maximum likelihood estimation for multivariate mixture observations of markov chains corresp to use probabilistic functions of a markov chain to model certain parameterizations of the speech signal we extend an estimation technique of liporace to the eases of multivariate mixtures such as gaussian sums and products of mixtures we also show how these problems relate to liporace s original framework

numerical analysis of the learning of fuzzified neural networks from fuzzy if then rules the main aim of this paper is to clearly show how fuzzified neural networks are trained by back propagation type learning algorithms for approximately realizing fuzzy if then rules our fuzzified neural network is a three layer feedforward neural network where connection weights are fuzzy numbers a set of fuzzy if then rules is used as training data for the learning of our fuzzified neural network that is inputs and targets are linguistic values such as small and large in this paper we first demonstrate that the fuzziness in training data propagates backward in our fuzzified neural network next we examine the ability of our fuzzified neural network to approximately realize fuzzy if then rules in computer simulations we compare four types of connection weights i e real numbers symmetric triangular fuzzy numbers asymmetric triangular fuzzy numbers and asymmetric trapezoidal fuzzy numbers in terms of the fitting ability to training data and the computation time we also examine a partially fuzzified neural network in our partially fuzzified neural network connection weights and biases to output units are fuzzy numbers while those to hidden units are real numbers simulation results show that such a partially fuzzified neural network is a good hybrid architecture of fully fuzzified neural networks and neural networks with non fuzzy connection weights

a comparison of fuzzy regression methods for the estimation of the implied volatility smile function the information content of option prices on the underlying asset has a special importance in finance in particular with the use of option implied trees market participants may price other derivatives estimate and forecast volatility see e g the volatility index vix or higher moments of the underlying asset distribution a crucial input of option implied trees is the estimation of the smile implied volatility as a function of the strike price which boils down to fitting a function to a limited number of existing knots however standard techniques require a one to one mapping between volatility and strike price which is not met in the reality of financial markets where to a given strike price two different implied volatilities are usually associated coming from different types of options call and put in this paper we compare the widely used methodology of discarding some implied volatilities and interpolating the remaining knots with cubic splines to a fuzzy regression approach which does not require an a priori choice of implied volatilities to this end we first extend some linear fuzzy regression methods to a polynomial form and we apply them to the financial problem the fuzzy regression methods used range from the possibilistic regression method of tanaka et al 28 to the least squares fuzzy regression method of savic and pedrycz 27 and to the hybrid method of ishibuchi and nii 11

the spectrum of eigenparameter dependent discrete sturm liouville equations let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1y n 1 b ny n a ny n email protected n email protected n 0 2 c email protected email protected y 1 b email protected email protected y 0 0 where a n and b n email protected n are complex sequences c i b email protected c i 0 1 and l is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if email protected n exp en d 1 a n b n 0 and email protected email protected 1

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

spectral properties of non selfadjoint sturm liouville operator with operator coefficient abstract in this paper we consider the sturm liouville operator l generated in l 2 r h by the differential expression l y y q x y 0 x with the boundary condition y 0 0 where q x is a non selfadjoint completely continuous operator in h for every x 0 here h is a separable hilbert space b h denotes the space of bounded operators in h and l 2 r h denotes the space of square integrable strongly measurable vector valued functions defined on 0 in particular we find some special solutions of the equation l y 2 y including jost solution then investigate the point spectrum of l under certain conditions on q x we obtain the resolvent of l if q x is quasi selfadjoint i e there exists p b h such that p 1 b h p is positive and q x p q x p 1 for every x 0 we also show that l has a finite number of spectral singularities

fuzzy regression based on asymmetric support vector machines this paper presents a modified framework of support vector machines which is called asymmetric support vector machines asvms and is designed to evaluate the functional relationship for fuzzy linear and nonlinear regression models in earlier works in order to cope with different types of input output patterns strong assumptions were made regarding linear fuzzy regression models with symmetric and asymmetric triangular fuzzy coefficients excellent performance is achieved on some linear fuzzy regression models however the nonlinear fuzzy regression model has received relatively little attention because such nonlinear fuzzy regression models having certain limitations this study modifies the framework of support vector machines in order to overcome these limitations the principle of asvms is applying an orthogonal vector into the weight vector in order to rotate the support hyperplanes the prime merits of the proposed model are in its simplicity understandability and effectiveness consequently experimental results and comparisons are given to demonstrate that the basic idea underlying asvms can be effectively used for parameter estimation

