0.7219 0.1795 0.294 0.2623 0.0646]); fis=addmfIntervalType3(fis,'output',1,'g21','linear',[0.2882 0.3678 0.0902 0 0 0]);

70

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

fis=addmfIntervalType3(fis,'output',1,'g31','linear',[0.3103 0.1533 0.0526 0.1567 0.1934 0.0304]);

% Output 2 fis=addvarIntervalType3(fis,'output','y2',[0 10]); fis=addmfIntervalType3(fis,'output',2,'g12','linear',[1.0414 0.8488 0.215 0.3224 0.2495 0.063]); fis=addmfIntervalType3(fis,'output',2,'g22','linear',[0.357 0.443 0.1033 0.0264 0 0]); fis=addmfIntervalType3(fis,'output',2,'g32','linear',[0.2607 0.0771 0.0547 0.1199 0.1786 0.02]); % % Add Rules % see addRule. % ruleList=[ ...

1 1, 1 1 1 1 2 2, 2 2 1 1 3 2, 3 3 1 1]; % % Add the rules to the FIS. fis=addruleIntervalType3(fis,ruleList); % % write it3fis % writefisIntervalType3(fis,' singletonTSKit3fls223');

4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS)

71

Now we show the singletonTSKit3fls223.fis file that contains the sequence of line commands and Matlab® functions for defining the Interval type-3 fuzzy inference system: [System] Name='singletonTSKit3fls223' Type='sugeno' InputsFuzzifier='singleton' Version=1.0 NumInputs=2 NumOutputs=2 NumRules=3

AndMethod='prod' OrMethod='probor' ImpMethod='prod' AggMethod='sum' DefuzMethod='wtaver' TypeReductionMethod='km' [Input1] Name='x1' Range=[0 10]

72

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

NumMFs=3 MF1='F11':'ScaleGaussScaleGaussIT3MF',{{[1.3633 5.4931]},0.9,[0.2]} MF2='F21':'ScaleGaussScaleGaussIT3MF',{{[1.0384 4.5038]},0.9,[0.2]} MF3='F31':'ScaleGaussScaleGaussIT3MF',{{[0.9766 6.2121]},0.9,[0.2]}

[Input2] Name='x2' Range=[0 10] NumMFs=3 MF1='F12':'ScaleTrapScaleGaussIT3MF',{{[1.1706 2.9989]},0.9,[0.2]} MF2='F22':'ScaleTrapScaleGaussIT3MF',{{[1.1185 5.9512]},0.9,[0.2]} MF3='F32':'ScaleTrapScaleGaussIT3MF',{{[1.4871 5.1395]},0.9,[0.2]}

[Output1] Name='y1' Range=[0 10] NumMFs=3 MF1='g11':'linear',[0.8311 0.7219 0.1795 0.294 0.2623 0.0646] MF2='g21':'linear',[0.2882 0.3678 0.0902 0 0 0] MF3='g31':'linear',[0.3103 0.1533 0.0526 0.1567 0.1934 0.0304]

4.3 Singleton TSK Interval Type-3 Fuzzy Logic System (STSKIT3FLS)

[Output2] Name='y2' Range=[0 10] NumMFs=3 MF1='g12':'linear',[1.0414 0.8488 0.215 0.3224 0.2495 0.063] MF2='g22':'linear',[0.357 0.443 0.1033 0.0264 0 0] MF3='g32':'linear',[0.2607 0.0771 0.0547 0.1199 0.1786 0.02]

[Rules] 1 1, 1 1 (1) : 1 2 2, 2 2 (1) : 1 3 3, 3 3 (1) : 1

2.

Evaluating the TSKit3fls223 singleton system.

% demo01_singletonTSKit3fls223.m clearvars; fis = readfisIntervalType3('singletonTSKit3fls223');

73

74

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

% Linguistic variables and linguistic values figure; subplot(1,2,1); plotIntervalType3MF(fis,'input',1); xlabel('x_1'); view(0,90);

subplot(1,2,2); plotIntervalType3MF(fis,'input',2); xlabel('x_2'); view(0,90);

% Overall input-output surface figure; subplot(1,2,1); gensurfIntervalType3(fis,[1 2],1,20); grid on; box on subplot(1,2,2); gensurfIntervalType3(fis,[1 2],2,20); grid on; box on

3.

Results of evaluating the TSKit3fls223 singleton. Figure 4.6 shows the membership functions of the system. Figure 4.7 shows the nonlinear surface of the model.

4.4 Approach to Evaluate a Singleton Mamdani Interval …

75

Fig. 4.6 Linguistic variables of the STSKIT3FLS223 system

Fig. 4.7 Nonlinear Surface of the TSKit3fls223 singleton model

4.4 Approach to Evaluate a Singleton Mamdani Interval Type-3 Fuzzy Logic System (SMAMIT3FLS) by Using α-planes The proposed approach to calculate the Interval type-3 fuzzy model, is based on the horizontal cuts representation because it allows to use all the knowledge of the Interval type-2 and general type-2 fuzzy systems. A fuzzy system resulting from the horizontal cut is analogous to an Interval type-2 fuzzy logic system (IT2 FLS) where all the calculations of the IT2 FS are performed for each horizontal slice. In the same way, a general type-2 fuzzy system is the aggregation of the horizontal cut fuzzy systems where the aggregation ocurrs by means of deffuzification. The idea of aggregating the horizontal cut fuzzy systems was originally proposed by Wagner and Hagras, and presented in Mendel [1]. This is based on the decomposition of the horizontal cuts of a T2 FS, where the α-planes of the membersip function of the T2 FSs should be equal to the function applied to the α-planes of these T2

76

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

FSs. In the following we show the inference process using the decomposition by the α-planes: 1.

