extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

computation of the gamma digamma and trigamma functions this paper gives an approximation for the gamma function that while different has the same form as one by lanczos siam j numer anal b1 1964 pp 86 96 both approximations correct stirlin

successive identification of a fuzzy model and its applications to prediction of a complex system abstract a successive identification method of a fuzzy model is suggested the identification mechanism consists of two levels one is the supervisor level and the other is the adjustment level the supervisor level determines a policy of parameter adjustment using a set of fuzzy adjustment rules the adjustment rules are derived from the fuzzy implications of a fuzzy model and are extended to fuzzy adjustment rules by using an extended concept of zadeh s contrast intensification the adjustment level executes the policy of parameter adjustment determined with the fuzzy adjustment rules the parameter adjustment consists of premise parameter adjustment and consequent parameter adjustment both of them are realized by the weighted recursive least square algorithm finally it is shown from two examples that the method is very useful for modeling complex systems

a high order recurrent neuro fuzzy system with internal dynamics application to the adaptive noise cancellation a high order recurrent neuro fuzzy system ho rnfs is suggested in this paper suitable for modeling highly complex nonlinear temporal processes feedback connections are introduced in the network including the context and the feedback nodes that serve as a means to memorize the firing history the feedback paths in the firing loop are implemented through finite impulse response fir synaptic filters leading to a higher order network with enhanced temporal capabilities the inference mechanism of the ho rnfs is implemented by means of dynamic fuzzy rules where multiple steps ahead predictions are provided for the internal variables at the consequent part its structure is organized in an on line fashion using a concurrent structure and parameter algorithm structure learning generates dynamically the input and output clusters of the rules while parameter learning adjusts the network weights the ho rnfs is compared to the recurrent self organizing neural fuzzy inference network rsonfin being a special case of the suggested network the experimental setup includes a benchmark temporal system and the adaptive noise cancellation problem extensive experimentation reveals that ho rnfs exhibits superior speech enhancement performance as contrasted to rsonfin when complex noise passages are considered

a ga based neural fuzzy system for temperature control abstract this paper addresses the structure and an associated learning algorithm of a feedforward multilayered connectionist network for realizing the basic elements and functions of a traditional fuzzy logic controller the genetic algorithm based neural fuzzy system ga nfs is based on takagi sugeno kang tsk type model possessing a neural network s learning ability a hybrid learning algorithm is proposed for parameters learning the proposed algorithm combines the genetic algorithm ga and the least squares estimate lse method to construct the ga nfs the genetic algorithm is used to tune membership functions at the precondition part of fuzzy rules while the lse method is used to tune parameters at the consequent part of fuzzy rules the performance of the ga nfs is compared to that of the traditional pid controller and fuzzy logic controller on the water bath temperature control system

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

min max linear estimation of band limited sequences in the presence of random noise corresp a problem in band limited signal estimation is presented the signal is assumed to be a band limited sequence subject to an energy constraint the observed signal components are assumed to be corrupted by the components of a discrete time stochastic noise process with zero mean and known antocorrelations the error metric is the energy in a fixed linear combination of the error components linear estimators are considered and an explicit expression for the optimal in a min max sense estimation scheme is derived an interesting feature of the optimal estimation scheme derived in this correspondence is that it is identical to one based on stochastic models for both the signal and the noise

min max linear estimation of band limited sequences from noisy observations a deterministic approach the representation of discrete time waveforms as finite energy sequences is a useful mathematical abstraction a special class of finite energy sequences of particular interest to communication theorists is the class of band limited sequences a linear estimation of technique for these sequences is presented the observations are assumed to be corrupted by the components of a finite energy band limited noise sequence not necessarily band limited to the same frequencies as the signal sequence the error metric for the estimation errors is assumed to be the energy in a linear functional of the error components and the error criterion by which the merit of an estimation scheme is judged is the maximal error over a class of band limited and energy constrained signal and noise sequences an interesting finding is that the min max estimation schemes described here have a structure similar to that of the familiar minimum mean square error mmse schemes which are designed to minimize some average error

