transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

ensembles of fuzzy linear model trees for the identification of multioutput systems we address the task of discrete time modeling of nonlinear dynamic systems with multiple outputs using measured data in the area of control engineering this task is typically converted into a set of classical regression problems one for each output which can then be solved with any nonlinear regression approach fuzzy models in the takagi sugeno form are popular in this context we use lolimot which is a tree learning method to build fuzzy linear model trees in this paper we propose implement and empirically evaluate three extensions of fuzzy linear model trees first we consider and evaluate multioutput models second we propose to use ensembles of such models third we investigate the use of a search heuristic based on simulation error as opposed to one step ahead prediction error specific to the context of modeling dynamic systems finally we perform an empirical evaluation and compare these approaches on six multioutput case studies using both measured and simulated data with noise the case studies include modeling of the inverse dynamics of a robot arm as well as five additional process industry systems ensembles improve the performance of both single and multioutput trees while the heuristic specific to modeling dynamic systems only improves performance very slightly multioutput model trees exhibit comparable or worse predictive performance to a set of single output models while providing a more compact model overall we can recommend the use of bagging of single output lolimot models learned by using the simulation error as a search heuristic

error analysis and stochastic differentiability in subsurface flow modeling in the stochastic analysis of steady aquifer flow the log hydraulic conductivity is the random input and the piezometric potential is the random output their joint behavior is governed by a differential equation that in view of the random nature of its dependent piezometric potential and independent log hydraulic conductivity variables represents a stochastic differential equation the analysis of the distributional properties of the piezometric potential field involves the product of the gradients of log hydraulic conductivity and of the piezometric potential previous research in this field assumes that such product of gradients is small in some sense this paper derives a closed form expression for the standard deviation and hence the order of magnitude of the product of the random gradients of log hydraulic conductivity and of piezometric potential it was found in this research that for statistically homogeneous log hydraulic conductivity fields 1 the product of the random gradients may or may not have a zero mean depending on whether the specific discharge is a constant or a random quantity respectively 2 under joint normality of the log hydraulic conductivity and the piezometric potential fields their random gradients are statistically independent if the specific discharge is constant but are dependent when the specific discharge is random 3 the standard deviation of the product of random gradients is proportional to the variance of log hydraulic conductivity times a term involving three quantities the covariance of the piezometric potential the covariance of the log hydraulic conductivity and the cross covariance of the latter two fields 4 a necessary and sufficient condition for the smallness of the product of random gradients is that the second derivatives of the covariance of the log hydraulic conductivity the covariance of the piezometric potential and the cross covariance of the latter two random fields be finite and that the variance of log hydraulic conductivity be much less than one this paper also reviews some fundamental principles on the stochastic analysis of random fields and their importance to the modeling of log hydraulic conductivity fields and to the analysis of subsurface flow specifically the paper highlights the role of gaussian distributional assumptions in deriving key results of stochastic groundwater flow via perturbation analysis

training anfis as an identifier with intelligent hybrid stable learning algorithm based on particle swarm optimization and extended kalman filter this paper proposes a novel hybrid learning algorithm with stable learning laws for adaptive network based fuzzy inference system anfis as a system identifier the proposed hybrid learning algorithm is based on the particle swarm optimization pso for training the antecedent part and the extended kalman filter ekf for training the conclusion part lyapunov stability theory is used to study the stability of the proposed algorithm comparison results of the proposed approach pso algorithm for training the antecedent part and recursive least squares rlss or ekf algorithm for training the conclusion part with the other classical approaches such as gradient descent resilient propagation quick propagation levenberg marquardt for training the antecedent part and rlss algorithm for training the conclusion part are provided moreover it is shown that applying pso a powerful optimizer to optimally train the parameters of the membership function on the antecedent part of the fuzzy rules in anfis system is a stable approach which results in an identifier with the best trained model stability constraints are obtained and different simulation results are given to validate the results also the stability of levenberg marquardt algorithms for anfis training is analyzed

