The design of membership functions in fuzzy systems often requires satisfying domain, semantic, and relational constraints. Existing methods, while effective at enforcing parameter bounds, often lack flexibility or fail to address complex relational constraints. To overcome these limitations, this paper introduces the Constraints Always Satisfied Parameters (CASPs) framework, which inherently satisfies constraints during optimisation. Three variants of the CASPs are proposed, each balancing design flexibility, performance, and interpretability differently. Experimental evaluations on the Electricity and Laser datasets demonstrate consistent constraint satisfaction across all runs, with CASPs-Single prioritising interpretability, CASPs-Free excelling in RMSE performance, and CASPs-Adapted offering a balanced approach. The results highlight the potential of CASPs to enhance the design and optimisation of fuzzy systems.

This article introduces the fuzzy-system kernel machines —a class of machine learning models based on the connection between fuzzy inference systems and kernel machines. For the connection, we observed a relationship between the representer theorem of kernel methods and the functional representation of nonsingleton fuzzy systems. We found that the nonsingleton kernel on fuzzy sets —a kernel defined in this article—is the core element allowing this two-way connection perspective. Consequently, a fuzzy system trained with the kernel method can be regarded as a kernel machine, whereas a kernel machine trained with a nonsingleton kernel on fuzzy sets can be interpreted as a fuzzy system. We conducted several experiments in supervised classification to understand the generalization power and properties of the proposed fuzzy-system kernel machines.

Computing with Words (CWW) methodology has been used to design intelligent systems which make decisions by manipulating the linguistic information, like human beings. Human beings naturally understand (and express) themselves linguistically, and hence can reason (and make decision) just with linguistic information without any numerical measure. Perceptual Computing makes use of type 2 fuzzy sets for modeling the words in the CWW paradigm. This use of type-2 fuzzy sets enables better representation of the inherent uncertainty in the fuzzy linguistic semantics on numerous problems. To realise the potential of Perceptual Computing, its MATLAB implementation has been made freely available to the end-users/ researchers, and MATLAB is a proprietary development environment. Therefore, this contribution aims at proposing a python implementation of the Perceptual Computing, or its main processing element the perceptual computer that consists of three components viz., encoder, CWW engine and decoder. Our python implementation provides the end user with a seamless blending amongst all three components, which does not exist yet, to the best of our knowledge.

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.

This article is about explainable artificial intelligence (XAI) for rule-based fuzzy systems [that can be expressed generically, as$y({{\bf x}}) = f({{\bf x}})$]. It explains why it isnot validto explain the output of Mamdani or Takagi–Sugeno–Kang rule-based fuzzy systems using IF-THEN rules, and why itis validto explain the output of such rule-based fuzzy systems as anassociationof the compound antecedents of a small subset of the original larger set of rules, using a phrase such as “these linguistic antecedents aresymptomaticof 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.

The articles in this special section are dedicated to the memory of Prof. Robert John, one of the pioneers of type-2 fuzzy sets and systems, who passed away during the preparation of this issue.

This paper explores the connection between fuzzy inference systems and kernel methods. Particularly, we explore the connection between the fuzzy-basis-function expansion of non-singleton fuzzy systems and regularized kernel methods with positive-definite kernels. We show that positive-definite kernel on fuzzy sets, i.e., a real-valued function defined on the set of fuzzy sets, is the core concept enabling such a connection. Furthermore, we rewrite the fuzzy-basis-function (FBF) expansion of non-singleton fuzzy systems in terms of a kernel-basis-function expansion with positive-definite kernels on fuzzy sets. The consequence of doing this is that fuzzy systems can be trained by regularized kernel methods, and the advantages of doing this are: 1) fuzzy systems can have more resistance to the curse of dimensionality, 2) they can explicitly regularize the function being approximated, 3) they can satisfy the Representer Theorem of kernel methods, which bounds the number of learned rules to the number of training observations, and, 4) they can learn functions in the functional space induced by the kernel on fuzzy sets. Consequently, from this perspective fuzzy systems can be regarded as kernel methods. Different from previous works, this connection relates the FBF-expansion directly to kernel methods without dropping the denominator of the expansion; also, thanks to the use of kernels on fuzzy sets, we provide an algorithm that makes it possible to drop the restriction of having fuzzy sets that share some of their antecedent parameters, e.g., Gaussian membership functions with the same variance in rule antecedents as is usually done in related works.

In 2015 fuzzy sets and fuzzy logic celebrated their golden anniversary, but (arguably) unfortunately they have not been and are not being used or considered by most technical people outside of the fuzzy community. I believe that this is, arguably, to a large extent due to the negative connotation of the name fuzzy and believe that it is time for a replacement of that word. Fuzzy may be okay to describe a soft teddy bear, a cuddly pet, or a peach but for it to be used for mathematics and its applications is a red flag. In this article I propose Zadeh set/logic as replacements of Fuzzy set/logic. After 55 years, it is time to honor Prof. Zadeh by using his name as a proper adjective. Replacing fuzzy by Zadeh is universal in that it needs no translation in any language, whereas fuzzy does. This article also illustrates how the use of these replacement terms will provide everyone with a way to describe what they are working on when they are in different situations, without encountering the derision or worse that frequently occurs when fuzzy is used.

This paper proposes a systematic approach to solve the design problem of General Type-2 (GT2) Fuzzy Logic Controllers (FLCs) with an online scheduling mechanism for performance enhancements. We firstly suggest constructing the GT2-FLC over its baseline type-1 and interval type-2 FLCs, and then to tune a single design parameter which defines the shape of the secondary membership functions. We present how the shape of the secondary membership function changes with respect to the design parameter and show resulting effect on the control surface generation. The presented comparative analysis on the control surfaces show that aggressive and smooth control surfaces can be easily generated by tuning the design parameter. We suggest tuning the design parameter by providing a tradeoff between robustness and performance of the control system. Also, to achieve satisfactory control performances for various steady-state points, we propose an online scheduling mechanism that tunes the design parameter with respect to the operating points. We perform a simulation study on a nonlinear system to validate our analyses and proposed design methods. The simulation results show that GT2-FLC has a potential to improve overall system performances in comparison to its type-1 and interval type-2 counterparts, while the developed scheduling mechanism provides an opportunity to achieve satisfactory results for various operating points.

Alpha-cut representation of fuzzy sets has been used as a basis for fuzzy numbers ranking in some applications but rarely used for defuzzification of rule-based systems or fuzzy controllers . Moreover, such alpha-cut defuzzification (called ACD here) is not yet formally linked to the membership function (MF) or to the common MF-based defuzzification methods, namely the centroid . The ACD can be considered as a generalisation of the similar algorithms in fuzzy numbers to any fuzzy set. A close-form formula for ACD is developed that involves both MF and its derivative, which shows that ACD reflects both static and dynamic aspects of a fuzzy set. Moreover, formal links between ACD and some MF-based defuzzification methods are shown. Through two groups of experiments, the utility of the new method is compared with centroid defuzzification. Particularly, we examined how the ACD significantly outperforms the centroid for noisy time-series prediction. Finally, the computation complexity of ACD is shown to be about the same as the centroid method, for convex fuzzy sets . Our tests suggest that ACD can be considered as a viable alternative defuzzification method for fuzzy system designers.

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.

This article provides new application-independent perspectives about the performance potential of an intuitionistic (I-) fuzzy system over a (classical) Takagi-Sugeno-Kang (TSK) fuzzy system. It does this by extending sculpting the state-space works from a TSK fuzzy system to an I-fuzzy system. It demonstrates that, for piecewise-linear membership functions (trapezoids and triangles), an I-fuzzy system always has significantly more first-order rule partitions of the state space-the coarse sculpting of the state space-than does a TSK fuzzy system, and that some I-fuzzy systems also have more second-order rule partitions of the state space-the fine sculpting of the state space-than does a TSK fuzzy system. It is the author's conjecture that for piecewise-linear membership functions (trapezoids and triangles): it is the always significantly greater coarse (and possibly fine) sculpting of the state space that provides an I-fuzzy system with the potential to outperform a TSK fuzzy system, and that a type-1 I-fuzzy system has the potential to outperform an interval type-2 fuzzy system.

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 O(w*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/).

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.

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.