Descomposition by the α-planes of the antecedents and consequents of the rules # Fik = ∫ μFik (xi ) (u)/xi = ∫ xi ∈X i

#

= ∫

$ sup α/Fik (xi )α /xi

xi ∈X i α∈[0,1]

$  k sup α/ F ik (xi )α , F i (xi )α /xi

xi ∈X i α∈[0,1]

where  F ik (xi )α = a ik (xi )α , bik (xi )α a ik (xi )α = inf { u|u ∈ [0, 1], μF (xi , u) ≥ α} bik (xi )α = sup { u|u ∈ [0, 1], μF (xi , u) ≥ α}   k k F i (xi )α = a ik (xi )α , bi (xi )α a ik (xi )α = inf { u|u ∈ [0, 1], μF (xi , u) ≥ α} k

bi (xi )α = sup { u|u ∈ [0, 1], μF (xi , u) ≥ α} # Gkj

sup = ∫ μ ( y j ) (u)/y j = ∫ y j ∈Y j y j ∈Y j α∈[0,1] # $    k  k sup α/ G j y j α , G j y j α /y j = ∫ Gkj

α/Gkj

$   y j /y j

y j ∈Y j α∈[0,1]

where         G kj y j α = ckj y j α , d ik y j α     ckj y j α = inf { u|u ∈ [0, 1], μG y j , u ≥ α}     d ik y j α = sup { u|u ∈ [0, 1], μG y j , u ≥ α}     k  k  G j y j α = ckj y j α , d j y j α

4.4 Approach to Evaluate a Singleton Mamdani Interval …

77

    ckj y j α = inf { u|u ∈ [0, 1], μG y j , u ≥ α}   k  d j x j α = sup { u|u ∈ [0, 1], μG y j , u ≥ α} 2.

˜ k (x)α . using the α-planes Evaluation of the firing strength          ˜ k x  = φk x , φk x  α α α   n n n = i=1 μFik (xi )α = i=1 μ F ik (xi )α , i=1 μ F k (xi )α i

        φ k x  α = ϕ lk x  α , ϕ rk x  α    n k n n = i=1 μ F ik xi = Ti=1 a i (xi )α , Ti=1 bik (xi )α α

where T is the T-Norm   n ϕ lk x  α = Ti=1 a ik (xi )α   n ϕ rk x  α = Ti=1 bik (xi )α and      k  φ x  α = ϕ lk x  α , ϕ rk x  α     k n n n μ F k xi = Ti=1 a ik (xi )α , Ti=1 bi (xi )α = i=1 α

i

  n ϕ lk x  α = Ti=1 a ik (xi )α   k n ϕ rk x  α = Ti=1 bi (xi )α 3.

Evaluation of the level of activation of the rule using the α-planes            μBkj y j  x  α = μ B kj y j  x  α , μ B k y j  x  α j     k ˜ =  x α μGkj y j α                  μ B kj y j  x  α = φ k x  α μG kj y j α = ϕ lk x  α ckj y j α , ϕ rk x  α d ik y j α              k  k  μ B k y j  x  α = φ x  α μG k y j α = ϕ lk x  α ckj y j α , ϕ rk x  α d j y j α j

4.

j

Aggregation of the levels of activation of the rules by the α-planes

78

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

           μB j y j  x  α = μ B j y j  x  α , μ B j y j  x  α       ˜ k x  μGk y j = ∪rk=1  α α j          μ B j y j  x  α = ∪rk=1 φ k x  α μG kj y j α           = ⊕rk=1 ϕ lk x  α ckj y j α , ⊕rk=1 ϕ rk x  α d ik y j α  k       μ B j y j |x  α = ∪rk=1 φ x  α μG k y j α j     k     k  r k = ⊕k=1 ϕ l x α c j y j α , ⊕rk=1 ϕ rk x  α d j y j α where ⊕ is the S-Norm operator. 5.

Type-Reduction 

     y Bα j , y B−αj = t ype Reduction y j , μ B j y j |x  α "nlevel

αk k=1 αk y B j "nlevel α k k=1 "nlevel α k=1 αk y B j y rj = " k=1 αk       y αB , y αB = t ype Reduction y j , μ B j y j |x  α

y lj

=

"nlevel

α k=1 αk y = "nlevel B k=1 αk "nlevel α k=1 αk y y rj = "nlevel B k=1 αk

y lj

6.

Defuzzifier  yˆ j =

   y lj + y rj + y lj + y rj 4

4.5 Approach to Evaluate a Singleton TSK Interval Type-2 Fuzzy …

79

4.5 Approach to Evaluate a Singleton TSK Interval Type-2 Fuzzy Logic System Using α-planes The structure of the k-th IF–THEN generic rule for the Takagi–Sugeno-Kang (TSK) fuzzy system is: RTk S K : I F x1 is Fk1 and . . . and xi is Fik and . . . and xn is Fkn T H E N y1 is g˜ 1k , . . . , y j is g˜ kj , . . . , ym is g˜ mk where i = 1,…, n (number of inputs), j = 1,…, m (number of outputs) and k = 1,…, r (number of rules). The functions of the rule consequents are linear interval equations, that is j

j

j

j

g˜ kj (x) = Ck,1 x1 + · · · + Ck,i xi + · · · + Ck,n xn + Ck,0 =

n !

j

j

Ck,i xi + Ck,0

i=1

  j j j j j in which the coefficients Ck,i ∈ ck,i − sk,i , ck,i + sk,i are type-1 fuzzy interval numbers (intervals), the IT3 TSK fuzzy system that uses these rules is called (A3C1), which means the antecedents are IT3 FS while the consequents are T1 FSs. The j j j coefficient ck,i is the center and sk,i is the radius of uncertainty of the interval Ck,i . By simplifying g˜ kj (x), we can obtain the linear Interval functions of the TSK rules,   g˜ kj (x) = g kj (x), g kj (x) , where g kj (x) = g kj (x) =

n !

j

n !

i=1

i=1

n !

j

n !

j

ck,i xi + ck,0 − j

ck,i xi + ck,0 +

i=1

j

j

j

j

sk,i |xi | − sk,0 sk,i |xi | + sk,0

i=1

In the following we show the inference process by using descomposition by the α-planes. 1.

