Search

In this paper, we extend the concept of Dempster-Shafer Belief Structures to the case of Linguistic Belief Structures, whose focal elements and probability mass assignments are linguistic, i.e. words modeled by Interval Type-2 Fuzzy Sets. We show that Linguistic Weighted Averages are pertinent tools for derivation of lower and upper probabilities from such Belief Structures, especially when words describing probability masses and focal elements are modeled by Interval Type-2 Fuzzy Sets synthesized by collecting data from subjects. We moreover introduce methods for performing operations on Linguistic Belief Structures as well as combining them. We demonstrate how Linguistic Belief Structures can be used to represent uncertainties in natural languages and present methods for inference from them.

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.

Non-singleton fuzzification is used in rule-based fuzzy systems when the measurements that activate them are imperfect or uncertain or when their inputs are words. It models such measurements or words as fuzzy numbers or more general fuzzy sets so that, regardless of the cause of a measurement’s or word’s imperfections or uncertainties, they are treated within the framework of fuzzy sets and systems. Since 2011, there has been a resurgence of interest in both type-1 and interval type-2 non-singleton fuzzy systems. This paper removes a computational bottleneck associated with computing the firing level or firing interval for such fuzzy systems, by providing closed-form formulas for them when the involved fuzzy sets are trapezoidal or triangular, which are widely used fuzzy sets. This is done for both the minimum and product t-norms. It is also demonstrated that a non-singleton fuzzy system that uses the product t-norm has the potential to outperform a non-singleton fuzzy system that uses the minimum t-norm. The results in this paper greatly simplify non-singleton fuzzy systems, which should make them much more popular.

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/ ).

Improving the efficiency of type-reduction algorithms continues to attract research interest. Recently, there has been some new type-reduction approaches claiming that they are more efficient than the well-known algorithms such as the enhanced Karnik–Mendel (EKM) and the enhanced iterative algorithm with stopping condition (EIASC). In a previous paper, we found that the computational efficiency of an algorithm is closely related to the platform, and how it is implemented. In computer science, the dependence on languages is usually avoided by focusing on the complexity of algorithms (using big O notation). In this article, the main contribution is the proposal of two novel type-reduction algorithms. Also, for the first time, a comprehensive study on both existing and new type-reduction approaches is made based on both algorithm complexity and practical computational time under a variety of programming languages. Based on the results, suggestions are given for the preferred algorithms in different scenarios depending on implementation platform and application context.

Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They have demonstrated superior performance in many applications. However, the operation of an IT2 fuzzy system is more complex than that of its type-1 counterpart. There are many questions to be answered in designing an IT2 fuzzy system: Should singleton or non-singleton fuzzifier be used? How many membership functions (MFs) should be used for each input? Should Gaussian or piecewise linear MFs be used? Should Mamdani or Takagi–Sugeno–Kang (TSK) inference be used? Should minimum or product t -norm be used? Should type-reduction be used or not? How to optimize the IT2 fuzzy system? These questions may look overwhelming and confusing to IT2 beginners. In this paper we recommend some representative starting choices for an IT2 fuzzy system design, which hopefully will make IT2 fuzzy systems more accessible to IT2 fuzzy system designers.

This paper proposes a novel top-down approach to rule-based fuzzy systems, one that begins with the product—an equation—and then addresses the unique features of the product, without requiring the reader to know anything about fuzzy sets and systems. The "products" are adaptive variable-structure basis function expansions , where "adaptive variable-structure" means that different subsets of its basis functions are active (non-zero) in different regions of the state space, something that occurs automatically by virtue of the structure of the basis functions, so that the products can be said to "adapt" to locations in the state space. These products are novel candidates for machine learning . Unique features of all products are: (1) Number of basis functions is no longer a variable and is established locally through type-1 or type-2 uncertainty partitioning of each variable; (2) Both coarse and fine sculpting of the state space are achieved, and are described in terms of first- and second-order partitions of the state space, respectively; and (3) Linguistic interpretability is obtained, which may be of value to an end-user. Learning about rule-based fuzzy systems can be greatly compressed by using the top-down approach that is described in this paper.

Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a non-membership function, and a hesitancy function, is a generalization of a fuzzy set. There are various intuitionistic fuzzy hybrid weighted aggregation operators to deal with multi-attribute decision making problems which consider the importance degrees of the arguments and their ordered positions simultaneously. However, these existing hybrid weighed aggregation operators are not monotone with respect to the total order on intuitionistic fuzzy values (AIFVs), which is undesirable. Based on the Łukasiewicz triangular norm, we propose an intuitionistic fuzzy hybrid weighted arithmetic mean, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a company intends to select a project manager to illustrate the validity and applicability of the proposed aggregation operator. Moreover, we extend this kind of hybrid weighted arithmetic mean to the interval-valued intuitionistic fuzzy environments.

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.

The special issue of IEEE Transactions On Fuzzy Systems s dedicated to the memory of Prof. Robert John, one of the pioneers of type-2 fuzzy sets and systems, who passed away during its preparation. This special issue focuses on type-2 model-based control and demonstrates the broadening of the type-2 fuzzy logic controller field. It is demonstrated that type-2 FMB control framework can be established by adopting type-2 fuzzy sets into type-1 fuzzy-model-based (FMB) designs. Besides the parameter uncertainty, the type-2 fuzzy sets have also demonstrated their potential to deal with the intrinsic uncertainty caused by specific control mechanisms, such as the sample-data control mechanism and event-triggered control mechanism.

This paper provides a new methodology for calibrating the fuzzy sets that are used in fsQCA, one that is based on clearly distinguishing between a linguistic variable and the linguistic terms for that variable, and that allows for uncertainties about those terms to be included in the calibration method. Each resulting fuzzy set, called an approximated reduced-information level 2 fuzzy set (RI L2 fuzzy set), is equivalent to a standard type-1 fuzzy set, but is for the linguistic variable, and, it has an S-shape, the kind of shape that is so widely used by fsQCA scholars, and is so important to fsQCA. This new calibration methodology is applied to Ragin's Breakdown of Democracy example, using new data provided by him, and demonstrates that his earlier solutions are also obtained using our approximated RI L2 fuzzy sets, something that should be reassuring to fsQCA scholars. Additionally, because the S-shaped membership functions are derived from footprints of uncertainty for all of the linguistic variable's terms, this paper shows how to obtain more precise statements of fsQCA causal combinations for their best instances, something that may be of added value to practitioners of fsQCA. Finally, we explain how different data-driven calibration robustness studies can be performed, something that may also be of great value to fsQCA practitioners.

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.

There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This article proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: first, train an initial global model using all training data; second, identify from the initial global model the patches that contribute the most to the learning error, and train a (local) patch model for each such patch; and, third, update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on one-dimensional (1-D)/2-D/3-D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.

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 paper is about some important changes in type-2 fuzzy set (T2 FS) definitions and notations that have occurred during the past 16 years. It summarizes the evolution of how the primary membership (Jx) has been used in both the mathematical descriptions of a T2 FS and its footprint of uncertainty (FOU); discusses notational problems associated with the secondary membership function; explains why and when one should distinguish between the FOU and the domain of uncertainty (DOU); explains why no errors have been introduced into T2 FS computations because of notation; and, it provides recommendations notational changes that can be used by all authors.

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 letter is a supplement to the previous paper “A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm”. In the previous paper, the enhanced iterative algorithm with stop condition (EIASC) was shown to be the most inefficient in R. Such outcome is apparently different from the results in another paper in which EIASC was illustrated to be the most efficient in MATLAB. An investigation has been made into this apparent inconsistency and it can be confirmed that both the results in R and MATLAB are valid for the EIASC algorithm. The main reason for such phenomenon is the efficiency difference of loop operations in R and MATLAB. It should be noted that the efficiency of an algorithm is closely related to its implementation in practice. In this letter, we update the comparisons of the three algorithms in the previous paper, based on optimized implementations under five programming languages (MATLAB, R, Python, C, and Java). From this, we conclude that results in one programming language cannot be simply extended to all languages.

This paper presents a very efficient method for establishing nonlinear combinations of variables from small to big data for use in later processing (e.g., regression, classification, etc.). Variables are first partitioned into subsets each of which has a linguistic term (called a causal condition ) associated with it. Our Causal Combination Method uses fuzzy sets to model the terms and focuses on interconnections ( causal combinations ) of either a causal condition or its complement, where the connecting word is AND which is modeled using the minimum operation. Our Fast Causal Combination Method is based on a novel theoretical result, leads to an exponential speedup in computation and lends itself to parallel and distributed processing; hence, it may be used on data from small to big.

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.