radial basis function based adaptive fuzzy systems and their applications to system identification and prediction abstract in this paper we describe a neuro fuzzy system with adaptive capability to extract fuzzy if then rules from input and output sample data through learning the proposed system called radial basis function rbf based adaptive fuzzy system afs employs the gaussian functions to represent the membership functions of the premise part of fuzzy rules three architectural deviations of the rbf based afs are also presented according to different consequence types such as constant first order linear function and fuzzy variable these provide versatility of the network to handle arbitrary fuzzy inference schemes we present examples of system identification and time series prediction to illustrate how to solve these problems and to demonstrate its validity and effectiveness using the rbf based afs

data driven fuzzy modeling for takagi sugeno kang fuzzy system this paper presents a systematic approach to design first order tagaki sugeno kang tsk fuzzy systems this approach attempts to obtain the fuzzy rules without any assumption about the structure of the data the structure identification and parameter optimization steps in this approach are carried out automatically and are capable of finding the optimal number of the rules with an acceptable accuracy starting with an initial structure the system first tries to improve the structure and then as soon as an improved structure is found it fine tunes its rules parameters then it goes back to improve the structure again to find a better structure and re fine tune the rules parameters this loop continues until a satisfactory solution tsk model is found the proposed approach has successfully been applied to well known benchmark datasets and real world problems the obtained results are compared with those obtained with other methods from the literature experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules finally as a case study the proposed approach is applied to the desulfurization process of a real steel industry comparing the proposed approach with some other fuzzy systems and neural networks it is shown that the developed tsk fuzzy system exhibits better results with higher accuracy and smaller size of architecture

fuzzy rule based combination of linear and switching state feedback controllers this paper presents a fuzzy rule base combined controller which is a fuzzy rule based combination of linear and switching state feedback controllers for nonlinear systems subject to parameter uncertainties the switching state feedback controller is employed to drive the system states toward the origin when the system state approaches the origin the linear state feedback controller will gradually replace the switching state feedback controller the smooth transition between the linear and switching state feedback controllers is governed by the fuzzy rules by using the fuzzy rule based combination technique the proposed fuzzy rule base combined controller integrates the advantages of both the linear and switching state feedback controllers but eliminates their disadvantages as a result the proposed fuzzy controller provides good performance during the transient period and the chattering effect is removed when the system state approaches the origin stability conditions will be derived to guarantee the system stability furthermore a saturation function is employed to replace the switching component to alleviate the chattering during the transient period by using the proposed fuzzy rule based combination technique the steady state error introduced by the saturation function can be eliminated application examples will be given to show the merits of the proposed approach

mathematical and numerical results on the sensitivity of the pod approximation relative to the burgers equation from mathematical and numerical points of view we study the sensitivity with respect to parametric evolutions of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition pod basis determined once and for all with association to a fixed parameter more precisely we will be considering the one dimensional burgers equation with a source term

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

genetic programming for model selection of tsk fuzzy systems this paper compares a genetic programming gp approach with a greedy partition algorithm lolimot for structure identification of local linear neuro fuzzy models the crisp linear conclusion part of a takagi sugeno kang tsk fuzzy rule describes the underlying model in the local region specified in the premise the objective of structure identification is to identify an optimal partition of the input space into gaussian axis orthogonal fuzzy sets the linear parameters in the rule consequent are then estimated by means of a local weighted least squares algorithm lolimot is an incremental tree construction algorithm that partitions the input space by axis orthogonal splits in each iteration it greedily adds the new model that minimizes the classification error gp performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub optimal intermediate split decisions we compare the performance of both methods for function approximation of a highly non linear two dimensional test function and an engine characteristic map

