optimization of type 2 fuzzy systems based on bio inspired methods a concise review a review of the optimization methods used in the design of type 2 fuzzy systems which are relatively novel models of imprecision has been considered in this work the fundamental focus of the work has been based on the basic reasons of the need for optimizing type 2 fuzzy systems for different areas of application recently bio inspired methods have emerged as powerful optimization algorithms for solving complex problems in the case of designing type 2 fuzzy systems for particular applications the use of bio inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems in this review we consider the application of genetic algorithms particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type 2 fuzzy systems we also provide a comparison of the different optimization methods for the case of designing type 2 fuzzy systems

asymmetric subsethood product fuzzy neural inference system asupfunis this work presents an asymmetric subsethood product fuzzy neural inference system asupfunis that directly extends the supfunis model by permitting signal and weight fuzzy sets to be modeled by asymmetric gaussian membership functions the asymmetric subsethood product network admits both numeric as well as linguistic inputs input nodes which act as tunable feature fuzzifiers fuzzify numeric inputs with asymmetric gaussian fuzzy sets and linguistic inputs are presented as is the antecedent and consequent labels of standard fuzzy if then rules are represented as asymmetric gaussian fuzzy connection weights of the network the model uses mutual subsethood based activation spread and a product aggregation operator that works in conjunction with volume defuzzification in a gradient descent learning framework despite the increase in the number of free parameters the proposed model performs better than supfunis on various benchmarking problems both in terms of the performance accuracy and architectural economy and compares excellently with other various existing models with a performance better than most of them

quantifying the effect of the surrogate marker by information gain statistical validation of a surrogate marker has been studied for more than a decade recently alonso et al 2004 biometrics 60 724 728 proposed a quantity called the likelihood reduction factor lrf to evaluate the validity of a surrogate marker however as pointed out in the present article the lrf may not correctly validate a surrogate marker therefore a new quantity the proportion of information gain pig using the kullback leibler information is proposed simulations show that under some model assumptions the pig precisely reflects the role of a surrogate marker

complete curves in moduli spaces of stable bundles on surfaces in this paper we give a partial answer to a question posed by hirschowitz and hulek in hi hu for moduli spaces of stable rank two vector bundles on the projective plane they asked for the maximal dimension of a complete subvariety of such a moduli space in fact they showed that for the projective plane these moduli spaces contain complete curves in almost every case for many cases they even contain complete surfaces cp hu lp and hu s the main theorem we will prove here is the following cp 3 8

computing derivatives in interval type 2 fuzzy logic systems 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

mpeg vbr video traffic modeling and classification using fuzzy technique 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

a recurrent self evolving interval type 2 fuzzy neural network for dynamic system processing this paper proposes a recurrent self evolving interval type 2 fuzzy neural network rseit2fnn for dynamic system processing an rseit2fnn incorporates type 2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system the antecedent parts in each recurrent fuzzy rule in the rseit2fnn are interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type with interval weights the antecedent part of rseit2fnn forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself the tsk type consequent part is a linear model of exogenous inputs the rseit2fnn initially contains no rules all rules are learned online via structure and parameter learning the structure learning uses online type 2 fuzzy clustering for the parameter learning the consequent part parameters are tuned by a rule ordered kalman filter algorithm to improve learning performance the antecedent type 2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm the rseit2fnn is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise free and noisy conditions comparisons with type 1 recurrent fuzzy neural networks validate the performance of the rseit2fnn

fuzzy sets of rules for system identification the synthesis of fuzzy systems involves the identification of a structure and its specialization by means of parameter optimization in doing this symbolic approaches which encode the structure information in the form of high level rules allow further manipulation of the system to minimize its complexity and possibly its implementation cost while all parametric methodologies often achieve better approximation performance in this paper we rely on the concept of a fuzzy set of rules to tackle the rule induction problem at an intermediate level an online adaptive algorithm is developed which almost surely learns the extent to which inclusion of a rule in the rule set significantly contributes to the reproduction of the target behavior then the resulting fuzzy set of rules can be defuzzified to give a conventional rule set with similar behavior comparisons with high level and low level methodologies show that this approach retains the most positive features of both

enhanced interval approach for encoding words into interval type 2 fuzzy sets and its convergence analysis 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