Similarity measures are very important concepts in fuzzy sets (FSs) theory. There are many different definitions of similarity measures for both type-1 (T1) FSs and interval type-2 (IT2) FSs. In this paper, one similarity measure for IT2 FSs, which is extended from the Jaccard similarity measure for T1 FSs, is reviewed; then, based on the α -plane representation for a general type-2 (GT2) FS, this similarity measure is generalized to such T2 FSs. Some examples that demonstrate how to compute the similarity measures for different T2 FSs are given.

This issue of "Historical Perspectives" is the third in a series of remembrances from past presidents of the IE Control Systems Society.

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.

In this paper, some properties of Novel Weighted Averages that are related to the concepts of possibility theory are examined. It is shown that Novel Weighted Averages have certain interpretations in terms of addition of interactive interval or fuzzy constraints. To do this, alternative forms of Novel Weighted Averages are provided. In particular, an alternative form of Novel Weighted Averages represented by the Extension Principle is determined. It is shown that, when fuzzy set models of words are obtained by collecting data from subjects in a Computing with Words setting, interactive addition of fuzzy sets is not a well-defined method, and the optimization problems related to it may have no solutions, although interactive addition is recommended in the literature for solving multicriteria decision making problems and for dealing with uncertain probabilities. On the other hand, Novel Weighted Averages perform a specific normalization that guarantees that they always exist.

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.

Fuzzy Set Qualitative Comparative Analysis (fsQCA) is a methodology for obtaining linguistic summarizations from data that are associated with cases. It was developed by the eminent social scientist Prof. Charles C. Ragin, but has, as of this date, not been applied by engineers or computer scientists. Unlike more quantitative methods that are based on correlation, fsQCA seeks to establish logical connections between combinations of causal conditions and an outcome, the result being rules that summarize the sufficiency between subsets of all of the possible combinations of the causal conditions (or their complements) and the outcome. The rules are connected by the word OR to the output. Each rule is a possible path from the causal conditions to the outcome. This paper, for the first time, explains fsQCA in a very quantitative way, something that is needed if engineers and computer scientists are to use fsQCA.

Removing Mixed Gaussian and Impulse Noise (MGIN) is considered to be one of the most essential topics in the domain of image restoration, and it is much more challenging than to remove pure Gaussian or impulse noise separately. Therefore, relatively fewer works have been published in this area. This paper proposes a new integrated approach for MGIN removal that is based on a Non-Singleton Interval Type-2 (NS-IT2) Fuzzy Logic System (FLS), and explains how to design such a NS-IT2 FLS using a Quantum-behaved Particle Swarm Optimization (QPSO) algorithm. Then the paper goes on to introduce two supplementary components, a Block-Matching 3-Dimensional Discrete Cosine Transformation (BM3D DCT) filter and a contrast scaling filter, which augment the overall performance of the NS-IT2 FLS. Finally, the paper shows that this proposed approach indeed provides both quantitatively and visually much better results compared to other often-used non-fuzzy techniques as well as its Type-1 (T1) and singleton IT2 (S-IT2) counterparts.

For the fuzzy weighted average (FWA), despite various discrete solution algorithms and their improvements, attempts at analytical solutions are very rare. This paper provides an analytical solution method for the FWA based on the conclusions of the Karnik-Mendel (KM) algorithm. Compared with the two current popular kinds of @a-cut based computational methods for the FWA (mathematical programming transformations and direct iterate computations), our method is precise, and, has a concise structure, efficient computation process, and sound theoretical proofs. We propose two algorithms for computing the analytical solution of the FWA. Two numerical examples illustrate our proposed approach.

Computing the centroid of an interval type-2 fuzzy set is an important operation in a type-2 fuzzy logic system, and is usually implemented by Karnik-Mendel (KM) iterative algorithms. By connecting KM algorithms and continuous KM algorithms together, this paper gives theoretical explanations on the initialization methods of KM and Enhanced Karnik-Mendel (EKM) algorithms, proposes exact methods for centroid computation of an interval type-2 fuzzy set, and extends the Enhanced Karnik-Mendel (EKM) algorithms to three different forms of weighted EKM (WEKM) algorithms. It shows that EKM algorithms become a special case of the WEKM algorithms when the weights of the latter are constant value. It also shows that, in general, the weighted EKM algorithms have smaller absolute error and faster convergence speed than the EKM algorithms which make them very attractive for real-time applications of fuzzy logic system. Four numerical examples are used to illustrate and analyze the performance of WEKM algorithms.

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.