This paper provides application-independent perspectives on why improved performance usually occurs as one goes from an interval type-2 (IT2) fuzzy system to a general type-2 (GT2) fuzzy system. This is achieved by using the horizontal-slice representation of a GT2 fuzzy set and GT2 fuzzy system and by examining first- and second-order rule partitions as well as novelty partitions for the horizontal slices. It demonstrates that, for triangle and trapezoid secondary membership functions, the numbers of first- and second-order rule partitions are exactly the same for IT2 and GT2 fuzzy systems, but that a maximum amount of change always occurs in every second-order rule partition of a GT2 fuzzy system. This does not always occur in such partitions of an IT2 fuzzy system. Furthermore, when type reduction (TR) is used in a GT2 fuzzy system, the total number of novelty partitions is directly proportional to the number of horizontal slices; consequently, there are many more such partitions in a GT2 fuzzy system that uses TR than occur in an IT2 fuzzy system that also uses TR. It is the author's conjecture that it is the maximum changes that occur in every second-order rule partition, as well as the greater number of novelty partitions when TR is used, that provide a GT2 fuzzy system with the potential to outperform an IT2 fuzzy system.

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 α-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.

Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision-making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov’s intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the Łukasiewicz triangular norm, 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 consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.

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 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 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 provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is obtained by using the Begian-Melek-Mendel (BMM) formula compares to the one obtained by using center-of-sets type-reduction followed by defuzzification (COS TR + D). This is made possible by reformulating the structural solutions of the two optimization problems that are associated with COS TR, and then expanding each of them using a Maclaurin series expansion. As a result of doing this, we show that BMM is the zero-order approximation to COS TR + D. Additionally, by retaining the zero-order and first-order terms from the Maclaurin series expansions, we provide a new Enhanced BMM, one that is non-iterative, has a closed form and is much faster than using the EKM algorithms for COS TR. Although the Enhanced BMM formula is slower than BMM, we demonstrate, by means of extensive simulations, that it is from 5% to 50% more accurate than is BMM for achieving the same numerical solution that is obtained from COS TR + D; and, it is at least 94% faster than when EKM is used for COS TR +D, which makes the Extended BMM a very strong candidate for use in real time applications of IT2 FLSs.

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.

This letter makes some observations about “Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wide view on their relationship,”IEEE Trans. Fuzzy Systemsthat further support the distinction between an interval type-2 fuzzy set (IT2 FS) and an interval-valued fuzzy set (IV FS), points out that all operations, methods, and systems that have been developed and published about IT2 FSs are, so far, only valid in the special case when IT2 FS = IVFS, and suggests some research opportunities.

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.

Variable Structure Regression (VSR) is a new kind of non-linear regression model, which simultaneously determines the exact mathematical structure of non-linear regressors and how many regressors there are, thereby freeing the end user from trial and error time-consuming studies to determine these. The results are based on an iterative procedure for optimizing parameters and automatically identifying the structure of the VSR model. A novel feature of this new model is it not only uses a linguistic term for a variable but it also uses the complement of that term. It also provides the end user with a physical understanding of the regressors. A Monte Carlo study shows the practical accuracy of VSR model on the classical Gas Furnace time-series prediction problem. VSR ranked #1 compared to five other methods.

This paper provides a new methodology for determining a word's interval type-2 fuzzy set model using only one subject, a Person FOU. It uses interval end-point uncertainty intervals instead of only the end-point intervals. Such uncertainty intervals are relatively easy to collect and they do not introduce methodological uncertainties during the data-collection process. This new method is applied to ten probability words. Person FOUs are obtained for these words, and the robustness of this new method to the choice of the probability distribution that is assigned to the interval end-point uncertainty intervals is examined and demonstrated.

Interval type-2 fuzzy logic systems (IT2 FLSs) have become increasingly popular in the last decade, and have demonstrated superior performance in a number of applications. However, the computations in an IT2 FLS are more complex than those in a type-1 FLS, and there are many choices to be made in designing an IT2 FLS, including the shape of membership functions (Gaussian or trapezoidal), number of membership functions, type of fuzzifier (singleton or non-singleton), kind of rules (Mamdani or Takagi-Sugeno-Kang), type of i-norm (minimum or product), method to compute the output (type-reduction or not), and methods for tuning the parameters (gradient-based methods or evolutionary computation algorithms; one-step or two-step). While these choices give an experienced IT2 FLS researcher extensive freedom to design the optimal IT2 FLS, they may look overwhelming and confusing to IT2 beginners. Such a beginner may make an inappropriate choice, obtain unexpected results, and lose interest, which will hinder the wider applications of IT2 FLSs. In this paper we try to help IT2 beginners navigate through the maze by recommending some representative choices for an IT2 FLS design. We also clarify two myths about IT2 FLSs. This paper will make IT2 FLSs more accessible to IT2 beginners.

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.

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 computations or multiple criteria decision analysis under uncertainty.

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.

In this paper, we present solutions to an Advanced Computing with Words problem that is equivalent to one of Zadeh's challenge problems on linguistic probabilities. We use a syllogism based on the entailment principle to interpret the problem so that it yields two linguistic belief structures. Then we perform an addition of those linguistic belief structures to obtain a belief structure on the variable about which linguistic probabilities have to be inferred. We show that pessimistic (lower) and optimistic (upper) probabilities can be inferred from such a belief structure using Linguistic Weighted Averages and pessimistic and optimistic compatibility measures. Then, we choose vocabularies for linguistic attributes (lifetimes of products) and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, and map them into words present in the vocabulary of linguistic probabilities, so that the results can be comprehended by a human.

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.

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.

Type-2 fuzzy logic systems that use general type-2 fuzzy sets (GT2 FSs) are now receiving renewed attention; hence, it is important to be able to communicate effectively about such fuzzy sets. While it is possible to generate the 3D membership functions (MFs) for these fuzzy sets using computer programs, this is not very useful for real-time on-the-fly discussions. Additionally, for most of us it is very difficult, if not impossible, to sketch or plot 3D MFs by hand. This short educational paper provides detailed instructions on how to plot different kinds of so-called 2 ½ D plots for a GT2 FS. Using such 2 ½ D plots should greatly enhance the communication about GT2 FSs and will therefore make them more accessible to the fuzzy logic community.

Karnik-Mendel (KM) algorithms used for type reduction (TR) followed by defuzzification are commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Up until now no closed-form solution has existed for KM+defuzzification. Our paper gives, for the first time, a Taylor-series approximation to KM+defuzzification. The first term in the approximation is the Nie-Tan (NT) formula, and the second-term (which is actually the third-order term in the Taylor series) is a correction to it. One very interesting fact is that the Taylor series approximation is itself iterative (as is the KM Algorithm) in that the second term depends on the first term. Simulations show that the first term is a very good approximation to KM+defuzzification.

In this paper, we present solutions to Zadeh's challenge problem on calculating linguistic probabilities. First, we argue that Zadeh's solution to this problem via the Generalized Extension Principle is very difficult to implement. Then, we use a syllogism based on the entailment principle to interpret the problem so that it can be solved by calculation of pessimistic (lower) and optimistic (upper) probabilities via Linguistic Weighted Averages. We use a pessimistic and an optimistic compatibility measure to calculate such probabilities. Then, we choose vocabularies for heights and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, which naturally would be interval type-2 fuzzy sets. Finally, we map the pessimistic and optimistic probabilities into linguistic probabilities present in the vocabularies, so that the results can be comprehended by a human. We investigate viable alternatives for the pessimistic and optimistic compatibility measures, and also solve a similar problem with a different hypothesis.

Certain classes of fuzzy rule-based systems have been shown to be universal approximators, capable of approximating any continuous mapping on a compact subset of the domain. In previous cases, this property has been proven for fuzzy systems employing t-norms to determine the firing level of each rule. In this paper, we prove that use of the weighted power mean to determine rule firing levels also results in a universal approximator. While it is only a t-norm in certain special cases, the weighted power mean is a more general aggregation operator, capable of providing greater logical flexibility in a rule. This will be of particular interest in computing with words (CWW) applications, where such flexibility is needed the better to mimic human reasoning.

