Towards an efficient type-reduction method for interval type-2 fuzzy logic systems
01 Jun 2008-pp 1425-1432
TL;DR: Results from a simulated coupled tank experiment demonstrated that IT2 FLCs that employ the proposed type reduction algorithm share similar robustness properties as FLC's based on the Karnik-Mendel type reducer.
Abstract: This paper introduces an alternative type-reduction method for interval type-2 (IT2) fuzzy logic systems (FLSs), with either continuous or discrete secondary membership function. Unlike the Karnik-Mendel type reducer which is based on the wavy-slice representation of a type-2 fuzzy set, the proposed type reduction algorithm is developed using the vertical-slice representation. One advantage of the approach is the output of the type reducer can be expressed in closed form, thereby providing a tool for the theoretical analysis of IT2 FLSs. The computational complexity of the proposed method is also lower than the uncertainty bounds method and the enhanced Karnik-Mendel method. To assess the feasibility of the proposed type-reducer, it is used to calculate the output of an IT2 fuzzy logic controller (FLCs). Results from a simulated coupled tank experiment demonstrated that IT2 FLCs that employ the proposed type reduction algorithm share similar robustness properties as FLCs based on the Karnik-Mendel type reducer.
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TL;DR: This paper explains two fundamental differences between IT2 and T1 FLCs: Adaptiveness and Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval.
Abstract: Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.
253 citations
TL;DR: This tutorial paper explains four different mathematical representations for general type-2 fuzzy sets (GT2 FS) and demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious.
Abstract: The purpose of this tutorial paper is to make general type-2 fuzzy logic systems (GT2 FLSs) more accessible to fuzzy logic researchers and practitioners, and to expedite their research, designs, and use. To accomplish this, the paper 1) explains four different mathematical representations for general type-2 fuzzy sets (GT2 FSs); 2) demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious; 3) shows how to obtain set theoretic and other operations for GT2 FSs using type-1 (T1) FS mathematics (α- cuts play a central role); 4) reviews Mamdani and TSK interval type-2 (IT2) FLSs so that their mathematical operations can be easily used in a GT2 FLS; 5) provides all of the formulas that describe both Mamdani and TSK GT2 FLSs; 6) explains why center-of sets type-reduction should be favored for a GT2 FLS over centroid type-reduction; 7) provides three simplified GT2 FLSs (two are for Mamdani GT2 FLSs and one is for a TSK GT2 FLS), all of which bypass type reduction and are generalizations from their IT2 FLS counterparts to GT2 FLSs; 8) explains why gradient-based optimization should not be used to optimally design a GT2 FLS; 9) explains how derivative-free optimization algorithms can be used to optimally design a GT2 FLS; and 10) provides a three-step approach for optimally designing FLSs in a progressive manner, from T1 to IT2 to GT2, each of which uses a quantum particle swarm optimization algorithm, by virtue of which the performance for the IT2 FLS cannot be worse than that of the T1 FLS, and the performance for the GT2 FLS cannot be worse than that of the IT2 FLS.
238 citations
Additional excerpts
...1) Interval Type-2 Fuzzy Logic System Results: Nie and Tan [54] defuzzify an IT2 FS B̃ by computing the COG of the average of its lower and upper MFs, i....
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...Such an IT2 FS is not computed for COS TR. 2) Proposed General Type-2 Fuzzy Logic System Results: The gist of how the Nie–Tan (NT) simplification can be used for a GT2 FLS is as follows: 1) For each of the M rules, compute its firing interval Fsα (x ′) for level α, exactly as in (50)....
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...1) Interval Type-2 Fuzzy Logic System Results: Nie and Tan [54] defuzzify an IT2 FS B̃ by computing the COG of the average of its lower and upper MFs, i.e., yNT(x′) = COG { 1 2 [ μ B̃ (y|x′) + μ̄B̃ (y|x′) ]} = ∑N i=1 yi [ μ B̃ (yi |x′) + μ̄B̃ (yi |x′) ] ∑N i=1 [ μ B̃ (yi |x′) + μ̄B̃ (yi |x′) ] (76) Recently, Mendel and Liu [45] have proven that yNT(x′) is a first-order approximation to the actual defuzzified value of B̃,mB̃ (x ′), where mB̃ (x ′) = [cl(B̃|x′) + cr (B̃|x′)]/2 (77) and cl(B̃|x′) and cr (B̃|x′) are the left and right end-points of the actual centroid....
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TL;DR: An overview and comparison of three categories of methods to reduce their computational cost will help researchers and practitioners on IT2 FLSs choose the most suitable structure and type-reduction algorithms, from a computational cost perspective.
Abstract: Interval type-2 fuzzy logic systems (IT2 FLSs) have demonstrated better abilities to handle uncertainties than their type-1 (T1) counterparts in many applications; however, the high computational cost of the iterative Karnik-Mendel (KM) algorithms in type-reduction means that it is more expensive to deploy IT2 FLSs, which may hinder them from certain cost-sensitive real-world applications. This paper provides a comprehensive overview and comparison of three categories of methods to reduce their computational cost. The first category consists of five enhancements to the KM algorithms, which are the most popular type-reduction algorithms to date. The second category consists of 11 alternative type-reducers, which have closed-form representations and, hence, are more convenient for analysis. The third category consists of a simplified structure for IT2 FLSs, which can be combined with any algorithms in the first or second category for further computational cost reduction. Experiments demonstrate that almost all methods in these three categories are faster than the KM algorithms. This overview and comparison will help researchers and practitioners on IT2 FLSs choose the most suitable structure and type-reduction algorithms, from a computational cost perspective. A recommendation is given in the conclusion.
225 citations
Cites background or methods from "Towards an efficient type-reduction..."
...Eleven of them [2], [8], [10], [13], [16], [27], [28], [37], [44], [56], [61] are introduced and compared in this section....
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...Observe that the Nie–Tan (NT) method does not require {yn} to be sorted, and it is a special case of the WT method when hni (x) = 0.5....
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...Nie and Tan [37] proposed another closed-form TR and defuzzification method, where the output of an IT2 FLS is computed as...
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...F. Nie–Tan Method Nie and Tan [37] proposed another closed-form TR and defuzzification method, where the output of an IT2 FLS is computed as y = ∑N n=1 y n (fn + f n ) ∑N n=1(f n + f n ) ....
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...D. Wu–Tan Method Wu and Tan [56] proposed a closed-form TR and defuzzification method by making use of the equivalent T1 membership grades [57]....
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27 Jun 2011
TL;DR: This paper proposes a new algorithm which improves over the latest results and is the most efficient one to use in practice, when the number of elements in type-reduction is smaller than 100, which is true in most practical type- reduction computations.
Abstract: Type-reduction algorithms are very important for type-2 fuzzy sets and systems. The earliest one, and also the most popular one, is the Karnik-Mendel Algorithm, which is iterative and computationally intensive. In the last a few years researchers have proposed several other more efficient type-reduction algorithms. In this paper we also propose a new algorithm which improves over the latest results. Experiments show that it is the most efficient one to use in practice. Particularly, when the number of elements in type-reduction is smaller than 100, which is true in most practical type-reduction computations, our proposed algorithm can save over 50% computational cost over the Karnik-Mendel Algorithms. We also give the Matlab implementation of our most efficient algorithm in the Appendix. It includes preprocessing steps to eliminate numerical problems, and also improved testing criteria to prevent possible infinite loops. This program will be very helpful in promoting the popularity of type-2 fuzzy sets and systems.
199 citations
TL;DR: Some theoretical analyses of the Nie-Tan direct defuzzification method are provided and it is suggested that the NT method is a very good way to simplify an interval type-2 fuzzy set.
Abstract: Type reduction (TR) followed by defuzzification is commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Because of the iterative nature of TR, it may be a computational bottleneck for the real-time applications of an IT2 FLS. This has led to many direct approaches to defuzzification that bypass TR, the simplest of which is the Nie-Tan direct defuzzification method (NT method). This paper provides some theoretical analyses of the NT method that answer the question “Why is the NT method good to use?” This paper also provides a direct relationship between TR followed by defuzzification (using KM algorithms) and the NT method. It also provides an improved NT method. Numerical examples illustrate our theoretical results and suggest that the NT method is a very good way to simplify an interval type-2 fuzzy set.
174 citations
Cites background or methods from "Towards an efficient type-reduction..."
...Index Terms—Defuzzification, interval type-2 fuzzy set (IT2 FS), Karnik–Mendel (KM) algorithms, Nie–Tan (NT) method, type reduction (TR)....
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...No applications have been given to support our recommendation in this paper, because this has already been provided in [27], which is the main motivator for this paper....
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...C. Nie–Tan Method In the NT method, one first computes the average, i.e., ci , of the LMF and UMF of ˜A at each xi , namely ci = 1 2 ( μ ˜A (xi) + μ ˜A (xi) ) , i = 1, . . . , N. (17) Each ci is a spike that is located at x = xi ....
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...Wu and Tan [31] proposed a closed-form TR and defuzzification method (referred to below as WT) that makes use of equivalent T1 membership grades....
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...Section III provides theoretical analyses of the NT method, establishes a new and fundamental relation between KM + defuzzification and the NT method, puts forward an improved NT (INT) method, and provides the computational 1A reviewer of this paper pointed out that Nie and Tan refer to this T1 FS as a type-reduced set, and therefore, their approach does not bypass TR....
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References
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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
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.
2,382 citations
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
Additional excerpts
...and ability to handle uncertainties [2]....
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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
"Towards an efficient type-reduction..." refers methods in this paper
...As vertical-slice representation is useful for computation [7], it is the inspiration for the alternative type-reduction strategy....
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TL;DR: This work derives inner- and outer-bound sets for the type-reduced set of an interval type-2 fuzzy logic system (FLS), based on a new mathematical interpretation of the Karnik-Mendel iterative procedure for computing thetype-reducing set, and demonstrates that the resulting system can operate without type- Reduction and can achieve similar performance to one that uses type- reduction.
Abstract: We derive inner- and outer-bound sets for the type-reduced set of an interval type-2 fuzzy logic system (FLS), based on a new mathematical interpretation of the Karnik-Mendel iterative procedure for computing the type-reduced set. The bound sets can not only provide estimates about the uncertainty contained in the output of an interval type-2 FLS, but can also be used to design an interval type-2 FLS. We demonstrate, by means of a simulation experiment, that the resulting system can operate without type-reduction and can achieve similar performance to one that uses type-reduction. Therefore, our new design method, based on the bound sets, can relieve the computation burden of an interval type-2 FLS during its operation, which makes an interval type-2 FLS useful for real-time applications.
506 citations