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

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

an efficient immune based symbiotic particle swarm optimization learning algorithm for tsk type neuro fuzzy networks design in this paper we propose a new learning algorithm that can be used to design tsk type neuro fuzzy networks though there has been a great deal of interest in the use of immune algorithms ias for computer science and engineering in terms of fundamental methodologies they are not dramatically different from other algorithms in order to enhance the ia performance we propose the immune based symbiotic particle swarm optimization ispso for use in tsk type neuro fuzzy networks for solving the prediction and skin color detection problems the proposed ispso embeds the symbiotic evolution scheme in an ia and utilizes particle swarm optimization pso to improve the mutation mechanism in order to avoid trapping in a local optimal solution and to ensure the search capability of a near global optimal solution mutation plays an important role therefore we employed the advantages of pso to improve the mutation mechanism and used a method that introduces chaotic mapping with certainty ergodicity and the stochastic property into pso to improve global convergence unlike the ia that uses each individual in a population as a full solution to a problem symbiotic evolution assumes that each individual in a population represents only a partial solution to a problem complex solutions combine several individuals in the population

a characteristic point based fuzzy inference system aimed to minimize the number of fuzzy rules this paper presents a characteristic point based fuzzy inference system cpfis for fuzzy modeling from training data the aim of the cpfis is not only satisfactory precision performance but also to employ as few purely linguistic fuzzy rules as possible by using a minimization based systematic training method characteristic points cps are defined as the few data points among the original training data which when they are directly mapped to fuzzy rules and thus form the entire rule base allow the underlying system to be effectively modeled three minimization based algorithms in a sequence are proposed to train the cpfis a gradient projection method a gauss jordan elimination based column elimination and back propagation the cps are determined by iterative computations of the first two minimization algorithms after which the resulting fuzzy sets are further fine tuned by the third algorithm experiments conducted on three benchmark problems showed that the cpfis used one of the smallest number of fuzzy rules among the reported results for other methods the gaussian membership functions in both the input and output fuzzy sets and the small number of fuzzy rules make the rule interpretation of the cpfis much easier than that of other methods thus enhancing human computer cooperation in knowledge discovery

a maximizing discriminability based self organizing fuzzy network for classification problems a maximizing discriminability based self organizing fuzzy network mdsofn that can classify highly confusable patterns is proposed in this paper the underlying notion of the proposed mdsofn is to split the generation of fuzzy rules into linear discriminant analysis lda and gaussian mixture model gmm in lda the weights are updated by seeking directions that are efficient for discrimination in gmm parameter learning adopts the gradient descent method to reduce the cost function since lda derived fuzzy rules increase the discriminative capability among different classes the proposed mdsofn can classify highly confusable patterns the effectiveness of the proposed mdsofn is demonstrated by two classification problems a detailed comparative performance analysis for the fuzzy networks using lda principal component analysis pca and the support vector machine svm with various noise types is presented experimental results and theoretical analysis indicate that the lda derived fuzzy network performs better than the pca based fuzzy network and the svm based fuzzy network

ts fuzzy system based support vector regression this paper proposes a new regression model the takagi sugeno fuzzy system based support vector regression tsfs svr the tsfs svr is motivated by ts type fuzzy rules and its parameters are learned by a combination of fuzzy clustering and linear svr in contrast to a kernel based svr the tsfs svr has a smaller number of parameters while retaining the svr s good generalization ability in tsfs svr a one pass clustering algorithm clusters the input training data a new ts kernel which corresponds to a ts type fuzzy rule is then constructed by the product of a cluster output and a linear combination of input variables the tsfs svr output is a linear weighted sum of the ts kernels to achieve high generalization ability tsfs svr weights are learned through linear svr this paper demonstrates the capabilities of tsfs svr by conducting simulations in clean and noisy function approximations and signal prediction this paper also compares simulation results from the tsfs svr with gaussian kernel based svr and other learning models

