comparing the performance potentials of interval and general type 2 rule based fuzzy systems in terms of sculpting the state space 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

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

interval type 2 fuzzy voter design for fault tolerant systems a voting scheme constitutes an essential component of many fault tolerant systems two types of voters are commonly used in applications of real valued systems the inexact majority and the amalgamating voters the inexact majority voter effectively isolates erroneous modules and is capable of reporting benign outputs when a significant disagreement is detected however an application specific voter threshold must be provided on the other hand amalgamating voter such as the weighted average voter reduces the influence of faulty modules by averaging the input values together unlike the majority voters amalgamating voters are not capable of producing benign outputs in the past a type 1 t1 fuzzy voting scheme was introduced allowing for both smooth amalgamation of voter inputs and effective signalization of benign outputs the presented paper proposes an extension to the fuzzy voting scheme via incorporating interval type 2 it2 fuzzy logic the it2 fuzzy logic allows for an improved handling of uncertain assumptions about the distributions of noisy and erroneous inputs which are essential for correct design of the fuzzy voting scheme the proposed voter design features robust performance when the uncertainty assumptions dynamically change over time the it2 fuzzy voter architecture was compared against the average voter inexact majority voter and the t1 fuzzy voter using a refined experimental harness the reported results demonstrate improved availability safety and reliability of the presented it2 fuzzy voting scheme

asymptotic behaviour of a class of stochastic approximation procedures letb be a real banach space considering stochastic approximation procedures of robbins monro and kiefer wolfowitz type one is led to recursion formulas of the form u k 1 u k k 1 gamma k u k k rho v k k eta t k and positive real numbers uk vk tkb valued random variables k random linear operatorsb b where in addition to some further assumptions k almost surely andtk t say almost surely letting y k n k 1 n n n be an array ofb valued random variables such that the partial sums sum limits k 1 m v k and sum limits k 1 m y k n do not differ too much uniformly inm 1 n we shall investigate the distance of u m m 1 n and u n m m 1 n where u k 1 left n right u k left n right k 1 gamma u k left n right k rho y k n k eta t fork 1 n 1 the proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence un

stability analysis and control of discrete type 1 and type 2 tsk fuzzy systems part i stability analysis this paper introduces sufficient conditions for the exponential stability of type 1 and type 2 takagi sugeno kang tsk fuzzy systems a major advantage of the new conditions is that they do not require the existence of a common lyapunov function and are therefore applicable to systems with unstable consequents in addition our results include two classes of type 2 tsk systems with type 1 consequents for which no stability tests are available the use of the conditions in stability testing is demonstrated using simple numerical examples that include cases where methods that are based on a common lyapunov function fail the application of the stability test to develop new controller design methodologies is presented in a separate paper i e part ii

fn topsis fuzzy networks for ranking traded equities fuzzy systems consisting of networked rule bases called fuzzy networks capture various types of imprecision inherent in financial data and in the decision making processes on them this paper introduces a novel extension of the technique for ordering of preference by similarity to ideal solution topsis method and uses fuzzy networks to solve multicriteria decision making problems where both benefit and cost criteria are presented as subsystems thus the decision maker evaluates the performance of each alternative for portfolio optimization and further observes the performance for both benefit and cost criteria this approach improves significantly the transparency of the topsis methods while ensuring high effectiveness in comparison with established approaches the proposed method is further tested to solve the problem of selection ranking of traded equity covering developed and emergent financial markets the ranking produced by the method is validated using spearman rho rank correlation based on the case study the proposed method outperforms the existing topsis approaches in terms of ranking performance

asynchronously switched control of discrete impulsive switched systems with time delays this paper is concerned with the stabilization problem for a class of uncertain discrete impulsive switched delay systems under asynchronous switching the so called asynchronous switching means that the switches between the candidate controllers and system modes are asynchronous by using the average dwell time adt approach sufficient conditions for the existence of an asynchronously switched controller is derived such that the resulting closed loop system is exponentially stable the desired controller gains and the admissible switching signals are obtained in terms of a set of matrix inequalities a numerical example is given to illustrate the effectiveness of the proposed method