Construction of interval type-2 fuzzy set models is the first step in the perceptual computer, which is an implementation of computing with words. The interval approach (IA) has, so far, been the only systematic method to construct such models from data intervals that are collected from a survey. However, as pointed out in this paper, it has some limitations, and its performance can be further improved. This paper proposes an enhanced interval approach (EIA) and demonstrates its performance on data that are collected from a web survey. The data part of the EIA has more strict and reasonable tests than the IA, and the fuzzy set part of the EIA has an improved procedure to compute the lower membership function. We also perform a convergence analysis to answer two important questions: 1) Does the output interval type-2 fuzzy set from the EIA converge to a stable model as increasingly more data intervals are collected, and 2) if it converges, then how many data intervals are needed before the resulting interval type-2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals? We show that the EIA converges in a mean-square sense, and generally, 30 data intervals seem to be a good compromise between cost and accuracy.

Recently, a centroid-flow (CF) algorithm has been proposed to compute the centroid of a type-2 fuzzy set Ã. This algorithm utilizes the Karnik-Mendel (KM) or the enhanced KM (EKM) algorithm only at the α = 0 α-level of Ã̃ and then lets its result “flow” upward to the α = 1 α-level of Ã. It avoids having to apply the KM/EKM algorithms at every α-level, which significantly improves its computational efficiency; however, the CF algorithm approximation errors will gradually accumulate as the algorithm “flows” upward, and in some cases, this can cause the centroid of the α = 1 α-level of à to differ from its theoretical value. This paper introduces an improved version of the CF algorithm, which is called enhanced CF algorithm, that reduces such accumulative errors by half and, therefore, greatly improves the computational accuracy.

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 ($\alpha$-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.

The union and intersection of general type-2 fuzzy sets (T2 FSs) are fundamental computations for such FSs. In the past, algorithms were developed for the union and intersection computations using vertical-slice and horizontal-slice representations of T2 FSs. The vertical-slice representation of a T2 FS traces its origins back to Zadeh [32] and requires computing the join or meet, whereas the horizontal-slice representation of a T2 FS is very recent, traces its origins to Liu [14], and requires computing the join and meet only for interval T2 FSs that are raised to level alpha, for which closed-form formulas are available [15]. In this paper, by studying the join and meet geometrically for general T2 FSs, we show that for many situations closed-form formulas exist for them. We also show that the formulas for computing the union and intersection of general T2 FSs that were derived from the horizontal-slice representation of a T2 FS [19] can also be obtained directly from geometrical interpretations of formulas that were derived by Karnik and Mendel [12] for the vertical-slice representation of a T2 FS.

In this paper, we present a solution to Zadeh's Magnus challenge problem on linguistic probabilities. First, we implement Zadeh's solution to this problem. Then, we use the intersection-product syllogism and a syllogism based on the entailment principle to interpret the problem so that it can be solved via Linguistic Weighted Averages. We show that the problem can be solved by calculation of pessimistic (lower) and optimistic (upper) probabilities via Linguistic Weighted Averages. Then, we choose vocabularies for quantifiers and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, which naturally would be interval type-2 fuzzy sets. Finally, we map the pessimistic and optimistic probabilities to linguistic probabilities present in the vocabularies, so that the results can be comprehended by a human.

Removing Mixed Gaussian and Impulse Noise (MGIN) is considered to be very important in the domain of image restoration, but it is a somewhat more challenging topic than removing pure Gaussian or impulse noise. Therefore, relatively fewer works have been published in this area. This paper pro poses a Non-Singleton Interval Type-2 (IT2) Fuzzy Logic System (FLS) for MGIN removal, explains how it can be designed based on a Quantum-behaved Particle Swarm Optimization algorithm, and shows that it provides both quantitatively and visually much better results compared to other often-used non-fuzzy techniques as well as its Type-1 and singleton IT2 counterparts.

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.

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 “IFXis large andYis medium, THENZis 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).

Previous studies have shown that the centroid of a general type-2 fuzzy set (T2 FS) à can be obtained by taking the union of the centroids of all the α-planes (each raised to level α) of Ã. 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 à 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 variablexand α -plane are discretized finely enough.

The Interval Approach (IA) [4] is a method for synthesizing an interval type-2 fuzzy set (IT2 FS) model for a word from data that are collected from a group of subjects. A key assumption made by the IA is: each person's data interval is random and uniformly distributed. This means, of course, that the IT2 FS model for the word is random. Consequently, one can question whether or not the IT2 FS model for the word converges in a stochastic sense. This paper focuses on this question. As a part of our study, we have had to modify some steps of the IA, the resulting being an Enhanced IA (EIA). The paper shows by means of some simulations, that the IT2 FS word models that are obtained from the EIA are converging in a mean-square sense. This provides substantial credence for using the EIA to obtain T2 FS word models.

The international logistics centers choice problem is very important in International logistics. The location choice problem usually involves both numbers and words in which all of the criteria are weighted using words and the performance evaluations for all sub-criteria are either numbers or words. How to aggregate all of these data without losing information is a very daunting task using a type-1 fuzzy set (T1 FS) approach. This paper applies a new methodology - Perceptual Computer (Per-C)-to help solve this hierarchical multi-criteria decision making problem. Per-C has three components: encoder, computing with words (CWW) engine and decoder. First, the interval approach (IA) is used to obtain the interval type-2 fuzzy set (IT2 FS) word model for the words in a pre-specified vocabulary. Second, a linguistic weighted average (LWA) is used to aggregate all the data including numbers and words modeled by IT2 FSs. Finally, a centroid-based ranking method is used to rank the location choices, and a similarity measure is used to obtain similarities of the location choices. The decision-maker decides the winning location choice as the one with the highest ranking, smallest uncertainty band, and least similarity to other locations.

The question “When will an IT2 FLS outperform a T1 FLS?” has been asked by many, and is acknowledged by many experts to be arguably the most important unanswered question in the T2 field. As a research problem, this question turns into: Establish when and by how much a type-2 fuzzy logic system (T2 FLS) will outperform a type-1 (T1) FLS. This paper provides first results on solving this problem. Its approach is novel because it does not focus immediately on a specific application, but instead focuses on the common component to all performance analyses, thereby providing results that can be used by others in their application-based performance comparisons. The Wu-Mendel minimax uncertainty bounds, which in the past have been used to approximate the type-reduced set, and to also act as a starting point for designs of IT2 FLSs, play the key role in this paper. Although comparing an IT2 FLS to a T1 FLS seems like a daunting task, because of the complicated nature of the equations that describe them, this paper shows that when an IT2 FLS is expanded about a T1 FLS-itself a new concept-, surprisingly, very simple first results are obtained.

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 using Karnik-Mendel type-reduction and center-of-sets defuzzification, we derive the conditions under which continuous and discontinuous input-output mappings can be obtained. Our results will be very useful in selecting 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 ordered weighted average (OWA) operator has been widely used in decision-making. In many situations, however, providing crisp numbers for either the sub-criteria or the weights is problematic (there could be uncertainties about them), and it is more meaningful to provide intervals, type-1 fuzzy sets (T1 FSs), interval type-2 fuzzy sets (IT2 FSs), or a mixture of all of these, for the sub-criteria and weights. Two fuzzy extensions of the OWA, ordered fuzzy weighted averages for T1 FSs and ordered linguistic weighted averages for IT2 FSs, as wells as procedures for computing them, are introduced in this paper. They are compared with Zhou et al.'s T1 and IT2 fuzzy extensions of the OWA. Examples show that our extensions may give different results from Zhou et al.'s extensions when the legs of the FSs have intersections. Because our extensions coincide with the intuition of “FS in its entirety,” they are the suggested ones to use.

The Perceptual Computer (Per-C) is an architecture for making subjective judgments by computing with words. An application of the Per-C to a social judgment is described in this paper. First, a vocabulary is established for the social judgment and its words are modeled by interval type-2 fuzzy sets (IT2 FSs). Surveys are then designed to establish a structure of the rulebase and to obtain a rule-consequent histograms. After pre-processing to remove bad responses and outliers, perceptual reasoning (PR) is used to simplify the rulebase. Once the rulebase is established, PR is also used to infer the output IT2 FSs for new inputs. Finally, the output IT2 FSs are mapped back into words in the codebook using a similarity measure. So, from a user's point of view, he or she is interacting with the Per-C using only words from a vocabulary. The techniques introduced in this paper should be applicable to many rule-based decision-making situations.

