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

a vectorization optimization method based type 2 fuzzy neural network for noisy data classification this paper proposes a vectorization optimization method vom based type 2 fuzzy neural network vom2fnn for noisy data classification in handling problems with uncertainties such as noisy data type 2 fuzzy systems usually outperform their type 1 counterparts hence type 2 fuzzy sets are adopted in the antecedent parts to model the uncertainty to consider the classification problems the discriminative capability is crucial to determine the performance therefore a vom is proposed in the consequent parts to increase the discriminability and reduce the parameters compared with other existing fuzzy neural networks the novelty of the proposed vom2fnn is its consideration of both uncertainty and discriminability the effectiveness of the proposed vom2fnn is demonstrated by three classification problems experimental results and theoretical analysis indicate that the proposed vom2fnn performs better than the other fuzzy neural networks

fuzzy wavelet neural network models for prediction and identification of dynamical systems this paper presents fuzzy wavelet neural network fwnn models for prediction and identification of nonlinear dynamical systems the proposed fwnn models are obtained from the traditional takagi sugeno kang fuzzy system by replacing the then part of fuzzy rules with wavelet basis functions that have the ability to localize both in time and frequency domains the first and last model use summation and multiplication of dilated and translated versions of single dimensional wavelet basis functions respectively and in the second model then parts of the rules consist of radial function of wavelets gaussian type of activation functions are used in if part of the fuzzy rules a fast gradient based training algorithm i e the broyden fletcher goldfarb shanno method is used to find the optimal values for unknown parameters of the fwnn models simulation examples are also given to compare the effectiveness of the models with the other known methods in the literature according to simulation results we see that the proposed fwnn models have impressive generalization ability

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

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

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

a species based improved electromagnetism like mechanism algorithm for tsk type interval valued neural fuzzy system optimization this paper proposes a novel tsk type interval valued neural fuzzy system with asymmetric membership functions tivnfs a which is optimized by a species based improved electromagnetism like mechanism siem algorithm the proposed tivnfs a uses interval valued asymmetric fuzzy membership functions and the tsk type consequent part to improve the approximation performance and to reduce the number of fuzzy rules in addition the corresponding type reduction procedure is integrated in the adaptive network layers to reduce the amount of computation in the system to train the tivnfs as the siem combines the advantages of electromagnetism like mechanism em algorithm and gradient descent technique to obtain faster convergence and lower computational complexity based on a similarity measurement the swarm of the siem has a dynamic population size and its population is also divided dynamically into subpopulations for optimization the gradient descent method is applied to the dominant particle in each species for a neighborhood search finally the nonlinear system identification and classification problems are shown to demonstrate the performance and effectiveness of the proposed tivnfs a with the siem

can a breakpoint graph be decomposed into none other than 2 cycles abstract breakpoint graph has been widely used as a key data structure in algorithm design for genome rearrangements the problem of breakpoint graph cycle decomposition which asks for a largest collection of edge disjoint cycles is crucial in computing rearrangement distances between genomes this problem is np hard and can be approximated to 1 4193 it is still open for deciding whether a breakpoint graph can admit a cycle decomposition with none other than 2 cycles in this paper we present a linear time algorithm to detect whether a breakpoint graph can be decomposed into none other than 2 cycles

fuzzy set based models of neurons and knowledge based networks we will introduce and study different fuzzy set oriented computational models of neurons the generic topologies of the neurons emerging there are significantly influenced by basic logic operators and or not encountered in the theory of fuzzy sets the logical flavor of the proposed constructs is expressed in terms of operators used in their formalization and a way of their superposition in the neurons the two broad categories of neurons embrace basic aggregation neurons named and and or neurons and referential processing units such as matching dominance inclusion neurons the specific features of the neurons are flexibly modeled with the aid of triangular norms the inhibitory and excitatory characteristics are captured by embodying direct and complemented negated input signals we will propose various topologies of neural networks put together with the use of these neurons and demonstrate straightforward relationships coming off between the problem specificity and the resulting architecture of the network this limpid way of mapping the domain knowledge onto the structure of the network contributes significantly toward enhancements in learning processes in the network and substantially facilitates its interpretation