aggregation operators on type 2 fuzzy sets abstract cubillo et al in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset partially ordered set in particular on m set of fuzzy membership degrees of t2fss which are the functions from 0 1 to 0 1 previously takac in 2014 had applied zadeh s extension principle to obtain a set of operators on m which are under some conditions aggregation operators on l the set of strongly normal and convex functions of m in this paper we introduce a more general set of operators on m than were given by takac and we study among other properties the conditions required to satisfy the axioms of the aggregation operator on l set of normal and convex functions on m which includes the set l

weighted means in stochastic approximation of minima weighted averages of kiefer wolfowitz type procedures which are driven by larger step lengths than usual can achieve the optimal rate of convergence a priori knowledge of a lower bound on the smallest eigenvalue of the hessian matrix is avoided the asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a newton type adaptive algorithm several different gradient estimates are considered one of them leads to a vanishing asymptotic bias this gradient estimate applied with the weighted averaging algorithm usually yields a better asymptotic mean squared error than applied with the standard algorithm

the sampling method of defuzzification for type 2 fuzzy sets experimental evaluation for generalised type 2 fuzzy sets the defuzzification process has historically been slow and inefficient this has hampered the development of type 2 fuzzy inferencing systems for real applications and therefore no advantage has been taken of the ability of type 2 fuzzy sets to model higher levels of uncertainty the research reported here provides a novel approach for improving the speed of defuzzification for discretised generalised type 2 fuzzy sets the traditional type reduction method requires every embedded type 2 fuzzy set to be processed the high level of redundancy in the huge number of embedded sets inspired the development of our sampling method which randomly samples the embedded sets and processes only the sample the paper presents detailed experimental results for defuzzification of constructed sets of known defuzzified value the sampling defuzzifier is compared on aggregated type 2 fuzzy sets resulting from the inferencing stage of a fis in terms of accuracy and speed with other methods including the exhaustive and techniques based on the a planes representation the results indicate that by taking only a sample of the embedded sets we are able to dramatically reduce the time taken to process a type 2 fuzzy set with very little loss in accuracy

on km algorithms for solving type 2 fuzzy set problems 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

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

nonparametric sequential estimation of zeros and extrema of regression functions let x y x l y l x 2 y 2 cdots be independent identically distributed pairs of random variables and let m x e y x x be the regression curve of y on x the estimation of zeros and extrema of the regression curve via stochastic approximation methods is considered consistency results of some sequential procedures are presented and termination rules are defined providing fixed width confidence intervals for the parameters to be estimated

classification of type 2 fuzzy sets represented as sequences of vertical slices granulation of information by using type 2 fuzzy sets is receiving more attention nowadays this is due to the superior capability of type 2 fuzzy sets in handling the data uncertainty from a theoretical perspective a set containing type 2 fuzzy sets has no trivial geometric structure therefore a proper metric cannot be easily defined as a consequence common pattern recognition systems which in one way or another rely on some geo metric structure of the input space are not easily applicable to a space of type 2 fuzzy sets in this paper we study the problem of designing a classifier in the input space of type 2 fuzzy sets type 2 fuzzy sets are hence interpreted as granular patterns forming a given input dataset by decomposing a type 2 fuzzy set into a sequence of simpler lower type fuzzy sets we explore the possibility of defining and building dissimilarity and kernel based classification systems on input spaces of type 2 fuzzy sets such an interpretation provided in terms of sequences allows us to conceive an effective sequence matching strategy which can be suitably embedded into well established pattern recognition systems we support the methodological developments by performing experiments on synthetically generated classification problems for datasets composed of type 2 fuzzy sets with adjustable and controlled level of difficulty results are promising and suggest to further investigate on the possibility of interpreting type 2 fuzzy sets as input patterns of a given data driven inference system

observer design of discrete time impulsive switched nonlinear systems with time varying delays this paper investigates the problem of observer design for discrete time impulsive switched nonlinear systems with time varying delays firstly based on the average dwell time approach and the lyapunov krasovskii functional technique a delay dependent exponential stability criterion for the discrete time impulsive switched nonlinear systems with time varying delays is derived in terms of a set of linear matrix inequalities lmis then sufficient conditions for the existence of an observer that guarantees the exponential stability of the corresponding error system are proposed finally two numerical examples are given to illustrate the effectiveness of the proposed method