identification and prediction of dynamic systems using an interactively recurrent self evolving fuzzy neural network this paper presents a novel recurrent fuzzy neural network called an interactively recurrent self evolving fuzzy neural network irsfnn for prediction and identification of dynamic systems the recurrent structure in an irsfnn is formed as an external loops and internal feedback by feeding the rule firing strength of each rule to others rules and itself the consequent part in the irsfnn is composed of a takagi sugeno kang tsk or functional link based type the proposed irsfnn employs a functional link neural network flnn to the consequent part of fuzzy rules for promoting the mapping ability unlike a tsk type fuzzy neural network the flnn in the consequent part is a nonlinear function of input variables an irsfnns learning starts with an empty rule base and all of the rules are generated and learned online through a simultaneous structure and parameter learning an on line clustering algorithm is effective in generating fuzzy rules the consequent update parameters are derived by a variable dimensional kalman filter algorithm the premise and recurrent parameters are learned through a gradient descent algorithm we test the irsfnn for the prediction and identification of dynamic plants and compare it to other well known recurrent fnns the proposed model obtains enhanced performance results

multiobjective tracking control design of t s fuzzy systems in this study a multiobjective fuzzy control design method is introduced for nonlinear dynamic systems to guarantee the optimal h 2 and h reference tracking performance simultaneously first the takagi and sugeno t s fuzzy model is used to represent the nonlinear dynamic system then based on the t s fuzzy model multiobjective h 2 h tracking control design problem is formulated as a multiobjective problem mop to minimize the h 2 tracking error and h disturbance attenuation level for the fuzzy system at the same time since it is not easy to solve this mop directly an indirect method is proposed for the multiobjective h 2 h tracking control design finally in order to achieve the simultaneous optimization of the mop linear matrix inequality lmi based multiobjective evolution algorithm lmi based moea is developed based on non dominating sorting scheme to efficiently search the set of pareto optimal solutions for the mop from which designer can select one design according to his own preference further the multiobjective h 2 h fuzzy control design problem based on the weighted sum method is also solved as an alternative choice finally a simulation example of a robotic system is given to illustrate the design procedure and to confirm the robust and optimal tracking performance of the proposed method

ten years of genetic fuzzy systems current framework and new trends fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains currently there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities two of the most successful approaches to hybridise fuzzy systems with learning and adaptation methods have been made in the realm of soft computing neural fuzzy systems and genetic fuzzy systems hybridise the approximate reasoning method of fuzzy systems with the learning capabilities of neural networks and evolutionary algorithms the objective of this paper is to provide an account of genetic fuzzy systems with special attention to genetic fuzzy rule based systems after a brief introduction to models and applications of genetic fuzzy systems the field is overviewed new trends are identified a critical evaluation of genetic fuzzy systems for fuzzy knowledge extraction is elaborated and open questions that remain to be addressed in the future are raised the paper also includes some of the key references required to quickly access implementation details of genetic fuzzy systems

criteria for the validation of surrogate endpoints in randomized experiments the validation of surrogate endpoints has been studied by prentice 1989 statistics in medicine 8 431 440 and freedman graubard and schatzkin 1992 statistics in medicine 11 167 178 we extended their proposals in the cases where the surrogate and the final endpoints are both binary or normally distributed letting t and s be random variables that denote the true and surrogate endpoint respectively and z be an indicator variable for treatment prentice s criteria are fulfilled if z has a significant effect on t and on s if s has a significant effect on t and if z has no effect on t given s freedman relaxed the latter criterion by estimating pe the proportion of the effect of z on t that is explained by s and by requiring that the lower confidence limit of pe be larger than some proportion say 0 5 or 0 75 this condition can only be verified if the treatment has a massively significant effect on the true endpoint a rare situation we argue that two other quantities must be considered in the validation of a surrogate endpoint re the effect of z on t relative to that of z on s and gamma z the association between s and t after adjustment for z a surrogate is said to be perfect at the individual level when there is a perfect association between the surrogate and the final endpoint after adjustment for treatment a surrogate is said to be perfect at the population level if re is 1 a perfect surrogate fulfills both conditions in which case s and t are identical up to a deterministic transformation fieller s theorem is used for the estimation of pe re and their respective confidence intervals logistic regression models and the global odds ratio model studied by dale 1986 biometrics 42 909 917 are used for binary endpoints linear models are employed for continuous endpoints in order to be of practical value the validation of surrogate endpoints is shown to require large numbers of observations