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

quadratic eigenparameter dependent discrete sturm liouville equations with spectral singularities abstract let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1 y n 1 b n y n a n y n 1 y n n n 0 2 0 1 2 2 y 1 0 1 2 2 y 0 0 where a n and b n n n are complex sequences i i c i 0 1 2 and is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if sup n n exp e n 1 a n b n for some e 0 and 1 2 1

support vector interval regression networks for interval regression analysis in this paper the support vector interval regression networks svirns are proposed for the interval regression analysis the svirns consist of two radial basis function networks one network identifies the upper side of data interval and the other network identifies the lower side of data intervals because the support vector regression svr approach is equivalent to solving a linear constrained quadratic programming problem the number of hidden nodes and the initial values of adjustable parameters can be easily obtained since the selection of a parameter e in the svr approach may seriously affect the modeling performance a two step approach is proposed to properly select the e value after the svr approach with the selected e an initial structure of svirns can be obtained besides outliers will not significantly affect the upper and lower bound interval obtained through the proposed two step approach consequently a traditional back propagation bp learning algorithm can be used to adjust the initial structure networks of svirns under training data sets without or with outliers due to the better initial structure of svirns are obtained by the svr approach the convergence rate of svirns is faster than the conventional networks with bp learning algorithms or with robust bp learning algorithms for interval regression analysis four examples are provided to show the validity and applicability of the proposed svirns

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

iterative inversion of fuzzified neural networks the inversion of a neural network is a process of computing inputs that produce a given target when fed into the neural network the inversion algorithm of crisp neural networks is based on the gradient descent search in which a candidate inverse is iteratively refined to decrease the error between its output and the target in this paper we derive an inversion algorithm of fuzzified neural networks from that of crisp neural networks first we present a framework of learning algorithms of fuzzified neural networks and introduce the idea of adjusting schemes for fuzzy variables next we derive the inversion algorithm of fuzzified neural networks by applying the adjusting scheme for fuzzy variables to total inputs in the input layer finally we make three experiments on the parity three problem examine the effect of the size of training sets on the inversion and investigate how the fuzziness of inputs and targets of training sets affects the inversion

eigenparameter dependent discrete dirac equations with spectral singularities let us consider the boundary value problem bvp for the discrete dirac equations 0 1 a n 1y n 1 2 b ny n 2 p ny n 1 email protected n 1 a n 1y n 1 1 b ny n 1 q ny n 2 email protected n 2 email protected n 0 2 c email protected email protected y 1 2 b email protected email protected y 0 1 0 where a n b n p n and q n email protected n are complex sequences c i b email protected c i 0 1 and l is a eigenparameter discussing the eigenvalues and the spectral singularities we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if n 1 exp en d 1 a n 1 b n p n q n 0 and 12 d 1

fuzzy regression analysis using neural networks abstract in this paper we propose simple but powerful methods for fuzzy regression analysis using neural networks since neural networks have high capability as an approximator of nonlinear mappings the proposed methods can be applied to more complex systems than the existing lp based methods first we propose learning algorithms of neural networks for determining a nonlinear interval model from the given input output patterns a nonlinear interval model whose outputs approximately include all the given patterns can be determined by two neural networks next we show two methods for deriving nonlinear fuzzy models from the interval model determined by the proposed algorithms nonlinear fuzzy models whose h level sets approximately include all the given patterns can be derived last we show an application of the proposed methods to a real problem