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

exact and analytic numerical solution of coupled diffusion problems in a semi infinite medium in this paper we consider coupled semi infinite diffusion problems of the form ut x t a2uxx x t x 0 t 0 subject to u 0 t b and u x 0 0 where a is a matrix in cr r and u x t and b are vectors in cr given an admissible error and a bounded domain d x0 x1 t0 x t 0 0 an analytic numerical solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in d x0 x1 t0

two efficient twin elm methods with prediction interval in the operational optimization and scheduling problems of actual industrial processes such as iron and steel and microelectronics the operational indices and process parameters usually need to be predicted however for some input and output variables of these prediction models there may exist a lot of uncertainties coming from themselves the measurement error the rough representation and so on in such cases constructing a prediction interval pi for the output of the corresponding prediction model is very necessary in this paper two twin extreme learning machine telm models for constructing pis are proposed first we propose a regularized asymmetric least squares extreme learning machine rals elm method in which different weights of its squared error loss function are set according to whether the error of the model output is positive or negative in order that the above error can be differentiated in the parameter learning process and tikhonov regularization is introduced to reduce overfitting then we propose an asymmetric bayesian extreme learning machine ab elm method based on the bayesian framework with the asymmetric gaussian distribution ab elm in which the weights of its likelihood function are determined as the same method in rals elm and the type ii maximum likelihood algorithm is derived to learn the parameters of ab elm based on rals elm and ab elm we use a pair of weights following the reciprocal relationship to obtain two nonparallel regressors including a lower bound regressor and an upper bound regressor respectively which can be used for calculating the pis finally some discussions are given about how to adjust the weights adaptively to meet the desired pi how to use the proposed telms for nonlinear quantile regression and so on results of numerical comparison on data from one synthetic regression problem three university of california irvine benchmark regression problems and two actual industrial regression problems show the effectiveness of the proposed models

exact and analytic numerical solution of coupled parabolic mixed problems in a semi infinite medium abstract this paper deals with the construction of accurate analytic numerical approximations of coupled parabolic mixed systems using sine fourier transforms an exact expression of the exact solution is given then given an admissible error e and a time bounded subdomain an analytic numerical solution is constructed so that the approximation error is less than e in

on characterization of generalized interval type 2 fuzzy rough sets this paper presents a systematic study of interval type 2 fuzzy rough sets integrating rough set theory with interval type 2 fuzzy set theory using constructive and axiomatic approaches from the perspective of a constructive approach a pair of lower and upper interval type 2 fuzzy rough approximation operators with respect to an interval type 2 fuzzy relation is defined the basic properties of the interval type 2 fuzzy rough approximation operators are studied using cut sets of interval type 2 fuzzy sets classical representations of interval type 2 fuzzy rough approximation operators are then presented and the connections between special interval type 2 fuzzy relations and interval type 2 fuzzy rough approximation operators are investigated adopting an axiomatic approach an operator oriented characterization of interval type 2 fuzzy rough sets is proposed in other words interval type 2 fuzzy rough approximation operators are characterized by axioms different axiom sets of interval type 2 fuzzy set theoretic operators guarantee the existence of different types of interval type 2 fuzzy relations that produce the same operators finally the relationship between interval type 2 fuzzy rough sets and interval type 2 fuzzy topological spaces is examined we obtain sufficient and necessary conditions for the conjecture that an interval type 2 fuzzy interior closure operator derived from an interval type 2 fuzzy topological space can associate with an interval type 2 fuzzy reflexive and transitive relation such that the corresponding lower upper interval type 2 fuzzy rough approximation operator is the interval type 2 fuzzy interior closure operator

robust h_ infty based synchronization of the fractional order chaotic systems by using new self evolving nonsingleton type 2 fuzzy neural networks in this paper a novel h infty based adaptive fuzzy control is presented for the synchronization of fractional order chaotic systems a self evolving nonsingleton type 2 fuzzy neural network se nst2fnn is proposed for the estimation of the unknown functions in the dynamics of the system the effects of the approximation error and the external disturbances are eliminated by designing an adaptive compensator such that the h infty norm of the synchronization error is minimized and asymptotically stability is achieved the consequent parameters of se nst2fnn are tuned based on the adaptation laws that are derived from lyapunov stability analysis the antecedent part and the rule database of se nst2fnn are optimized based on a clustering method and the modified invasive weed optimization algorithm respectively the effectiveness of proposed control scheme is verified by simulation examples

reduced interval type 2 neural fuzzy system using weighted bound set boundary operation for computation speedup and chip implementation this paper proposes a reduced interval type 2 neural fuzzy system using weighted bound set boundaries rit2nfs wb for the simplification of type reduction operations the objective of this simplification is to reduce the system training time in software implementation and chip size in hardware implementation especially when the number of rules is large the antecedent part in the rit2nfs wb uses interval type 2 fuzzy sets it2fss and the consequent part can be of the takagi sugeno kang tsk or mamdani type the rit2nfs wb is built through an online structure and parameter learning to improve model accuracy in addition the interpretability of the rit2nfs wb is improved by considering distributions of the it2fss in input variables a distinguishability oriented cost function is used in parameter learning to generate distinguishable it2fss and improve semantics based interpretability for highly overlapped it2fss they are merged to reduce the number of it2fss and improve complexity based interpretability the software implemented tsk type rit2nfs wb is hardware implemented on a field programmable gate array chip to accelerate the chip execution speed the chip utilizes not only the parallel execution properties of fuzzy rules and bound set boundaries but the pipeline technique as well in particular the flexibility of the chip is considered so that no redesign of the circuits is required when the rit2nfs wb is applied to different problems the characteristics of the software and hardware implemented rit2nfs wb are verified through various examples and comparisons with various type 1 and interval type 2 fuzzy models