Descomposition by α-planes of the antecedents and consequents of the rules # Fik

= ∫ μFik (xi ) (u)/xi = ∫ xi ∈X i

= ∫

#

$  k k sup α/ F i (xi )α , F i (xi )α /xi

xi ∈X i α∈[0,1]

where

sup

xi ∈X i α∈[0,1]

α/Fik (xi )α

$ /xi

80

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

 F ik (xi )α = a ik (xi )α , bik (xi )α a ik (xi )α = inf { u|u ∈ [0, 1], μF (xi , u) ≥ α} bik (xi )α = sup { u|u ∈ [0, 1], μF (xi , u) ≥ α}   k k F i (xi )α = a ik (xi )α , bi (xi )α a ik (xi )α = inf { u|u ∈ [0, 1], μF (xi , u) ≥ α} k

bi (xi )α = sup { u|u ∈ [0, 1], μF (xi , u) ≥ α} 2.

  ˜ k x  using the α-planes Evaluation of the firing strength  α           n ˜ k x = φk x , φk x k x  =  μ i i=1 Fi α α α      α  n n = i=1 μ F ik xi , i=1 μ F k xi α

i

α

          n φ k x  α = ϕ lk x  α , ϕ rk x  α = i=1 μ F ik xi α  n k n = Ti=1 a i (xi )α , Ti=1 bik (xi )α where T is the T-Norm,   n a ik (xi )α ϕ lk x  α = Ti=1   n ϕ rk x  α = Ti=1 bik (xi )α        k  n φ x  α = ϕ lk x  α , ϕ rk x  α = i=1 μ F k xi i α   k k n n = Ti=1 a i (xi )α , Ti=1 bi (xi )α   n ϕ lk x  α = Ti=1 a ik (xi )α   k n ϕ rk x  α = Ti=1 bi (xi )α 3.

  Evaluation of the functions of the consequents of the rules, g˜ kj x  =      g kj x  , g kj x  , where

4.5 Approach to Evaluate a Singleton TSK Interval Type-2 Fuzzy … n n   !   !  j j j   j g kj x  = ck,i xi + ck,0 − sk,i xi  − sk,0 i=1

i=1

i=1

i=1

n n   !   !  j j j   j g kj x  = ck,i xi + ck,0 + sk,i xi  + sk,0

4.

Type-reduction by the Normalized Weighted Average (wta)         y αgk = le f t point g kj x  , ϕ lk x  α , ϕ rk x  α j

    "     ϕ rk x  α g kj x  + rk=L+1 ϕ lk x  α g kj x  =     "L "r k k k=1 ϕ r x α + k=L+1 ϕ l x α "nlevel α k=1 αk y g k j l y j = "nlevel α k=1 k         y αgk = right point g kj x  , ϕ lk x  α , ϕ rk x  α "L

y αgk j

k=1

j

    "     ϕ lk x  α g kj x  + rk=R+1 ϕ rk x  α g kj x      "R "r k k k=1 ϕ l x α + k=R+1 ϕ r x α "nlevel α k=1 αk y g k j r y j = "nlevel α k=1 k       = le f t point g kj (x), ϕ lk x  α , ϕ rk x  α

"R y αgk j

k=1

=

y αgk j

    "     ϕ rk x  α g kj x  + rk=L+1 ϕ lk x  α g kj x      "r "L k k k=1 ϕ r x α + k=L+1 ϕ l x α "nlevel α k=1 αk y g k j l y j = "nlevel α k=1 k       k = right point g j (x), ϕ lk x  α , ϕ rk x  α

"L

k=1

y αgk = j

y αgk j

"R y αgk j

=

k=1

    "     ϕ lk x  α g kj x  + rk=R+1 ϕ rk x  α g kj x      "R "r k k k=1 ϕ l x α + k=R+1 ϕ r x α "nlevel α k=1 αk y g kj r y j = "nlevel k=1 αk

81

82

5.

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

Defuzzifier  yˆ j =

   y lj + y rj + y lj + y rj 4

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) Using the Principle of Justifiable Granularity The design of a complex fuzzy inference system (FIS) with a great number of inputs and membership functions (MF) is a big problem due to the high number of parameters in the membership functions and rules. To design a FIS of this type, we could utilize an approach based on data to learn the rules and tune the parameters of the FIS. During the training, the optimization algorithm generates sets of parameters of the candidate FISs. The fuzzy system is updated with each set of parameters and then is evaluated with the input training data. If there is training input/output data, the cost of each solution is calculated as a function of the difference between the output of the fuzzy system and the expected outputs of the training data. In the literature there are various tuning methods (genetic algorithms, particle swarm optimization, scatter search, simulated annealing and adaptive neuro-fuzzy inference systems) to find the optimal parameters of the FIS. The global optimization methods, such as the genetic algorithms and the particle swarm optimization, perform better for parameter adjustment in large search spaces. These algorithms are useful for the stages of learning the rules and parameter adjustment in the FIS optimization. On the other hand, the local search methods, such as the scatter search and simulated annealing, perform better in small search spaces. If a FIS is generated from a training data set or if a rule base has been added to a FIS using training data, then these algorithms can produce a faster convergence in comparison with the global optimization methods. The problem of overfitting is a common issue in the process of parameter optimization for the FISs. When the problem of overfitting is present, the adjusted FIS produces optimized results for the training data set but a defficient performance on the testing data set. To overcome this overfitting problem, an adjustment process can be stopped before time based on an unbiased separate validation data. In addition, when adjustments are done, the overfitting problem can be avoided by using the k-fold crossvalidation. To improve the performance of fuzzy systems with adjustment methods, we can take into account the following: • Use several phases in the process of ajustment. For example, first learn the rules of the fuzzy system and then adjust the parameters of the input/output membership functions usign the learned rules. • Increase the number of iterations in the phases of learning the rules and parameter adjustment. This will increase the time for the optimization process and can also