a generalized type 2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design abstract in this work a generalized type 2 fuzzy logic system gt2fls approach for dynamic parameter adaptation in metaheuristics and for optimal fuzzy controller design is presented in these two cases the efficiency of the gt2fls approach is verified with simulation results in the first case the gt2fls provides an approach to dynamically find the optimal values of the heuristic parameters that are a critical part of the bee colony optimization bco algorithm performance in the second case the gt2fls approach provides the basis for building a generalized type 2 fuzzy logic controller gt2flc which can be optimized with the traditional bco specifically to find the optimal design of the membership functions mfs in the fuzzy controller in both cases the gt2fls approach shows advantages in the optimization of the solutions to the problems for both cases we can considered them as hybrid systems combining gt2fls and bco although the combination is made in a different way and it can be noted that a gt2fls presents better stability in the minimization of the errors when applied to benchmark control problems simulation results illustrate that the implementation of the generalized type 2 fuzzy logic controller gt2flc approach improves its performance when using the bco algorithm and the stability of the fuzzy controller is better when compared with respect to a type 1 fuzzy logic controller t1flc and an interval type 2 fuzzy logic controller it2flc

linguistic summarization using if then rules and interval type 2 fuzzy sets linguistic summarization ls is a data mining or knowledge discovery approach to extract patterns from databases many authors have used this technique to generate summaries like most senior workers have high salary which can be used to better understand and communicate about data however few of them have used it to generate if then rules like if x is large and y is medium then z is small which not only facilitate understanding and communication of data but can also be used in decision making in this paper an ls approach to generate if then rules for causal databases is proposed both type 1 and interval type 2 fuzzy sets are considered five quality measures the degrees of truth sufficient coverage reliability outlier and simplicity are defined among them the degree of reliability is especially valuable for finding the most reliable and representative rules and the degree of outlier can be used to identify outlier rules and data for close up investigation an improved parallel coordinates approach for visualizing the if then rules is also proposed experiments on two datasets demonstrate our ls and rule visualization approaches finally the relationships between our ls approach and the wang mendel wm method perceptual reasoning and granular computing are pointed out

on the robustness of type 1 and interval type 2 fuzzy logic systems in modeling research on the robustness of fuzzy logic systems flss an imperative factor in the design process is very limited in the literature specifically when a system is subjected to small deviations of the sampling points operating points it is of great interest to find the maximum tolerance of the system which we refer to as the system s robustness in this paper we present a methodology for the robustness analysis of interval type 2 flss it2 flss that also holds for t1 flss hence making it more general a procedure for the design of robust it2 flss with a guaranteed performance better than or equal to their t1 counterparts is then proposed several examples are performed to demonstrate the effectiveness of the proposed methodologies it was concluded that both t1 and it2 flss can be designed to achieve robust behavior in various applications and preference one or the other in general is application dependant it2 flss having a more flexible structure than t1 flss exhibited relatively small approximation errors in the several examples investigated the methodologies presented in this paper lay the foundation for the design of flss with robust properties that will be very useful in many practical modeling and control applications

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

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

structure identification in complete rule based fuzzy systems the identification of a model is one of the key issues in the field of fuzzy system modeling and function approximation theory there are numerous approaches to the issue of parameter optimization within a fixed fuzzy system structure but no reliable method to obtain the optimal topology of the fuzzy system from a set of input output data this paper presents a reliable method to obtain the structure of a complete rule based fuzzy system for a specific approximation accuracy of the training data i e it can decide which input variables must be taken into account in the fuzzy system and how many membership functions mfs are needed in every selected input variable in order to reach the approximation target with the minimum number of parameters

estimating the proportion of treatment effect explained by a surrogate marker in this paper we measure the extent to which a biological marker is a surrogate endpoint for a clinical event by the proportional reduction in the regression coefficient for the treatment indicator due to the inclusion of the marker in the cox regression model we estimate this proportion by applying the partial likelihood function to two cox models postulated on the same failure time variable we show that the resultant estimator is asymptotically normal with a simple variance estimator one can construct confidence intervals for the proportion by using the direct normal approximation to the point estimator or by using fieller s theorem extensive simulation studies demonstrate that the proposed methods are appropriate for practical use we provide applications to hiv aids clinical trials

black box identification of a class of nonlinear systems by a recurrent neurofuzzy network this brief presents a structure for black box identification based on continuous time recurrent neurofuzzy networks for a class of dynamic nonlinear systems the proposed network catches the dynamics of a system by generating its own states using only input and output measurements of the system the training algorithm is based on adaptive observer theory the stability of the network the convergence of the training algorithm and the ultimate bound on the identification error as well as the parameter error are established experimental results are included to illustrate the effectiveness of the proposed method