Research on the robustness of fuzzy logic systems (FLSs), an imperative factor in the design process, is very limited in the literature. Specifically, when a system is subjected to small deviations of the sampling points (operating points), it is of great interest to find the maximum tolerance of the system, which we refer to as the system's robustness. In this paper, we present a methodology for the robustness analysis of interval type-2 FLSs (IT2 FLSs) that also holds for T1 FLSs, hence, making it more general. A procedure for the design of robust IT2 FLSs with a guaranteed performance better than or equal to their T1 counterparts is then proposed. Several examples are performed to demonstrate the effectiveness of the proposed methodologies. It was concluded that both T1 and IT2 FLSs can be designed to achieve robust behavior in various applications, and preference one or the other, in general, is application-dependant. IT2 FLSs, having a more flexible structure than T1 FLSs, exhibited relatively small approximation errors in the several examples investigated. The methodologies presented in this paper lay the foundation for the design of FLSs with robust properties that will be very useful in many practical modeling and control applications.

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.

Five uncertainty measures have previously been defined for interval Type-2 fuzzy sets (IT2 FSs), namely centroid, cardinality, fuzziness, variance and skewness. Based on a recently developed @a-plane representation for a general T2 FS, this paper generalizes these definitions to such T2 FSs and, more importantly, derives a unified strategy for computing all different uncertainty measures with low complexity. The uncertainty measures of T2 FSs with different shaped Footprints of Uncertainty and different kinds of secondary membership functions (MFs) are computed and are given as examples. Observations and summaries are made for these examples, and a Summary Interval Uncertainty Measure for a general T2 FS is proposed to simplify the interpretations. Comparative studies of uncertainty measures for Quasi-Type-2 (QT2), IT2 and T2 FSs are also performed to examine the feasibility of approximating T2 FSs using QT2 or even IT2 FSs.

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).

Type-2 fuzzy logic systems have recently been utilized in many control processes due to their ability to model uncertainties. This paper proposes a novel inference mechanism for an interval type-2 Takagi-Sugeno-Kang fuzzy logic control system (IT2 TSK FLCS) when antecedents are type-2 fuzzy sets and consequents are crisp numbers (A2-C0). The proposed inference mechanism has a closed form which makes it more feasible to analyze the stability of this FLCS. This paper focuses on control applications for the following cases: 1) Both plant and controller use A2-C0 TSK models, and 2) the plant uses type-1 Takagi-Sugeno (TS) and the controller uses IT2 TS models. In both cases, sufficient stability conditions for the stability of the closed-loop system are derived. Furthermore, novel linear-matrix-inequality-based algorithms are developed for satisfying the stability conditions. Numerical analyses are included which validate the effectiveness of the new inference methods. Case studies reveal that an IT2 TS FLCS using the proposed inference engine clearly outperforms its type-1 TSK counterpart. Moreover, due to the simple nature of the proposed inference engine, it is easy to implement in real-time control systems. The methods presented in this paper lay the mathematical foundations for analyzing the stability and facilitating the design of stabilizing controllers of IT2 TSK FLCSs and IT2 TS FLCSs with significantly improved performance over type-1 approaches.

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.

This paper, which is tutorial in nature, demonstrates how the Embedded Sets Representation Theorem (RT) for a general type-2 fuzzy set (T2 FS), when specialized to an interval (I)T2 FS, can be used as the starting point to solve many diverse problems that involve IT2 FSs. The problems considered are: set theoretic operations, centroid, uncertainty measures, similarity, inference engine computations for Mamdani IT2 fuzzy logic systems, linguistic weighted average, person membership function approach to type-2 fuzzistics, and Interval Approach to type-2 fuzzistics. Each solution obtained from the RT is a structural solution but is not a practical computational solution, however, the latter are always found from the former. It is this author's recommendation that one should use the RT as a starting point whenever solving a new problem involving IT2 FSs because it has had such great success in solving so many such problems in the past, and it answers the question ''Where do I start in order to solve a new problem involving IT2 FSs?''

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.

Ranking methods, similarity measures and uncertainty measures are very important concepts for interval type-2 fuzzy sets (IT2 FSs). So far, there is only one ranking method for such sets, whereas there are many similarity and uncertainty measures. A new ranking method and a new similarity measure for IT2 FSs are proposed in this paper. All these ranking methods, similarity measures and uncertainty measures are compared based on real survey data and then the most suitable ranking method, similarity measure and uncertainty measure that can be used in the computing with words paradigm are suggested. The results are useful in understanding the uncertainties associated with linguistic terms and hence how to use them effectively in survey design and linguistic information processing.