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

robust exponential attractors for singularly perturbed hodgkin huxley equations abstract here we consider a singular perturbation of the hodgkin huxley system which is derived from the lieberstein s model we study the associated dynamical system on a suitable bounded phase space when the perturbation parameter e i e the axon specific inductance is sufficiently small we prove the existence of bounded absorbing sets as well as of smooth attracting sets we deduce the existence of a smooth global attractor a e finally we prove the main result that is the existence of a family of exponential attractors e e which is holder continuous with respect to e

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

continuous numerical solutions and error bounds for time dependent systems of partial differential equations mixed problems abstract the aim of this paper is to construct continuous numerical solutions with a prefixed accuracy in a bounded domain t 0 t 1 0 p t 0 t 1 for mixed problems of the type u t x t d t u xx x t 0 0 x p t 0 subject to u 0 t u p t 0 and u x 0 f x here u x t and f x are r component vectors and d t is a c r r valued two times continuously differentiable function so that d t 1 d t 2 d t 2 d t 1 for t 2 t 1 0 and there exists a positive number such that every eigenvalue z of d t d h t 2 with t 0 is bigger than

on constructing parsimonious type 2 fuzzy logic systems via influential rule selection type 2 fuzzy systems are increasing in popularity and there are many examples of successful applications while many techniques have been proposed for creating parsimonious type 1 fuzzy systems there is a lack of such techniques for type 2 systems the essential problem is to reduce the number of rules while maintaining the system s approximation performance in this paper four novel indexes for ranking the relative contribution of type 2 fuzzy rules are proposed which are termed r values c values omega1 values and omega2 values the r values of type 2 fuzzy rules are obtained by applying a qr decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model the c values rank rules based on the effects of rule consequents while the omega1 values and omega2 values consider both the rule base structure via firing strength matrices and the output contribution of fuzzy rule consequents two procedures for utilizing these indexes in fuzzy rule selection termed ldquoforward selectionrdquo and ldquobackward eliminationrdquo are described experiments are presented which demonstrate that by using the proposed methodology the most influential type 2 fuzzy rules can be effectively retained in order to construct parsimonious type 2 fuzzy models

analytic and numerical solution of coupled implicit semi infinite diffusion problems abstract this paper deals with singular coupled implicit semi infinite mixed diffusion problems by application of the sine fourier transform existence conditions and an analytic closed form solution is first obtained given an admissible error and a rectangular bounded closed domain analytic numerical approximations whose error with respect to the exact solution is less than the admissible error in the bounded domain are constructed an algorithm and an illustrative example are included

on the general linear coupled system for diffusion in media with two diffusivities abstract for the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity simple uniqueness criteria and a method of solution of boundary value problems are established the equations studied retain the so called cross terms which have been neglected in all previous investigations moreover these equations arise as generalizations of a number of existing theories for example heat flow in heterogeneous multicomponent systems flow of water in fissured rocks and a model of an arms race the simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants the solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems the paper emphasizes the general equations without special reference to particular physical applications or boundary value problems

sequential adaptive fuzzy inference system safis for nonlinear system identification and prediction in this paper a sequential adaptive fuzzy inference system called safis is developed based on the functional equivalence between a radial basis function network and a fuzzy inference system fis in safis the concept of influence of a fuzzy rule is introduced and using this the fuzzy rules are added or removed based on the input data received so far if the input data do not warrant adding of fuzzy rules then only the parameters of the closest in a euclidean sense rule are updated using an extended kalman filter ekf scheme the performance of safis is compared with several existing algorithms on two nonlinear system identification benchmark problems and a chaotic time series prediction problem results indicate that safis produces similar or better accuracies with less number of rules compared to other algorithms