estimating the functional relationships for quality function deployment under uncertainties product planning is one of four important processes in new product development npd using quality function deployment qfd which is a widely used customer driven approach in our opinion the first problem to be solved is how to incorporate both qualitative and quantitative information regarding relationships between customer requirements crs and engineering characteristics ecs as well as those among ecs into the problem formulation owing to the typical vagueness or imprecision of functional relationships in a product product planning is becoming more difficult particularly in a fuzzy environment in this paper an asymmetric fuzzy linear regression approach is proposed to estimate the functional relationships for product planning based on qfd firstly by integrating the least squares regression into fuzzy linear regression a pair of hybrid linear programming models with asymmetric triangular fuzzy coefficients are developed to estimate the functional relationships for product planning under uncertainties secondly using the basic concept of fuzzy regression asymmetric triangular fuzzy coefficients are extended to asymmetric trapezoidal fuzzy coefficients and another pair of hybrid linear programming models with asymmetric trapezoidal fuzzy coefficients is proposed the main advantage of these hybrid programming models is to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression next the illustrated example shows that trapezoidal fuzzy number coefficients have more flexibility to handle a wider variety of systematic uncertainties and ambiguities that cannot be modeled efficiently using triangular number fuzzy coefficients both asymmetric triangular and trapezoidal fuzzy number coefficients can be applicable to a much wider variety of design problems where uncertain qualitative and fuzzy relationships are involved than when symmetric triangular fuzzy numbers are used finally future research direction is also discussed

maximum likelihood estimation for multivariate observations of markov sources parameter estimation for multivariate functions of markov chains a class of versatile statistical models for vector random processes is discussed the model regards an ordered sequence of vectors as noisy multivariate observations of a markov chain mixture distributions are a special case the foundations of the theory presented here were established by baum petrie soules and weiss a powerful representation theorem by fan is employed to generalize the analysis of baum em et al to a larger class of distributions

difference equations of second order with spectral singularities abstract in this paper using the uniqueness theorem of analytic functions we investigated the eigenvalues and the spectral singularities of the difference equation a n 1 y n 1 b n y n a n y n 1 y n n z 0 1 2 where t a n n z b n n z are complex sequences and is a spectral parameter

non selfadjoint matrix sturm liouville operators with spectral singularities in this paper we investigate discrete spectrum of the non selfadjoint matrix sturm liouville operator l generated in l 2 r s by the differential expression y y q x y x r 0 and the boundary condition y 0 0 under the conditionsupx r exp ex q x 0using the uniqueness theorem of analytic functions we prove that l has a finite number of eigenvalues and spectral singularities with finite multiplicities

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

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

an inverse problem for the matrix schrodinger equation abstract by modifying and generalizing some old techniques of n levinson a uniqueness theorem is established for an inverse problem related to periodic and sturm liouville boundary value problems for the matrix schrodinger equation

an estimate for the resolvent of a non self adjoint differential operator on the half line we consider the operator defined by t0y x y q x y x 0 on the domain dom t0 f l2 0 fty f l2 0 fty f 0 0 here q x p x ib x where p x and b x are real functions satisfying the following conditions b x is bounded on 0 there exists the limit b0 limx b x and b x b0 l2 0 fty in addition infxp x supx b1 x we derive an estimate for the norm of the resolvent of t0 as well as prove that t0 ib0i 1 is a sum of a normal operator and a quasinilpotent one and these operators have the same invariant subspaces

jost solution and the spectral properties of the matrix valued difference operators abstract in this paper we find polynomial type jost solution of the selfadjoint matrix valued difference equation of second order then we investigate analytical properties and asymptotic behaviour of the jost solution using the weyl compact perturbation theorem we prove that the selfadjoint operator l generated by the matrix valued difference expression of second order has the continuous spectrum filling the segment 2 2 we also study the eigenvalues of l and prove that it has a finite number of simple real eigenvalues