identifying fuzzy models utilizing genetic programming abstract fuzzy models offer a convenient way to describe complex nonlinear systems moreover they permit the user to deal with uncertainty and vagueness due to these advantages fuzzy models are employed in various fields of applications e g control forecasting and pattern recognition nevertheless it has to be emphasized that the identification of a fuzzy model is a complex optimization task with many local minima genetic programming provides a way to solve such complex optimization problems in this work the use of genetic programming to identify the input variables the rule base and the involved membership functions of a fuzzy model is proposed for this purpose several new reproduction operators are introduced

supervised hierarchical clustering in fuzzy model identification this paper presents a new supervised hierarchical clustering algorithm suhiclust for fuzzy model identification the presented algorithm solves the problem of global model accuracy together with the interpretability of local models as valid linearizations of the modeled nonlinear system the algorithm combines the advantages of supervised hierarchical algorithms which are based on heuristic tree construction algorithms together with the advantages of fuzzy product space clustering the high flexibility of the validity functions that is obtained by fuzzy clustering combined with supervised learning results in an efficient partitioning algorithm which is independent of initialization and results in a parsimonious fuzzy model furthermore the usability of suhiclust is very undemanding because it delivers in contrast with many other methods reproducible results in order to get reasonable results the user only has to set either a threshold for the maximum number of local models or a value for the maximum allowed global model error as a termination criterion for fine tuning the interpolation smoothness controls the degree of regularization the performance is illustrated on both analytical examples and benchmark problems from the literature

a stochastic inverse solution for transient groundwater flow parameter identification and reliability analysis in this paper a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow adjoint state equations are derived for the stochastic partial differential equation spde relating transient head and log hydraulic conductivity perturbations the derived equations can be used to calculate the covariance matrix of head observations and the cross covariance matrix between head observations and log hydraulic conductivity measurements for each observation time as well as the covariance and cross covariance matrices between different observation times the number of simulation runs required for obtaining these matrices is equal to the number of head observations the reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate the reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies a numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem the example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially commonly known as the quasi steady approach the example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations

stability analysis and performance design for fuzzy model based control system under imperfect premise matching this paper investigates the stability analysis and performance design of nonlinear systems to facilitate the stability analysis the takagi sugeno t s fuzzy model is employed to represent the nonlinear plant under the imperfect premise matching in which t s fuzzy model and fuzzy controller do not share the same membership functions a fuzzy controller with enhanced design flexibility and robustness property is proposed to control the nonlinear plant however the nice characteristic given by the perfect premise matching leading to conservative stability conditions vanishes in this paper under the imperfect premise matching information of membership functions of the fuzzy model and controller are considered in stability analysis with the introduction of slack matrices relaxed linear matrix inequality lmi based stability conditions are derived using lyapunov based approach furthermore lmi based performance conditions are provided to guarantee system performance simulation examples are given to illustrate the effectiveness of the proposed approach

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

neuro fuzzy reasoning for occluded object recognition abstract to tackle the problem of occluded object recognition first we give a new interpretation to the multidimensional fuzzy reasoning and then realize that new interpretation through backpropagation type neural network at the learning stage of the neural network fuzzy linguistic statements are used once learned the nonfuzzy features of an occluded object can be classified at the time of classification of the nonfuzzy features of an occluded object we use the concept of fuzzy singleton an effective approach to recognize an unknown scene which consists of a set of occluded objects is to detect a number of significant local features on the boundary of the unknown scene thus the major problems fall into the selection of the appropriate set of features local for representing the object in the training stage as well as in the detection of these features in the recognition process the features should be invariant to scale orientation and minor distortions in boundary shape the performance of the proposed scheme is tested through several examples

karhunen loeve expansion revisited for vector valued random fields scaling errors and optimal basis due to scaling effects when dealing with vector valued random fields the classical karhunen loeve expansion which is optimal with respect to the total mean square error tends to favorize the components of the random field that have the highest signal energy when these random fields are to be used in mechanical systems this phenomenon can introduce undesired biases for the results this paper presents therefore an adaptation of the karhunen loeve expansion that allows us to control these biases and to minimize them this original decomposition is first analyzed from a theoretical point of view and is then illustrated on a numerical example