Subsethood and similarity measures are important concepts in fuzzy set (FS) theory. There are many different definitions of them, for both type-1 (T1) FSs and interval type-2 (IT2) FSs. In this paper, Rickard et al.'s definition of IT2 FS subsethood measure, extended from Kosko's T1 FS subsethood measure using the Representation Theorem, and Nguyen and Kreinovich's IT2 FS similarity measure, extended from the Jaccard similarity measure for T1 FSs, are introduced. Efficient algorithms for computing them are also proposed. Simulations demonstrate that our proposed algorithms outperform existing algorithms in the literature.

Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. It has been studied by many researchers; however, none of them has used it to generate IF-THEN rules, which can be added to a knowledge base for better understanding of the data, or be used in Perceptual Reasoning to infer the outputs for new scenarios. In this paper LS using IF-THEN rules is proposed. Five quality measures for such summaries are defined. Among them, the degree of usefulness 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. An example verifies the effectiveness of our approach. The relationship between LS and the Wang-Mendel method is also discussed.

The centroid of a general type-2 fuzzy set (T2 FS) à can be obtained by taking the union of the centroids of all the α-planes (each raised to level α) of Ã. Karnik-Mendel (KM) or the Enhanced Karnik-Mendel (EKM) algorithms are used for computing 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 à 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% to 80% and 50% to 75% compared to KM and EKM algorithms, respectively, and still maintains satisfactory computation accuracy for various T2 FSs when the primary variable x and ar-plane are discretized finely enough.

This comment points out a misnomer and two errors in a previous paper by the author (IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1189-1207, Oct. 2009), and because one of the errors relates the term "α-plane" to the term "z-slice," it also connects these two terms more correctly.

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.

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.

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., PR using firing intervals is proposed in [8], [9], [16], and similarity-based PR is proposed in this paper. Both approaches satisfy the constraint on a CWW engine, i.e., 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 the output FOUs from similarity-based PR more closely resemble the three kinds of FOUs in a codebook, and the resulting linguistic descriptions are more intuitive; so, similarity-based PR is a better choice for a CWW engine.

Interval type-2 fuzzy logic control systems (IT2 FLCSs) have the potential of handling uncertainties better than type-1 FLCSs. However, lack of systematic design methodology of IT2 FLCSs limits their utility. This paper presents systematic methods to design interval IT2 Takagi-Sugeno-Kang (TSK) FLCSs that are PD-type and PI-type fuzzy controllers to satisfy certain desired transient response. We adopt the MacVicar-Whelan rule-base system and present general schemes for the design of IT2 TSK FLCSs, that include the design of the TSK consequent parameters. To validate the performance of the proposed controllers, some nonlinear plants have been considered. Results show that the IT2 TSK FLCSs satisfy the desired performance measures in terms of a set point tracking. Moreover, they reveal remarkable improvements in comparison to their type-1 counterparts for the plants considered in this paper.

Recently a simple and practical type-2-fuzzistics methodology called an interval approach (IA) was presented for obtaining interval type-2 fuzzy set (IT2 FS) models for words using data collected from a group of subjects. There may be times, however, when a group of subjects is not available. This paper proposes a way to obtain IT2 FS models from words collected from a single subject using an IA, and is called an individual IA (IIA). Two methods are presented for doing this. Both use end-point and uncertainty data that are collected from an individual, assume a probability distribution on each interval, map them into pre-specified T1 membership functions (MF), interpret the MFs as nine embedded T1 FSs of an IT2 FS, and then aggregate the FSs using union to obtain the footprint of uncertainty (FOU) for the word. This approach not only captures the strong points of the previously developed IA but simplifies it. Experiments show that the IIA is easy to implement and the resulting FOUs match our intuition.

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 ¿-plane representation technique, this paper generalizes these definitions to general 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 (FOU) and different triangular secondary membership functions are computed and are given as examples.

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.

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.

Type-2 fuzzy systems have successfully been applied in control applications. Due to the complicated structure of type-2 systems, they lack systematic control design and hence the stability of the system is not guaranteed. This paper presents stability analysis of dynamic type-2 Takagi-Sugeno-Kang (TSK) fuzzy systems. Novel inference mechanisms for type-2 TSK systems for the case when antecedents are type-2 and consequents are crisp numbers (A2-C0) are developed and utilized in fuzzy model generation. Owing to the simple nature of the proposed methods, they are easy to implement in real-time applications. One of the proposed inference mechanisms is used and the sufficient stability conditions for these systems are derived. It is shown that the criteria obtained herein must satisfy some linear matrix inequalities (LMI) and an algorithm is also presented to solve the obtained LMI. Two numerical examples are provided that detail the design method. The methodology presented proves to be an efficient approach to systematically design stable dynamic type-2 TSK fuzzy systems.

This paper provides an answer to the question that the type-2 fuzzy logic community is now asking: “What comes after interval type-2 fuzzy logic systems (TT2 FLSs)?” It demonstrates, through a geometrical understanding of the type-reduced set, that logical next steps in the progression from type-1 to interval type-2 to type-2 FLSs are quasi-T2 FLSs, either an interconnection of a T1 FLS and an IT2 FLS, or an interconnection of two IT2 FLSs, in which both FLSs are designed simultaneously. The quasi-T2 FLSs overcome the computational difficulties that are associated with set theoretic operations and type-reduction (TR) for general T2 FSs and FLSs, because all set theoretic operations can be performed as in existing T1 or IT2 FLSs, and because TR for an IT2 FLS can be performed using existing KM Algorithms.

Perceptual Reasoning (PR) is an Approximate Reasoning mechanism that can be used as a Computing with Words (CWW) Engine, i.e., given input words, PR can infer the output from a rulebase. When the input words and the words in the rulebase are modeled by interval type-2 fuzzy sets (IT2 FSs), the output of PR, ỸPR, is also an IT2 FS, and it will be mapped to a word in a codebook. For accurate mapping, we need to ensure that ỸPRresembles the IT2 FSs in the codebook. The concept of PR using IT2 FSs was originally proposed in [10]. In this paper, the procedures to compute PR are introduced, and the properties of PR are studied in more detail. More specifically, we show under what conditions ỸPRcan be a shoulder or interior footprint of uncertainty.

By connecting work from two different problems-the fuzzy weighted average (FWA) and the generalized centroid of an interval type-2 fuzzy set-a new alpha-cut algorithm for solving the FWA problem has been obtained, one that is monotonically and superexponentially convergent. This new algorithm uses the Karnik-Mendel (KM) algorithms to compute the FWA -cut end-points. It appears that the KM -cut algorithms approach for computing the FWA requires the fewest iterations to date, and may therefore be the fastest available FWA algorithm to date.

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 ofcomputingwithwords(CWW). A specific architecture for making subjective judgments using CWW was proposed by Mendel in 2001. It is called aPerceptualComputer(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 calledPerceptualReasoning, 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.

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.

In this paper, a new and simple approach, calledinterval approach, to type-2 fuzzistics is presented, one that captures the strong points of both the person-MF and interval end-points approaches. It uses interval end-point data that are collected from a group of subjects, assumes a probability distribution for each person's data and maps the mean and standard deviation of that distribution into the parameters of an iteratively specified type-1 person MF. These type-1 person MFs are then aggregated using the union leading to the FOU for a word. Experiments show that this approach is easy to implement and the derived interval type-2 word models match our intuitions, i.e., the FOUs of the small-sounding words are located to the left, the FOUs of the medium-sounding words are located in the middle, and the FOUs of the large-sounding words are located to the right.

Fuzzy logic is frequently used incomputing with words(CWW). When input words to a CWW engine are modeled by interval type-2 fuzzy sets (IT2 FSs), the CWW engine's output can also be an IT2 FS,Ã, which needs to be mapped to a linguistic label so that it can be understood. Because each linguistic label is represented by an IT2 FSB̃i, there is a need to compare the similarity ofÃandB̃ito find theB̃imost similar toÃ. In this paper, a vector similarity measure (VSM) is proposed for IT2 FSs, whose two elements measure the similarity in shape and proximity, respectively. A comparative study shows that the VSM gives more reasonable results than all other existing similarity measures for IT2 FSs.