an eigenfunction expansion for a quadratic pencil of a schrodinger operator with spectral singularities in this paper we consider the operatorlgenerated inl2 r by the differential expressionl y y q x 2 p x 2 y x r 0 and the boundary conditiony 0 0 wherepandqare complex valued functions andpis continuously differentiable onr we derive a two fold spectral expansion ofl in the sense of keldysh 1951 soviet math dokl 77 11 14 1971 russian math survey26 15 44 engl transl in terms of the principal functions under the conditionslimx p x 0 supx r eex q x p x 0 taking into account the spectral singularities also we investigate the convergence of the spectral expansion

the likelihood ratio test for the two component normal mixture problem power and sample size analysis we find through simulation and modeling an approximation to the alternative distribution of the likelihood ratio test for two component mixtures in which the components have different means but equal variances we consider the range of mixing proportions from 0 5 through 95 our simulation results indicate a dependence of power on the mixing proportion when pi less than 2 and pi greater than 80 our model results indicated that the alternative distribution is approximately noncentral chi square possibly with 2 degrees of freedom using this model we estimate a sample of 40 is needed to have 50 power to detect a difference between means equal to 3 0 for mixing proportions between 2 and 8 the sample size increases to 50 when the mixing proportion is 90 or 1 and 82 when the mixing proportion is 95 or 05 this paper contains a complete table of sample sizes needed for 50 80 and 90 power

a fuzzy adaptive network approach to parameter estimation in cases where independent variables come from an exponential distribution in a regression analysis it is assumed that the observations come from a single class in a data cluster and the simple functional relationship between the dependent and independent variables can be expressed using the general model y f x email protected however a data cluster may consist of a combination of observations that have different distributions that are derived from different clusters when faced with issues of estimating a regression model for fuzzy inputs that have been derived from different distributions this regression model has been termed the switching regression model and it is expressed with y l f l x email protected l email protected i 1 pl i here l i indicates the class number of each independent variable and p is indicative of the number of independent variables j r jang anfis adaptive network based fuzzy inference system ieee transaction on systems man and cybernetics 23 3 1993 665 685 m michel fuzzy clustering and switching regression models using ambiguity and distance rejects fuzzy sets and systems 122 2001 363 399 e q richard a new approach to estimating switching regressions journal of the american statistical association 67 338 1972 306 310 in this study adaptive networks have been used to construct a model that has been formed by gathering obtained models there are methods that suggest the class numbers of independent variables heuristically alternatively in defining the optimal class number of independent variables the use of suggested validity criterion for fuzzy clustering has been aimed in the case that independent variables have an exponential distribution an algorithm has been suggested for defining the unknown parameter of the switching regression model and for obtaining the estimated values after obtaining an optimal membership function which is suitable for exponential distribution

spectrum and spectral singularities of a quadratic pencil of a schrodinger operator with a general boundary condition abstract in this article we investigate the spectrum and the spectral singularities of the quadratic pencil of schrodinger operatorlgenerated inl2 r by the differential expression l y y q x 2 p x 2 y x r 0 and the boundary condition 0 k x y x dx y 0 y 0 0 wherep q and k are complex valued functions p is continuously differentiable onr k l2 r and c with 0 discussing the spectrum we prove that l has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions lim x p x 0 sup x r e e x q x p x k x 0 later we investigate the properties of the principal functions corresponding to the spectral singularities moreover some results about the spectrum oflare applied to non selfadjoint sturm liouville and klein gordons wave operators

an intelligent stock trading decision support system through integration of genetic algorithm based fuzzy neural network and artificial neural network the stock market which has been investigated by various researchers is a rather complicated environment most research only concerned the technical indexes quantitative factors instead of qualitative factors e g political effect however the latter plays a critical role in the stock market environment thus this study develops a genetic algorithm based fuzzy neural network gfnn to formulate the knowledge base of fuzzy inference rules which can measure the qualitative effect on the stock market next the effect is further integrated with the technical indexes through the artificial neural network ann an example based on the taiwan stock market is utilized to assess the proposed intelligent system evaluation results indicate that the neural network considering both the quantitative and qualitative factors excels the neural network considering only the quantitative factors both in the clarity of buying selling points and buying selling performance

incorporating fuzzy membership functions into the perceptron algorithm the perceptron algorithm one of the class of gradient descent techniques has been widely used in pattern recognition to determine linear decision boundaries while this algorithm is guaranteed to converge to a separating hyperplane if the data are linearly separable it exhibits erratic behavior if the data are not linearly separable fuzzy set theory is introduced into the perceptron algorithm to produce a fuzzy algorithm which ameliorates the convergence problem in the nonseparable case it is shown that the fuzzy perceptron like its crisp counterpart converges in the separable case a method of generating membership functions is developed and experimental results comparing the crisp to the fuzzy perceptron are presented

a note on the spectral singularities of non selfadjoint matrix valued difference operators let l denote the non selfadjoint operator generated in 2 n e by the matrix difference expression y n a n 1y n 1 b ny n a ny n 1 n n and the boundary condition y 0 0 in this paper we investigate the jost solution the continuous spectrum the eigenvalues and the spectral singularities of l

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

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

trace formulas and borg type theorems for matrix valued jacobi and dirac finite difference operators abstract borg type uniqueness theorems for matrix valued jacobi operators h and supersymmetric dirac difference operators d are proved more precisely assuming reflectionless matrix coefficients a b in the self adjoint jacobi operator h as a s b with s the right left shift operators on the lattice z and the spectrum of h to be a compact interval e e e e we prove that a and b are certain multiples of the identity matrix an analogous result which however displays a certain novel nonuniqueness feature is proved for supersymmetric self adjoint dirac difference operators d with spectrum given by e 1 2 e 1 2 e 1 2 e 1 2 0 e e our approach is based on trace formulas and matrix valued exponential herglotz representation theorems as a by product of our techniques we obtain the extension of flaschka s borg type result for periodic scalar jacobi operators to the class of reflectionless matrix valued jacobi operators

mixture autoregressive hidden markov models for speech signals in this paper a signal modeling technique based upon finite mixture autoregressive probabilistic functions of markov chains is developed and applied to the problem of speech recognition particularly speaker independent recognition of isolated digits two types of mixture probability densities are investigated finite mixtures of gaussian autoregressive densities gam and nearest neighbor partitioned finite mixtures of gaussian autoregressive densities pgam in the former gam the observation density in each markov state is simply a stochastically constrained weighted sum of gaussian autoregressive densities while in the latter pgam it involves nearest neighbor decoding which in effect defines a set of partitions on the observation space in this paper we discuss the signal modeling methodology and give experimental results on speaker independent recognition of isolated digits we also discuss the potential use of the modeling technique for other applications

can fuzzy neural nets approximate continuous fuzzy functions abstract we first argue that regular fuzzy neural nets based on standard fuzzy arithmetic and the extension principle cannot be universal approximators this negative result is surprising since regular neural nets are universal approximators we then argue that hybrid fuzzy neural nets not necessarily based on standard fuzzy arithmetic can be universal approximators

possibilistic linear regression analysis for fuzzy data abstract fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined recently it has become important to deal with fuzzy data in connection with expert knowledge three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data since our formulations can be reduced to linear programming problems the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters this approach can be regarded as a fuzzy interval analysis in a fuzzy environment

numerical relationships between neural networks continuous functions and fuzzy systems abstract in this survey paper we first discuss computational equivalence between continuous functions regular neural nets fuzzy controllers and discrete fuzzy expert systems we also discuss how to build hybrid neural nets numerically identical to a fuzzy controller or a discrete fuzzy expert system we then discuss computational equivalence between continuous fuzzy functions regular fuzzy neural nets and fuzzy systems we also discuss how to construct hybrid fuzzy neural nets to approximate continuous fuzzy functions

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

extended support vector interval regression networks for interval input output data in many applications it is natural to use interval data because of uncertainty existence in the measurements variability in defining terms such as the temperature during a given day description for extremely behavior such as the maximum wind speed in a given area etc in order to handle such interval data a novel approach called the interval support vector interval regression networks isvirns is proposed the isvirns is extended from our previous work the support vector interval regression networks svirns it is easy to find that svirns can handle interval output data but for input data they must be crisp in this study the hausdorff distance is employed as the distance measure of interval data and is incorporated into the kernel functions of svirns to determine the initial structure of isvirns because the proposed approach can provide a better initial structure for isvirns it can have a faster convergent speed the experimental results with real data sets show the validity of the proposed isvirns

structure identification of generalized adaptive neuro fuzzy inference systems this paper presents a method to identify the structure of generalized adaptive neuro fuzzy inference systems ganfiss the structure of ganfis consists of a number of generalized radial basis function grbf units the radial basis functions are irregularly distributed in the form of hyper patches in the input output space the minimum number of grbf units is selected based on a heuristic using the fuzzy curve for structure identification a new criterion called structure identification criterion sic is proposed sic deals with a trade off between performance and computational complexity of the ganfis model the computational complexity of gradient descent learning is formulated based on simulation study three methods of initialization of ganfis viz fuzzy curve fuzzy c means in x spl times y space and modified mountain clustering have been compared in terms of cluster validity measure akaike s information criterion aic and the proposed sic

max min fuzzy hopfield neural networks and an efficient learning algorithm abstract we set up a dynamical fuzzy neural network system i e the so called max min fuzzy hopfield network in the paper and prove the lyapunov stability of the equilibrium points attractor of the system also we discuss the uniform stability of the system and show some sufficient conditions with which the given fuzzy pattern is the attractor of the system moreover we obtain a nontrivially attractive basin of the attractor therefore our models have good fault tolerance with an analytic method we design an efficient learning algorithm for connected weights of the networks finally simulation examples demonstrate our conclusions

a unified approach to define fuzzy integrals recently the two most general and well known fuzzy integrals sugeno s integral and choquet s integral were characterized it was also shown how both integrals fitted the same formal model and they only differed in the operators used in this paper we try to analyze whether it is possible to improve those characterization theorems and we also study the possibility of defining other fuzzy integrals following the same model but using other operators in a recent paper campos and bolafios 6 characterized two general and well known fuzzy integrals sugeno s integral 11 and choquet s integral 7 the last one also called monotone expectation 2 4 5 6 the two characterization theorems are based on the same formal model of four properties which only differ in the operators used sum and product for choquet s integral maximum and minimum for sugeno s integral two important questions remained open one involved the characterization theorems of these integrals in an attempt to further refine them the second question was to investigate the possibility of defining other fuzzy integrals using different operators but following the same model in this paper we give answers to those problems section 2 is devoted to describing fuzzy integrals defined on sugeno s fuzzy measures particularly sugeno s and choquet s integrals and their characterizations in section 3 we show how the characterization of choquet s integral can be refined whereas the same cannot be made for sugeno s integral finally other fuzzy integrals fitting our formal model are obtained in section 4

an architecture of neural networks with interval weights and its application to fuzzy regression analysis abstract in this paper we first propose an architecture of neural networks that have interval weights and interval biases a neural network with the proposed architecture maps an input vector of real numbers to an output interval the target output is also given as an interval next we define a cost function using the interval output from the neural network and the corresponding target output a learning algorithm is derived from the cost function in a similar manner as the bp back propagation algorithm we also show two variations of the learning algorithm for the proposed architecture which lead to the inclusion relation between the interval output and the interval target last we apply the learning algorithms to fuzzy regression analysis where target outputs are given as fuzzy numbers