multidimensional a priori hyper reduction of mechanical models involving internal variables this paper concerns the adaptation of reduced order models during simulations of series of elastoviscoplastic problems in continuation with previous works this paper aimed at extending the a priori hyper reduction method aphr method for nonlinear thermal problems to nonlinear mechanical problems involving internal variables this method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced order models the recent extension of the hyper reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points a multi level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced order models

parameter conditions for monotonic takagi sugeno kang fuzzy system this paper presents sufficient conditions on the parameters of the takagi sugeno kang tsk fuzzy system under which the output of the tsk fuzzy system becomes monotonic with respect to its input the output of the tsk fuzzy system is first differentiated with respect to its input and the parameter conditions are derived so as to make the derivative nonnegative the parameter conditions are developed for both single input and multi input tsk fuzzy systems where the involved fuzzy membership functions are differentiable everywhere or bar some finite points the derived monotonicity conditions consist of two parts the conditions on the consequent parts and the conditions on the fuzzy membership functions they are further characterized for the most important types of fuzzy membership functions

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

fuzzy network model based fuzzy state controller design the performance of model based controller design relies heavily on the quality and suitability of the utilized process model this contribution proposes a fuzzy network based nonlinear controller design methodology fuzzy networks are a model approach combining high approximation quality with high interpretability the input output i o models commonly used for identification are transformed to fuzzy state space models transferring and adjusting methods from linear state space theory a control concept consisting of a fuzzy state controller and an adaptive set point filter for nonlinear dynamic processes is deduced the capability of the method is demonstrated for a hydraulic drive

rspop rough set based pseudo outer product fuzzy rule identification algorithm system modeling with neuro fuzzy systems involves two contradictory requirements interpretability verses accuracy the pseudo outer product pop rule identification algorithm used in the family of pseudo outer product based fuzzy neural networks popfnn suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high dimensional data this decreases the interpretability of the popfnn in linguistic fuzzy modeling this article proposes a novel rough set based pseudo outer product rspop algorithm that integrates the sound concept of knowledge reduction from rough set theory with the pop algorithm the proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes as many possible reducts exist in a given rule set an objective measure is developed for popfnn to correctly identify the reducts that improve the inferred consequence experimental results are presented using published data sets and real world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the popfnn using compositional rule of inference and singleton fuzzifier popfnn cri s architecture results showed that the proposed rough set based pseudo outer product algorithm reduces computational complexity improves the interpretability of neuro fuzzy systems by identifying significantly fewer fuzzy rules and improves the accuracy of the popfnn

stochastic models of subsurface flow log linearized gaussian models are exact for covariances stochastic solutions for steady state infinite domain flow models are studied under the assumptions that log conductivity and head are second order stationary jointly gaussian processes it is shown that the covariance relationships previously derived are exact solutions for the nonlinearized equations no matter how large the input variance is

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 priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

a fully implicit finite difference scheme for two dimensional burgers equations the two dimensional burgers equations are discretized in fully implicit finite difference form this scheme leads to a system of nonlinear difference equations to be solved at each time step newton s method is used to solve this nonlinear system the linear system is solved by a direct method at each iteration of newton s method the accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results the present method performs well

stability analysis of polynomial fuzzy model based control systems with mismatched premise membership functions this paper investigates the stability of polynomial fuzzy model based pfmb control system which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop to enhance the design flexibility the number of rules and the shape of premise membership functions of the polynomial fuzzy controller are considered to be chosen freely and are different from those of the polynomial fuzzy model however which make the stability analysis more difficult and potentially lead to conservative stability analysis result a sum of squares sos based stability analysis approach using the lyapunov stability theory is proposed to investigate the stability of the pfmb control systems and synthesize the polynomial fuzzy controller to facilitate the stability analysis and relax the stability analysis result the property of the membership functions and the boundary information of the membership grades and premise variables are taken into account in the stability analysis and incorporated into the sos based stability conditions a simulation example is given to illustrate the effectiveness of the proposed approach

