stationary fuzzy fokker planck learning for derivative free optimization stationary fuzzy fokker planck learning sffpl is a recently introduced computational method that applies fuzzy modeling to solve optimization problems this study develops a concept of applying sffpl based computations for nonlinear constrained optimization we consider the development of sffpl based optimization algorithms which do not require derivatives of the objective function and of the constraints the sequential penalty approach was used to handle the inequality constraints it was proved under some standard assumptions that the carefully designed sffpl based algorithms converge asymptotically to the stationary points the convergence proofs follow a simple mathematical approach and invoke mean value theorem the algorithms were evaluated on the test problems with the number of variables up to 50 the performance comparison of the proposed algorithms with some of the standard optimization algorithms further justifies our approach the sffpl based optimization approach due to its novelty could possibly be extended to several research directions

tsk fuzzy modeling based on spl epsiv insensitive learning in this paper new learning methods tolerant to imprecision are introduced and applied to fuzzy modeling based on the takagi sugeno kang fuzzy system the fuzzy modeling has an intrinsic inconsistency it may perform thinking tolerant to imprecision but learning methods are zero tolerant to imprecision the proposed methods make it possible to exclude this intrinsic inconsistency of a fuzzy modeling where zero tolerance learning is used to obtain fuzzy model tolerant to imprecision these new methods can be called spl epsiv insensitive learning or spl epsiv learning where in order to fit the fuzzy model to real data the spl epsiv insensitive loss function is used this leads to a weighted or fuzzified version of vapnik s support vector regression machine this paper introduces two approaches to solving the spl epsiv insensitive learning problem the first approach leads to the quadratic programming problem with bound constraints and one linear equality constraint the second approach leads to a problem of solving a system of linear inequalities two computationally efficient numerical methods for the spl epsiv insensitive learning are proposed the spl epsiv insensitive learning leads to a model with the minimal vapnik chervonenkis dimension which results in an improved generalization ability of this model and its outliers robustness finally numerical examples are given to demonstrate the validity of the introduced methods

fuzzy theoretic approach to signals and systems static systems abstract fuzzy theoretic approach to signals and systems assumes all system variables and parameters as uncertain i e being characterized by membership functions develops a mathematical theory for analytically determining the membership functions on system variables and parameters derives algorithms for estimating the parameters of membership functions and establishes robustness and convergence properties of the estimation algorithms the membership functions are analytically determined via solving a variational optimization problem that maximizes the over uncertainties averaged log membership of the observed data around an initial guess this paper develops the analytical fuzzy theory for the particular case of a multi input single output static system affected by noises the theory facilitates designing an adaptive filtering algorithm the robustness of the adaptive filtering algorithm is proved theoretically via a mathematical analysis numerical experiments further demonstrate the robustness of the filtering algorithm a comparison of the algorithm with the state of art methods is made using the practical biomedical applications related to the modeling and analysis of heart rate signals for assessing the physiological state of an individual

stationary fuzzy fokker planck learning and stochastic fuzzy filtering the application of nonlinear optimization to the estimation of fuzzy model parameters is well known to do the reverse of this the concept of stationary fuzzy fokker planck learning sffpl is introduced i e sffpl applies the fuzzy modeling technique in nonlinear optimization problems sffpl is based on the fuzzy approximation of the stationary cumulative distribution function of a stochastic search process associated with the nonlinear optimization problem a carefully designed algorithm is suggested for sffpl to locate the optimum point this paper also considers the variational bayes vb based inference of a stochastic fuzzy filter whose consequents as well as antecedents are random variables the problem of vb inference of stochastic antecedents because of the nonlinearity of the likelihood function is analytically intractable the sffpl algorithm for high dimensional nonlinear optimization that does not require the derivative of the objective function can be used to numerically solve the stochastic fuzzy filtering problem

vertical fault detection from scattered data the purpose of this paper is to study the problem of detection of vertical faults in explicit surfaces first we establish a characterization of jump discontinuities for bivariate functions next we derive a detection method of vertical faults in surfaces which takes as input a set of scattered data points finally some numerical and graphical examples are given