new classes of threshold aggregation functions based upon the tsallis q exponential with applications to perceptual computing we introduce two new classes of single parameter aggregation functions based upon the tsallis q exponential qe function of nonextensive statistical mechanics these aggregation functions denoted qe aggregation facilitate simple modeling of the common human reasoning trait of threshold inference where either 1 at least one input must exceed a threshold in order to achieve a nonzero aggregation output or 2 if any one of the inputs exceeds a different threshold the aggregation output takes its maximum value we illustrate the thresholding behavior of these functions on interval type 2 fuzzy inputs using an example known in the literature as the investment judgment advisor we believe that the new qe class of aggregation operators will prove useful in extending the range of options available for the design of perceptual computing systems

simplified interval type 2 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

further results on the robust stability of linear systems with a single time delay abstract for linear systems with a single time varying delay time we propose a method to characterize how large the deviation of the delay time can be compared to the nominal zero value such that the system still preserves its asymptotic stability both cases of linear time delay systems with and without uncertainties are considered the criteria developed are seen to give less conservative results in simulation examples

multi central general type 2 fuzzy clustering approach for pattern recognitions this paper presents a new approach to general type 2 fuzzy clustering called multi central general type 2 fuzzy clustering model this model mainly focuses on uncertainty associated with the cluster centers in this model a set of points is considered as the center for each cluster the membership values to the clusters are defined as general type 2 fuzzy sets including primary and secondary variables primary variable indicates the degree of belonging to the central objects and the secondary variables indicate the degree of belonging of the central objects to the center of the cluster there is not any type reduction or defuzzification for updating cluster prototypes in the proposed clustering algorithm the compatible indexes with the proposed model are defined for validation and verification of the clustering process and results several experimental results are given to evaluate the performance of the proposed multi central general type 2 fuzzy clustering the results also compared with the results of the type 1 fuzzy clustering model

comment on toward general type 2 fuzzy logic systems based on zslices 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 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

from interval valued data to general type 2 fuzzy sets in this paper a new approach is presented to model interval based data using fuzzy sets fss specifically we show how both crisp and uncertain intervals where there is uncertainty about the endpoints of intervals collected from individual or multiple survey participants over single or repeated surveys can be modeled using type 1 interval type 2 or general type 2 fss based on zslices the proposed approach is designed to minimize any loss of information when transferring the interval based data into fs models and to avoid as much as possible assumptions about the distribution of the data furthermore our approach does not rely on data preprocessing or outlier removal which can lead to the elimination of important information different types of uncertainty contained within the data namely intra and inter source uncertainty are identified and modeled using the different degrees of freedom of type 2 fss thus providing a clear representation and separation of these individual types of uncertainty present in the data we provide full details of the proposed approach as well as a series of detailed examples based on both real world and synthetic data we perform comparisons with analogue techniques to derive fss from intervals namely the interval approach and the enhanced interval approach and highlight the practical applicability of the proposed approach

limit theorems for sums of weakly dependent banach space valued random variables we prove an estimate for the prohorov distance in the central limit theorem for strong mixing banach space valued random variables using a recent variant of an approximation theorem of berkes and philipp 1979 we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term t tfrac 1 2 gamma for strong mixing sequences we prove a strong invariance principle with error term o t log logt 1 2

an almost sure invariance principle for hilbert space valued martingales we obtain an almost sure approximation of a martingale with values in a real separable hilbert space h by a suitable h valued brownian motion

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems 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

note on on the extension of nullnorms and uninorms to fuzzy truth values fuzzy sets syst 352 2018 92 118 abstract xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type 2 nullnorms and type 2 uninorms on the algebra of fuzzy truth values in this note we show by counterexamples that there exist some flaws in a previous paper by xie fuzzy sets syst 352 2018 92 118 such as corollaries 2 1 3 1 4 1 and 4 2 lemmas 3 3 3 4 3 5 3 6 3 7 and 3 8 propositions 3 4 3 5 3 10 3 11 4 1 and 4 4 theorems 3 1 3 3 4 1 and 4 3 and then we provide the correct versions in addition proposition 3 7 in that paper is correct but its proof contains a flaw we point out the flaw and give a more general result than proposition 3 7 in that paper