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

identification of non linear system structure and parameters using regime decomposition abstract an off line algorithm for empirical modeling and identification of non linear dynamic systems is presented the minimal input to the algorithm is a sequence of empirical data and the model order using this information the algorithm searches for an optimal model structure and parameters within a rich non linear model set the model representation is based on the interpolation of a number of simple local models where each local model has a limited range of validity but the local models yield a complete global model when interpolated the method is illustrated using simulated data

h sub spl infin output feedback control of discrete time fuzzy systems with application to chaos control this paper presents an observer based h sub spl infin output feedback synthesis method for discrete time fuzzy dynamic systems based on a piecewise lyapunov function the basic idea of the approach is to design an observer based piecewise linear output feedback control law to guarantee the global stability with h sub spl infin performance of the resulting closed loop fuzzy control systems it is shown that the controller parameters can be obtained by solving a set of linear matrix inequalities lmis that are numerically feasible with commercially available software application to control chaotic systems is given to illustrate the effectiveness and advantages of the proposed method

continuous time stochastic analysis of groundwater flow in heterogeneous aquifers the problem of depth averaged groundwater flow in heterogeneous aquifers is looked at from a stochastic point of view the galerkin finite element version of the linearized stochastic equation governing the flow is solved analytically in time using an eigenvalue eigenvector technique the stochastic solution which is valid for small variance of aquifer log transmissivity relates spatial and temporal variabilities of aquifer head to aquifer heterogeneity stochastic recharge and random initial head the computational effort requires the evaluation of integrals of matrices whose elements are linear functions of the nodal mean heads the solutions are obtained for the exact i e continuous and quasi steady approximation of the mean head aquifer head temporal covariances are evaluated using standard matrix operations once the storage matrix is inverted and the associated generalized eigenvalue problem is solved implication of the level of spatial discretization on the performance of the solution is examined and the influence of aquifer heterogeneity on spatial and temporal variances of hydraulic heads is investigated in the presence of a semipervious boundary

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

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

author s reply to comments on a proposal to improve the accuracy of linguistic modelling in our opinion there are two main concerns in roubos and babugka s note that are summarized as follows 1 the kinds of problems used in our paper to test the algorithm proposed in a proposal to improve the accuracy of linguistic modeling and other studies the authors claim that they are very simple to be considered as benchmarks for nonlinear modeling techniques 2 the interpretability of the different kinds of models considered roubos and babuska think that there is no difference between the interpretability of fuzzy linguistic models takagi sugeno kang tsk fuzzy models and mathematical formulations linear models in this case we agree with some of the opinions of the authors of the note but not with some others

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

a clustering algorithm for fuzzy model identification the fuzzy model proposed by takagi and sugeno can represent highly nonlinear systems and is widely used for the representation of fuzzy rules in this paper the model is firstly modified to make its identification easier based on the fuzzy c partition space four criteria are proposed for optimization of the model parameters following that a clustering algorithm composed of fuzzy c linear functions clustering and like fuzzy c means clustering is developed for minimizing the four criteria an identification scheme for rule s premise and consequence parameters is deduced from the clustering algorithm in succession finally four examples are demonstrated to verify the effectiveness of the proposed algorithm

monotone fuzzy rule relabeling for the zero order tsk fuzzy inference system to maintain the monotonicity property of a fuzzy inference system a monotonically ordered and complete set of fuzzy rules is necessary however monotonically ordered fuzzy rules are not always available e g errors in human judgments lead to nonmonotone fuzzy rules the focus of this paper is on a new monotone fuzzy rule relabeling mfrr method that is able to relabel a set of nonmonotone fuzzy rules to meet the monotonicity property with reduced computation unlike the brute force approach which is susceptible to the combinatorial explosion problem the proposed mfrr method explores within a reduced search space to find the solutions therefore decreasing the computational requirements the usefulness of the proposed method in undertaking failure mode and effect analysis problems is demonstrated using publicly available information the results indicate that the mfrr method can produce optimal solutions with reduced computational time