an enhanced type reduction algorithm for type 2 fuzzy sets karnik and mendel proposed an algorithm to compute the centroid of an interval type 2 fuzzy set efficiently based on this algorithm liu developed a centroid type reduction strategy to carry out type reduction for type 2 fuzzy sets a type 2 fuzzy set is decomposed into a collection of interval type 2 fuzzy sets by cuts then the karnik mendel algorithm is called for each interval type 2 fuzzy set iteratively however the initialization of the switch point in each application of the karnik mendel algorithm is not a good one in this paper we present an improvement to liu s algorithm we employ the previously obtained result to construct the starting values in the current application of the karnik mendel algorithm convergence in each iteration except the first one can then speed up and type reduction for type 2 fuzzy sets can be carried out faster the efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results

continuous numerical solutions of coupled mixed partial differential systems using fer s factorization in this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time dependent systems of the type ut a t uxx b t u 0 0 0 u 0 t u p t 0 u x 0 f x 06 x6p after truncation of an exact series solution the numerical solution is constructed using fer s factorization given 0 and t0 t1 with 0 t0 t1 and d t0 t1 f x t 06x6p t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in d t0 t1 an algorithm is also included c 1999 elsevier science b v all rights reserved ams classication 65m15 34a50 35c10 35a50

a matrix approach to the analytic numerical solution of mixed partial differential systems e navarro e ponsoda and l jodar departamento de matem tica aplicada universidad polit cnica de valencia p o box 22012 valencia spain received and accepted march 1994 abstract in this paper analytic numerical solutions for initial boundary value problems related to systems of partial differential equations are proposed given an admissible error e 0 and a finite domain g a numerical approximation is constructed in terms of the data so that the error is uniformly upper bounded by e in g keywords partial differential system eigenvalues bound error bound schur decomposition matrix norms matrix functions

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

type 2 fuzzy neural networks with fuzzy clustering and differential evolution optimization in many real world problems involving pattern recognition system identification and modeling control decision making and forecasting of time series available data are quite often of uncertain nature an interesting alternative is to employ type 2 fuzzy sets which augment fuzzy models with expressive power to develop models which efficiently capture the factor of uncertainty the three dimensional membership functions of type 2 fuzzy sets offer additional degrees of freedom that make it possible to directly and more effectively account for model s uncertainties type 2 fuzzy logic systems developed with the aid of evolutionary optimization forms a useful modeling tool subsequently resulting in a collection of efficient if then rules the type 2 fuzzy neural networks take advantage of capabilities of fuzzy clustering by generating type 2 fuzzy rule base resulting in a small number of rules and then optimizing membership functions of type 2 fuzzy sets present in the antecedent and consequent parts of the rules the clustering itself is realized with the aid of differential evolution several examples including a benchmark problem of identification of nonlinear system are considered the reported comparative analysis of experimental results is used to quantify the performance of the developed networks

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

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

a sequential learning algorithm for complex valued self regulating resource allocation network csran this paper presents a sequential learning algorithm for a complex valued resource allocation network with a self regulating scheme referred to as complex valued self regulating resource allocation network csran the self regulating scheme in csran decides what to learn when to learn and how to learn based on the information present in the training samples csran is a complex valued radial basis function network with a sech activation function in the hidden layer the network parameters are updated using a complex valued extended kalman filter algorithm csran starts with no hidden neuron and builds up an appropriate number of hidden neurons resulting in a compact structure performance of the csran is evaluated using a synthetic complex valued function approximation problem two real world applications consisting of a complex quadrature amplitude modulation channel equalization and an adaptive beam forming problem since complex valued neural networks are good decision makers the decision making ability of the csran is compared with other complex valued classifiers and the best performing real valued classifier using two benchmark unbalanced classification problems from uci machine learning repository the approximation and classification results show that the csran outperforms other existing complex valued learning algorithms available in the literature

a generalized constraint neural network model associating partially known relationships for nonlinear regressions in an attempt to enhance the neural network technique so that it can evolve from a black box tool into a semi analytical one we propose a novel modeling approach of imposing generalized constraints on a standard neural network we redefine approximation problems by use of a new formalization with the aim of embedding prior knowledge explicitly into the model to the maximum extent a generalized constraint neural network gcnn model has therefore been developed which basically consists of two submodels one is constructed by the standard neural network technique to approximate the unknown part of the target function the other is formed from partially known relationships to impose generalized constraints on the whole model three issues arising after combination of the two submodels are discussed a the better approximation provided by the gcnn model compared with a standard neural network b the identifiability of parameters in the partially known relationships and c the discrepancy in the approximation due to removable singularities in the target function numerical studies of three benchmark problems show important findings that have not previously been reported in the literature significant benefits were observed from using the gcnn model in comparison with a standard neural network