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

multiobjective programming for type 2 hierarchical fuzzy inference trees this paper proposes a design of hierarchical fuzzy inference tree hfit an hfit produces an optimum tree like structure i e a natural hierarchical structure that accommodates simplicity by combining several low dimensional fuzzy inference systems fiss such a natural hierarchical structure provides a high degree of approximation accuracy the construction of the hfit takes place in two phases first a nondominated sorting based multiobjective genetic programming mogp is applied to obtain a simple tree structure a low complexity model with a high accuracy second the differential evolution algorithm is applied to optimize the obtained tree s parameters in the derived tree each node acquires a different input s combination where the evolutionary process governs the input s combination hence hfit nodes are heterogeneous in nature which leads to a high diversity among the rules generated by the hfit additionally the hfit provides an automatic feature selection because it uses mogp for the tree s structural optimization that accepts inputs only relevant to the knowledge contained in data the hfit was studied in the context of both type 1 and type 2 fiss and its performance was evaluated through six application problems moreover the proposed multiobjective hfit was compared both theoretically and empirically with recently proposed fiss methods from the literature such as mcit2fis tscit2fnn sit2fnn rit2fns wb et2fis mrit2nfs it2fnn svr etc from the obtained results it was found that the hfit provided less complex and highly accurate models compared to the models produced by the most of other methods hence the proposed hfit is an efficient and competitive alternative to the other fiss for function approximation and feature selection

noninteracting adaptive control of pmsm using interval type 2 fuzzy logic systems based on the type 2 fuzzy logic systems and adaptive control theory this paper develops a decentralized adaptive fuzzy controller for a permanent magnet synchronous motor pmsm the adaptive law is designed to compensate for the interconnection effects and the reconstruction errors using a sliding mode term the type 2 fuzzy logic systems are used to model the unknown functions based on lyapunov s stability theorem the stability of closed loop systems can be verified moreover the proposed overall control schemes guarantee that all the signals that are involved are bounded to demonstrate the effectiveness of the proposed methods simulation results are illustrated in this paper

type 2 fuzzy hidden markov models and their application to speech recognition this paper presents an extension of hidden markov models hmms based on the type 2 t2 fuzzy set fs referred to as type 2 fuzzy hmms t2 fhmms membership functions mfs of t2 fss are three dimensional and this new third dimension offers additional degrees of freedom to evaluate the hmms fuzziness therefore t2 fhmms are able to handle both random and fuzzy uncertainties existing universally in the sequential data we derive the t2 fuzzy forward backward algorithm and viterbi algorithm using t2 fs operations in order to investigate the effectiveness of t2 fhmms we apply them to phoneme classification and recognition on the timit speech database experimental results show that t2 fhmms can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical hmms

type 2 fuzzy markov random fields and their application to handwritten chinese character recognition in this paper we integrate type 2 t2 fuzzy sets with markov random fields mrfs referred to as t2 fmrfs which may handle both fuzziness and randomness in the structural pattern representation on the one hand the t2 membership function mf has a 3 d structure in which the primary mf describes randomness and the secondary mf evaluates the fuzziness of the primary mf on the other hand mrfs can represent patterns statistical structurally in terms of neighborhood system and clique potentials and thus have been widely applied to image analysis and computer vision in the proposed t2 fmrfs we define the same neighborhood system as that in classical mrfs to describe uncertain structural information in patterns we derive the fuzzy likelihood clique potentials from t2 fuzzy gaussian mixture models the fuzzy prior clique potentials are penalties for the mismatched structures based on prior knowledge because chinese characters have hierarchical structures we use t2 fmrfs to model character structures in the handwritten chinese character recognition system the overall recognition rate is 99 07 which confirms the effectiveness of the proposed method