quadratic stability analysis of the takagi sugeno fuzzy model the nonlinear dynamic takagi sugeno fuzzy model with offset terms is analyzed as a perturbed linear system a sufficient criterion for the robust stability of this nominal system against nonlinear perturbations guarantees quadratic stability of the fuzzy model the criterion accepts a convex programming formulation of reduced computational cost compared to the common lyapunov matrix approach parametric robust control techniques suggest synthesis tools for stabilization of the fuzzy system application examples on fuzzy models of nonlinear plants advocate the efficiency of the method the examples demonstrate reduced conservatism compared to norm based criteria

a priori reduction method for solving the two dimensional burgers equations abstract the two dimensional burgers equations are solved here using the a priori reduction method this method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved the method is called a priori because it does not need any prior knowledge of the solution which is not the case if the standard karhunen loeve decomposition is used the accuracy of the apr method is compared with the standard newton raphson scheme and with results from the literature the apr basis is also compared with the karhunen loeve basis

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

universal fuzzy controllers for a class of nonlinear systems in this paper we address the issues of universal fuzzy controllers we first present a universal function approximation theorem based on a fuzzy dynamic model then we show the results of universal fuzzy controllers for a large class of nonlinear systems

evolutionary fuzzy modeling this study is concerned with a general methodology of identification of fuzzy models unlike numeric models fuzzy models operate at a level of information granules fuzzy sets and this aspect brings up an important design requirement of transparency of the model we propose a three phase development framework by distinguishing between structural and parametric optimization processes the underlying topology of the model dwells on fuzzy neural networks architectures governed by fuzzy logic and equipped with parametric flexibility two general optimization mechanisms are explored the structural optimization is realized via genetic programming whereas for the ensuing detailed parametric optimization we proceed with gradient based learning the main advantages of this approach are discussed in detail the study is illustrated with the aid of a numeric example that provides a detailed insight into the performance of the fuzzy models and quantifies crucial design issues

mining of protein protein interfacial residues from massive protein sequential and spatial data it is a great challenge to process big data in bioinformatics in this paper we addressed the problem of identifying protein protein interfacial residues from massive protein structural data a protein set comprising 154 993 residues was analyzed we applied the three dimensional alpha shape modeling to the search of surface and interfacial residues in this set and adopted the spatially neighboring residue profiles to characterize each residue these residue profiles which revealed the sequential and spatial information of proteins translated the original data into a large matrix after vertically and horizontally refining this matrix we comparably implemented a series of popular learning procedures including neuro fuzzy classifiers nfcs cart neighborhood classifiers necs extreme learning machines elms and naive bayesian classifiers nbcs to predict the interfacial residues aiming to investigate the sensitivity of these massive structural data to different learning mechanisms as a consequence elms cart and nfcs performed better in terms of computational costs nfcs nbcs and elms provided favorable prediction accuracies overall nfcs nbcs and elms are favourable choices for fastly and accurately handling this type of data more importantly the marginal differences between the prediction performances of these methods imply the insensitivity of this type of data to different learning mechanisms

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

cluster centric fuzzy modeling in this study we propose a cluster oriented development of fuzzy models an overall design process is focused on an efficient usage of fuzzy clustering fuzzy c means fcm in particular to form information granules clusters that are used in the construction of the fuzzy model fuzzy models are regarded as mappings from information granules expressed in the input and output spaces this position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules this study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages the augmented fcm method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized it is shown that the cluster oriented modeling gives rise to the mamdani like fuzzy rules and a zero order takagi sugeno model under a certain decoding scheme we identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding decoding or granulation degranulation encountered in processing fuzzy sets and granular computing in general furthermore refinements of zero order fuzzy models are investigated leading to first order fuzzy models with linear functions standing in the conclusions of the rules a series of experiments is reported where we used synthetic and real world data using which an issue of generalization capabilities is elaborated in detail

fuzzy radial basis function neural networks with information granulation and its parallel genetic optimization fuzzy modeling of complex systems is a challenging task which involves important problems of dimensionality reduction and calls for various ways of improving the accuracy of modeling the ig frbfnn a hybrid architecture of the ig fis fuzzy inference system and frbfnn fuzzy radial basis function neural networks is proposed to address these problems the paper is concerned with the analysis and design of ig frbfnns and their optimization by means of the hierarchical fair competition based parallel genetic algorithm hfc pga in the proposed network the membership functions of the premise part of the fuzzy rules of the ig based frbfnn model directly rely on the computation of the relevant distance between data points and the use of four types of polynomials such as constant linear quadratic and modified quadratic are considered for the consequent part of fuzzy rules moreover the weighted least square wls learning is exploited to estimate the coefficients of the polynomial forming the conclusion part of the rules since the performance of the ig rbfnn model is affected by some key design parameters such as a specific subset of input variables the fuzzification coefficient of the fcm the number of rules and the order of polynomial of the consequent part of fuzzy rules it becomes beneficial to carry out both structural as well as parametric optimization of the network in this study the hfc pga is used as a comprehensive optimization vehicle the performance of the proposed model is illustrated by means of several representative numerical examples

incremental evolutionary design of tsk fuzzy controllers this paper proposes a novel method for the incremental design and optimization of first order tagaki sugeno kang tsk fuzzy controllers by means of an evolutionary algorithm starting with a single linear control law the controller structure is gradually refined during the evolution structural augmentation is intertwined with evolutionary adaptation of the additional parameters with the objective not only to improve the control performance but also to maximize the stability region of the nonlinear system from the viewpoint of optimization the proposed method follows a divide and conquer approach additional rules and their parameters are introduced into the controller structure in a neutral fashion such that the adaptations of the less complex controller in the previous stage are initially preserved the proposed scheme is evaluated at the task of tsk fuzzy controller design for the upswing and stabilization of a rotational inverted pendulum in the first case the objective is a time optimal controller that upswings the pendulum in to the upper equilibrium point in shortest time the stabilizing controller is designed as a state optimal controller in a second application the optimization method is applied to the design of a fuzzy controller for vision based mobile robot navigation the results demonstrate that the incremental scheme generates solutions that are similar in control performance to pure parameter optimization of only the gains of a tsk system even more important whereas direct optimization of control systems with more than 35 rules fails to identify a stabilizing control law the incremental scheme optimizes fuzzy state space partitions and gains for hundreds of rules

proper generalized decomposition method for incompressible navier stokes equations with a spectral discretization proper generalized decomposition pgd is a method which consists in looking for the solution to a problem in a separate form this approach has been increasingly used over the last few years to solve mathematical problems the originality of this work consists in the association of pgd with a spectral collocation method to solve transfer equations as well as navier stokes equations in the first stage the pgd method and its association with spectral discretization is detailed this approach was tested for several problems the poisson equation the darcy problem navier stokes equations the taylor green problem and the lid driven cavity in the navier stokes problems the coupling between velocity and pressure was performed using a fractional step scheme and a p n p n 2 discretization for all problems considered the results from pgd simulations were compared with those obtained by a standard solver and or with the results found in the literature the simulations performed showed that pgd is as accurate as standard solvers pgd preserves the spectral behavior of the errors in velocity and pressure when the time step or the space step decreases moreover for a given number of discretization nodes pgd is faster than the standard solvers

method for higher order polynomial sugeno fuzzy inference systems a method for higher order polynomial sugeno fuzzy inference systems formation is presented compared to other existing higher order sugeno implementations it uses fewer parameters and compared to zero and 1st order sugeno fuzzy systems it has overall improved model performance while best models are not always obtained via a higher order representation in our proposed method it is possible to choose the polynomial order which best fits the desired data its input is a previously established model found by a clustering algorithm subtractive algorithm in this case afterward parameters of all higher order polynomials are adjusted using recursive least square algorithm for experimental validation multiple benchmark datasets are tested using hold out and k fold validation as well as data forecasting various performance measures are used although akaike information criterion is used as a primary measure to demonstrate that our proposed higher order polynomials have overall better model performance over zero and 1st order polynomials

logic minimization as an efficient means of fuzzy structure discovery established methods of boolean minimization have previously unseen potential as an efficient and unrestricted means of fuzzy structure discovery becoming particularly useful within a design methodology for the automatic development of fuzzy models traditionally used in digital systems design logic minimization tools allow us to exploit the fundamental links between binary two valued and fuzzy multivalued logic in this paper we show how logic optimization plays an integral role in a two phase fuzzy model design process adaptive logic processing is realized as the discovered boolean structures are augmented with fuzzy granules and then refined by adjusting connections of fuzzy neurons helping to further capture the numeric details of the target systems behavior accurate and highly interpretable fuzzy models are the result of the entire development process

a trainable transparent universal approximator for defuzzification in mamdani type neuro fuzzy controllers a novel technique of designing application specific defuzzification strategies with neural learning is presented the proposed neural architecture considered as a universal defuzzification approximator is validated by showing the convergence when approximating several existing defuzzification strategies the method is successfully tested with fuzzy controlled reverse driving of a model truck the transparent structure of the universal defuzzification approximator allows us to analyze the generated customized defuzzification method using the existing theories of defuzzification the integration of universal defuzzification approximator instead of traditional methods in mamdani type fuzzy controllers can also be considered as an addition of trainable nonlinear noise to the output of the fuzzy rule inference before calculating the defuzzified crisp output therefore nonlinear noise trained specifically for a given application shows a grade of confidence on the rule base providing an additional opportunity to measure the quality of the fuzzy rule base the possibility of modeling a mamdani type fuzzy controller as a feedforward neural network with the ability of gradient descent training of the universal defuzzification approximator and antecedent membership functions fulfil the requirement known from multilayer preceptrons in finding solutions to nonlinear separable problems

a simple but powerful heuristic method for generating fuzzy rules from numerical data abstract in this paper we propose a simple but powerful heuristic method for automatically generating fuzzy if then rules from numerical data fuzzy if then rules with nonfuzzy singletons i e real numbers in the consequent parts are generated by the proposed heuristic method the main advantage of the proposed heuristic method is its simplicity i e it involves neither time consuming iterative learning procedures nor complicated rule generation mechanisms we also suggest a linguistic representation method for deriving linguistic rules from fuzzy if then rules with consequent real numbers the proposed linguistic approximation method consists of two linguistic rule tables which can realize exactly the same nonlinear mapping as an original system based on fuzzy if then rules with consequent real numbers using computer simulations on rice taste data we demonstrate the high performance of the proposed heuristic method and illustrate the proposed linguistic representation method

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

on the dissipativity of nonlinear systems fuzzy control approach a controller with large dissipativity is designed and implemented the stability margin is increased in order to tolerate the uncertainties and the disturbance in the nonlinear system in this study the dissipativity of a nonlinear system is facilitated via fuzzy control method first the nonlinear system is represented by an takagi and sugeno fuzzy model if the state variables of the nonlinear system are available then a state feedback fuzzy control is developed to guarantee that the nonlinear control system is dissipative this design problem becomes a linear matrix inequality lmi problem the maximum dissipativity design is obtained by solving an eigenvalue problem evp if the state variables are not available a fuzzy observer based state feedback controller will be used to solve the dissipative problem of the nonlinear system simulation examples are given to illustrate the design procedures and performances of the proposed method