functional identities in superalgebras abstract the aim of this paper is to establish the theory of functional identities in superalgebras as an application of this theory k supercommuting maps in superalgebras are investigated

vertical and oblique fault detection in explicit surfaces the purpose of this paper is to study the problem of detection of vertical and oblique faults in explicit surfaces first we characterize the finite jump discontinuities of a univariate function in terms of the divergence of sequences related to the slopes of least squares polynomial approximations of the function then we propose an algorithm to locate the finite jump discontinuities of a univariate function and its first derivative from a finite set of scattered data values of the function as a consequence we derive a method to detect vertical and oblique faults in explicit surfaces when the data sets are distributed along lines we finally present some numerical and graphical examples

completeness and consistency conditions for learning fuzzy rules the completeness and consistency conditions were introduced in order to achieve acceptable concept recognition rules in real problems we can handle noise affected examples and it is not always possible to maintain both conditions moreover when we use fuzzy information there is a partial matching between examples and rules therefore the consistency condition becomes a matter of degree in this paper a learning algorithm based on soft consistency and completeness conditions is proposed this learning algorithm combines in a single process rule and feature selection and it is tested on different databases

on the global convergence of derivative free methods for unconstrained optimization in this paper starting from the study of the common elements that some globally convergent direct search methods share a general convergence theory is established for unconstrained minimization methods employing only function values the introduced convergence conditions are useful for developing and analyzing new derivative free algorithms with guaranteed global convergence as examples we describe three new algorithms which combine pattern and line search approaches

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

fuzzy identification using fuzzy neural networks with stable learning algorithms in general fuzzy neural networks cannot match nonlinear systems exactly unmodeled dynamic leads parameters drift and even instability problem according to system identification theory robust modification terms must be included in order to guarantee lyapunov stability this paper suggests new learning laws for mamdani and takagi sugeno kang type fuzzy neural networks based on input to state stability approach the new learning schemes employ a time varying learning rate that is determined from input output data and model structure stable learning algorithms for the premise and the consequence parts of fuzzy rules are proposed the calculation of the learning rate does not need any prior information such as estimation of the modeling error bounds this offer an advantage compared to other techniques using robust modification

flexfis a robust incremental learning approach for evolving takagi sugeno fuzzy models in this paper we introduce a new algorithm for incremental learning of a specific form of takagi sugeno fuzzy systems proposed by wang and mendel in 1992 the new data driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents but also the incremental learning of premise parameters appearing in the membership functions fuzzy sets together with a rule learning strategy in sample mode a modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules premise parts the modifications include an automatic generation of new clusters based on the nature distribution and quality of new data and an alternative strategy for selecting the winning cluster rule in each incremental learning step antecedent and consequent learning are connected in a stable manner meaning that a convergence toward the optimal parameter set in the least squares sense can be achieved an evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high dimensional data recorded at engine test benches and at rolling mills for the latter the obtained data driven fuzzy models are even compared with an analytical physical model furthermore a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three input nonlinear function approximation example

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

robust tsk fuzzy modeling for function approximation with outliers the takagi sugeno kang tsk type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications most approaches for modeling tsk fuzzy rules define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions besides training data sets algorithms often contain outliers which seriously affect least square error minimization clustering and learning algorithms a robust tsk fuzzy modeling approach is presented in the approach a clustering algorithm termed as robust fuzzy regression agglomeration rfra is proposed to define fuzzy subspaces in a fuzzy regression manner with robust capability against outliers to obtain a more precision model a robust fine tuning algorithm is then employed various examples are used to verify the effectiveness of the proposed approach from the simulation results the proposed robust tsk fuzzy modeling indeed showed superior performance over other approaches