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) …

83

increase the validation error due to possible overfitting of the system parameters. To avoid the overfitting problem, we can perform training of the system with the k-fold crossvalidation. • Change the clustering technique. Depending on the clustering technique, the generated rules can be different in their way to represent the training data. As a consequence, the use of different clustering techniques can affect the performance of the adjustment. • Change the properties of the FIS. We can change the number of inputs, the number of MFs for inputs and outputs, the types of MFs and the number of rules. A TSK system has less parameters in the output (assuming constant functions) and a faster defuzzification. As a consequence, for fuzzy systems with a high number of inputs, a FIS of the TSK form would generally converge faster than the Mamdani FIS. A small number of MFs and rules reduces the number of parameters to tune which produces a faster tuning process. In addition, a high number of rules could overfit the training data. • Modify the configuration of adjustable parameters for the MFs and rules. For example, we can adjust the support of a triangular MF without changing the core. This reduces the number of adjustable of parameters and could produce a faster adjustment for specific applications. To improve the results of adjustment for fuzzy trees, we can take into account the following: • The parameters of each FIS can be adjusted separately in a tree of the FIS. Then, we can adjust all the fuzzy systems together to generalize the parameters. • Change the properties of the tree of FISs, such as the number of fuzzy systems and connections between the fuzzy systems. • Different classification and clustering algorithms can be used for the inputs of the tree of FISs. In this work we are presenting a more robust approach based on data to learn the rules and adjust the parameters of the IT3FIS model using granular computing. The evaluation and optimization of this method are guided by the principle of justifiable granularity that involves the criterion of coverage and specificity and it is done with the help of the global and local optimization. In other words, we use clustering techniques to find the partition of data to characterize the justifiable information granules represented by the IT3 FS fuzzy sets with a justifiable footprint of uncertainty, where the partition of fuzzy granules generates the rule base of the fuzzy model. The approach is a process in which the parameters and rules of the model should be adjusted. This should be a natural adjustment to build the model with experimental justifiable evidence, and at the same time exhibit high generalization capabilities. In the following we show the methodology for the design of the Interval type-3 fuzzy models with the Mamdani and Takagi–Sugeno-Kang reasoning schemes using the principle of justifiable granularity.

84

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

4.6.1 Methodology for the Design of Singleton Mamdani Interval Type-3 Fuzzy Logic Systems (SMAMIT3FLS) Using the Principle of Justifiable Granularity Step 1: Given a collection of q pairs of training data of n-inputs and m-outputs,  {(X:T)}, where X = x p,i is the input vector of size q × n, and T = t p, j is an output vector of size q × m. Step 2: Cluster the set of q pairs of training data for the input–output variables of the system using the fuzzy c-means (FCM) algorithm with κ prototypes, fuzzification coefficient of 2 (m = 2), máximum number of iterations (maxIter = 200) and final tolerance for interations (tol = 1 × 10−5 ) to extract the data of the partition matrix (U) for each cluster. The essence of the FCM clustering algorithm can be outlined symbolically as: [V, U ] = FC M([X, T ], κ, [m, max I ter, tol]  where m is the  fuzzification coefficient, U = μk, p is partition matrix of size κ × q and V = vk,l is the prototype matrix (medians) of size κ × (n + m) for the input–output variables. In the FCM, the fuzzification coefficient (m) controls the degree of shared membership between the fuzzy clusters. There is no theoretical evidence for an optimal m value, only experimental evidence, however, there can be consistent results for m ∈ [1.5, 3.0]. The most used value in the literature [2–5] is m = 2, and this value is the one used in this work. Step 3: Calculate the following components of the box diagram: inferior extremum (WLO), confidence interval of the inferior median (MLO), median (M), confidence interval of the superior median (MHI) and superior extremum (WHI) in each κ cluster of data in the partition matrix of the partition U for the input–output variables. Step 4: Calculate the Interval of the justifiable experimental evidence [a, b] in each κ cluster of data in the partition matrix U for the input–output variables. Step 5: Parameterize the IT3 FS in the antecedents and consequents of the rules of a MAMIT3FLS with the elements of the box diagram and the interval of justifiable experimental evidence. Step 6: Generate the structure of the MAMIT3FIS in the antecedents and consequents of the rules of the MAMIT3FLS with the elements of the box diagram and the interval of justifiable experimental evidence. Step 7: Evaluate the MAMIT3FLS system using the approximation of the α-planes (nlevel = 5 by default). Step 8: Evaluate the coverage and specificity indices of the MAMIT3FLS model. Case study: Gauss3 The framework of data for Gauss3 [42] has one predictor variable (x) and one response variable (y) with 250 observations that generate data with Gaussian peaks with an

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) …

85

exponential decaying. The data consists of two strongly intermixed Gaussians with noise of the zero mean and variance of 6.25. Structure of the MAMIT3FIS [System] Name='SingletonMamIT3FLS115Gauss3' Type='mamdani' InputsFuzzifier='singleton' Version=1.0 NumInputs=1 NumOutputs=1 NumRules=5 AndMethod='prod' OrMethod='probor' ImpMethod='prod' AggMethod='sum'

86

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

DefuzMethod='cs' TypeReductionMethod='km' [Input1] Name='x' Range=[1 250] NumMFs=5 MF1='F11':'ScaleGaussScaleGaussIT3MF',{{[25.4654 46.9729]},0.8,0.3} MF2='F21':'ScaleGaussScaleGaussIT3MF',{{[20.5829 190.3973]},0.8,0.3} MF3='F31':'ScaleGaussScaleGaussIT3MF',{{[26.7360 133.5465]},0.8,0.3} MF4='F41':'ScaleGaussScaleGaussIT3MF',{{[9.9745 133.9419]},0.8,0.3} MF5='F51':'ScaleGaussScaleGaussIT3MF',{{[26.9152 21.3485]},0.8,0.3}

[Output1] Name='y' Range=[1.182678 135.1278] NumMFs=5 MF1='G11':'ScaleGaussScaleGaussIT3MF',{{[23.5268 34.2499]},0.8,0.3} MF2='G21':'ScaleGaussScaleGaussIT3MF',{{[29.3761 7.1650]},0.8,0.3}