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

an interval type 2 neural fuzzy classifier learned through soft margin minimization and its human posture classification application this paper proposes an interval type 2 neural fuzzy classifier learned through soft margin minimization it2nfc smm and applies it to human body posture classification the it2nfc smm consists of interval type 2 zero order takagi sugeno t s fuzzy rules established through online structure learning the antecedent part of the it2nfc smm uses interval type 2 fuzzy sets to decrease the number of rules and manage noisy data for parameter learning the consequent parameters are learned through a linear support vector machine svm for soft margin minimization to improve the generalization ability the proposed svm based learning addresses the problem that the orders of the fuzzy rules in computing the outputs of an interval type 2 fuzzy system depend on the consequent values that are unknown in advance to address this problem the it2nfc smm uses weighted bound set boundaries to simplify the type reduction operation and a novel crisp to interval linear svm learning algorithm based on the soft margin minimization the antecedent parameters are tuned using the gradient descent algorithm the it2nfc smm is applied to a vision based human body posture classification system the system uses two cameras and novel classification features extracted from a silhouette of the human body to classify the four postures of standing bending sitting and lying the classification performance of the it2nfc smm is verified through results in clean and noisy classification examples and through the posture classification problem as well as through comparisons with various type 1 and type 2 fuzzy classifiers the overall result shows that the it2nfc smm achieves higher classification rates with a smaller or similar model size than the classifiers used for comparison especially for noisy classification problems

an interval type 2 neural fuzzy system for online system identification and feature elimination we propose an integrated mechanism for discarding derogatory features and extraction of fuzzy rules based on an interval type 2 neural fuzzy system nfs in fact it is a more general scheme that can discard bad features irrelevant antecedent clauses and even irrelevant rules high dimensional input variable and a large number of rules not only enhance the computational complexity of nfss but also reduce their interpretability therefore a mechanism for simultaneous extraction of fuzzy rules and reducing the impact of or eliminating the inferior features is necessary the proposed approach namely an interval type 2 neural fuzzy system for online system identification and feature elimination it2nfs sife uses type 2 fuzzy sets to model uncertainties associated with information and data in designing the knowledge base the consequent part of the it2nfs sife is of takagi sugeno kang type with interval weights the it2nfs sife possesses a self evolving property that can automatically generate fuzzy rules the poor features can be discarded through the concept of a membership modulator the antecedent and modulator weights are learned using a gradient descent algorithm the consequent part weights are tuned via the rule ordered kalman filter algorithm to enhance learning effectiveness simulation results show that it2nfs sife not only simplifies the system architecture by eliminating derogatory irrelevant antecedent clauses rules and features but also maintains excellent performance

solving coupled systems of linear second order differential equations knowing a part of the spectrum of the companion matrix abstract in this paper an explicit closed form solution of initial value problems for coupled systems of time invariant second order differential equations is given without computing the exponential of the associated companion matrix

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

state space black box modelling via markov parameters based on evolving type 2 neural fuzzy inference system for nonlinear multivariable dynamic systems abstract in this paper an approach for state space interval type 2 neural fuzzy identification of multivariable dynamic systems in an evolving and incremental learning context is proposed in combination with type 2 fuzzy systems the adopted methodology regards the following aspects recursive learning of the footprint of uncertainty and recursive learning of the rule consequents part by a recursive linear state space approaches where markov parameters are incrementally and robustly learnt in sample wise manner the efficiency and applicability of the proposed methodology are demonstrated through experimental and computational results and compared with prominent learning algorithms of the literature

nonlinear systems design by a novel fuzzy neural system via hybridization of electromagnetism like mechanism and particle swarm optimisation algorithms in this paper a hybrid of algorithms for electromagnetism like mechanisms em and particle swarm optimisation pso called hempso is proposed for use in designing a functional link based petri recurrent fuzzy neural system flprfns for nonlinear system control the flprfns has a functional link based orthogonal basis function fuzzy consequent and a petri layer to eliminate the redundant fuzzy rule for each input calculation in addition the flprfns is trained by the proposed hybrid algorithm the main innovation is that the random neighbourhood local search is replaced by a pso algorithm with an instant update strategy for particle information each particle updates its information instantaneously and in this way receives the best current information thus hempso combines the advantages of multiple agent based searching global optimisation and rapid convergence simulation results confirm that hempso can be used to perform global optimisation and offers the advantage of rapid convergence they also indicate that the flprfns exhibits high accuracy

interval type 2 mutual subsethood fuzzy neural inference system it2msfunis this paper presents an interval type 2 mutual subsethood fuzzy neural inference system it2msfunis a mutual subsethood measure between two interval type 2 fuzzy sets it2 fs has been derived and has been used in determining the similarity between the it2 fs inputs and it2 fs antecedents the consequent weights are taken to be interval sets the inputs to the system are fuzzified into it2 fss with gaussian primary membership function having fixed center and uncertain variance aggregation of type 2 mutual subsethood based activation spreads is performed using product operator the output is obtained using simplified type reduction followed by defuzzification the system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space the mathematical modeling and empirical studies of it2msfunis bring forth its efficacy in problems pertaining to function approximation time series prediction control and classification comparisons with other type 1 and type 2 neuro fuzzy systems verify that it2msfunis compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy empirical studies indicate the intelligent decision making capability of the proposed model the main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between it2 fss and to find out its applicability in diverse application domains the improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of it2 fss the integration of mutual subsethood with interval type 2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well known problems reported in the literature

a metacognitive complex valued interval type 2 fuzzy inference system this paper presents a complex valued interval type 2 neuro fuzzy inference system cit2fis and derive its metacognitive projection based learning pbl algorithm metacognitive cit2fis mc cit2fis consists of a cit2fis which realizes takagi sugeno kang type inference mechanism as its cognitive component a pbl with self regulation is its metacognitive component the rules of cit2fis employ interval type 2 q gaussian membership functions that can represent different radial basis functions for different values of q as each sample is presented to the network the metacognitive component monitors the hinge loss error and class specific knowledge potential of the current sample to efficiently decide on what to learn when to learn and how to learn it when a new rule is added or existing rules are updated the optimal parameters of cit2fis corresponding to the minimum of the hinge loss error function are computed using a pbl algorithm derived using the wirtinger calculus the performance of mc cit2fis is evaluated on a set of benchmark real valued classification problems from the uci machine learning repository a circular transformation is used to convert the real valued features to the complex valued features in these problems the performance comparison and statistical study clearly show the superior classification ability of mc cit2fis finally the proposed complex valued network is used to solve a practical human action recognition problem that is represented by complex valued optical flow based feature set and a human emotion recognition problem represented using complex valued gabor filter based features the performance results on these problems substantiate the superior classification ability of mc cit2fis

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

developing a local least squares support vector machines based neuro fuzzy model for nonlinear and chaotic time series prediction local modeling approaches owing to their ability to model different operating regimes of nonlinear systems and processes by independent local models seem appealing for modeling identification and prediction applications in this paper we propose a local neuro fuzzy lnf approach based on the least squares support vector machines lssvms the proposed lnf approach employs lssvms which are powerful in modeling and predicting time series as local models and uses hierarchical binary tree hbt learning algorithm for fast and efficient estimation of its parameters the hbt algorithm heuristically partitions the input space into smaller subdomains by axis orthogonal splits in each partitioning the validity functions automatically form a unity partition and therefore normalization side effects e g reactivation are prevented integration of lssvms into the lnf network as local models along with the hbt learning algorithm yield a high performance approach for modeling and prediction of complex nonlinear time series the proposed approach is applied to modeling and predictions of different nonlinear and chaotic real world and hand designed systems and time series analysis of the prediction results and comparisons with recent and old studies demonstrate the promising performance of the proposed lnf approach with the hbt learning algorithm for modeling and prediction of nonlinear and chaotic systems and time series

an online self constructing neural fuzzy inference network and its applications a self constructing neural fuzzy inference network sonfin with online learning ability is proposed in this paper the sonfin is inherently a modified takagi sugeno kang tsk type fuzzy rule based model possessing neural network learning ability there are no rules initially in the sonfin they are created and adapted as online learning proceeds via simultaneous structure and parameter identification in the structure identification of the precondition part the input space is partitioned in a flexible way according to an aligned clustering based algorithm as to the structure identification of the consequent part only a singleton value selected by a clustering method is assigned to each rule initially afterwards some additional significant terms selected via a projection based correlation measure for each rule will be added to the consequent part incrementally as learning proceeds the combined precondition and consequent structure identification scheme can set up an economic and dynamically growing network a main feature of the sonfin in the parameter identification the consequent parameters are tuned optimally by either least mean squares or recursive least squares algorithms and the precondition parameters are tuned by a backpropagation algorithm to enhance the knowledge representation ability of the sonfin a linear transformation for each input variable can be incorporated into the network so that much fewer rules are needed or higher accuracy can be achieved

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

design of interval type 2 fuzzy sliding mode controller in this paper an interval type 2 fuzzy sliding mode controller it2fsmc is proposed for linear and nonlinear systems the proposed it2fsmc is a combination of the interval type 2 fuzzy logic control it2flc and the sliding mode control smc which inherits the benefits of these two methods the objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed loop system the lyapunov stability method is adopted to verify the stability of the interval type 2 fuzzy sliding mode controller system the design procedure of the it2fsmc is explored in detail a typical second order linear interval system with 50 parameter variations an inverted pendulum with variation of pole characteristics and a duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method the simulation results show that the it2fsmc achieves the best tracking performance in comparison with the type 1 fuzzy logic controller t1flc the it2flc and the type 1 fuzzy sliding mode controller t1fsmc

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

simulation of the bird age structured population growth based on an interval type 2 fuzzy cellular structure in this paper an age structured population growth model based on a fuzzy cellular structure is proposed an age structured population growth model enables a better description of population dynamics in this paper the dynamics of a particular bird species is considered the dynamics is governed by the variation of natality mortality and emigration rates which in this work are evaluated using an interval type 2 fuzzy logic system the use of type 2 fuzzy logic enables handling the effects caused by environment heterogeneity on the population a set of fuzzy rules about population growth are derived from the interpretation of the ecological laws and the bird life cycle the proposed model is formulated using discrete mathematics within the framework of a fuzzy cellular structure the fuzzy cellular structure allows us to visualize the evolution of the population s spatial dynamics the spatial distribution of the population has a deep effect on its dynamics moreover the model enables not only to estimate the percentage of occupation on the cellular space when the species reaches its stable equilibrium level but also to observe the occupation patterns

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

finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed hodgkin huxley systems abstract in an effort to show that the standard hodgkin huxley system approximates the hyperbolic hodgkin huxley system we view the hyperbolic system as a singular perturbation of the standard system we establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and hausdorff dimensions we then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

data based system modeling using a type 2 fuzzy neural network with a hybrid learning algorithm we propose a novel approach for building a type 2 neural fuzzy system from a given set of input output training data a self constructing fuzzy clustering method is used to partition the training dataset into clusters through input similarity and output similarity tests the membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster then a type 2 fuzzy takagi sugeno kang if then rule is derived from each cluster to form a fuzzy rule base a fuzzy neural network is constructed accordingly and the associated parameters are refined by a hybrid learning algorithm which incorporates particle swarm optimization and a least squares estimation for a new input a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type 2 fuzzy set which is then defuzzified by applying a refined type reduction algorithm experimental results are presented to demonstrate the effectiveness of our proposed approach

an initial boundary value problem of physiological significance for equations of nerve conduction a model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid loligo of hodgkin and huxley 5 this nonlinear parabolic partial differential equation pde when considered as part of a particular initial boundary value problem ibvp models the electrical activity of an axon under conditions found both in nature and in the laboratory this ibvp for the hodgkin huxley equations is proved to be well posed in the sense of hadamard 3 and a priori bounds on the solution are derived

robustness of interval valued fuzzy inference since interval valued fuzzy set intuitively addresses not only vagueness lack of sharp class boundaries but also a feature of uncertainty lack of information interval valued fuzzy reasoning plays a vital role in intelligent systems including fuzzy control classification expert systems and so on to utilize interval valued fuzzy inference better it is very important to study the fundamental properties of interval valued fuzzy inference such as robustness in this paper we first discuss the robustness of interval valued fuzzy connectives and then investigate the robustness of interval valued fuzzy reasoning in terms of the sensitivity of interval valued fuzzy connectives and maximum perturbation of interval valued fuzzy sets these results reveal that the robustness of interval valued fuzzy reasoning is directly linked to the selection of interval valued fuzzy connectives

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

a three domain fuzzy wavelet system for simultaneous processing of time frequency information and fuzziness traditional wavelet system is a two domain time and frequency domains wavelet system 2dws which works only in time and frequency domains the 2dws is not able to treat time frequency information and fuzziness simultaneously for this reason a three domain fuzzy time and frequency domains fuzzy wavelet system 3dfws is proposed where the three domain mechanism provides a solution to handle fuzzy uncertainties and time frequency information together the major advantage of 3dfws is able to use the prior knowledge via the novel fuzzy domain to analyze uncertain data and signals which will enhance the potentials of 2dws experimental and simulation studies show that the performance of the proposed 3dfws is superior to the traditional one for simultaneous processing of time frequency and fuzziness