statistical inference of rough set dependence and importance analysis statistical inference about dependence degree dd and importance degree id of variables in an information system is crucial for variables appraisement and model reconstruction however in rough set data analysis rsda the literature is restricted to validate independence or test whether the degree is significantly big while the fixed value test and interval estimation for related measurements have been ignored because these important issues have not been addressed we cannot determine whether the data support expert opinions and compare the features in depth to enhance the integrity of statistical inference for dd and id in an rsda fixed value tests and interval estimations of dd and id are presented in this paper with multinomial distribution as the carrier for statistical information in the databases the fixed value test of dd is successfully transformed into a restricted estimation of multinomial distribution and a goodness of fit test for distributions the fixed value test and interval estimation algorithms for dd and id are then presented in detail and illustrated with examples explicit expressions for the dd and id interval estimation dd confidence curves and the limit theory for dd and id are shown furthermore the effectiveness and discrimination of the proposed algorithms are validated using the car evaluation tic tac toe endgame and fisher s iris databases

edge detection methods based on rbf interpolation abstract edge detection is a widely used tool in signal image processing with the aim of identifying abrupt changes or discontinuities in a signal digital image for the detection of jump discontinuities in 1d problems we present an iterative method based on interpolation with variably scaled kernels vsks this is shown to outperform an existing iterative edge detection method based on multiquadric mq radial basis function interpolation to extend our purely one dimensional edge detector to any dimension we then introduce an innovative non iterative technique that detects jumps edges by identifying the local maxima of the normalized absolute values of the rbf interpolant coefficients the rbf interpolant is built upon the compactly supported c 2 wendland function and exploits its advantageous properties to provide a robust and low cost method numerical examples in 1d and 2d are included to illustrate its effectiveness and efficiency in the context of edge detection for digital images comparisons with canny method are also presented

kernel based adaptive approximation of functions with discontinuities one of the basic principles of approximation theory is that the quality of approximations increase with the smoothness of the function to be approximated functions that are smooth in certain subdomains will have good approximations in those subdomains and these sub approximations can possibly be calculated efficiently in parallel as long as the subdomains do not overlap this paper proposes an algorithm that first calculates sub approximations on non overlapping subdomains then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain consequently there will be no gibbs phenomenon along the boundaries of the subdomains the method detects faults and gradient faults with good accuracy throughout the algorithm works for fixed scattered input data of the function itself not on spectral data and it does not resample

near derivations in lie algebras abstract let l be a lie algebra we call a linear map f l l a near derivation if there exists a linear map g l l such that ad x f g ad x is a derivation for every x l the paper is devoted to describing the structure of near derivations in certain lie algebras arising from associative ones

generalized derivations of lie algebras abstract suppose l is a finite dimensional lie algebra with multiplication l l l let l denote the set of triples f f f with f f f hom l l such that f i l i l f f we consider the lie algebra gender l f hom l l f f f f f l well researched subalgebras of gender l include the derivation algebra der l f hom l l f f f l and the centroid c l f hom l l f 0 f l we now study the subalgebra qder l f hom l l f f f f l and the subspace qc l f hom l l f f 0 l in characteristic 2 gender l qder l qc l and we are concerned with the inclusions der l qder l and c l qc l qder l if z l 0 then c l qc l qder l and under reasonable conditions on lie algebras with toral cartan subalgebras we show qder l der l c l if l is a parabolic subalgebra of a simple lie algebra of rank 1 in characteristic 0 then we even have gender l ad l i l in general qc l is not closed under composition or lie bracket however if z l 0 then qc l is a commutative associative algebra and we describe conditions that force qc l c l or equivalently gender l qder l we show that in characteristic 0 gender l preserves the radical of l thus generalizing the classical result for der l we also discuss some applications of the main results to the study of functions f hom l l such that f or f i l defines a lie multiplication

training fuzzy systems with the extended kalman filter the generation of membership functions for fuzzy systems is a challenging problem we show that for mamdani type fuzzy systems with correlation product inference centroid defuzzification and triangular membership functions optimizing the membership functions can be viewed as an identification problem for a nonlinear dynamic system this identification problem can be solved with an extended kalman filter we describe the algorithm and compare it with gradient descent and with adaptive neuro fuzzy inference system anfis based optimization of fuzzy membership functions the methods discussed in this paper are illustrated on a fuzzy filter for motor winding current estimation and are compared with butterworth filtering we demonstrate that the kalman filter can be an effective tool for improving the performance of a fuzzy system