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

general type 2 fuzzy c means algorithm for uncertain fuzzy clustering pattern recognition in real world data is subject to various sources of uncertainty that should be appropriately managed the focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms type 2 fuzzy sets t2 fss have received increased research interest over the past decade primarily due to their potential to model various uncertainties however because of the computational intensity of the processing of general t2 fuzzy sets gt2 fss only their constrained version i e the interval t2 it2 fss were typically used fortunately the recently introduced concepts of planes and zslices allow for efficient representation and computation with gt2 fss following this recent development this paper presents a novel approach for uncertain fuzzy clustering using the general type 2 fuzzy c means gt2 fcm algorithm the proposed method builds on top of the previously published it2 fcm algorithm which is extended via the planes representation theorem the fuzzifier parameter of the fcm algorithm can be expressed using linguistic terms such as small or high which are modeled as t1 fss this linguistic fuzzifier value is then used to construct the gt2 fcm cluster membership functions the linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters the gt2 fcm algorithm was found to balance the performance of t1 fcm algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present

computing the centroid of a general type 2 fuzzy set by means of the centroid flow algorithm previous studies have shown that the centroid of a general type 2 fuzzy set t2 fs a can be obtained by taking the union of the centroids of all the planes each raised to level of a karnik mendel km or the enhanced km ekm algorithms are used to compute the centroid of each plane the iterative features in km ekm algorithms can be time consuming especially when the algorithms have to be repeated for many planes this paper proposes a new method named centroid flow cf algorithm to compute the centroid of a without having to apply km ekm algorithms for every plane extensive simulations have shown that the cf algorithm can reduce the computation time by 75 80 and 50 75 compared with km and ekm algorithms respectively and still maintains satisfactory computation accuracy for various t2 fss when the primary variable x and plane are discretized finely enough

accelerated randomized stochastic optimization we propose a general class of randomized gradient estimates to be employed in a recursive search for the minimum of an unknown multivariate regression function here only two observations per iteration step are used special cases include random direction stochastic approximation kushner and clark simultaneous perturbation stochastic approximation spall and a special kernel based stochastic approximation method polyak and tsybakov if the unknown regression is p smooth p 2 at the point of minimum these methods achieve the optimal rate of convergence o n p 1 2p for both the classical stochastic approximation scheme kiefer and wolfowitz and the averaging scheme ruppert and polyak the related asymptotic distributions are computed

combinatorial iterative algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval type 2 fuzzy set it2 fs is an important type reduction method the aim of this paper is to develop a new method to calculate the centroid of an it2 fs when the problems of centroid computation of an it2 fs are continuous for the continuous centroid computation problems the structures of optimal solutions are strictly proven from mathematics for the first time in this paper furthermore we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal i e if there are two optimal solutions f 1 x and f 2 x the lebesgue measure of x f 1 x neq f 2 x is equal to 0 subsequently a combinatorial iterative ci method is proposed to solve the roots of the sufficiently differentiable objective functions it is proven that the convergence of the proposed iterative method is at least sixth order based on the proposed iterative method two algorithms called ci algorithms are devised to compute the centroid of an it2 fs the efficiencies of ci algorithms are demonstrated by comparing the continuous karnik mendel algorithms and the hallye s methods with the ci algorithms through three numerical examples

delay dependent robust stability and stabilization of uncertain linear delay systems a linear matrix inequality approach this paper considers the problems of robust stability analysis and robust control design for a class of uncertain linear systems with a constant time delay the uncertainty is assumed to be norm bounded and appears in all the matrices of the state space model we develop methods for robust stability analysis and robust stabilization the proposed methods are dependent on the size of the delay and are given in terms of linear matrix inequalities

robust stabilization of uncertain impulsive switched systems with delayed control in this paper stability criteria and switching controllers design problems for uncertain impulsive switched systems with input delay are investigated by using the receding horizon method some lmi conditions are derived to guarantee asymptotical stability of an impulsive switched system under a certain designed delayed controller finally a numerical example is presented to illustrate the effectiveness of the results obtained