Computing the centroid of an interval T2 FS is an important operation in a type-2 fuzzy logic system (where it is called type-reduction), but it is also a potentially time-consuming operation. The Karnik-Mendel (KM) iterative algorithms are widely used for doing this. In this paper, we prove that these algorithms converge monotonically and super-exponentially fast. Both properties are highly desirable for iterative algorithms and explain why in practice the KM algorithms have been observed to converge very fast, thereby making them very practical to use

In Part 1 of this two-part paper, we bounded the centroid of a symmetric interval type-2 fuzzy set (T2 FS), and consequently its uncertainty, using geometric properties of its footprint of uncertainty (FOU). We then used these bounds to solve forward problems, i.e., to go from parametric interval T2 FS models to data. The main purpose of the present paper is to formulate and solve inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models, which we call type-2 fuzzistics. Given interval data collected from people about a phrase, and the inherent uncertainties associated with that data, which can be described statistically using the first- and second-order statistics about the end-point data, we establish parametric FOUs such that their uncertainty bounds are directly connected to statistical uncertainty bounds. These results should find applicability in computing with words

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 solveforward 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.

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

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.

In this paper we introduce embedded type-2 fuzzy relations and provide an alternative proof of the extended sup-star composition that makes use of these embedded type-2 fuzzy relations. The paper commences by giving a background to some terms and definitions used in type-2 fuzzy logic. Definitions are given for a type-2 fuzzy set, a secondary membership function and an embedded set. Additionally the representation theorem is described along with an explanation of type-2 fuzzy relations. These preliminary sections provide the basis for the remainder of the paper, which introduces the concept of embedded fuzzy relations and makes use of this in order to provide an alternative proof of the extended sup-star composition.

Computing the centroid of an interval T2 FS is an important operation in a type-2 fuzzy logic system (where it is called type-reduction), but it is also a potentially time-consuming operation. The Karnik-Mendel (KM) iterative algorithms are widely used for doing this. In this paper we prove that these algorithms converge monotonically and super-exponentially fast. Both properties are highly desirable for iterative algorithms and explain why in practice the KM algorithms have been observed to converge very fast, thereby making them very practical to use.

The focus of this paper is the linguistic weighted average (LWA), which is a generalization of the fuzzy weighted average (FWA) that is obtained by replacing the type-1 fuzzy inputs in the FWA by interval type-2 fuzzy sets (IT2 FSs). Consequently, the output of the LWA is an IT2 FS. In this paper, the relations between the LWA and the FWA are studied. It is shown that finding the LWA can be decomposed into finding two FWAs, where alpha-cuts and KM algorithms are used. Hence, the computational cost of a LWA is about twice that of a FWA. A flowchart for computing the LWA is also provided.

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

Interval type-2 fuzzy sets (T2 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). In this two-part paper, we focus on symmetric interval T2 FSs for which the centroid (which is an interval type-1 FS) provides a measure of its uncertainty. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, will be associated with the amount of uncertainty in such a T2 FS. The main purpose of this paper (Part 1) is to demonstrate that our intuition is correct and to quantify the centroid of a symmetric interval T2 FS, and consequently its uncertainty, with respect to such geometric properties. It is then possible, for the first time, to formulate and solve forward problems, i.e., to go from parametric interval T2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems. These solutions are used in Part 2 to solve some inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models (T2 fuzzistics)

The centroid of an interval type-2 fuzzy set (IT2 FS) provides a measure of the uncertainty of such a FS. Its calculation is very widely used in interval type-2 fuzzy logic systems. In this paper, we present properties about the centroid of an IT2 FS. We also illustrate many of the general results for a T2 fuzzy granule (FG) in order to develop some understanding about the uncertainty of the FG in terms of its vertical and horizontal dimensions. At present, the T2 FG is the only IT2 FS for which fit is possible to obtain closed-form formulas for the centroid, and those formulas are in this paper

Interval type-2 fuzzy sets (T2 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 interval T2 FS, which is an interval T1 FS, provides a measure of the uncertainty in the interval T2 FS. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, associated with the amount of uncertainty in an interval T2 FS. The main purpose of this paper is to demonstrate that our intuition is correct and to quantify the centroid of an interval T2 FS with respect to these geometric properties of its FOU. It is then possible to formulate and solve inverse problems, i.e., going from data to parametric T2 FS models.

We explore the reduction of a fuzzy classifier designed to perform a binary classification of tracked or wheeled vehicles based on acoustic data. A genetic algorithm is used to explore the design space of the classifier, with variations performed on the number of antecedents included in the final fuzzy system. Besides the original individual set generated by the GA, we define a subset of it with a small number of antecedents as a filtered set. A novel method of extracting important system components, known as open product analysis, is applied to these two sets, yielding systems that perform well with a small number of antecedents. The fuzzy classifier we reduced performs well using only 20 to 30% of the antecedents that were originally used for classification.

We investigate ten compensatory operators and SOWA operators in the framework of Mamdani interval type-2 fuzzy logic systems (FLS) so that for the first time the uncertainties originating from descriptive words, connector words and data can be simultaneously modeled. Our investigations show that: 1) for a Mamdani singleton interval type-2 FLS all the ten operators can be implemented and optimized; and 2) for a Mamdani non-singleton interval type-2 FLS the multiplicative compensatory operator that uses the product t-norm and maximum t-conorm, /spl Phi//sub p//sup MCA/, can be implemented and optimized. We apply /spl Phi//sub p//sup MCA/ to chaotic time-series prediction where the observations are corrupted by non-stationary noise. Our experimental results show that by incorporating /spl Phi//sub p//sup MCA/ into a Mamdani interval type-2 FLS it may take less time to train an interval type-2 FLS to achieve a satisfactory performance, and the resulting system is more robust to noise.

This paper makes type-2 fuzzy logic systems much more accessible to fuzzy logic system designers, because it provides mathematical formulas and computational flowcharts for computing the derivatives that are needed to implement steepest-descent parameter tuning algorithms for such systems. It explains why computing such derivatives is much more challenging than it is for a type-1 fuzzy logic system. It provides derivative calculations that are applicable to any kind of type-2 membership functions, since the calculations are performed without prespecifying the nature of those membership functions. Some calculations are then illustrated for specific type-2 membership functions.

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 begins with a delineation of two approaches to fuzzy sets, abstract mathematics and models for words. It demonstrates, by using Karl Popper's Falsificationism, the present approach to fuzzy sets (FSs) for words is scientifically incorrect. A new theory of fuzzy sets is then presented for words that is based on collecting data from people -person MFs-that reflect intra- and inter-levels of uncertainties about a word, and defines a word FS as the union of all such person fuzzy sets. It also demonstrates that intra-uncertainty about a word can be modeled using type-2 person fuzzy sets, and that inter-uncertainty about a word can be modeled by means of an equally weighted union of each person's type-2 fuzzy set. Finally, it proposes a methodology for obtaining a parsimonious parametric type-2 fuzzy set approximation to the aggregated type-2 person FSs. This new theory of fuzzy sets for words is testable and is therefore subject to refutation.

A Modulated-Reasoning fuzzy rule operator is one involving a product of Logical-Reasoning and Mamdani fuzzy rule operators, in which each operator is raised to a power and the sum of the powers equals one. In this paper we establish which of 30 possible Modulated-Reasoning fuzzy rule operators lead to fired-rule output fuzzy sets (or a property of such a set) for a fuzzy logic system (FLS) that can be expressed as closed form mathematical expressions that are (piecewise-) differentiable with respect to membership function (MF) parameters, since such operators could then be used in the back-propagation design of a FLS. We do this for singleton fuzzification and two kinds of defuzzifiers, combined-centroid and property-centroid. We show that Modulated Reasoning is practical for singleton fuzzification, but if Gaussian (or any other exponential) MFs are used there is no difference between it and Mamdani Reasoning.

We examine ten antecedent connector models in the framework of a singleton or non-singleton 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 (MF) parameters and also the parameter characterizing that connector model. The. multiplicative compensatory and model that uses the product t-norm and maximum t-conorm, /spl Phi//sub p//sup MCA/, is shown to be usable for both singleton and non-singleton Mamdani-product FLSs. We also show, by examples, that the parameter of /spl Phi//sub p//sup MCA/ 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.

We demonstrate that some of the so-called uncertainties that may be present in a rule-based fuzzy logic system (FLS) are not uncertainties, but are instead choices that must be made as a result of the rich variety of mathematical models associated with the various elements of a FLS. We have established a hierarchy for the uncertainties and the choices, one that will hopefully guide us in developing FLS models that can better account for all sources of uncertainties than do present models. It seems that, at the very least, we must account for the uncertainties present in all rule-words, including connector words. This can be accomplished by using type-2 fuzzy sets for antecedent and consequent words and parametric operators for connector words.

Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems. However, they are difficult to understand for a variety of reasons which we enunciate. In this paper, we strive to overcome the difficulties by: (1) establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, (2) presenting a new representation for type-2 fuzzy sets, and (3) using this new representation to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use the Extension Principle.

We derive inner- and outer-bound sets for the type-reduced set of an interval type-2 fuzzy logic system (FLS), based on a new mathematical interpretation of the Karnik-Mendel iterative procedure for computing the type-reduced set. The bound sets can not only provide estimates about the uncertainty contained in the output of an interval type-2 FLS, but can also be used to design an interval type-2 FLS. We demonstrate, by means of a simulation experiment, that the resulting system can operate without type-reduction and can achieve similar performance to one that uses type-reduction. Therefore, our new design method, based on the bound sets, can relieve the computation burden of an interval type-2 FLS during its operation, which makes an interval type-2 FLS useful for real-time applications.

Our thesis is that computing with words needs to account for the uncertainties associated with the meanings of words, and that these uncertainties require using type-2 fuzzy sets. Doing this leads to a proposed architecture for making judgments by means of computing with words - a perceptual computer Per-C. The Per-C includes an encoder, a type-2 rule-based fuzzy logic system, and a decoder. It lets all human-computer interactions be performed using words. In this paper, a quantitative language is established for the Per-C, and many open issues about the perceptual computer are described.

In this paper, we derive inner- and outer-bound sets for the type-reduced set of an interval type-2 fuzzy logic system, based on a new mathematical interpretation of the Karnik-Mendel (2001) iterative procedure. The bound sets can not only provide estimates about the uncertainty contained in the output, but can also be used to design an interval type-2 fuzzy logic system. We demonstrate, by means of a simulation experiment, that the resulting system can operate without type reduction and that it can achieve similar performance to one that uses type reduction. Therefore, our new design method, based on the bound sets, can relieve the computational burden of an interval type-2 fuzzy logic system during its operation.

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.

We propose an efficient and simplified method to compute the input and antecedent operations for interval type-2 FLSs using the concept of upper and lower membership functions (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 non-stationary time-series is corrupted by additive noise where SNR is uncertain, and demonstrate improved performance over type-1 FLSs.

Presents a decision feedback equalizer (DFE) that uses type-2 fuzzy adaptive filters (FAFs). These FAFs are realized using an unnormalized type-2 TSK fuzzy logic system. We apply our DFE to 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. Our DFE vastly reduces computational complexity as compared to a transversal equalizer. Simulation results show that our type-2 FAF-based DFE performs much better than nearest neighbor classifiers or a DFE based on type-1 FAFs.

A type-2 fuzzy logic system (FLS) can handle numerical and linguistic uncertainties, but, like a type-1 FLS, rule explosion is one of its major disadvantages. In this paper, we present a design method which can tremendously reduce rule number for interval type-2 fuzzy logic systems using an SVD-QR method. The SVD-QR method is performed after extracting two fuzzy basis function expansions from the interval type-2 FLS. We evaluate this method by applying it to a time-series forecasting problem in conjunction with back-propagation training, and demonstrate that tremendous rule number reduction ratio is achieved with very little performance degradation. © 2000 John Wiley & Sons, Inc.

We apply the maximum-likelihood (ML) method to the classification of digital quadrature modulations. We show that under an ideal situation, the I-Q domain data are sufficient statistics for modulation classification and obtain a generic formula for the error probability of a ML classifier. Our study of asymptotic performance shows that the ML classifier is capable of classifying any finite set of distinctive constellations with zero error rate when the number of available data symbols goes to infinity.

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.

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.

A new neuron activation function based on a property found in physical systems--hysteresis--is proposed. We incorporate this neuron activation in a fully connected dynamical system to form the hysteretic Hopfield neural network (HHNN). We then present an analog implementation of this architecture and its associated dynamical equation and energy function.We proceed to prove Lyapunov stability for this new model, and then solve a combinatorial optimization problem (i.e., the N-queen problem) using this network. We demonstrate the advantages of hysteresis by showing increased frequency of convergence to a solution, when the parameters associated with the activation function are varied.

This paper presents a connection admission control (CAC) method that uses a type-2 fuzzy logic system (FLS). Type-2 FLSs can handle linguistic uncertainties. The linguistic knowledge about CAC is obtained from 30 computer network experts. A methodology for representing the linguistic knowledge using type-2 membership functions and processing surveys using type-2 FLS is proposed. The type-2 FLS provides soft decision boundaries, whereas a type-1 FLS provides a hard decision boundary. The soft decision boundaries can coordinate the cell loss ratio (CLR) and bandwidth utilization, which is impossible for the hard decision boundary.

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 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.

Type-reduction in a type-2 fuzzy logic system (FLS) is an "extended" version of the defuzzification operation in a type-1 FLS. In this paper, we briefly review the structure of a type-2 FLS and describe type-reduction in detail. We focus on a center-of-sets type-reducer, and provide some examples to illustrate it. We also provide some practical approximations to type-reduction computations for certain type-2 membership functions.

Prospective authors are requested to submit new, unpublished manuscripts for inclusion in the upcoming event described in this call for papers.

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).

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.

We initiate an investigation of the use of nonsingleton fuzzy logic systems (NSFLSs) in forecasting of financial markets. The abilities of NSFLSs to approximate arbitrary functions, and to effectively deal with noise and uncertainty, are used to analyze several time series. First we show how to construct NSFLSs and train them using recursive least squares, or backpropagation. Then we use them to build predictive models of discrete and continuous chaotic time series corrupted by additive noise. Finally, we present an example of how NSFLSs can be used to produce predicted estimates of future values of commodities, and baseline our results with linear regression. Our NSFLS outperforms the linear regression results.

We propose an optimum cumulant-based blind beamforming method for signal recovery in coherent signal environments. Our approach is applicable to any array configuration having arbitrary and unknown response. There is no need to estimate the directions of arrival. A comparable result does not exist using second-order statistics.

A fuzzy logic system (FLS) is unique in that it is able to simultaneously handle numerical data and linguistic knowledge. It is a nonlinear mapping of an input data (feature) vector into a scalar output, i.e., it maps numbers into numbers. Fuzzy set theory and fuzzy logic establish the specifics of the nonlinear mapping. This tutorial paper provides a guided tour through those aspects of fuzzy sets and fuzzy logic that are necessary to synthesize an FLS. It does this by starting with crisp set theory and dual logic and demonstrating how both can be extended to their fuzzy counterparts. Because engineering systems are, for the most part, causal, we impose causality as a constraint on the development of the FLS. After synthesizing a FLS, we demonstrate that it can be expressed mathematically as a linear combination of fuzzy basis functions, and is a nonlinear universal function approximator, a property that it shares with feedforward neural networks. The fuzzy basis function expansion is very powerful because its basis functions can be derived from either numerical data or linguistic knowledge, both of which can be cast into the forms of IF-THEN rules.>

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.>

The authors propose a cumulant-based method to perform voice-unvoiced decision and pitch period estimation. The approach is based on the nature of excitation for different states of speech. The authors accomplished this goal by analyzing cumulant-related time sequences obtained via nonlinear processing of the speech signal. Experimental results indicating the performance of the proposed method, especially in the pitch estimation problem in which there are high power harmonics are presented.>

The authors answer the following questions regarding solving the harmonic retrieval problem using second- or fourth-order statistics: (1) How do cumulant-based results compare against correlation-based results?; (2) How robust are the cumulant-based methods to tones of different local SNRs?; (3) How close can the two tones be brought together before cumulant-based methods break down?; (4) Does one have to use a high-resolution method, such as MUSIC or Minimum Norm, or can the Pisarenko method be used?; and (5) What is the 'footprint' of success for the cumulant-based methods? Extensive simulations are used to provide the answers.>

A novel approach based on fuzzy set theory is developed for recognizing handwritten rotated characters. This fuzzy approach consists of four steps: (1) generating crisp sets for reference characters rotated through different degrees; (2) fuzzifying these crisp sets; (3) determining the degrees of a given character to the fuzzy sets; and (4) classifying the given character based on an average rule or a maximum rule. Simulation results show that the fuzzy approach correctly classified 94% to 100% of a small test set of characters.>

Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules.