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) …

87

MF3='G31':'ScaleGaussScaleGaussIT3MF',{{[11.6602 150.2119]},0.8,0.3} MF4='G41':'ScaleGaussScaleGaussIT3MF',{{[10.7506 69.0593]},0.8,0.3} MF5='G51':'ScaleGaussScaleGaussIT3MF',{{[12.5653 127.9498]},0.8,0.3}

[Rules] 1, 1 (1) : 1 2, 2 (1) : 1 3, 3 (1) : 1 4, 4 (1) : 1 5, 5 (1) : 1

Partition of the input–output linguistic variables Figure 4.8 shows the input linguistic variables. Surface of the model MAMIT3FIS (nlevel = 5 by default) Figure 4.9 shows the solution of the model as a surface. Coverage and specificity of the model SingletonMAMIT3FLS115Gauss3. The results are: RMSE = 3.9720 R = 0.9911

Fig. 4.8 Input linguistic variables of SingletonMAMIT3FLS115Gauss3 model

88

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

Fig. 4.9 Solution of the model SingletonMamIT3FLS115Gauss3

cov = 0.8600 spc = 0.9447

4.6.2 Methodology for the Design of Singleton TSK Interval Type-3 Fuzzy Logic Systems (STSKIT3FLS) Using the Principle of Justifiable Granularity Step 1: Given a collection of q pairs of training data of n-inputs and m-outputs,  {(X:T)}, where X = x p,i is the input vector of size q × n, and T = t p, j is an output vector of size q × m. Step 2: Cluster the set of q pairs of training data for the input–output variables of the system using the fuzzy c-means (FCM) algorithm with κ prototypes, fuzzification coefficient of 2 (m = 2), maximum number of iterations (maxIter = 200) and final tolerance for interations (tol = 1 × 10−5 ) to extract the data of the partition matrix (U) for each cluster. The essence of the FCM clustering algorithm can be shown as: [V, U ] = FC M([X, T ], κ, [m, max I ter, tol]  where m is the  fuzzification coefficient, U = μk, p is partition matrix of size κ × q and V = vk,l is the prototype matrix (medians) of size κ × (n + m) for the input–output variables. Step 3: Calculate the following components of a box diagram: inferior extremum (WLO), confidence interval of the inferior median (MLO), median (M), confidence interval of the superior median (MHI) and superior extremum (WHI) in each κ cluster of data in the partition matrix of the partition U for the input–output variables.

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) …

89

Step 4: Calculate the Interval of justifiable experimental evidence [a, b] in each κ cluster of data in the partition matrix U for the input–output variables. Step 5: Parameterize the IT3 FS in the antecedents and consequents of the rules of the TSKIT3FLS with the elements of the box diagram and the Interval of justifiable experimental evidence. j Step 6: Calculate the intervals of the parameters, Ck,i , of the linear functions, g˜ kj (x), in the consequents of the rules of the TSKIT3FLS using linear recursive regression (RLS) in each κ cluster of data in the partition matrix U of the output variables. Step 7: Generate the structure of the TSKIT3FIS in the antecedents and consequents of the rules of the TSKIT3FLS with the elements of the box diagram and the interval of justifiable experimental evidence, the same thing with the parameters of the consequents evaluated with the RLS. Step 8: Evaluate the TSKIT3FLS system using the approximation of α-planes (nlevel = 5 by default). Step 9: Evaluate the coverage and specificity indices of the TSKIT3FLS model. Case study: Gauss3 The framework of data for Gauss3 [42] has one predictor variable (x) and one response variable (y) with 250 observations that generate data with Gaussian peaks with an exponential decay. The data consists of two strongly intermixed Gaussians with noise of the zero mean and variance of 6.25. Structure of the TSKIT3FIS [System] Name='SingletonTSKIT3FLS115Gauss3' Type='sugeno' InputsFuzzifier='singleton' Version=1.0

90

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

NumInputs=1 NumOutputs=1 NumRules=5 AndMethod='prod' OrMethod='probor' ImpMethod='prod' AggMethod='sum' DefuzMethod='wtaver' TypeReductionMethod='km'

[Input1] Name='x' Range=[0 250] NumMFs=5 MF1='F11':'tsGaussScaleGaussIT3MF',{{[7.1176 12.8824],[19.0668 21.9332],[7.1176 12.8824]},0.8,[0.3]} MF2='F21':'tsGaussScaleGaussIT3MF',{{[9.7942 16.2058],[64.8659 68.1341],[9.7942 16.2058]},0.8,[0.3]} MF3='F31':'tsGaussScaleGaussIT3MF',{{[7.9217 14.0783],[179.514 182.486],[7.9217 14.0783]},0.8,[0.3]} MF4='F41':'tsGaussScaleGaussIT3MF',{{[8.8988 15.1012],[224.93 228.07],[8.8988 15.1012]},0.8,[0.3]} MF5='F51':'tsGaussScaleGaussIT3MF',{{[13.3421 20.6579],[124.1451 127.8549],[13.3421 20.6579]},0.8,[0.3]}

4.6 Design of Interval Type-3 Fuzzy Logic Systems (IT3FLS) …

91

[Output1] Name='y' Range=[1.17 135.13] NumMFs=5 MF1='g11':'linear',[-0.8362 97.2199 0.01 8.6855] MF2='g21':'linear',[-0.2277 68.0271 0.02 4.5105] MF3='g31':'linear',[-0.8866 182.1392 0 6.2079] MF4='g41':'linear',[-0.2018 54.7467 0.01 2.0454] MF5='g51':'linear',[-0.5382 183.7977 0 9.6962]

[Rules] 1, 1 (1) : 1 2, 2 (1) : 1 3, 3 (1) : 1 4, 4 (1) : 1 5, 5 (1) : 1

Plot of the partition of the input linguistic variables We show in Fig. 4.10 the membership functions of the input linguistic variables of the model. Surface of the model TSKIT3FIS (nlevel = 5 by default) We show in Fig. 4.11 the surface of the model. Regression measures (RMSE, R), Coverage (cov) and Specificity (spc) of the model TSKIT3FIS RMSE = 3.8440 R = 0.9955 cov = 0.9640 spc = 0.8696

92

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

Fig. 4.10 Input linguistic variables of the Singleton model TSKIT3FLS115Gauss3

Fig. 4.11 Surface of the model SingletonTSKIT3FLS115Gauss3

4.7 Comparison of Results of Type-3 Versus Type-2 Fuzzy Systems We present in this section simulation results for seven bechmark data sets to compare the results of using the type-3 fuzzy models versus the results of using the type-2 fuzzy models. The main characteristics of the data sets are shown in Table 4.1. The experimental results are obtained by random subsampling, dividing the datasets in 60% for training and 40% for testing, and using 30 executions. We use 2 two measures for performance evaluation, R-squared adjusted (Rad j ) that penalizes for additional indepdendent variables added to the model and helps avoiding overtraining, and the Root Mean Squared Error ( R M S E) that helps interpret the results.

4.7 Comparison of Results of Type-3 Versus Type-2 Fuzzy Systems

93

Table 4.1 Characteristics for the applied regression benchmark datasets Dataset

# Samples

# Inputs

# Outputs

Synthetic complex curve SNR30dB [8]

1500

1

1

Gauss3 [6]

250

1

1

Price [9]

1000

2

1

Engine behavior [8]

1199

2

2

Gas furnace [9] Concrete compressive strength [9] Energy efficiency [8]

296

4

1

1030

8

1

768

8

2

2 Tables 4.2 and 4.3 show the obatined results for Rad j and R M S E, respectively. For the comparison of results the models: Singleton MAM IT3FLS+FCM+RLS, Singleton MAM IT2FLS+FCM+RLS and MAM T2FLS+FCM+RLS are selected, using the basic clustering and regression techniques, this means, using the fuzzy c-means (FCM) clustering to obtain the parameters of the antecedent membership functions and the recursive least squares (RLS) for the consequent membership functions parameters, and granular computing to evaluate for a justifiable footprint of uncertainty (FOU) [7]. The idea is to provide a better understanding of the achieved performance and model interpretability. 2 Tables 4.4 and 4.5 show the results for the Rad j and RMSE, respectively, using the training of 10 epochs for adjusting the parameters of the antecedent and consequent upper membership functions in the rules of the fuzzy models. The training is done for the paramters of the models: Singleton MAM IT2FLS+FCM+RLS, MAM T2FLS+FCM+RLS y MAM IT3FLS+FCM+RLS using the Levenberg–Marquardt algorithm for minimizing the MSE cost with the maximum likelihood, by using random subsampling, dividing the datasets in 60% for training and 40% for testing, and using 30 executions. We use in the antecedents and consequents of the Interval type-2 membership functions (IT2 MF) gaussScaleIntervalType2MF(x,{{[s m]},lowerScale,lowerLag}) for the IT2FLS model, generalized type-2 membership functions (T2 MF) ScaleGaussGaussT2MF(x,u,{{[s m]},lowerScale,lowerLag}) for the T2FLS model and Interval type-3 membership functions (IT3 MF) ScaleGaussScaleGaussIT3MF(x,u,{{[s m]},lowerScale,lowerLag}) for the IT3 FLS model.

4.8 Summary We have provided in this chapter the basic definitions for building the Interval type-3 fuzzy systems and we have illustrated their implementaion with some examples and case studies. We have covered the singleton and non-singleton, as well as Mamdani and Takagi—Sugeno—Kang Interval type-3 fuzzy systems, which in our opinion will be very useful in many areas of applications, such as in intelligent control, robotics,

3

7

5

Gas furnace

Concrete compressive strength

Energy efficiency

0.8159 0.8255

0.8023

0.3863

0.7865

0.3044

0.8339

0.8231

0.7939

0.8036

0.9276

0.5656

0.9411

0.9146

0.9066

0.4987

0.8191

0.8943

0.8440

0.8471

0.4660

0.8673

0.8505

0.9455

0.6119

0.9727

0.9293

0.0112

0.0140

0.0417

0.0152

0.0113

0.0106

0.0237

0.0339

0.0093

0.8019

0.7861

0.3055

0.8040

0.8140

0.9055

0.5007

0.8192

0.8870

0.8263

0.8160

0.4017

0.8347

0.8327

0.9271

0.5748

0.9420

0.9170

Mean

0.8466

0.8473

0.4813

0.8711

0.8897

0.9460

0.6118

0.9745

0.9295

Max

Min

Std

Min

Max

2 Rad j

2 Rad j

Mean

MAM T2 FLS+FCM+RLS

MAM IT2FLS+FCM+RLS

0.0117

0.0143

0.0425

0.0157

0.0132

0.0137

0.0241

0.0328

0.0083

Std

0.8102

0.7866

0.3033

0.8033

0.8135

0.9048

0.5005

0.8195

0.8860

Min

2 Rad j

0.8265

0.8163

0.4016

0.8355

0.8325

0.9268

0.5745

0.9425

0.9187

Mean

0.8475

0.8477

0.4901

0.8812

0.8901

0.9463

0.6117

0.9773

0.9298

Max

MAM IT3FLS+FCM+RLS

0.0115

0.0151

0.0431

0.0167

0.0139

0.0166

0.0245

0.0321

0.0088

Std

2 is used as a performance index and the bold values indicate the best values, the green is the best value from the worst ones in normal color In this case, Rad j among the algorithms

3

3

5

Gauss3

Engine behavior

7

Synthetic complex curve SNR30dB

Price

# Rules

Dataset

Table 4.2 Experimental results for the proposed MAM IT3FLS+FCM+RLS and comparison with MAM IT2FLS+FCM+RLS and MAM T2FLS+FCM+RLS