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

improved uncertainty capture for nonsingleton fuzzy systems in nonsingleton fuzzy logic systems nsflss input uncertainties are modeled with input fuzzy sets in order to capture input uncertainty e g sensor noise the performance of nsflss in handling such uncertainties depends on both the appropriate modeling in the input fuzzy sets of the uncertainties present in the system s inputs and on how the input fuzzy sets and their inherent model of uncertainty interact with the antecedent and thus affect the inference within the remainder of the nsfls this paper proposes a novel development on the latter specifically an alteration to the standard composition method of type 1 fuzzy relations is proposed and applied to build a new type of nsfls the proposed approach is based on employing the centroid of the intersection of input and antecedent sets as origin of the firing degree rather than the traditional maximum of their intersection thus making the nsfls more sensitive to changes in the input s uncertainty characteristics the traditional and novel approach to nsflss are experimentally compared for two well known problems of mackey glass and lorenz chaotic time series predictions where the nsflss inputs have been perturbed with different levels of gaussian noise experiments are repeated for system training under noisy and noise free conditions analyses of the results show that the new method outperforms the traditional approach moreover it is shown that while formally more complex in practice the new method has no significant computational overhead compared with the standard approach

memoryless stabilization of uncertain linear systems including time varying state delays the problem of stabilizing a class of uncertain time delay systems via memoryless linear feedback is examined the systems under consideration are linear systems with time varying state delays they also contain uncertain parameters possibly time varying whose values are known only to within a prescribed compact bounding set the main contribution given is to enlarge the class of time delay systems for which one can construct a stabilizing memoryless linear feedback controller within this framework a novel notion of robust memoryless stabilizability is first introduced via the method of lyapunov functionals then a sufficient condition for the stabilizability is proposed it is shown that solvability of a parameterized riccati equation can be used to determine whether the time delay system satisfies the sufficient condition if there exists a positive definite symmetric solution satisfying the riccati equation a suitable memoryless linear feedback law can be derived

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

toward general type 2 fuzzy logic systems based on zslices higher order fuzzy logic systems flss such as interval type 2 flss have been shown to be very well suited to deal with the high levels of uncertainties present in the majority of real world applications general type 2 flss are expected to further extend this capability however the immense computational complexities associated with general type 2 flss have until recently prevented their application to real world control problems this paper aims to address this problem by the introduction of a complete representation framework which is referred to as zslices based general type 2 fuzzy systems the proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type 2 flss while it offers the capability to represent complex general type 2 fuzzy sets as a proof of concept application we have implemented a zslices based general type 2 fls for a two wheeled mobile robot which operates in a real world outdoor environment we have evaluated the computational performance of the zslices based general type 2 fls which is suitable for multiprocessor execution finally we have compared the performance of the zslices based general type 2 fls against type 1 and interval type 2 flss and a series of results is presented which is related to the different levels of uncertainty handled by the different types of flss

experimental study of intelligent controllers under uncertainty using type 1 and type 2 fuzzy logic uncertainty is an inherent part in control systems used in real world applications the use of new methods for handling incomplete information is of fundamental importance type 1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems type 2 fuzzy sets that are used in type 2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom this paper deals with the design of control systems using type 2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise etc the experimental results are divided in two classes in the first class simulations of a feedback control system for a non linear plant using type 1 and type 2 fuzzy logic controllers are presented a comparative analysis of the systems response in both cases was performed with and without the presence of uncertainty for the second class a non linear identification problem for time series prediction is presented based on the experimental results the conclusion is that the best results are obtained using type 2 fuzzy systems

on clarifying some definitions and notations used for type 2 fuzzy sets as well as some recommended changes this paper is about some important changes in type 2 fuzzy set t2 fs definitions and notations that have occurred during the past 16 years it summarizes the evolution of how the primary membership jx has been used in both the mathematical descriptions of a t2 fs and its footprint of uncertainty fou discusses notational problems associated with the secondary membership function explains why and when one should distinguish between the fou and the domain of uncertainty dou explains why no errors have been introduced into t2 fs computations because of notation and it provides recommendations notational changes that can be used by all authors