temporal problems solved by dynamic fuzzy network based on genetic algorithm with variable length chromosomes abstract in this paper a dynamic fuzzy network and its design based on genetic algorithm with variable length chromosomes is proposed first the dynamic fuzzy network constituted from a series of dynamic fuzzy if then rules is proposed one characteristic of this network is its ability to deal with temporal problems then the proposed genetic algorithm with variable length chromosomes is adopted into the design process as a means of allowing the application of the network in situations where the actual desired output is unavailable in the proposed genetic algorithm the length of each chromosome varies with the number of rules coded in it using this algorithm no pre assignment of the number of rules in the dynamic fuzzy network is required since it can always help to find the most suitable number of rules all free parameters in the network including the spatial input partition consequent parameters and feedback connection weights are tuned concurrently to further promote the design performance genetic algorithm with variable length chromosomes and relative based mutated reproduction operation is proposed in this algorithm the elite individuals are directly reproduced to the next generation only when their averaged similarity value is smaller than a similarity threshold otherwise the elites are mutated to the next generation to show the efficiency of this dynamic fuzzy network designed by genetic algorithm with variable length chromosomes and relative based mutated reproduction operation two temporal problems are simulated the simulated results and comparisons with recurrent neural and fuzzy networks verify the efficacy and efficiency of the proposed approach

sugeno type controllers are universal controllers abstract we show that given any continuous real valued function on a compact set there is a sugeno type controller that will approximate this function to any degree of accuracy an application of this result to fuzzy control is presented which shows that this type of sugeno controller is a universal controller

a self evolving interval type 2 fuzzy neural network with online structure and parameter learning this paper proposes a self evolving interval type 2 fuzzy neural network seit2fnn with online structure and parameter learning the antecedent parts in each fuzzy rule of the seit2fnn are interval type 2 fuzzy sets and the fuzzy rules are of the takagi sugeno kang tsk type the initial rule base in the seit2fnn is empty and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space to avoid generating highly overlapping fuzzy sets in each input variable an efficient fuzzy set reduction method is also proposed this method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated for parameter learning the consequent part parameters are tuned by the rule ordered kalman filter algorithm for high accuracy learning performance detailed learning equations on applying the rule ordered kalman filter algorithm to the seit2fnn consequent part learning with rules being generated online are derived the antecedent part parameters are learned by gradient descent algorithms the seit2fnn is applied to simulations on nonlinear plant modeling adaptive noise cancellation and chaotic signal prediction comparisons with other type 1 and type 2 fuzzy systems in these examples verify the performance of the seit2fnn

adaptive fuzzy output feedback tracking backstepping control of strict feedback nonlinear systems with unknown dead zones in this paper an adaptive fuzzy backstepping control approach is considered for a class of nonlinear strict feedback systems with unknown functions unknown dead zones and immeasurable states fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a fuzzy filters state observer is designed to estimate the immeasurable states by using the adaptive backstepping recursive design technique and constructing the dead zone inverse a new adaptive fuzzy backstepping output feedback control approach is developed it is mathematically proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin by appropriate choice of design parameters the proposed approach cannot only solve the problem of the dead zones but also cancel the restrictive assumption in the previous literature that the states are all available for measurement two simulation examples are provided to show the effectiveness of the proposed approach

a neural network of smooth hinge functions smooth hinging hyperplane shh has been proposed as an improvement over the well known hinging hyperplane hh by the fact that it retains the useful features of hh while overcoming hh s drawback of nondifferentiability this paper introduces a formal characterization of smooth hinge function shf which can be used to generate shh as a neural network a method for the general construction of shf is also given furthermore the work proves that shh is better than hh in functional approximation i e the optimal error of shh approximating a general function is always smaller or equal to that of hh particularly in the case that the shf is generated via the integration of a class of sigmoidal functions it is further proven that the corresponding shh of the 2m shfs would outperform a neural network with m of the sigmoidal function from which the shf is derived any upper bound established on the approximation error of a neural network of m sigmoidal activation functions can hence be translated to the shh of m shfs by replacing m with m 2 the work also includes an algorithm for the identification of shh making use of its differentiability property simulation experiments are presented to validate the theoretical conclusions to possible extent