94 4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

3

Gas furnace

Concrete 7 compressive strength

9.5881

194.53

181.96

4.3126

3.9459

4.0189

3.7147

12.943

1.3550

147.25

21.533

125.61

19.985

0.8418

Mean

4.2776

4.7283

13.952

1.6202

204.23

171.21

22.674

17.129

0.9191

Max

126.17

18.235

6.4863

0.7541

0.1408

0.1651

0.4177

0.0788

3.7512

4.0234

11.332

1.2201

3.9167

4.2178

12.413

1.3490

191.35

148.32

19.765

9.4135

0.8201

Mean

4.2637

4.6512

13.824

1.6199

201.65

170.98

23.125

16.567

0.8817

Max

126.67

18.361

6.2001

0.7588

Min

0.1407

0.1701

0.4201

0.0872

3.6357

3.7812

12.378

1.2501

5.3307 179.98

11.189

0.8023

2.0147

0.0332

Std

3.8897

4.1101

12.878

1.3389

191.89

148.67

20.002

9.2887

0.7951

Mean

4.2412

4.6202

13.798

1.6037

201.05

169.78

23.435

16.115

0.8193

Max

RMSE

Min

RMSE

5.3190 178.03

11.072

0.7571

2.3134

0.0350

Std

MAM IT3FLS+FCM+RLS

MAM T2FLS+FCM+RLS

0.1413

0.1723

0.4403

0.0898

5.4589

12.001

0.9501

1.9873

0.0301

Std

In this case, R M S E is used as a performance index and the bold values indicate the best values, the green is the best value from the worst ones in normal color among the algorithms

5

12.150

3

Engine behavior

Energy efficiency

1.2352

3

Price

6.6486

5

Gauss3

0.7653

7

Min

# MAM IT2FLS+FCM+RLS Rules R M S E

Synthetic complex curve SNR30dB

Dataset

Table 4.3 Experimental results for the proposed MAM IT3FLS+FCM+RLS and comparison with MAM IT2FLS+FCM+RLS and MAM T2FLS+FCM+RLS

4.8 Summary 95

3

7

5

Gas furnace

Concrete compressive strength

Energy efficiency

0.8887 0.8835

0.8689

0.9045

0.8643

0.8728

0.9929

0.9101

0.8920

0.9910

0.9913

0.9628

0.9953

0.9377

0.9890

0.9191

0.9947

0.9281

0.9028

0.9065

0.9210

0.9958

0.9251

0.9898

0.9706

0.9973

0.9413

0.0089

0.0115

0.0107

0.0019

0.0074

0.0006

0.0073

0.0004

0.0038

0.8685

0.8638

0.8730

0.9911

0.8924

0.9850

0.9195

0.9945

0.9315

0.8844

0.8891

0.9050

0.9932

0.9109

0.9902

0.9638

0.9957

0.9398

Mean

0.9032

0.9071

0.9212

0.9961

0.9255

0.9921

0.9702

0.9971

0.9489

Max

Min

Std

Min

Max

2 Rad j

2 Rad j

Mean

MAM T2FLS+Tuning

MAM IT2FLS+Tuning

0.0085

0.0117

0.0109

0.0017

0.0071

0.0005

0.0078

0.0008

0.0042

Std

0.8692

0.8647

0.8727

0.9908

0.8921

0.9848

0.9192

0.9951

0.9313

Min

2 Rad j

0.8856

0.8895

0.9048

0.9936

0.9105

0.9899

0.9630

0.9968

0.9424

Mean

0.9034

0.9074

0.9213

0.9966

0.9260

0.9922

0.9700

0.9974

0.9506

Max

MAM IT3FLS+Tuning

0.0082

0.0119

0.0111

0.0015

0.0075

0.0008

0.0081

0.0005

0.0040

Std

2 is used as a performance index and the bold values indicate the best values, the green is the best value from the worst ones in normal color In this case, Rad j among the algorithms

3

3

5

Gauss3

Engine behavior

7

Synthetic complex curve SNR30dB

Price

# Rules

Dataset

Table 4.4 Experimental results for the proposed MAM IT3FLS+Tuning and comparison with MAM IT2FLS+Tuning and MAM T2FLS+Tuning using R2

96 4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

7

5

3

3

3

7

5

Synthetic complex curve SNR30dB

Gauss3

Price

Engine behavior

Gas furnace

Concrete compressive strength

Energy efficiency

9.1375

2.1897

5.4197

6.4001

2.0287

0.7691

0.0818

74.371

6.7153

0.0015

211.19

4.3198

2.4201

0.7628

8.4007

0.9725

0.7521

0.0055

7.7502

11.202

17.898

0.2571

344.92

434.33

7.9738

3.4897

1.8478

2.1541

2.2702

6.1508

0.0655

86.121

171.41

2.3324

0.7734

0.6036

2.0303

4.4208

0.7686

0.0011

6.6994

8.4098

0.9680

0.7525

0.0049

2.1862

9.1360

6.3904

0.0811

74.181

211.77

4.3122

2.4197

0.7625

Mean

Min

Std

Min

Max

RMSE

RMSE

Mean

MAM T2FLS+Tune

MAM IT2FLS+Tune

7.7412

11.187

17.891

0.2562

344.87

434.01

7.9753

3.4900

1.8473

Max

2.1533

2.2697

6.1506

0.0645

86.081

171.68

2.3341

0.7741

0.6045

Std

2.0278

5.4174

0.7698

0.0021

6.7104

8.4007

0.9701

0.7517

0.0052

Min

RMSE

7.7367

9.1352

6.3997

0.0806

74.201

211.19

4.3178

2.4183

0.7619

Mean

MAM IT3FLS+Tune

11.628

11.094

17.794

0.2557

344.39

433.76

7.9788

3.4885

1.8462

Max

2.1529

2.2689

6.1511

0.0638

86.281

172.03

2.3355

0.7737

0.6041

Std

In this case, R M S E is used as a performance index and the bold values indicate the best values, the green is the best value from the worst ones in normal color among the algorithms

# Rules

Dataset

Table 4.5 Experimental results for the proposed MAM IT3FLS+Tuning and comparison with MAM IT2FLS+Tuning and MAM T2FLS+Tuning using RMSE

4.8 Summary 97

98

4 Interval Type-3 Fuzzy Logic Systems (IT3FLS)

pattern recognition, classification, time series prediction and diagnosis. We expect that in the future these techniques will be used for many real-world problems in the above mentioned areas.

References 1. Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper-Saddle River, NJ (2001) 2. Bezdek, J.C., Ehrlich, R., Full, W.: FCM: the fuzzy c-means clustering algorithm. Comput. Geosci. 10, 191–203 (1984) 3. Pal, N.R., Bezdek, J.C.: On cluster validity for the fuzzy c-means model. IEEE Trans. Fuzzy Syst. 3(3), 370–379 (1995) 4. Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE Trans. Pattern Anal. Mach. Intell 13(8), 841–847 (1991) 5. Yu, J., Cheng, Q., Huang, H.: Analysis of the weighting exponent in the fcm. IEEE Trans. Syst. Man Cybernetics-Part B 34(1), 634–639 (2004) 6. B. Rust, StRD Dataset Gauss3 (1996) 7. Cao, Y., Raise, A., Mohammadzadeh, A., et al.: Deep learned recurrent type-3 fuzzy system: application for renewable energy modeling/prediction. Energy Reports (2021) 8. Mathworks, Inc., Natick, Massachusetts, Matlab Release 2013b (2013) 9. Dheeru, D., Karra Taniskidou, E.: {UCI} machine learning repository, Univ. Calif. Irvine Sch. Inf. (2017)

Chapter 5

Conclusions of Type-3 Fuzzy Systems

In this chapter the conclusions about the presented work in the area of the type-3 fuzzy systems are presented. The basic concepts and methods for the type-2 fuzzy sets, membership functions, inference and fuzzy systems are proposed in this book. Also, possible future trends that we can envision based on the review of the research work done in this area are presented. It is well-known that designing optimal fuzzy systems is a difficult task, and this is especially true in the case of the type-2 fuzzy systems. In the case of designing the type-3 fuzzy systems the problem is more complicated due to a higher number of parameters to consider, and we envision the use of bio-inspired optimization techniques in the future for the optimal design of this sort of systems, as it has already been the case for the type-1 and type-2 fuzzy systems. The main idea of this work is to introduce in a systematic way the concepts and methods of the interval type-3 fuzzy systems that with their higher uncertainty handling capabilities can be expected to be able to solve in a better way more difficult problems. One of the main aspects of the focus of this work is the defintion of the Footprint of Uncertainty (FOU) based on data (in many works the FOU is arbitrary or results from expert experience), and in this point we proposed different alternatives for obtaining of the FOU. To this end, the principle of justifiable granularity is proposed. Some of the contributions of the work comprise the definitions of the mathematical concepts and operations of the type-3 fuzzy sets, relations and systems, in addition to showing examples to illustrate these concepts and operations. In particular, the membership functions and their footprints of uncertainty are presented. In addition, the problem of designing the Interval type-3 fuzzy systems based on data is presented and solved at the end of Chap. 4. As future works, we consider that it will be interesting to build type-3 fuzzy systems for specific problems in different areas of application, such as intelligent control, robotics, pattern recognition, time series prediction, and diagnosis.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0_5

99

100

5 Conclusions of Type-3 Fuzzy Systems

Also, it would be intersting and useful to explore hybrid approaches with modular architectures or hierarchical approaches, for example type-3 fuzzy neural networkss. There are also bio-inspired or nature-inspired techniques that at the moment have not been applied to the optimization of type-3 fuzzy systems that may be worth mentioning. For example, membrane computing, harmony computing, electromagnetism based computing, and other similar approaches have not been so far applied to the optimization of the type-3 fuzzy systems. It is expected that these approaches and similar ones could be applied in the near future in the area of type-3 fuzzy system optimization. In addition, another fruitful area of research has been the use of fuzzy systems (type-1 and type-2 ones) for dynamic parameter adaptation in metaheuristics, and we expect that a natural extension will be to explore the use of the type-3 fuzzy systems in this area, possibly achieving even better optimization results for the metaheuristics. Finally, on the theoretical side, as this book has only considered the interval type-3 fuzzy systems, and we also expect the extension of the theory and applications to the general type-3 fuzzy systems.

Index

A Adaptive neuro-fuzzy inference systems, 82 A-planes, 75–77, 79, 80, 84, 89 Antecedent, 6, 7, 9, 42, 45–47, 58, 66, 76, 79, 84, 89, 93

C Centroid, 9, 60, 61, 68 Complement, 30, 33, 38, 40 Consequents, 6, 42, 45, 66, 79 Core, 83 Cuts horizontal, 28, 29, 37, 38, 75 vertical, 21, 22, 25, 26, 28–33, 40, 42

Type-3, 1, 2, 10, 13, 40, 41, 45, 56, 68

L Lower membership function, 5, 14, 19, 20, 28–30

M Meet operation, 32, 33 Membership functions Gaussian, 21, –27, 84, 89 generalized, 93 triangular, 83

D Defuzzification, 55, 83

N Non-Singleton Type 3 FLS, 45, 48, 49, 56, 93

E Extension principle, 30, 39

O Operations of Type 3 Fuzzy Systems, 99

F Footprint of uncertainty, 5, 19–21, 83, 93, 99 Fuzzy inference system Mamdani, 45, 46, 48, 56, 57, 59, 75, 83, 85, 93 Sugeno, 66, 69, 71, 79, 83, 89, 93 Fuzzy relations, 39, 46, 48 Fuzzy system Type-2, 1, 2, 5, 8, 75

P Parametrization, 20, 21, 25 Primary membership, 5, 14, 19, 29 Principle extension, 39 of justifiable granularity, 82, 83, 99

S Secondary membership, 5, 14, 18, 30, 39

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 O. Castillo et al., Interval Type-3 Fuzzy Systems: Theory and Design, Studies in Fuzziness and Soft Computing 418, https://doi.org/10.1007/978-3-030-96515-0

101

102 Support, 19, 20, 83

U Union, 8, 14, 16, 18, 28–31, 38, 47, 48, 58 Upper membership function, 5, 14, 15, 19, 20, 29, 30, 93

Index V Vertical cut, 16, 21, 25, 26, 40 Vertical-slices, 16, 28

W Weighted average, 67, 81