generalized type 2 fuzzy weight adjustment for backpropagation neural networks in time series prediction in this paper the comparison of a proposed neural network with generalized type 2 fuzzy weights nngt2fw with respect to the monolithic neural network nn and the neural network with interval type 2 fuzzy weights nnit2fw is presented generalized type 2 fuzzy inference systems are used to obtain the generalized type 2 fuzzy weights and are designed by a strategy of increasing and decreasing an epsilon variable for obtaining the different sizes of the footprint of uncertainty fou for the generalized membership functions the proposed method is based on recent approaches that handle weight adaptation using type 1 and type 2 fuzzy logic the approach is applied to the prediction of the mackey glass time series and results are shown to outperform the results produced by other neural models gaussian noise was applied to the test data of the mackey glass time series for finding out which of the presented methods in this paper shows better performance and tolerance to noise

on computing normalized interval type 2 fuzzy sets this paper explains how to compute normalized interval type 2 fuzzy sets in closed form and explains how the results reduce to well known results for type 1 fuzzy sets and interval sets such normalized interval type 2 fuzzy sets may be needed in linguistic probability computa tions or multiple criteria decision analysis under uncertainty index terms fuzzy weighted average fwa interval type 2 fuzzy sets it2 fss interval weighted average iwa linguistic probability linguistic weighted average lwa normalized interval type 2 fuzzy sets

almost sure approximations to the robbins monro and kiefer wolfowitz processes with dependent noise we study a recursive algorithm which includes the multidimensional robbins monro and kiefer wolfowitz processes the assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence it is shown that the algorithm can be represented as a weighted average of the disturbances this representation can be used to prove asymptotic results for stochastic approximation procedures as an example we approximate the one dimensional kiefer wolfowitz process almost surely by brownian motion and as a byproduct obtain a law of the iterated logarithm

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

general type 2 fuzzy logic systems made simple a tutorial 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 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

study on enhanced karnik mendel algorithms initialization explanations and computation improvements computing the centroid of an interval type 2 fuzzy set is an important operation in a type 2 fuzzy logic system and is usually implemented by karnik mendel km iterative algorithms by connecting km algorithms and continuous km algorithms together this paper gives theoretical explanations on the initialization methods of km and enhanced karnik mendel ekm algorithms proposes exact methods for centroid computation of an interval type 2 fuzzy set and extends the enhanced karnik mendel ekm algorithms to three different forms of weighted ekm wekm algorithms it shows that ekm algorithms become a special case of the wekm algorithms when the weights of the latter are constant value it also shows that in general the weighted ekm algorithms have smaller absolute error and faster convergence speed than the ekm algorithms which make them very attractive for real time applications of fuzzy logic system four numerical examples are used to illustrate and analyze the performance of wekm algorithms

hybrid learning mechanism for interval a2 c1 type 2 non singleton type 2 takagi sugeno kang fuzzy logic systems a proposed learning methodology based on a hybrid mechanism for training interval a2 c1 type 2 non singleton type 2 takagi sugeno kang fuzzy logic systems uses a recursive square root filter to tune the type 1 consequent parameters and the steepest descent method to tune the interval type 2 antecedent parameters the proposed hybrid learning algorithm changes the interval type 2 model parameters adaptively to minimize some criterion function as new information becomes available and to match desired input output data pairs its antecedent sets are type 2 fuzzy sets its consequent sets are type 1 fuzzy sets and its inputs are interval type 2 non singleton fuzzy numbers with uncertain standard deviations as reported in the literature the performance indices of hybrid models have proved to be better than those of the individual training mechanisms used alone comparison with non hybrid interval a2 c1 type 2 takagi sugeno kang fuzzy logic systems and with non hybrid a1 c0 type 1 takagi sugeno kang fuzzy logic systems shows that the proposed hybrid mechanism is a well performing non linear adaptive method that enables the interval type 2 fuzzy model to match an unknown non linear mapping and to converge very fast experiments were carried out involving the application of the hybrid interval a2 c1 type 2 non singleton type 2 takagi sugeno kang fuzzy logic systems for modeling and prediction of the scale breaker entry temperature in a hot strip mill for three different types of coils the results demonstrate how the interval type 2 fuzzy system learns from selected input output data pairs and improves its performance as hybrid training progresses