multiobjective optimization and comparison of nonsingleton type 1 and singleton interval type 2 fuzzy logic systems singleton interval type 2 fuzzy logic systems flss have been widely applied in several real world applications where it was shown that the singleton interval type 2 flss outperform their singleton type 1 counterparts in applications with high uncertainty levels however one of the main criticisms of singleton interval type 2 flss is the fact that they outperform singleton type 1 flss solely based on their use of extra degrees of freedom extra parameters and that type 1 flss with a sufficiently large number of parameters may provide the same performance as interval type 2 flss in addition most works on type 2 flss only compare their results with singleton type 1 flss but fail to consider nonsingleton type 1 systems in this paper we aim to directly address and investigate this criticism in order to do so we will perform a comparative study between optimized singleton type 1 nonsingleton type 1 and singleton interval type 2 flss under the presence of noise we will also present a multiobjective evolutionary algorithm moea for the optimization of singleton type 1 nonsingleton type 1 and singleton interval type 2 fuzzy systems for function approximation problems the moea will aim to satisfy two objectives to maximize the accuracy of the fls and minimize the number of rules in the fls thus improving its interpretability furthermore we will present a methodology to obtain optimal consequents for the flss hence this paper has two main contributions first it provides a common methodology to learn the three types of flss i e singleton type 1 nonsingleton type 1 and singleton interval type 2 flss from data samples the second contribution is the creation of a common framework for the comparison of type 1 and type 2 flss that allows us to address the aforementioned criticism we provide details of a series of experiments and include statistical analysis showing that the type 2 fls is able to handle higher levels of noise than its nonsingleton and singleton type 1 counterparts

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

on the continuity of type 1 and interval type 2 fuzzy logic systems 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

analytical structure and characteristics of symmetric karnik mendel type reduced interval type 2 fuzzy pi and pd controllers this paper presents the analytical structure of a class of interval type 2 it2 fuzzy proportional derivative pd and proportional integral pi controllers that have symmetrical rule base and symmetrical consequent sets two assumptions are made 1 the zadeh and operator is employed as the t norm operator 2 type reduction is performed by the karnik mendel km type reduction method the main contributions are the methodology that identifies the input conditions where the km algorithm uses a new switch point to compute the bounds of the type reduced set the closed form expressions that relate the inputs and output of an it2 fuzzy controller and insights into the potential performance improvement because of the inclusion of the footprint of uncertainty fou compared with its t1 counterpart two additional fou parameters generate 31 extra local regions each providing a unique relationship between the inputs and output signals the generation of a relatively large number of local regions at the cost of two extra design parameters indicates that an it2 fuzzy controller may be able to provide better performance furthermore by comparing the analytical structure with the corresponding t1 counterpart the potential advantages to use the it2 over the t1 fuzzy controller are studied four interesting characteristics are identified and they provide insights into why the it2 fuzzy controller may better balance the conflicting aims of fast rise time and small overshoot

nonsingleton fuzzy logic systems theory and application 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

a fuzzy logic method for modulation classification in nonideal environments 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

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

an interval type 2 fuzzy neural network with support vector regression for noisy regression problems this paper proposes an interval type 2 fuzzy neural network with support vector regression it2fnn svr for noisy regression problems the antecedent part in each fuzzy rule of an it2fnn svr uses interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type the use of interval type 2 fuzzy sets helps improve the network s noise resistance the network inputs may be numerical values or type 1 fuzzy sets with the latter being used for further improvements in robustness it2fnn svr learning consists of both structure learning and parameter learning the structure learning algorithm is responsible for online rule generation the parameters are optimized for structural risk minimization using a two phase linear svr algorithm in order to endow the network with high generalization ability it2fnn svr performance is verified through comparisons with type 1 and type 2 fuzzy logic systems and other regression models on noisy regression problems

fuzzy control with fuzzy inputs this paper is concerned with the use of fuzzy inputs in fuzzy logic controllers a precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed the proposed concepts are applied to control a first order process with a pi controller this simple process is chosen to clearly illustrate the behavior of the closed loop system using fuzzy inputs for fuzzy reference and fuzzy measurement finally a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed loop fuzzy control

fuzzy basis functions universal approximation and orthogonal least squares learning 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

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

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

dynamic non singleton fuzzy logic systems for nonlinear modeling 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