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

an energy gain bounding approach to robust fuzzy identification a novel method for the robust identification of interpretable fuzzy models based on the criterion that identification errors are least sensitive to data uncertainties and modelling errors is suggested the robustness of identification errors towards unknown disturbances data uncertainties modelling errors etc is achieved by bounding i e minimizing the maximum possible value of energy gain from disturbances to the identification errors the solution of energy gain bounding problem being robust shows an improved performance of the identification method the flexibility of the proposed framework is shown by designing the variable learning rate identification algorithms in both deterministic and stochastic frameworks

a min max approach to fuzzy clustering estimation and identification this study for any unknown physical process y f x1 xn is concerned with the 1 fuzzy partition of n dimensional input space x x1timesmiddotmiddotmiddottimesxn into k different clusters 2 estimating the process behavior ycirc f xcirc for a given input xcirc xcirc1 middotmiddotmiddot xcircn isinx and 3 fuzzy approximation of the process with uncertain input output identification data x k plusmndeltaxk y k plusmnvk k 1 using a sugeno type fuzzy inference system a unified min max approach that attempts to minimize the worst case effect of data uncertainties and modeling errors on estimation performance is suggested to provide robustness against data uncertainties and modeling errors the proposed method of min max fuzzy parameters estimation does not make any assumption and does not require a priori knowledge of upper bounds statistics and distribution of data uncertainties and modeling errors to show the feasibility of the approach simulation studies and a real world application of physical fitness classification based on the fuzzy interpretation of physiological parameters have been provided

recursive stochastic algorithms for global optimization in r d an algorithm of the form x k 1 x k a k nabla u x k xi k b k w k where u cdot is a smooth function on mathbb r d xi k is a sequence of mathbb r d valued random variables w k is a sequence of independent standard d dimensional gaussian random variables a k a k and b k sqrt b sqrt k log log k for k large is considered an algorithm of this type arises by adding slowly decreasing white gaussian noise to a stochastic gradient algorithm it is shown under suitable conditions on u cdot xi k a and b that x k converges in probability to the set of global minima of u cdot no prior information is assumed as to what bounded region contains a global minimum the analysis is based on the asymptotic behavior of the related diffusion process dy t nabla u y t dt c t dw t where w cdot is a standard d dimensional wiener process and c t sqrt c sqrt log t for t large

the detection and recovery of discontinuity curves from scattered data we present a numerical method for the detection of faults and gradient faults of two dimensional functions when uniformly scattered and noisy data are given the method recognizes the presence of a discontinuity curve by a suitable statistic moreover we discuss the recovery of such curves and provide a computational bound for the pointwise error and convergence properties

on the estimation of parameters of takagi sugeno fuzzy filters this study derives a class of filtering algorithms for takagi sugeno fuzzy models via solving a nonlinear parameters estimation problem the considered estimation problem is related to the problem of minimizing the expected value of the exponential of filtering errors energy under some stochastic assumptions the filtering criteria which involve an expectation operator are replaced by the deterministic quadratic optimization problems whose solutions provide a class of fuzzy filtering algorithms from a viewpoint of errors in the estimation of linear parameters of the fuzzy filter the derived filtering algorithms were analyzed with emphasis on stability robustness and steady state error issues the stability and robustness analyses have been made deterministically without making any assumption

variational bayes for a mixed stochastic deterministic fuzzy filter this study under the variational bayes vb framework infers the parameters of a takagi sugeno fuzzy filter having deterministic antecedents and stochastic consequents the aim of this study is to take advantages of the vb framework to design fuzzy filtering algorithms which include an automated regularization incorporation of statistical noise models and model comparison capability the vb method can be easily applied to the linear in parameters models this paper applies the vb method to the nonlinear fuzzy filters without using taylor expansion for a linear approximation of some nonlinear function it is assumed that the nonlinear parameters i e antecedents of the fuzzy filter are deterministic while linear parameters are stochastic the vb algorithm by maximizing a strict lower bound on the data evidence makes the approximate posterior of linear parameters as close to the true posterior as possible the nonlinear deterministic parameters are tuned in a way to further increase the lower bound on data evidence the vb paradigm can be used to design an algorithm that automatically selects the most suitable fuzzy filter out of the considered finite set of fuzzy filters this is done by fitting the observed data as a stochastic combination of the different takagi sugeno fuzzy filters such that the individual filters compete with one another to model the data

