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This paper is about explainable AI (XAI) for rule-based fuzzy systems [that can be expressed generically, as y(x) = f(x)]. It explains why it is not valid to explain the output of Mamdani or TSK rule-based fuzzy systems using IF-THEN rules, and why it is valid to explain the output of such rule-based fuzzy systems as an association of the compound antecedents of a small subset of the original larger set of rules, using a phrase such as These linguistic antecedents are symptomatic of this output. Importantly, it provides a novel multi-step approach to obtain such a small subset of rules for three kinds of fuzzy systems, and illustrates it by means of a very comprehensive example. It also explains why the choice for antecedent membership function shapes may be more critical for XAI than before XAI, why Linguistic Approximation and similarity are essential for XAI, and, it provides a way to estimate the quality of the explanations.

This article aims to provide a new perspective on how the deployment of general type-2 (GT2) fuzzy sets affects the mapping of a class of fuzzy logic controllers (FLCs). It is shown that an α -plane represented a GT2-FLC is easily designed via baseline type-1 and interval type-2 FLCs and two design parameters (DPs). The DPs are the total number of α planes and the tuning parameter of the secondary membership function that are interpreted as sensitivity and shape DPs, respectively. We provide a clear understanding and interpretation of the sensitivity and shape DPs on controller performance through various comparative analyses. We present design approaches on how to tune the shape DP by providing a tradeoff between robustness and performance. We also propose two online scheduling mechanisms to tune the shape DP. We explore the effect of the sensitivity DP on the GT2-FLC and provide practical insights on how to tune the sensitivity DP. We present an algorithm for tuning the sensitivity DP that provides a compromise between computational time and sensitivity. We validate our analyses, interpretations, and design methods with experimental results conducted on a drone. We believe that this article provides clear explanations on the role of DPs on the performance, robustness, sensitivity, and computational time of GT2-FLCs.

This paper provides a novel and better understanding of the performance potential of a nonsingleton (NS) fuzzy system over a singleton (S) fuzzy system. It is done by extending sculpting the state space works from S to NS fuzzification and demonstrating uncertainties about measurements, modeled by NS fuzzification: first, fire more rules more often, manifested by a reduction (increase) in the sizes of first-order rule partitions for those partitions associated with the firing of a smaller (larger) number of rules—the coarse sculpting of the state space; second, this may lead to an increase or decrease in the number of type-1 (T1) and interval type-2 (IT2) first-order rule partitions, which now contain rule pairs that can never occur for S fuzzification—a new rule crossover phenomenon —discovered using partition theory; and third, it may lead to a decrease, the same number, or an increase in the number of second-order rule partitions, all of which are system dependent—the fine sculpting of the state space. The authors' conjecture is that it is the additional control of the coarse sculpting of the state space, accomplished by prefiltering and the max–min (or max-product) composition, which provides an NS T1 or IT2 fuzzy system with the potential to outperform an S T1 or IT2 system when measurements are uncertain.

Present-day handheld battery-enabled devices such as smartphones and tablets attract rich user experience but are often criticized for their short battery lives. Battery life is a subjective term and depends on a user's perceptions. A novel work to achieve power optimization for these devices, according to users’ perceptions, was the design of user-satisfaction-aware power management approach, perceptual computer power management approach (Per-C PMA). But we have found that the design of Per-C PMA requires collection of data intervals from a group of subjects. This limits the practical viability of Per-C PMA for highly personal handheld battery-enabled devices such as smartphones and tablets. So, here we propose a user-satisfaction-aware PMA called Per-C for Personalized Power Management Approach or “Per-C PPMA,” one that achieves significant reductions in power consumption compared to existing PMAs and noticeable improvements in the overall user satisfaction. Per-C PPMA uses the mathematical technique of person footprint of uncertainty (FOU) to process users’ linguistic opinions. Person FOU can either use an interval approach (IA) or Hao–Mendel approach (HMA) for data processing. The recommendations generated using IA and HMA are the same. However, IA takes a much higher computational time than HMA, even though both have the same asymptotic complexity of $\boldsymbol{O}({\boldsymbol{w}*\boldsymbol{n}})$ . We strongly believe that Per-C PPMA is a novel technique and our work is the first such application of Person FOU on any hardware platform. An important outcome of this study is a ready-to-use mobile app “Per-C PPMA” (currently freely available on the website http://www.sau.int/∼cilab/ ).

The similarity between two fuzzy sets (FSs) is an important concept in fuzzy logic. As the research interest on general type-2 (GT2) FSs has increased recently, many similarity measures for them have also been proposed. This paper gives a comprehensive overview of existing similarity measures for GT2 FSs, points out their limitations, and, by using an intuitive geometric explanation, proposes a Jaccard similarity measure for GT2 FSs that is an extension of the popular Jaccard similarity measure for type-1 and interval type-2 FSs. The fundamental difference between the proposed Jaccard similarity measure for GT2 FSs and all existing similarity measures is that the Jaccard similarity measure considers the overall geometries of two GT2 FSs and does not depend on a specific representation of the GT2 FSs, whereas all existing similarity measures for GT2 FSs depend either on the vertical slice representation or the $\alpha$ -plane representation. We show that the Jaccard similarity measure for GT2 FSs satisfies four properties of a similarity measure and demonstrate its reasonableness using two examples.

This paper provides some new and novel application-independent perspectives on why improved performance usually occurs as one goes from crisp, to type-1 (T1), and to interval type-2 (IT2) fuzzy systems, by introducing three kinds of partitions: (1) Uncertainty partitions that let us distinguish T1 fuzzy sets from crisp sets, and IT2 fuzzy sets from T1 fuzzy sets; (2) First-and second-order rule partitions that are direct results of uncertainty partitions, and are associated with the number of rules that fire in different regions of the state space, and, the changes in their mathematical formulae within those regions; and (3) Novelty partitions that can only occur in an IT2 fuzzy system that uses type-reduction. Rule and novelty partitions sculpt the state space into hyperrectangles within each of which resides a different nonlinear function. It is the author's conjecture that the greater sculpting of the state space by a T1 fuzzy system lets it outperform a crisp system, and the even greater sculpting of the state space by an IT2 fuzzy system lets it outperform a T1 fuzzy system. The latter can occur even when the T1 and IT2 fuzzy systems are described by the same number of parameters.

Present day portable devices such as laptops, smartphones, etc., offer their users fastest processors, advanced operating systems, and numerous applications. However, a large section of the users are critical to the available battery capacity and its lifetime. This is because performance of the battery and its lifetime as perceived by the users are quite subjective in nature. It depends directly on user satisfactions, which are usually expressed in terms of words. So, in this paper, we propose a user-satisfaction-aware energy management approach, called “perceptual computer power management approach (Per-C PMA),” based on the technique of perceptual computing. At the heart of our technique is the perceptual computer that processes the linguistic input of the users to aid in the selection of a suitable processor frequency, which plays a significant role in the overall energy consumption of the systems. The Per-C PMA minimizes the energy consumption, while still keeping the user satisfied with the perceived system performance. The Per-C PMA achieves 1) reductions of 42.26% and 10.84% in power consumption, and 2) improvements in the overall satisfaction ratings of 16% and 10%, when compared to other existing power-saving schemes such as ON-DEMAND and human and application-driven frequency scaling for processor power efficiency, respectively. Per-C PMA is the first such application of Per-C on any hardware platform. It is implemented as Ubuntu scripts for end users and can be downloaded from: http://sau.ac.in/∼cilab/ . We have also provided the MATLAB files so that interested researchers can use it in their research. For the ease of the users, the Ubuntu scripts and the MATLAB codes are given in the graphical user interface mode; a demo video on how to use the software is also provided on the webpage.

This paper focuses on an approach, called the HM Approach (HMA), to determine (for the first time) a normal interval type-2 fuzzy set model for a word that uses interval data about a word that are collected either from a group of subjects or from one subject. The HMA has two parts: 1) Data part, which is the same as the Data Part of the enhanced interval approach (EIA) [44] , and 2) Fuzzy Set Part, which is very different from the second part of the EIA, the most notable difference being that in the HMA, the common overlap of subject data intervals is interpreted to indicate agreement by all of the subjects for that overlap, and therefore, a membership grade of 1 is assigned to the common overlap. Another difference between the HMA and EIA is the way in which data intervals are collectively classified into either a Left-shoulder, Interior, or Right-shoulder footprint of uncertainty. The HMA does this more simply than does the EIA, and requires fewer probability assumptions about the intervals than does the EIA.

We propose a new mathematical framework for the evolution and propagation of opinions, called fuzzy opinion network, which is the connection of a number of Gaussian nodes, possibly through some weighted average, time delay, or logic operators, where a Gaussian node is a Gaussian fuzzy set with the center and the standard deviation being the node inputs and the fuzzy set itself being the node output. In this framework, an opinion is modeled as a Gaussian fuzzy set with the center representing the opinion itself and the standard deviation characterizing the uncertainty about the opinion. We study the basic connections of fuzzy opinion networks, including basic center, basic standard deviation (sdv), basic center-sdv, chain-in-center, and chain-in-sdv connections, and we analyze a number of dynamic connections to show how opinions and their uncertainties propagate and evolve across different network structures and scenarios. We explain what insights we might gain from these mathematical results about the formation and evolution of human opinions.

This letter provides a critical review of “A New Look at Type-2 Fuzzy Sets and Type-2 Fuzzy Logic Systems” IEEE Trans. Fuzzy Systems , and debunks its four claims.

This paper explains how to compute normalized interval type-2 fuzzy sets in closed form and explains how the results reduce to well-known results for type-1 fuzzy sets and interval sets. Such normalized interval type-2 fuzzy sets may be needed in linguistic probability computa- tions or multiple criteria decision analysis under uncertainty. Index Terms—Fuzzy weighted average (FWA), interval type-2 fuzzy sets (IT2 FSs), interval weighted average (IWA), linguistic probability, linguistic weighted average (LWA), normalized interval type-2 fuzzy sets.

In this paper, we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve Advanced Computing with Words (ACWW) problems. Such problems involve implicit assignments of linguistic truth, probability, and possibility. To begin, we establish the vocabularies of the words involved in the problems, and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the Interval Approach (IA) or the Enhanced Interval Approach (EIA). Next, the solutions of the ACWW problems, which involve the fuzzy set models of the words, are formulated using the Generalized Extension Principle. Because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically, we then develop a numerical method for their solution. Finally, the resulting fuzzy set solutions are decoded into natural language words using Jaccard's similarity measure. We explain how ACWW problems can solve some potential prototype engineering problems and connect the methodology of this paper with Perceptual Computing.

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.

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.

Computing the centroid and performing type-reduction for type-2 fuzzy sets and systems are operations that must be taken into consideration. Karnik-Mendel (KM) algorithms are the standard ways to do these operations; however, because these algorithms are iterative, much research has been conducted during the past decade about centroid and type-reduction computations. This tutorial paper focuses on the research that has been conducted to 1) improve the KM algorithms; 2) understand the KM algorithms, leading to further improved algorithms; 3) eliminate the need for KM algorithms; 4) use the KM algorithms to solve other (nonfuzzy logic system) problems; and 5) use (or not use) KM algorithms for general type-2 fuzzy sets and fuzzy logic systems.

Based on a new continuous Karnik-Mendel (KM) algorithm expression, this paper proves that the centroid computation of an interval type-2 fuzzy set using KM algorithms is equivalent to the Newton-Raphson method in root-finding, which reveals the mechanisms in KM algorithm computation. The theoretical results of KM algorithms are re-obtained. Different from current KM algorithms, centroid computation methods that use different root-finding routines are provided. Such centroid computation methods can obtain the exact solution and are different from the current approximate methods using sampled data. Further improvements and analysis of the centroid problem using root-finding and integral computation techniques are also possible.

Previous studies have shown that the centroid of a general type-2 fuzzy set (T2 FS) A can be obtained by taking the union of the centroids of all the α-planes (each raised to level α) of A. Karnik-Mendel (KM) or the enhanced KM (EKM) algorithms are used to compute the centroid of each α-plane. The iterative features in KM/EKM algorithms can be time-consuming, especially when the algorithms have to be repeated for many α-planes. This paper proposes a new method named centroid-flow (CF) algorithm to compute the centroid of A without having to apply KM/EKM algorithms for every α-plane. Extensive simulations have shown that the CF algorithm can reduce the computation time by 75%-80 % and 50% -75%, compared with KM and EKM algorithms, respectively, and still maintains satisfactory computation accuracy for various T2 FSs when the primary variable x and α -plane are discretized finely enough.

Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. Many authors have used this technique to generate summaries like “Most senior workers have high salary,” which can be used to better understand and communicate about data; however, few of them have used it to generate IF-THEN rules like “IF X is large and Y is medium, THEN Z is small,” which not only facilitate understanding and communication of data but can also be used in decision-making. In this paper, an LS approach to generate IF-THEN rules for causal databases is proposed. Both type-1 and interval type-2 fuzzy sets are considered. Five quality measures-the degrees of truth, sufficient coverage, reliability, outlier, and simplicity-are defined. Among them, the degree of reliability is especially valuable for finding the most reliable and representative rules, and the degree of outlier can be used to identify outlier rules and data for close-up investigation. An improved parallel coordinates approach for visualizing the IF-THEN rules is also proposed. Experiments on two datasets demonstrate our LS and rule visualization approaches. Finally, the relationships between our LS approach and the Wang-Mendel (WM) method, perceptual reasoning, and granular computing are pointed out.

This paper studies the continuity of the input-output mappings of fuzzy logic systems (FLSs), including both type-1 (T1) and interval type-2 (IT2) FLSs. We show that a T1 FLS being an universal approximator is equivalent to saying that a T1 FLS has a continuous input-output mapping. We also derive the condition under which a T1 FLS is discontinuous. For IT2 FLSs, we consider six type-reduction and defuzzification methods (the Karnik-Mendel method, the uncertainty bound method, the Wu-Tan method, the Nie-Tan method, the Du-Ying method, and the Begian-Melek-Mendel method) and derive the conditions under which continuous and discontinuous input-output mappings can be obtained. Guidelines for designing continuous IT2 FLSs are also given. This paper is to date the most comprehensive study on the continuity of FLSs. Our results will be very useful in the selection of the parameters of the membership functions to achieve a desired continuity (e.g., for most traditional modeling and control applications) or discontinuity (e.g., for hybrid and switched systems modeling and control).

The seven papers in this special section provide an overview of the main ideas and research directions for computing with words.

The perceptual computer (Per-C) is an architecture that makes subjective judgments by computing with words (CWWs). This paper applies the Per-C to hierarchical decision making, which means decision making based on comparing the performance of competing alternatives, where each alternative is first evaluated based on hierarchical criteria and subcriteria, and then, these alternatives are compared to arrive at either a single winner or a subset of winners. What can make this challenging is that the inputs to the subcriteria and criteria can be numbers, intervals, type-1 fuzzy sets, or even words modeled by interval type-2 fuzzy sets. Novel weighted averages are proposed in this paper as a CWW engine in the Per-C to aggregate these diverse inputs. A missile-evaluation problem is used to illustrate it. The main advantages of our approaches are that diverse inputs can be aggregated, and uncertainties associated with these inputs can be preserved and are propagated into the final evaluation.

Perceptual reasoning (PR) is an approximate reasoning method that can be used as a computing-with-words (CWW) engine in perceptual computing. There can be different approaches to implement PR, e.g., firing-interval-based PR (FI-PR), which has been proposed in J. M. Mendel and D. Wu, IEEE Trans. Fuzzy Syst., vol. 16, no. 6, pp. 1550-1564, Dec. 2008 and similarity-based PR (S-PR), which is proposed in this paper. Both approaches satisfy the requirement on a CWW engine that the result of combining fired rules should lead to a footprint of uncertainty (FOU) that resembles the three kinds of FOUs in a CWW codebook. A comparative study shows that S-PR leads to output FOUs that resemble word FOUs, which are obtained from subject data, much more closely than FI-PR; hence, S-PR is a better choice for a CWW engine than FI-PR.

This paper 1) reviews the alpha-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the alpha-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using alpha-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each alpha-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 alpha -planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.

The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and superexponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that, on average, the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time. An additional (at least) 23% computational cost can be saved if no sorting of the inputs is needed.

This paper presents a very practical type-2-fuzzistics methodology for obtaining interval type-2 fuzzy set (IT2 FS) models for words, one that is called an interval approach (IA). The basic idea of the IA is to collect interval endpoint data for a word from a group of subjects, map each subject's data interval into a prespecified type-1 (T1) person membership function, interpret the latter as an embedded T1 FS of an IT2 FS, and obtain a mathematical model for the footprint of uncertainty (FOU) for the word from these T1 FSs. The IA consists of two parts: the data part and the FS part. In the data part, the interval endpoint data are preprocessed, after which data statistics are computed for the surviving data intervals. In the FS part, the data are used to decide whether the word should be modeled as an interior, left-shoulder, or right-shoulder FOU. Then, the parameters of the respective embedded T1 MFs are determined using the data statistics and uncertainty measures for the T1 FS models. The derived T1 MFs are aggregated using union leading to an FOU for a word, and finally, a mathematical model is obtained for the FOU. In order that all researchers can either duplicate our results or use them in their research, the raw data used for our codebook examples, as well as a MATLAB M-file for the IA, have been put on the Internet at: http://sipi.usc.edu/ ~ mendel.

In 1996, Zadeh proposed the paradigm of computing with words (CWW). A specific architecture for making subjective judgments using CWW was proposed by Mendel in 2001. It is called a Perceptual Computer (Per-C), and because words can mean different things to different people, it uses interval type-2 fuzzy set (IT2 FS) models for all words. The Per-C has three elements: the encoder, which transforms linguistic perceptions into IT2 FSs that activate a CWW engine; the decoder, which maps the output of a CWW engine back into a word; and the CWW engine. Although di-fferent kinds of CWW engines are possible, this paper only focuses on CWW engines that are rule-based and the computations that map its input IT2 FSs into its output IT2 FS. Five assumptions are made for a rule-based CWW engine, the most important of which is: The result of combining fired rules must lead to a footprint of uncertainty (FOU) that resembles the three kinds of FOU that have previously been shown to model words (interior, left-shoulder, and right-shoulder FOUs). Requiring this means that the output FOU from a rule-based CWW engine will look similar in shape to an FOU in a codebook (i.e., a vocabulary of words and their respective FOUs) for an application, so that the decoder can therefore sensibly establish the word most similar to the CWW engine output FOU. Because existing approximate reasoning methods do not satisfy this assumption, a new kind of rule-based CWW engine is proposed, one that is called Perceptual Reasoning, and is proved to always satisfy this assumption. Additionally, because all IT2 FSs in the rules as well as those that excite the rules are either an interior, left-shoulder, or right-shoulder FOU, it is possible to carry out the sup-min calculations that are required by the inference engine, and those calculations are also in this paper. The results in this paper let us implement a rule-based CWW engine for the Per-C.

The focus of this paper is the linguistic weighted average (LWA), where the weights are always words modeled as interval type-2 fuzzy sets (IT2 FSs), and the attributes may also (but do not have to) be words modeled as IT2 FSs; consequently, the output of the LWA is an IT2 FS. The LWA can be viewed as a generalization of the fuzzy weighted average (FWA) where the type-1 fuzzy inputs are replaced by IT2 FSs. This paper presents the theory, algorithms, and an application of the LWA. It is shown that finding the LWA can be decomposed into finding two FWAs. Since the LWA can model more uncertainties, it should have wide applications in distributed and hierarchical decision-making.

Interval type-2 fuzzy sets (IT2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). The centroid of an IT2 FS, which is an IT1 FS, provides a measure of the uncertainty in the IT2 FS. The main purpose of this paper is to quantify the centroid of a non-symmetric IT2 FS with respect to geometric properties of its FOU. This is very important because interval data collected from subjects about words suggests that the FOUs of most words are non-symmetrical. Using the results in this paper, it is possible to formulate and solve forward problems, i.e., to go from parametric non-symmetric IT2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems for non-symmetrical triangular, trapezoidal, Gaussian and shoulder FOUs.

KM algorithms are widely used to perform type-reduction and to compute the centroid of type-2 fuzzy sets. Because KM algorithms are iterative, there has been some concern about their convergance time. Many people have observed, from simulations, that convergence occurs rapidly, i.e. under 10 iterations; but, it is one thing to observe this in simulations and another thing to mathematically prove super-exponential convergence, as is done in this paper. So, by mathematically proving superexponential convergence, Mendel and Liu have provided the type- 2 fuzzy set community with concrete evidence of the speed of the KM Algorithms.

In this paper, we demonstrate, through the multicategory classification of battlefield ground vehicles using acoustic features, how it is straightforward to directly exploit the information inherent in a problem to determine the number of rules, and subsequently the architecture, of fuzzy logic rule-based classifiers (FLRBC). We propose three FLRBC architectures, one non-hierarchical and two hierarchical (HFLRBC), conduct experiments to evaluate the performances of these architectures, and compare them to a Bayesian classifier. Our experimental results show that: 1) for each classifier the performance in the adaptive mode that uses simple majority voting is much better than in the non-adaptive mode; 2) all FLRBCs perform substantially better than the Bayesian classifier; 3) interval type-2 (T2) FLRBCs perform better than their competing type-1 (T1) FLRBCs, although sometimes not by much; 4) the interval T2 nonhierarchical and HFLRBC-series architectures perform the best; and 5) all FLRBCs achieve higher than the acceptable 80% classification accuracy

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

We examine ten antecedent connector models in the framework of a singleton or nonsingleton fuzzy logic system (FLS), to establish which models can be used. In this work, a usable connector model must lead to a separable firing degree that is a closed-form and piecewise-differentiable function of the membership function parameters and also the parameter characterizing that connector model. Our analysis shows that: for a singleton FLS where the Mamdani-product or Mamdani-minimum implication method is used, all ten antecedent connector models are usable; for a nonsingleton FLS where the Mamdani-product implication method is used, only one antecedent connector model is usable; and for a nonsingleton FLS where the Mamdani-minimum implication method is used, none of the ten antecedent connector models is usable. We also show, by examples, that the parameter of the antecedent connector model provides additional freedom in adjusting a FLS, so that the FLS has the potential to achieve better performance than a FLS that uses the traditional product or minimum t-norm for the antecedent connections.

This paper introduces the Representation Theorem for type-2 fuzzy sets opening up a new approach to investigating theoretical aspects of type-2 fuzzy systems. The Representation Theorem is used to prove join, meet and complement operations without the use of the extension principle. It has been cited 41 times in the ISI web of science and appears in the leading journal in fuzzy logic with an impact factor of 1.803. According to the ISI Web of Science, Essential Science Indicator, this article is in the top 1% of articles. This article has led the author to give an invited talk at the IEEE World Conference on Computational Intelligence 2006.

We present an approach for MPEG variable bit rate (VBR) video modeling and classification using fuzzy techniques. We demonstrate that a type-2 fuzzy membership function, i.e., a Gaussian MF with uncertain variance, is most appropriate to model the log-value of I/P/B frame sizes in MPEG VBR video. The fuzzy c-means (FCM) method is used to obtain the mean and standard deviation (std) of T/P/B frame sizes when the frame category is unknown. We propose to use type-2 fuzzy logic classifiers (FLCs) to classify video traffic using compressed data. Five fuzzy classifiers and a Bayesian classifier are designed for video traffic classification, and the fuzzy classifiers are compared against the Bayesian classifier. Simulation results show that a type-2 fuzzy classifier in which the input is modeled as a type-2 fuzzy set and antecedent membership functions are modeled as type-2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters.

Presents a kind of adaptive filter: type-2 fuzzy adaptive filter (FAF); one that is realized using an unnormalized type-2 Takagi-Sugeno-Kang (TSK) fuzzy logic system (FLS). We apply this filter to equalization of a nonlinear time-varying channel and demonstrate that it can implement the Bayesian equalizer for such a channel, has a simple structure, and provides fast inference. A clustering method is used to adaptively design the parameters of the FAF. Two structures are used for the equalizer: transversal equalizer (TE) and decision feedback equalizer (DFE). A decision tree structure is used to implement the decision feedback equalizer, in which each leaf of the tree is a type-2 FAF. This DFE vastly reduces computational complexity as compared to a TE. Simulation results show that equalizers based on type-2 FAFs perform much better than nearest neighbor classifiers (NNC) or equalizers based on type-1 FAFs.

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.

In the original paper (IEEE Trans. Fuzzy Syst., vol.6, p.1-11, 1998), Combs and Andrews proved the following logical equivalence (stated here for two antecedents p and q and one consequent r, but easily generalize to an arbitrary number of antecedents and consequents): [(p/spl and/q)/spl rArr/r]/spl hArr/[(p/spl rArr/r)V(q/spl rArr/r)]. This is a very significant result because it suggests that we can replace multi-antecedent rules with an interconnection of single antecedent rules, which eliminates the rule explosion that is associated with multi-antecedent rules. Combs and Andrews refer to the left-hand side of this equivalence as an intersection rule configuration (IRC) and to its right-hand side as a union rule configuration (URC). Their result gives rise to two distinctly different paths for the design of fuzzy logic systems; IRC, which leads to rule explosion, and URC, which does not. The authors discuss four points about the IRC/spl hArr/URC relation. The original authors reply, acknowledging some of the points and stating that they would present their results differently if starting now.

In this paper, we present a fuzzy logic modulation classifier that works in nonideal environments in which it is difficult or impossible to use precise probabilistic methods. We first transform a general pattern classification problem into one of function approximation, so that fuzzy logic systems (FLS) can be used to construct a classifier; then, we introduce the concepts of fuzzy modulation type and fuzzy decision and develop a nonsingleton fuzzy logic classifier (NSFLC) by using an additive FLS as a core building block. Our NSFLC uses 2D fuzzy sets, whose membership functions are isotropic so that they are well suited for a modulation classifier (MC). We establish that our NSFLC, although completely based on heuristics, reduces to the maximum-likelihood modulation classifier (ML MC) in ideal conditions, In our application of NSFLC to MC in a mixture of /spl alpha/-stable and Gaussian noises, we demonstrate that our NSFLC performs consistently better than the ML MC and it gives the same performance as the ML MC when no impulsive noise is present.

In the original paper (IEEE Trans. Fuzzy Syst., vol.6, p.1-11, 1998), Combs and Andrews proved the following logical equivalence (stated here for two antecedents p and q and one consequent r, but easily generalize to an arbitrary number of antecedents and consequents): [(p∧q)⇒r]⇔[(p⇒r)V(q⇒r)]. This is a very significant result because it suggests that we can replace multi-antecedent rules with an interconnection of single antecedent rules, which eliminates the rule explosion that is associated with multi-antecedent rules. Combs and Andrews refer to the left-hand side of this equivalence as an intersection rule configuration (IRC) and to its right-hand side as a union rule configuration (URC). Their result gives rise to two distinctly different paths for the design of fuzzy logic systems; IRC, which leads to rule explosion, and URC, which does not. The authors discuss four points about the IRC⇔URC relation. The original authors reply, acknowledging some of the points and stating that they would present their results differently if starting now

We introduce a type-2 fuzzy logic system (FLS), which can handle rule uncertainties. The implementation of this type-2 FLS involves the operations of fuzzification, inference, and output processing. We focus on "output processing," which consists of type reduction and defuzzification. Type-reduction methods are extended versions of type-1 defuzzification methods. Type reduction captures more information about rule uncertainties than does the defuzzified value (a crisp number), however, it is computationally intensive, except for interval type-2 fuzzy sets for which we provide a simple type-reduction computation procedure. We also apply a type-2 FLS to time-varying channel equalization and demonstrate that it provides better performance than a type-1 FLS and nearest neighbor classifier.

In the previous paper, we have proposed linguistic weighted average (LWA) algorithms that can be used in distributed and hierarchical decision making. The original LWA algorithms were completely based on the representation theorem for interval type-2 fuzzy sets (IT2 FSs). In later usage, we found that when the lower membership functions (LMFs) of the inputs and weights are of different heights, the LMF of the output IT2 FS may be nonconvex and discontinuous. In this letter, a correction to the original LWA algorithms is proposed. The new LWA algorithms are simpler and easier to understand; so, it should facilitate the applications of the LWAs.

We investigate dynamic versions of fuzzy logic systems (FLSs) and, specifically, their non-Singleton generalizations (NSFLSs), and derive a dynamic learning algorithm to train the system parameters. The history-sensitive output of the dynamic systems gives them a significant advantage over static systems in modeling processes of unknown order. This is illustrated through an example in nonlinear dynamic system identification. Since dynamic NSFLS's can be considered to belong to the family of general nonlinear autoregressive moving average (NARMA) models, they are capable of parsimoniously modeling NARMA processes. We study the performance of both dynamic and static FLSs in the predictive modeling of a NARMA process.

In this paper, we present a formal derivation of general nonsingleton fuzzy logic systems (NSFLSs) and show how they can be efficiently computed. We give examples for special cases of membership functions and inference and we show how an NSFLS can be expressed as a "nonsingleton fuzzy basis function" expansion and present an analytical comparison of the nonsingleton and singleton fuzzy logic systems formulations. We prove that an NSFLS can uniformly approximate any given continuous function on a compact set and show that our NSFLS does a much better job of predicting a noisy chaotic time series than does a singleton fuzzy logic system (FLS).

Fuzzy basis functions (FBF's) which have the capability of combining both numerical data and linguistic information, are compared with other basis functions. Because a FBF network is different from other networks in that it is the only one that can combine numerical and linguistic information, comparisons are made when only numerical data is available. In particular, a FBF network is compared with a radial basis function (RBF) network from the viewpoint of function approximation. Their architectural interrelationships are discussed. Additionally, a RBF network, which is implemented using a regularization technique, is compared with a FBF network from the viewpoint of overcoming ill-posed problems. A FBF network is also compared with Specht's probabilistic neural network and his general regression neural network (GRNN) from an architectural point of view. A FBF network is also compared with a Gaussian sum approximation in which Gaussian functions play a central role. Finally, we summarize the architectural relationships between all the networks discussed in this paper. >

First break picking is a pattern recognition problem in seismic signal processing, one that requires much human effort and is difficult to automate. The authors' goal is to reduce the manual effort in the picking process and accurately perform the picking. Feedforward neural network first break pickers have been developed using backpropagation training algorithms applied either to an encoded version of the raw data or to derived seismic attributes which are extracted from the raw data. The authors summarize a study in which they applied a backpropagation fuzzy logic system (BPFLS) to first break picking. The authors use derived seismic attributes as features, and take lateral variations into account by using the distance to a piecewise linear guiding function as a new feature. Experimental results indicate that the BPFLS achieves about the same picking accuracy as a feedforward neural network that is also trained using a backpropagation algorithm; however, the BPFLS is trained in a much shorter time, because there is a systematic way in which the initial parameters of the BPFLS can be chosen, versus the random way in which the weights of the neural network are chosen. >

Two fuzzy adaptive filters are developed: one uses a recursive-least-squares (RLS) adaptation algorithm, and the other uses a least-mean-square (LMS) adaptation algorithm. The RLS fuzzy adaptive filter is constructed through the following four steps: (1) define fuzzy sets in the filter input space Rn whose membership functions cover U; (2) construct a set of fuzzy IF-THEN rules which either come from human experts or are determined during the adaptation procedure by matching input-output data pairs; (3) construct a filter based on the set of rules; and (4) update the free parameters of the filter using the RLS algorithm. The design procedure for the LMS fuzzy adaptive filter is similar. The most important advantage of the fuzzy adaptive filters is that linguistic information (in the form of fuzzy IF-THEN rules) and numerical information (in the form of input-output pairs) can be combined in the filters in a uniform fashion. The filters are applied to nonlinear communication channel equalization problems. >

Type-2 fuzzy sets and systems are more general and potentially more powerful than their type-1 counterparts because they model membership function (MF) uncertainties, which can be very valuable in applications where uncertainties are prevalent. In an interval type-2 fuzzy set (IT2 FS) and fuzzy logic system (FLS), all of the MF uncertainties are uniformly weighted, whereas in a general type-2 fuzzy set (GT2 FS) and FLS, the MF uncertainties are nonuniformly weighted. Since the preparation of the first Special Issue of the IEEE Transactions on Fuzzy Systems that appeared in 2007 on such fuzzy sets and systems, much has happened. Since the 2007 Special Issue, more than 2000 journal and conference articles about T2 FSs and FLSs have appeared; there have been two special sections of the IEEE Computational Intelligence Magazine devoted to them: one in 2007 and the other in 2012; in addition, special or invited sessions about them now appear regularly at all conferences devoted to fuzzy sets and systems. It is therefore very appropriate to have a second Special Issue of these transactions about Type-2 fuzzy sets and systems. For the record, a total of 45 submissions were received in response to the open call. These papers were rigorously evaluated according to the normal reviewing process of the IEEE Transactions on Fuzzy Systems. As in the previous Special Issue, the evaluation process took into consideration factors pertaining to originality, technical quality, presentational quality, and overall contribution. In all, seven papers were accepted for publication. The editors provide here an overview of the technical articles presented in this issue.

Wagner and Hagras introduced a novel defuzzification formula in their recent paper and showed that it works very well within the framework of their general type-2 fuzzy logic systems based on zSlices ( α-plane representation). This letter aims to point out the hidden connection between the standard centroid defuzzification formula and Wagner and Hagras' new defuzzification formula, which leads to the proof of complete equivalence of the two.