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

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

comparing the performance potentials of singleton and non singleton type 1 and interval type 2 fuzzy systems in terms of sculpting the state space 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

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

a multiobjective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy rule based systems in this paper we propose the use of a multiobjective evolutionary approach to generate a set of linguistic fuzzy rule based systems with different tradeoffs between accuracy and interpretability in regression problems accuracy and interpretability are measured in terms of approximation error and rule base rb complexity respectively the proposed approach is based on concurrently learning rbs and parameters of the membership functions of the associated linguistic labels to manage the size of the search space we have integrated the linguistic two tuple representation model which allows the symbolic translation of a label by only considering one parameter with an efficient modification of the well known 2 2 pareto archived evolution strategy paes we tested our approach on nine real world datasets of different sizes and with different numbers of variables besides the 2 2 paes we have also used the well known nondominated sorting genetic algorithm nsga ii and an accuracy driven single objective evolutionary algorithm ea we employed these optimization techniques both to concurrently learn rules and parameters and to learn only rules we compared the different approaches by applying a nonparametric statistical test for pairwise comparisons thus taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective eas finally a data complexity measure which is typically used in pattern recognition to evaluate the data density in terms of average number of patterns per variable has been introduced to characterize regression problems results confirm the effectiveness of our approach particularly for possibly high dimensional datasets with high values of the complexity metric

an entropy based nonparametric test for the validation of surrogate endpoints we present a nonparametric test to validate surrogate endpoints based on measure of divergence and random permutation this test is a proposal to directly verify the prentice statistical definition of surrogacy the test does not impose distributional assumptions on the endpoints and it is robust to model misspecification our simulation study shows that the proposed nonparametric test outperforms the practical test of the prentice criterion in terms of both robustness of size and power we also evaluate the performance of three leading methods that attempt to quantify the effect of surrogate endpoints the proposed method is applied to validate magnetic resonance imaging lesions as the surrogate endpoint for clinical relapses in a multiple sclerosis trial copyright 2012 john wiley sons ltd

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

a ts type maximizing discriminability based recurrent fuzzy network for classification problems this work proposes a takagi sugeno ts type maximizing discriminability based recurrent fuzzy network mdrfn that can classify highly confusable patterns the discriminative capability plays a significant role in determining classification performance to increase the discriminative capability the proposed mdrfn considers minimum classification error mce and minimum training error mte in mce the weights are updated by maximizing the discrimination among different classes in mte the parameter learning adopts the gradient descent method to reduce the cost function therefore the novelty of mdrfn is that it not only minimizes the cost function but maximizes the discriminative capability as well the effectiveness of the proposed mdrfn is demonstrated by three temporal classification problems in the experiments other rfns including the singleton type recurrent neural fuzzy network srnfn ts type rfn trfn and simple rfn srfn are compared analysis results indicate that the proposed mdrfn exhibits excellent classification performance

kolmogorov s theorem is relevant we show that kolmogorov s theorem on representations of continuous functions of n variables by sums and superpositions of continuous functions of one variable is relevant in the context of neural networks we give a version of this theorem with all of the one variable functions approximated arbitrarily well by linear combinations of compositions of affine functions with some given sigmoidal function we derive an upper estimate of the number of hidden units

hybrid adaptive fuzzy identification and control of nonlinear systems we present a combined direct and indirect adaptive control scheme for adjusting an adaptive fuzzy controller and adaptive fuzzy identification model parameters first using adaptive fuzzy building blocks with a common set of parameters we design and study an adaptive controller and an adaptive identification model that have been proposed for a general class of uncertain structure nonlinear dynamic systems we then propose a hybrid adaptive ha law for adjusting the parameters the ha law utilizes two types of errors in the adaptive system the tracking error and the modeling error performance analysis using a lyapunov synthesis approach proves the superiority of the ha law over the direct adaptive da method in terms of faster and improved tracking and parameter convergence furthermore this is achieved at negligible increased implementation cost or computational complexity we prove a theorem that shows the properties of this hybrid adaptive fuzzy control system i e bounds for the integral of the squared errors and the conditions under which these errors converge asymptotically to zero are obtained finally we apply the hybrid adaptive fuzzy controller to control a chaotic system and the inverted pendulum system

applications of type 2 fuzzy logic systems to forecasting of time series abstract in this paper we begin with a type 1 fuzzy logic system fls trained with noisy data we then demonstrate how information about the noise in the training data can be incorporated into a type 2 fls which can be used to obtain bounds within which the true noisefree output is likely to lie we do this with the example of a one step predictor for the mackey glass chaotic time series m c mackey l glass oscillation and chaos in physiological control systems science 197 1977 287 280 we also demonstrate how a type 2 fls can be used to obtain better predictions than those obtained with a type 1 fls

information gain and a general measure of correlation summary given a parametric model of dependence between two random quantities x and y the notion of information gain can be used to define a measure of correlation this definition of correlation generalizes both the usual product moment correlation coeffi cient for the bivariate normal model and the multiple correlation coefficient in the standard linear regression model the use of this information based correlation in a descriptive statistical analysis is examined and several examples are given if the dependence between two random quantities x and y is modelled parametri cally then the concept of information gain can be used to define a measure of correlation this correlation coefficient can appear in two possible contexts depending on whether one models the joint distribution of x and y or just the conditional distribution of y given x an important motivating feature for this information based correlation coefficient is the fact that it generalizes both the usual product moment correlation coefficient for the bivariate normal model and the usual multiple correlation coefficient for the standard multiple regression model with normal errors our intuition is well developed for these usual correlation coefficients and hopefully our intuition will still be applicable for the information based correlation in more general modelling situations of parametric dependence further since our correlation coefficient is based on information gain we might hope to extend our intuition to interpret the information gain in any statistical modelling situation where we want to assess how much better a more complicated model is than a simpler model the concept of information gain for general statistical models is described in 2 this concept is then used to define an information based measure of correlation the joint case is covered in 3 and the conditional case in 4 estimation of the correlation coefficient is carried out by estimating the corresponding information gain see 5 7 the use of information gain for the purpose of model choice in a descriptive statistical analysis is discussed in 8 and a comparison between our approach and akaike s information criterion is given in 9 some examples of the use of this information based correlation are given in 10 and 11

a new methodology for the online adaptation of fuzzy self structuring controllers in this study a novel fuzzy controller which is able to self design from scratch while working online is proposed the controller does not use the information regarding the differential equations that govern the plant s behavior or any of their bounds the algorithm presented is able to determine the most adequate topology for the fuzzy controller based on the data obtained during the system s normal operation therefore the controller can start operating with an empty set of fuzzy rules and needs no offline training the proposed methodology comprises two phases adaptation of the consequents for every selected topology and online addition of new membership functions mfs some of the main advantages of this method are its robustness under changes on the plant s dynamics good performance in noisy situations and the ability to perform variable selection among a group of candidate variables unlike other online methods the modification of the topology is based on the analysis of the whole operating region of the plant thus providing higher robustness several simulation examples are used to show these features

an orthogonal least squares method for recurrent fuzzy neural modeling abstract this paper presents an orthogonal least squares ols based modeling method named dynamic ols d ols for generating recurrent fuzzy models a dynamic neuron based fuzzy neural network is proposed comprising generalized takagi sugeno kang tsk fuzzy rules whose consequent parts consist of dynamic neurons with local output feedback from an arbitrarily large set of candidate dynamic neurons the d ols method selects automatically the most important ones thus in the resulting model the consequent part of each fuzzy rule contains dynamic neurons with different time delays the proposed dynamic model equipped with the learning algorithm is applied to two temporal problems where the effectiveness of the suggested method as well as the advantages of the resulting dynamic model are demonstrated

powerful and flexible fuzzy algorithm for nonlinear dynamic system identification a new powerful and flexible fuzzy algorithm for nonlinear dynamic system identification is presented it is based on the identification of the derivative of the system state instead of the future system state the membership functions of the underlying static fuzzy model are two sided gaussian functions and the learning algorithm is a hybrid nested routine based on least squares quasi newton and simplex optimization methods moreover a simple clustering algorithm based on an additional higher level fuzzy model is proposed the application to the identification of the mackey glass chaotic time series is presented and compared with previous results in terms of maximum error and nondimensional error index finally the application to a test nonlinear dynamic system is presented to show the capabilities of the clustering algorithm the obtained results show that the proposed algorithm can find wide application in practical problems such as in nonlinear electronic circuit design

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

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