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

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

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

complete solution sets of inf interval valued fuzzy relation equations fuzzy relation equations play an important role in fuzzy set theory interval valued fuzzy set theory is an extension of fuzzy theory in which a closed subinterval of the unit interval is assigned with membership degree therefore it is very significant to study interval valued fuzzy relation equations from both the theoretical and practical viewpoints in this paper the solution sets of interval valued fuzzy relation equations with inf composition is investigated where is interval valued r s or ql implication necessary and sufficient conditions such that there exist solutions for these equations are first shown some sufficient conditions for existence of maximal solutions for these equations are represented and then it is shown that the complete solution sets of inf interval valued fuzzy relation equations can be determined by their maximal solutions finally the solution sets of linear interval valued fuzzy relation equations are described by a method similar to that in linear algebra

t2fela type 2 fuzzy extreme learning algorithm for fast training of interval type 2 tsk fuzzy logic system a challenge in modeling type 2 fuzzy logic systems is the development of efficient learning algorithms to cope with the ever increasing size of real world data sets in this paper the extreme learning strategy is introduced to develop a fast training algorithm for interval type 2 takagi sugeno kang fuzzy logic systems the proposed algorithm called type 2 fuzzy extreme learning algorithm t2fela has two distinctive characteristics first the parameters of the antecedents are randomly generated and parameters of the consequents are obtained by a fast learning method according to the extreme learning mechanism in addition because the obtained parameters are optimal in the sense of minimizing the norm the resulting fuzzy systems exhibit better generalization performance the experimental results clearly demonstrate that the training speed of the proposed t2fela algorithm is superior to that of the existing state of the art algorithms the proposed algorithm also shows competitive performance in generalization abilities

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

a neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

neuro fuzzy systems for function approximation abstract we present a neuro fuzzy architecture for function approximation based on supervised learning the learning algorithm is able to determine the structure and the parameters of a fuzzy system the approach is an extension to our already published nefcon and nefclass models which are used for control or classification purposes the proposed extended model which we call nefprox is more general and can be used for any application based on function approximation

polynomial operator matrices as semigroup generators the 2 2 case on considere des systemes d equations d evolution non lineaires ecrits sous la forme u t au t ou u est une fonction a valeurs dans un espace produit e n et a a ij n m est une matrice dont les a ij sont des operateurs lineaires sur e on etudie le cas ou les a ij sont des polynomes p ij a par rapport a un seul operateur a sur e et on se restreint au cas des matrices 2 2

a hybrid learning algorithm for a class of interval type 2 fuzzy neural networks in real life information about the world is uncertain and imprecise the cause of this uncertainty is due to deficiencies on given information the fuzzy nature of our perception of events and objects and on the limitations of the models we use to explain the world the development of new methods for dealing with information with uncertainty is crucial for solving real life problems in this paper three interval type 2 fuzzy neural network it2fnn architectures are proposed with hybrid learning algorithm techniques gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation at the antecedents layer a interval type 2 fuzzy neuron it2fn model is used and in case of the consequents layer an interval type 1 fuzzy neuron model it1fn in order to fuzzify the rule s antecedents and consequents of an interval type 2 takagi sugeno kang fuzzy inference system it2 tsk fis it2 tsk fis is integrated in an adaptive neural network in order to take advantage the best of both models this provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy if then rules in addition to handling uncertainty and imprecision on the other hand neural networks are highly adaptable with learning and generalization capabilities experimental results are divided in two kinds in the first one a non linear identification problem for control systems is simulated here a comparative analysis of learning architectures it2fnn and anfis is done for the second kind a non linear mackey glass chaotic time series prediction problem with uncertainty sources is studied finally it2fnn proved to be more efficient mechanism for modeling real world problems

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

analytic solution with a priori error bounds for a class of mixed coupled partial differential equations in this paper the existence of a sequence of stable solutions xn of the parametric matrix equation z2 az n2b 0 so that xnn converges to a stable matrix is studied the result is used for solving mixed problems related to the coupled second order partial differential system utt auxx but 0

an evolving interval type 2 neurofuzzy inference system and its metacognitive sequential learning algorithm in this paper we propose an evolving interval type 2 neurofuzzy inference system it2fis and its fully sequential learning algorithm it2fis employs interval type 2 fuzzy sets in the antecedent part of each rule and the consequent realizes takagi sugeno kang fuzzy inference mechanism in order to render the inference fast and accurate we propose a data driven interval reduction approach to convert interval type 1 fuzzy set in antecedent to type 1 fuzzy number in the consequent during learning the sequential algorithm learns a sample one by one and only once the it2fis structure evolves automatically and adapts its network parameters using metacognitive learning mechanism concurrently the metacognitive learning regulates the learning process by appropriate selection of learning strategies and helps the proposed it2fis to approximate the input output relationship efficiently an evolving it2fis employing a metacognitive learning algorithm is referred to as mcti2fis performance of metacognitive interval type 2 neurofuzzy inference system mcit2fis is evaluated using a set of benchmark time series problems and is compared with existing type 2 and type 1 fuzzy inference systems finally the performance of the proposed mcit2fis has been evaluated using a practical stock price tracking problem the results clearly highlight that mcit2fis performs better than other existing results in the literature