adaptive fuzzy filtering in a deterministic setting many real world applications involve the filtering and estimation of process variables this study considers the use of interpretable sugeno type fuzzy models for adaptive filtering our aim in this study is to provide different adaptive fuzzy filtering algorithms in a deterministic setting the algorithms are derived and studied in a unified way without making any assumptions on the nature of signals i e process variables the study extends in a common framework the adaptive filtering algorithms usually studied in signal processing literature and p norm algorithms usually studied in machine learning literature to semilinear fuzzy models a mathematical framework is provided that allows the development and an analysis of the adaptive fuzzy filtering algorithms we study a class of nonlinear lms like algorithms for the online estimation of fuzzy model parameters a generalization of the algorithms to the p norm is provided using bregman divergences a standard tool for online machine learning algorithms

a new approach to fuzzy modeling of nonlinear dynamic systems with noise relevance vector learning mechanism this paper presents a new fuzzy inference system for modeling of nonlinear dynamic systems based on input and output data with measurement noise the proposed fuzzy system has a number of fuzzy rules and parameter values of membership functions which are automatically generated using the extended relevance vector machine rvm the rvm has a probabilistic bayesian learning framework and has good generalization capability the rvm consists of the sum of product of weight and kernel function which projects input space into high dimensional feature space the structure of proposed fuzzy system is same as that of the takagi sugeno fuzzy model however in the proposed method the number of fuzzy rules can be reduced under the process of optimizing a marginal likelihood by adjusting parameter values of kernel functions using the gradient ascent method after a fuzzy system is determined coefficients in consequent part are found by the least square method examples illustrate effectiveness of the proposed new fuzzy inference system

metropolis type annealing algorithms for global optimization in r d the convergence of a class of metropolis type markov chain annealing algorithms for global optimization of a smooth function u cdot on mathbb r d is established no prior information is assumed as to what bounded region contains a global minimum the analysis contained herein is based on writing the metropolis type algorithm in the form of a recursive stochastic algorithm x k 1 x k a k nabla u x k xi k b k w k where w k is a standard white gaussian sequence xi k are random variables and a k a k b k sqrt b sqrt k log log k for k large convergence results for x k are then applied from our previous work siam journal on control and optimization 29 1991 pp 999 1018 since the analysis of x k is based on the asymptotic behavior of the related langevin type markov diffusion annealing algorithm dy t nabla u y t dt c t dw t where w cdot is a standard wiener process and c t

efficient self evolving evolutionary learning for neurofuzzy inference systems this study proposes an efficient self evolving evolutionary learning algorithm seela for neurofuzzy inference systems nfiss the major feature of the proposed seela is that it is based on evolutionary algorithms that can determine the number of fuzzy rules and adjust the nfis parameters the seela consists of structure learning and parameter learning the structure learning attempts to determine the number of fuzzy rules a subgroup symbiotic evolution is adopted to yield several variable fuzzy systems and an elite based structure strategy is adopted to find a suitable number of fuzzy rules for solving a problem the parameter learning is to adjust parameters of the nfis it is a hybrid evolutionary algorithm of cooperative particle swarm optimization cpso and cultural algorithm called cultural cpso ccpso the ccpso which uses cooperative behavior among multiple swarms can increase the global search capacity using the belief space experimental results demonstrate that the proposed method performs well in predicting time series and solving nonlinear control 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 robust design criterion for interpretable fuzzy models with uncertain data we believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation h sup 2 and h sup spl infin filtering techniques because of the fact that not only linear parameters consequents but also the nonlinear parameters membership functions attempt to identify the uncertain behavior of the unknown system however the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system s behavior is optimal in some sense this study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data we consider online identification of an interpretable fuzzy model based on the robust solution of a regularized least squares fuzzy parameters estimation problem the aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties the study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors a time domain feedback analysis of the proposed identification approach is carried out with emphasis on stability robustness and steady state issues the simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods