Type-2 fuzzy sets made simple
TL;DR: Establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, and presenting a new representation for type- 2 fuzzy sets, and using this new representation to derive formulas for union, intersection and complement of type-1 fuzzy sets without having to use the Extension Principle.
Abstract: Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems. However, they are difficult to understand for a variety of reasons which we enunciate. In this paper, we strive to overcome the difficulties by: (1) establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, (2) presenting a new representation for type-2 fuzzy sets, and (3) using this new representation to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use the Extension Principle.
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TL;DR: This paper demonstrates that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics.
Abstract: To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way
1,892 citations
Cites background from "Type-2 fuzzy sets made simple"
...This representation, which makes very heavy use of embedded IT2 FSs (Definition 8), was first presented in [19] for an arbitrary T2 FS, and is the bedrock for the rest of this paper....
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...Comment 2: A detailed proof of this theorem appears in [19]....
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TL;DR: A novel reactive control architecture for autonomous mobile robots that is based ontype-2 FLC to implement the basic navigation behaviors and the coordination between these behaviors to produce a type-2 hierarchical FLC is presented.
Abstract: Autonomous mobile robots navigating in changing and dynamic unstructured environments like the outdoor environments need to cope with large amounts of uncertainties that are inherent of natural environments. The traditional type-1 fuzzy logic controller (FLC) using precise type-1 fuzzy sets cannot fully handle such uncertainties. A type-2 FLC using type-2 fuzzy sets can handle such uncertainties to produce a better performance. In this paper, we present a novel reactive control architecture for autonomous mobile robots that is based on type-2 FLC to implement the basic navigation behaviors and the coordination between these behaviors to produce a type-2 hierarchical FLC. In our experiments, we implemented this type-2 architecture in different types of mobile robots navigating in indoor and outdoor unstructured and challenging environments. The type-2-based control system dealt with the uncertainties facing mobile robots in unstructured environments and resulted in a very good performance that outperformed the type-1-based control system while achieving a significant rule reduction compared to the type-1 system.
980 citations
Cites background from "Type-2 fuzzy sets made simple"
...The membership functions of type-2 fuzzy sets are three dimensional and include a footprint of uncertainty, it is the new third dimension of type-2 fuzzy sets and the footprint of uncertainty that provide additional degrees of freedom that make it possible to directly model and handle uncertainties…...
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...Finally, conclusions are presented in Section VII....
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...…dry ground may be different from “fast speed” on a rainy day with muddy ground as wheels may slip, etc. • Linguistic uncertainties as the meaning of words that are used in the antecedents and consequents linguistic labels can be uncertain as words mean different things to different people [18]....
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TL;DR: It is shown that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers, and the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction is looked at.
Abstract: We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π-i numbers. We investigate operations that are closed under Π-i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π-i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.
953 citations
Book•
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01 Jan 2006
TL;DR: The aim of the BSN is to provide a truly personalised monitoring platform that is pervasive, intelligent, and effective in patients with chronic diseases such as heart disease.
Abstract: Some patients, especially patients with chronic diseases such as heart disease, require continuous monitoring of their condition. Wearable devices for patient monitoring were introduced many years ago – for instance, wearable ambulatory ECG (electrocardiogram) recorders commonly known as Holter monitors (Figure 1) are used for monitoring cardiac patients. However, these monitors are quite bulky and can only record the signal for a limited time. Patients are often asked to wear a Holter monitor for a few days and then return to the clinic for diagnosis. This often overlooks transient but life-threatening events. In addition, we don’t know under what condition the signals are acquired, and this often leads to false alarms. For example, a sudden rise in heart rate may be caused by emotion, such as watching a horror movie, or by exercise, rather than by a heart condition. Body sensing To address these issues, the concept of Body Sensor Networks (BSN) was first proposed in 2002 by Prof. Guang-Zhong Yang from Imperial College London. The aim of the BSN is to provide a truly personalised monitoring platform that is pervasive, intelligent, and Figure 1 A patient wearing a Holter monitor
866 citations
Cites background or methods from "Type-2 fuzzy sets made simple"
...In case of implantable sensors this however, will only succeed if the tissue reaction is comparable in both sensors [85, 87]....
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...One of the main indications for an ICD is sudden cardiac death, which affects approximately 100,000 people annually in the UK, demonstrating the size of the patient population that may benefit from this device [87]....
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...Kapton-based K and pH flexible microelectrode ISE arrays have been described by Buck [87, 88] and have been used to record on a beating heart during ischaemia....
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TL;DR: This paper provides an introduction to and an overview of type-2 fuzzy sets (T2 FS) and systems by answering the following questions: What is a T2 FS and how is it different from a T1 FS.
Abstract: This paper provides an introduction to and an overview of type-2 fuzzy sets (T2 FS) and systems. It does this by answering the following questions: What is a T2 FS and how is it different from a T1 FS? Is there new terminology for a T2 FS? Are there important representations of a T2 FS and, if so, why are they important? How and why are T2 FSs used in a rule-based system? What are the detailed computations for an interval T2 fuzzy logic system (IT2 FLS) and are they easy to understand? Is it possible to have an IT2 FLS without type reduction? How do we wrap this up and where can we go to learn more?
802 citations
References
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TL;DR: Much of what constitutes the core of scientific knowledge may be regarded as a reservoir of concepts and techniques which can be drawn upon to construct mathematical models of various types of systems and thereby yield quantitative information concerning their behavior.
12,530 citations
"Type-2 fuzzy sets made simple" refers background or methods in this paper
...Then, Zadeh’s Extension Principle [ 44 ] allows us to induce from the type-1 fuzzy sets a type-1 fuzzy set on , through , i.e., , such that...
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...[ 44 ] introduced the concept of a type-2 fuzzy set as an extension of an ordinary fuzzy set, i.e., a type-1 fuzzy set....
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...... nature of type-2 fuzzy sets makes them very difficult to draw; (2) there is no simple collection of well-defined terms that let us effectively communicate about type-2 fuzzy sets, and to then be mathematically precise about them (terms do exist but have not been precisely defined2 ); (3) derivations of the formulas for the union, intersection, and complement of type-2 fuzzy sets all rely on using Zadeh’s Extension Principle [ 44 ], which in ......
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Book•
01 Jan 2011
TL;DR: This book effectively constitutes a detailed annotated bibliography in quasitextbook style of the some thousand contributions deemed by Messrs. Dubois and Prade to belong to the area of fuzzy set theory and its applications or interactions in a wide spectrum of scientific disciplines.
Abstract: (1982). Fuzzy Sets and Systems — Theory and Applications. Journal of the Operational Research Society: Vol. 33, No. 2, pp. 198-198.
5,861 citations
Book•
01 Jan 2001
TL;DR: This chapter discusses Type-2 Fuzzy Sets, a New Direction for FLSs, and Relations and Compositions on different Product Spaces on Different Product Spaces, as well as operations on and Properties of Type-1 Non-Singleton Type- 2 FuzzY Sets.
Abstract: (NOTE: Each chapter concludes with Exercises.) I: PRELIMINARIES. 1. Introduction. Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation. Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic. Primer on Fuzzy Sets. Primer on FL. Remarks. 2. Sources of Uncertainty. Uncertainties in a FLS. Words Mean Different Things to Different People. 3. Membership Functions and Uncertainty. Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation. 4. Case Studies. Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys. II: TYPE-1 FUZZY LOGIC SYSTEMS. 5. Singleton Type-1 Fuzzy Logic Systems: No Uncertainties. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Defuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation. 6. Non-Singleton Type-1 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation. III: TYPE-2 FUZZY SETS. 7. Operations on and Properties of Type-2 Fuzzy Sets. Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation. 8. Type-2 Relations and Compositions. Introduction. Relations in General. Relations and Compositions on the Same Product Space. Relations and Compositions on Different Product Spaces. Composition of a Set with a Relation. Cartesian Product of Fuzzy Sets. Implications. 9. Centroid of a Type-2 Fuzzy Set: Type-Reduction. Introduction. General Results for the Centroid. Generalized Centroid for Interval Type-2 Fuzzy Sets. Centroid of an Interval Type-2 Fuzzy Set. Type-Reduction: General Results. Type-Reduction: Interval Sets. Concluding Remark. Computation. IV: TYPE-2 FUZZY LOGIC SYSTEMS. 10. Singleton Type-2 Fuzzy Logic Systems. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. Defuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation. 11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation. 12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation. 13. TSK Fuzzy Logic Systems. Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation. 14. Epilogue. Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS. A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets. Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation. B. Properties of Type-1 and Type-2 Fuzzy Sets. Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets. C. Computation. Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs. References. Index.
2,555 citations
"Type-2 fuzzy sets made simple" refers background in this paper
...Type-2 Fuzzy Sets Made Simple Jerry M. Mendel and Robert I. Bob John Abstract—Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems....
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TL;DR: The usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle, and the practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis.
Abstract: A fuzzy number is a fuzzy subset of the real line whose highest membership values are clustered around a given real number called the mean value ; the membership function is monotonia on both sides of this mean value. In this paper, the usual algebraic operations on real numbers are extended to fuzzy numbers by the use of a fuzzification principle. The practical use of fuzzified operations is shown to be easy, requiring no more computation than when dealing with error intervals in classic tolerance analysis. The field of applications of this approach seems to be large, since it allows many known algorithms to be fitted to fuzzy data.
2,412 citations
"Type-2 fuzzy sets made simple" refers background in this paper
...It is now obvious, from the properties of a -norm, that: Lemma 1: Under the four properties just stated, is a -norm....
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TL;DR: An efficient and simplified method to compute the input and antecedent operations for interval type-2 FLSs: one that is based on a general inference formula for them is proposed.
Abstract: We present the theory and design of interval type-2 fuzzy logic systems (FLSs). We propose an efficient and simplified method to compute the input and antecedent operations for interval type-2 FLSs: one that is based on a general inference formula for them. We introduce the concept of upper and lower membership functions (MFs) and illustrate our efficient inference method for the case of Gaussian primary MFs. We also propose a method for designing an interval type-2 FLS in which we tune its parameters. Finally, we design type-2 FLSs to perform time-series forecasting when a nonstationary time-series is corrupted by additive noise where SNR is uncertain and demonstrate an improved performance over type-1 FLSs.
1,845 citations