learning control using fuzzified self organizing radial basis function network this note describes an approach to integrating fuzzy reasoning systems with radial basis function rbf networks and shows how the integrated network can be employed as a multivariable self organizing and self learning fuzzy controller in particular by drawing some equivalence between a simplified fuzzy control algorithm sfca and a rbf network we conclude that the rbf network can be interpreted in the context of fuzzy systems and can be naturally fuzzified into a class of more general networks referred to as fbfn with a variety of basis functions not necessarily globally radial synthesized from each dimension by fuzzy logical operators on the other hand as a result of natural generalization from rbf to sfca we claim that the fuzzy system like rbf is capable of universal approximation next the fbfn is used as a multivariable rule based controller but with an assumption that no rule base exists leading to a challenging problem of how to construct such a rule base directly from the control environment we propose a simple and systematic approach to performing this task by using a fuzzified competitive self organizing scheme and incorporating an iterative learning control algorithm into the system we have applied the approach to a problem of multivariable blood pressure control with a fbfn based controller having six inputs and two outputs representing a complicated control structure

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

correntropy based evolving fuzzy neural system in this paper a correntropy based evolving fuzzy neural system cefns is proposed for approximation of nonlinear systems different from the commonly used mean square error criterion correntropy has a strong outliers rejection ability through capturing the higher moments of the error distribution considering the merits of correntropy this paper brings contributions to build evolving fuzzy neural system efns based on the correntropy concept to achieve a more stable evolution of the rule base and update of the rule parameters instead of the commonly used mean square error criterion the correntropy efns cefns begins with an empty rule base and all rules are evolved online based on the correntropy criterion the consequent part parameters are tuned based on the maximum correntropy criterion where the correntropy is used as the cost function so as to improve the non gaussian noise rejection ability the steady state convergence performance of the cefns is studied through the calculation of the steady state excess mean square error emse in two cases gaussian noise and non gaussian noise finally the cefns is validated through a benchmark system identification problem a mackey glass time series prediction problem as well as five other real world benchmark regression problems under both noise free and noisy conditions compared with other efnss the simulation results show that the proposed cefns produces better approximation accuracy using the least number of rules and training time and also owns superior non gaussian noise handling capability

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

a new neuro fds definition for indirect adaptive control of unknown nonlinear systems using a method of parameter hopping the indirect adaptive regulation of unknown nonlinear dynamical systems is considered in this paper the method is based on a new neuro fuzzy dynamical system neuro fds definition which uses the concept of adaptive fuzzy systems afss operating in conjunction with high order neural network functions fhonnfs since the plant is considered unknown we first propose its approximation by a special form of an fds and then the fuzzy rules are approximated by appropriate honnfs thus the identification scheme leads up to a recurrent high order neural network rhonn which however takes into account the fuzzy output partitions of the initial fds the proposed scheme does not require a priori experts information on the number and type of input variable membership functions making it less vulnerable to initial design assumptions once the system is identified around an operation point it is regulated to zero adaptively weight updating laws for the involved honnfs are provided which guarantee that both the identification error and the system states reach zero exponentially fast while keeping all signals in the closed loop bounded the existence of the control signal is always assured by introducing a novel method of parameter hopping which is incorporated in the weight updating law simulations illustrate the potency of the method and comparisons with conventional approaches on benchmarking systems are given also the applicability of the method is tested on a direct current dc motor system where it is shown that by following the proposed procedure one can obtain asymptotic regulation