A general method is developed to generate fuzzy rules from numerical data. The method consists of five steps: divide the input and output spaces of the given numerical data into fuzzy regions; generate fuzzy rules from the given data; assign a degree of each of the generated rules for the purpose of resolving conflicts among the generated rules; create a combined fuzzy rule base based on both the generated rules and linguistic rules of human experts; and determine a mapping from input space to output space based on the combined fuzzy rule base using a defuzzifying procedure. The mapping is proved to be capable of approximating any real continuous function on a compact set to arbitrary accuracy. Applications to truck backer-upper control and time series prediction problems are presented.>

An autoregressive (AR) model which is excited by a non-zero-mean, independent and identically distributed stationary random process is investigated. As in the zero-mean case, a cumulant-based higher-order Yule-Walker equation is derived. By expanding the cumulants in terms of their moments, a higher-order moment formula is obtained. This formula not only relates the higher-order moment with the lower-order moment, but also makes it possible to estimate the AR parameters and the output mean simultaneously. The formula is computationally more efficient than the cumulant formula.>

Three Hopfield (1984, 1985) neural networks are developed to realize a new adaptive minimum prediction-error deconvolution procedure. The first neural network is developed to detect the reflectivity sequence. The second neural network is developed to determine the magnitudes of the detected reflections. The third neural network is developed to estimate the seismic wavelet. A block-component method is proposed for simultaneous reflectivity estimation and wavelet extraction based on these three neural networks. These three neural networks and the block-component method are simulated for a narrowband wavelet. Real seismic data are processed using the block-component method, and the results are compared with those using the minimum variance deconvolution (MVD) filter and the maximum-likelihood based SMLR detector.>

A compendium of recent theoretical results associated with using higher-order statistics in signal processing and system theory is provided, and the utility of applying higher-order statistics to practical problems is demonstrated. Most of the results are given for one-dimensional processes, but some extensions to vector processes and multichannel systems are discussed. The topics covered include cumulant-polyspectra formulas; impulse response formulas; autoregressive (AR) coefficients; relationships between second-order and higher-order statistics for linear systems; double C(q,k) formulas for extracting autoregressive moving average (ARMA) coefficients; bicepstral formulas; multichannel formulas; harmonic processes; estimates of cumulants; and applications to identification of various systems, including the identification of systems from just output measurements, identification of AR systems, identification of moving-average systems, and identification of ARMA systems.>

Two three-dimensional structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is to first represent a given equation-solving problem by a 3-D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem: then, train the 3-D structured network to match the desired pattern array using some training algorithms; and finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3-D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3-D structured networks.>

The problem of identifiability of a wind vector that is estimated from wind scatterometer measurements of the radar backscatter of the ocean's surface is addressed. The traditional wind estimation approach produces multiple estimates of the wind direction. A second processing step, known as dealiasing or ambiguity removal, is used to select a single wind estimate from these multiple solutions. Dealiasing is typically based on various ad hoc considerations. The traditional wind estimation approach results in multiple solutions associated with local minima in an objective function formed from the noisy backscatter measurements. The authors discuss the question of the uniqueness of the wind vector estimates resulting from this intuitive approach.>

A two-level three-layer structured network is developed to estimate the moving-average model parameters based on second-order and third-order cumulant matching. The structured network is a multilayer feedforward network composed of linear summers in which the weights of these summers have a clear physical meaning. The first level is composed of random access memory units, which are used to control the connectivities of the second-level summers. The second level is composed of three layers of linear summers in which the weight of any summer represents the moving-average parameter to be estimated. The connectivities among these summers are controlled by the first-level memory units in such a way that the outputs of the second-level structured network equal the desired second-order or third-order statistics if the summer weights equal their corresponding true moving-average parameter values. Each second-order and third-order cumulant is viewed as a pattern which the structured network needs to learn, and a steepest-descent algorithm is proposed for training the structured network. The author also presents extensions to particular sorts of estimation, and results of simulations.

A two-level, three-layer artificial neural network to estimate the MA (moving average) model parameters based on second- and third-order cumulant matching is developed. The first level is composed of some RAM units that are used to control the synaptic connectivities of the second-level neurons. The second level is composed of three layers of linear weighted-sum neurons in which the weight parameters of any neuron represent the MA parameter to be estimated. Each second- and third-order cumulant is viewed as a pattern the neural network needs to learn, and a steepest descent algorithm is proposed to train the neural network. The main advantage of this approach is that it uses a parallel architecture to represent the problem and a parallel to perform the estimation. A simulation is performed to demonstrate the performance of the neural network approach. Extension to ARMA (autoregressive moving-average) parameter estimation is discussed.>

A novel approach to a large variety of matrix algebra problems is proposed. The basic idea is to represent a given problem by a structured network architecture, train the structured network to match some desired patterns, and obtain the solution to the problem from the weights of the resulting structured network. The basic unit used to construct the network is a simple linear multi-input, single-output weighted summer. The training algorithms for the problems are either standard error back-propagation or the modified error back-propagation. Three detailed structured networks and the corresponding training algorithms are presented for matrix LU decomposition, linear equation solving and singular value decomposition, respectively. Extensions to other matrix algebra problems are straightforward. These new approaches use parallel architectures and algorithms, suitable for VLSI realizations; provide robust computations, with no divisions involved in all the calculations, so that they are free of the divide-by-zero problem; and are very general, suitable for most matrix computation and matrix equation-solving problems

Using a compact Kronecker-product-based representation for the cumulants of vector processes, the authors develop several techniques for estimating the parameters of a multichannel ARMA (autoregressive moving average) process, from sample cumulants of the output processes: (1) the AR parameters are estimated first; the MA parameters are then estimated from the AR compensated time series. (2) AR and IR (impulse response) parameters are estimated simultaneously; (3) an algorithm that handles causal as well as noncausal ARMA models, by transforming the ARMA parameter estimation problem to a pair of MA parameter estimation problems, is given. Order-determination techniques are also proposed. The algorithms are applicable to both stochastic and deterministic problems.>

Using the observable state-space realization corresponding to a given multi-input-multi-output autoregressive moving average (ARMA) model, the authors derive closed-form and lag-recursive expressions for the cumulants of the output process. Their approach involves the computation of cumulants of vector processes, which they define compactly in terms of Kronecker products, and leads to a unified treatment of multichannel, time-varying and nonstationary processes. Computational aspects are discussed in detail. A new cumulant-based identification method is proposed in which the matrices of the SSM are first estimated and then transformed to ARMA parameters.>

The authors present the class of semi-Markov random fields and deal, in particular, with the subclass of discrete-valued, nonsymmetric half-plane, unilateral Markov random fields. A hierarchical nonstationary-mean nonstationary-variance (NMNV) image model is proposed for the modeling of nonstationary and noisy images. This model seems to be advantageous as compared to a regular NMNV model because it statistically incorporates the correlation between pixels around the boundary of two adjacent regions. The hierarchical NMNV model leads to the development of an optimal algorithm that allows the simultaneous segmentation and model estimation of measured images. Although no theoretical result is available for the consistency of the estimated model parameters, the method seems to work sufficiently well for the examples considered.>

General problems and solutions are described for maximum a posteriori estimation of multichannel Bernoulli-Gaussian sequences, which are inputs to a linear discrete-time multivariable system. The authors first develop a separation principle, which indicates that one can estimate multichannel Gaussian amplitudes and Bernoulli events separately. They then discuss approaches for estimation of these quantities.>

The authors extrapolate, in the maximum-entropy (ME) sense, one-dimensional (1-D) cumulant statistics of a stationary random process which is the output of a linear, time-invariant (LTI) model excited by a non-Gaussian, independent and identically distributed input. The entropy rate of a linear process, is related with a special 1-D polyspectrum. Based on this relationship they derive 1-D polyspectral estimates that correspond to the most random time series whose cumulant sequence is consistent with the given finite set of 1-D cumulant statistics. The ME extension of the cumulant sequence of linear processes corresponds to that of an AR (autoregressive) process whose coefficients can be computed as the solution of a system of linear equations. The AR filter obtained using the ME cumulant extrapolation is applied to harmonic retrieval, and phase estimation of nonminimum-phase LTI systems.>

Given finite samples of a stationary, perhaps noisy, nonGaussian r-variate moving-average, MA(q) process, the authors study cumulant based identifiability conditions, under which the MA coefficient matrices, the input statistics, and the order q, can be uniquely determined. The selection of a unique representative from the equivalence class corresponding to a given cumulant structure involves less restrictions than that corresponding to a given covariance structure. They derive two algorithms for estimating the (possibly) nonminimum phase MA coefficient matrices.>

A lattice version of the recursive instrumental variable method for adaptive parameter identification of ARMA (autoregressive moving-average) processes is developed. Appropriate choice of the instrumental variables leads to cumulant-based AR parameter estimates. Cumulant-based normal equations may be obtained by using nonconventional orthogonality conditions in the linear prediction problem. The development leads to a pair of lattices, one excited by the observed process y(n), and the other by the instrumental process z(n). The lattices are coupled through order-update and time-update equations. The lattice structure yields the AR compensated residual time series. Hence, adaptive versions of cumulant-based MA parameter identification algorithms are directly applicable. Some convergence results are presented.>

A time-series consisting of sinusoids observed in additive i.i.d. noise or in additive colored Gaussian noise of unknown spectral density is considered. The number of harmonics, as well as their amplitudes and frequencies are determined using the one-dimensional diagonal slice of the fourth-order cumulant. Applications to the detection of cubic phase coupling are discussed.>

An optimal algorithm for the detection of noisy filtered discrete semi-Markov chains is presented. Estimation of the underlying model parameters is also considered. For a given path of the discrete semi-Markov chain the optimum estimates of the model parameters obtained by the maximum likelihood method are expressed as functions of the path. These functions are then used to derive a single maximum a posteriori criterion for the optimal detection of the unknown single path. The final optimization is carried out numerically by a combination of gradient, divide-and-conquer, and search techniques. This set of techniques is referred to as the integer most likely search detector. Experimental results, using synthetic data, demonstrate the potential of the algorithm.>

The Total Least Squares (TLS) method is a generalized least square technique to solve an overdetermined system of equationsAx\simeqb. The TLS solution differs from the usual Least Square (LS) in that it tries to compensate for arbitrary noise present in bothAandb. In certain problems the noise perturbations ofAandbare linear functions of a common "noise source" vector. In this case we obtain a generalization of the TLS criterion called the Constrained Total Least Squares (CTLS) method by taking into account the linear dependence of the noise terms inAandb. If the noise columns ofAandbare linearly related then the CTLS solution is obtained in terms of the largest eigenvalue and corresponding eigenvector of a certain matrix. The CTLS technique can be applied to problems like Maximum Likelihood Signal Parameter Estimation, Frequency Estimation of Sinusoids in white or colored noise by Linear Prediction and others.

One dimensional cumulant and auto-correlation output statistics are combined to form an overdetermined system of equations whose least-squares solution yields the coefficients of an ARMA model. The driving input noise is assumed to be non-Gaussian and white. The ARMA model is allowed to be non-minimum phase and even to contain all-pass factors. The special cases of AR and MA models are also included. The overdetermined nature of the method makes the solution practical for moderate output data lengths, when additive white Gaussian noise is considered. Simulations illustrate that our approach performs very well even at low signal-to-noise ratios.

Based on the Maximum-Likelihood principle, we develop a locally optimal method for detecting the location and estimating the amplitude of spikes in a sequence, which are considered the random input of a known ARMA model. A Bernoulli-Gaussian product model is adopted for the sparse-spike sequence, and the available data consist of a single, noisy, output record. By employing a Prediction-Error formulation our iterative algorithm guarantees the increase of a unique likelihood function used for the combined estimation/detection problem. Amplitude estimation is carried out with Kalman smoothing techniques, and event detection is performed in two ways, as an event adder and as an event remover. Synthetic examples verify that our algorithm is self-initialized, consistent, and fast.

In this paper we examine the problem of image segmentation of noisy images. We consider a doubly stochastic image model. The image is assumed to be the sum of the realizations of two independent random fields: the uncorrupted image and the noise field, consisting of independent, identically distributed, Gaussian random variables. The image segmentation technique employed here is a technique in which the image is represented by a semi-Markov random field corrupted by additive white noise. An adaptive Bayesian parameter estimation/image detection algorithm is developed. This algorithm allows us to estimate the unknown image and its underlying parameters in an optimal manner. We demonstrate the potential of the proposed algorithm in the case of the smoothing/segmentation of two 4-gray level real images.

In this paper we present a one-dimensional normal-incidence inversion procedure for reflection seismic data. A lossless layered system is considered which is characterized by reflection coefficients and traveltimes. A priori knowledge for the unknown parameters, in the form of statistics, is incorporated into a nonuniform layered system, and a maximum a posteriori estimation procedure is used for the estimation of the system's unknown parameters (i.e., we assume a random reflector model) from noisy and band-limited data. Our solution to the inverse problem includes a downward continuation procedure for estimation of the states of the system. The state sequences are composed of overlapping wavelets. We show that estimation of the unknown parameters of a layer is equivalent to estimation of the amplitude and detection of the time delay of the first wavelet in the upgoing state sequence of the layer. A suboptimal maximum-likelihood deconvolution procedure is employed to perform estimation and detection. The most desirable features of the proposed algorithm are its layer-recursive structure and its ability to process noisy and band-limited data.

We derive and implement a maximum-likelihood detection and estimation algorithm based on the same channel and statistical models used by Kormylo and Mendel [1], that leads to less computations than the approach presented by Chi, Mendel and Hampson [2]. We introduce a single generalized likelihood function and we develop the Multiple-Most-Likely Replacement (MMLR) detector. This detector is computationally faster compared with the Single-Most-Likely Replacement (SMLR) detector developed by Kormylo and Mendel [3]. We demonstrate good performance of our algorithm for a synthetic data example.

Recently, we observed zero phase and undershoot patterns in data processed by a minimum-variance deconvolution (MVD) filter. These observations motivated a careful analys is of the MVD filter, which, as we demonstrate in this paper, explains both the zero phase and undershoot patterns. This analysis also connects the MVD filter with the well-known prediction-error filter [6], and Berkhout's two-sided least-squares inverse filter [7]. We show that the performance of the MVD filter depends heavily on the bandwidth of the source wavelet, and signal-to-noise ratio, and only slightly on data length.

When a wavelet to be estimated is not spiky, then a single most likely replacement (SMLR) detector, which is used to detect randomly located impulsive events that have Gaussian-distributed amplitudes, may split a large spike into two smaller ones and may also detect some spikes at wrong locations, although these locations are very close to their true ones. Presented here are two new detection algorithms, namely a single-spike-shift (SSS) detector and an SSS-SMLR detector both of which help correct the SMLR detector's spike-splitting and shifting problem.

In this paper Mendel and Hsueh's [10] state-variable modeling technique for transforming a 1-D non-causal system into a causal system is extended to 2-dimensional systems which have half-plane (HP) supports. The 2-D impulse responses treated here are restricted to a class in which their associated 2-D z-transforms have separable denominators. The final state space model is a special type of so-called Roesser's model. It is a very low-order model, which is important from a computational point of view, when, for example, 2-D recursive estimation algorithms are applied to 2-D systems with HP supports.

In this paper we develop optimal recursive waveshaping filters in the framework of estimation theory and state-variable models. We develop a linear minimum-variance waveshaper and a nonlinear maximum-likelihood waveshaper. Both waveshapers are comprised of two components:(1) stochastic inversion and (2) waveshaping. The former is performed by means of minimum-variance deconvolution. Simulation results are given which illustrate results that can be obtained by both waveshapers. In retrospect, we view the minimum-variance results of this paper as the recursive counterparts to those presented by Treitel and Robinson (13), which are for finite-impulse response waveshaping.

A new detection algorithm, single most likely replacement (SMLR), for detecting randomly located impulsive events which have Gaussian-distributed amplitudes is presented. This detector is designed for the case of severely overlapping wavelets. Estimation of the probability of events also is consider. Experimental results and comparisons with other detectors, using synthetic data, are provided.

We consider plane-wave motion at normal incidence in a horizontally layered system. The system is assumed lossless, and only the compressional waves are treated. A procedure is introduced for determining the reflection coefficients of the layered system when the observed seismic data may contain random noise. No deconvolution of the measured seismic data is required by the procedure when the input is a narrow wavelet.