similarity measures for general type 2 fuzzy sets based on the α plane representation abstract similarity measures are very important concepts in fuzzy sets fss theory there are many different definitions of similarity measures for both type 1 t1 fss and interval type 2 it2 fss in this paper one similarity measure for it2 fss which is extended from the jaccard similarity measure for t1 fss is reviewed then based on the plane representation for a general type 2 gt2 fs this similarity measure is generalized to such t2 fss some examples that demonstrate how to compute the similarity measures for different t2 fss are given

monotone centroid flow algorithm for type reduction of general type 2 fuzzy sets recently type 2 fuzzy logic systems t2 flss have received increased research attention due to their potential to model and cope with the dynamic uncertainties ubiquitous in many engineering applications however because of the complex nature and the computational intensity of the inference process only the constrained version of t2 flss i e the interval t2 flss was typically used fortunately the very recently introduced concepts of planes and zslices allow for efficient representation as well as a computationally fast inference process with general t2 gt2 flss this paper addresses the type reduction phase in gt2 flss using gt2 fuzzy sets fss represented in the plane framework the monotone property of centroids of a set of planes is derived and leveraged toward developing a simple to implement but fast algorithm for type reduction of gt2 fss i e the monotone centroid flow mcf algorithm when compared with the centroid flow cf algorithm which was previously developed by zhai and mendel the mcf algorithm features the following advantages 1 the mcf algorithm computes numerically identical centroid as the karnik mendel km iterative algorithms unlike the approximated centroid which is obtained with the cf algorithm 2 the mcf algorithm is faster than the cf algorithm as well as the independent application of the km algorithms 3 the mcf algorithm is easy to implement unlike the cf algorithm which requires computation of the derivatives of the centroid and 4 the mcf algorithm completely eliminates the need to apply the km iterative procedure to any planes of the gt2 fs the performance of the algorithm is presented on benchmark problems and compared with other type reduction techniques that are available in the literature

simplified interval type 2 fuzzy neural networks this paper describes a self evolving interval type 2 fuzzy neural network fnn for various applications as type 1 fuzzy systems cannot effectively handle uncertainties in information within the knowledge base we propose a simple interval type 2 fnn which uses interval type 2 fuzzy sets in the premise and the takagi sugeno kang tsk type in the consequent of the fuzzy rule the tsk type consequent of fuzzy rule is a linear combination of exogenous input variables given an initially empty the rule base all rules are generated with on line type 2 fuzzy clustering instead of the time consuming k m iterative procedure the design factors ql and qr are learned to adaptively adjust the upper and lower positions on the left and right limit outputs using the parameter update rule based on a gradient descent algorithm simulation results demonstrate that our approach yields fewer test errors and less computational complexity than other type 2 fnns

enhanced centroid flow algorithm for computing the centroid of general type 2 fuzzy sets recently a centroid flow cf algorithm has been proposed to compute the centroid of a type 2 fuzzy set a this algorithm utilizes the karnik mendel km or the enhanced km ekm algorithm only at the 0 level of a and then lets its result flow upward to the 1 level of a it avoids having to apply the km ekm algorithms at every 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 1 level of a 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

learning of interval and general type 2 fuzzy logic systems using simulated annealing this paper reports the use of simulated annealing to design more efficient fuzzy logic systems to model problems with associated uncertainties simulated annealing is used within this work as a method for learning the best configurations of interval and general type 2 fuzzy logic systems to maximize their modeling ability the combination of simulated annealing with these models is presented in the modeling of four benchmark problems including real world problems the type 2 fuzzy logic system models are compared in their ability to model uncertainties associated with these problems issues related to this combination between simulated annealing and fuzzy logic systems including type 2 fuzzy logic systems are discussed the results demonstrate that learning the third dimension in type 2 fuzzy sets with a deterministic defuzzifier can add more capability to modeling than interval type 2 fuzzy logic systems this finding can be seen as an important advance in type 2 fuzzy logic systems research and should increase the level of interest in the modeling applications of general type 2 fuzzy logic systems despite their greater computational load

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

geometric type 1 and type 2 fuzzy logic systems this paper presents a novel approach to the representation of type 1 and type 2 fuzzy sets utilising computational geometry to achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic we provide new algorithms for various operations on type 1 and type 2 fuzzy sets and for defuzzification results of experiments indicate that this approach reduces the execution speed of these operations