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

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

a novel generic hebbian ordering based fuzzy rule base reduction approach to mamdani neuro fuzzy system there are two important issues in neuro fuzzy modeling 1 interpretability the ability to describe the behavior of the system in an interpretable way and 2 accuracy the ability to approximate the outcome of the system accurately as these two objectives usually exert contradictory requirements on the neuro fuzzy model certain compromise has to be undertaken this letter proposes a novel rule reduction algorithm namely hebb rule reduction and an iterative tuning process to balance interpretability and accuracy the hebb rule reduction algorithm uses hebbian ordering which represents the degree of coverage of the samples by the rule as an importance measure of each rule to merge the membership functions and hence reduces the number of the rules similar membership functions mfs are merged by a specified similarity measure in an order of hebbian importance and the resultant equivalent rules are deleted from the rule base the rule with a higher hebbian importance will be retained among a set of rules the mfs are tuned through the least mean square lms algorithm to reduce the modeling error the tuning of the mfs and the reduction of the rules proceed iteratively to achieve a balance between interpretability and accuracy three published data sets by nakanishi nakanishi turksen sugeno 1993 the pat synthetic data set pal mitra mitra 2003 and the traffic flow density prediction data set are used as benchmarks to demonstrate the effectiveness of the proposed method good interpretability as well as high modeling accuracy are derivable simultaneously and are suitably benchmarked against other well established neuro fuzzy models

building sugeno type models using fuzzy discretization and orthogonal parameter estimation techniques this paper develops a new approach to building sugeno type models the essential idea is to separate the premise identification from the consequence identification while these are mutually related in the previous methods a fuzzy discretization technique is suggested to determine the premise of the model and an orthogonal estimator is provided to identify the consequence of the model the orthogonal estimator can provide information about the model structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner the well known gas furnace data of box and jenkins is used to illustrate the proposed modeling approach and to compare its performance with other statistical and fuzzy modeling approaches it shows that the performance of the new approach compares favorably with these existing techniques

support vector based fuzzy neural network for pattern classification fuzzy neural networks fnns for pattern classification usually use the backpropagation or c cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data however such kinds of learning algorithms usually cannot minimize the empirical risk training error and expected risk testing error simultaneously and thus cannot reach a good classification performance in the testing phase to tackle this drawback a support vector based fuzzy neural network svfnn is proposed for pattern classification in this paper the svfnn combines the superior classification power of support vector machine svm in high dimensional data spaces and the efficient human like reasoning of fnn in handling uncertainty information a learning algorithm consisting of three learning phases is developed to construct the svfnn and train its parameters in the first phase the fuzzy rules and membership functions are automatically determined by the clustering principle in the second phase the parameters of fnn are calculated by the svm with the proposed adaptive fuzzy kernel function in the third phase the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method to investigate the effectiveness of the proposed svfnn classification it is applied to the iris vehicle dna satimage ijcnn1 datasets from the uci repository statlog collection and ijcnn challenge 2001 respectively experimental results show that the proposed svfnn for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions

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

a fuzzy system constructed by rule generation and iterative linear svr for antecedent and consequent parameter optimization this paper proposes a new fuzzy regression model i e the fuzzy system constructed by rule generation and iterative linear support vector regression fs rglsvr for structural risk minimization the fs rglsvr is composed of takagi sugeno ts type fuzzy if then rules these rules are automatically constructed by a self splitting rule generation algorithm that introduces the self splitting technique to the k means clustering algorithm this new algorithm regards a cluster as a fuzzy rule where no preassignment of the cluster rule number is necessary the cost function for parameter learning is defined based on structural risk instead of empirical risk minimization in order to achieve generalizability tuning all of the free parameters in the fs rglsvr using linear support vector regression svr is proposed to minimize the cost function each of the consequent and antecedent part parameters is expressed as a linear combination coefficient in a transformed input space so that the linear svr is applicable this paper introduces iterative linear svr to tune antecedent and consequent parameters this paper demonstrates the capabilities of fs rglsvr by two simulated and four practical regression examples comparisons with fuzzy systems with different types of learning algorithms verify the performance of the fs rglsvr

averages of central l values of hilbert modular forms with an application to subconvexity we use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change l functions averaged over hilbert modular forms of a fixed weight and level we apply these formulas to the subconvexity problem for these l functions we also establish an equidistribution result for the hecke eigenvalues weighted by these l values