fuzzy nonlinear regression analysis of dose response relationships abstract a general methodology for fuzzy regression is developed and illustrated by analyzing a dose response relationship fuzzy regression is used as an alternative to statistical regression analysis because the relationship between variables is imprecise data are uncertain and sample sizes are insufficient three goodness of fit criteria are used namely the maximum the average vagueness and the prediction vagueness criterion in the case of a nonlinear relationship the calculation of fuzzy regression parameters leads to a nonlinear programming problem this technique is illustrated by the relationship between the dose of a n nitroso compound and the probability of tumor development in rats this relationship is nonlinear imprecise and based on only six data points in the present case three common dose response models are fitted using nonlinear fuzzy regression techniques with the six data points regression parameters have been obtained by minimizing a nonlinear objective function under a set of specific nonlinear constraints the methodology is relatively simple and offers a viable alternative to statistical regression techniques

functional equivalence between radial basis function networks and fuzzy inference systems it is shown that under some minor restrictions the functional behavior of radial basis function networks rbfns and that of fuzzy inference systems are actually equivalent this functional equivalence makes it possible to apply what has been discovered learning rule representational power etc for one of the models to the other and vice versa it is of interest to observe that two models stemming from different origins turn out to be functionally equivalent

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

support vector interval regression machine for crisp input and output data support vector regression svr has been very successful in function estimation problems for crisp data in this paper we propose a robust method to evaluate interval regression models for crisp input and output data combining the possibility estimation formulation integrating the property of central tendency with the principle of standard svr the proposed method is robust in the sense that outliers do not affect the resulting interval regression furthermore the proposed method is model free method since we do not have to assume the underlying model function for interval nonlinear regression model with crisp input and output in particular this method performs better and is conceptually simpler than support vector interval regression networks svirns which utilize two radial basis function networks to identify the upper and lower sides of data interval five examples are provided to show the validity and applicability of the proposed method

generalized hidden markov models i theoretical frameworks in this paper we present the theoretical framework for the generalization of classical hidden markov models using fuzzy measures and fuzzy integrals the main characteristic of the generalization is the relaxation of the usual additivity constraint of probability measures fuzzy integrals are defined with respect to fuzzy measures whose key property is monotonicity with respect to set inclusion this property is far weaker than the usual additivity property of probability measures as a result of the new formulation the statistical independence assumption of the classical hidden markov models is relaxed two attractive properties of this generalization are the generalized hidden markov model reduces to the classical hidden markov model if we used the choquet fuzzy integral and probability measures and the establishment of a relation between the generalized hidden markov model and the classical nonstationary hidden markov model in which the transitional parameters vary with time

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

possibilistic linear regression with fuzzy data tolerance approach with prior information abstract we introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data both input and output the method is very general the only assumption is that cuts of the fuzzy data are efficiently computable we take into account possible prior restrictions of the parameters space we assume that the restrictions are given by linear and quadratic constraints the method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy valued model to an interval valued model on a given cut which is further reduced to a linear time method i e running in time o n p computing with endpoints of the intervals here n is the number of observations and p is the number of regression parameters the speed of computation makes the method applicable for huge datasets unlike various approaches based on mathematical programming formulations the tolerance based construction preserves central tendency of the resulting regression coefficients in addition we prove further properties if inputs are crisp and outputs are fuzzy then the construction preserves piecewise linearity and convex shape of fuzzy numbers we derive an o n 2 p algorithm for enumeration of breakpoints of the membership function of the estimated coefficients similar results are also derived for the fuzzy input and output model we illustrate the theory for the case of triangular and asymmetric gaussian fuzzy inputs and outputs of an inflation consumption model

fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks in this paper first we explain several versions of fuzzy regression methods based on linear fuzzy models with symmetric triangular fuzzy coefficients next we point out some limitations of such fuzzy regression methods then we extend the symmetric triangular fuzzy coefficients to asymmetric triangular and trapezoidal fuzzy numbers we show that the limitations of the fuzzy regression methods with the symmetric triangular fuzzy coefficients are remedied by such extension several formulations of linear programming problems are proposed for determining asymmetric fuzzy coefficients from numerical data finally we show how fuzzified neural networks can be utilized as nonlinear fuzzy models in fuzzy regression in the fuzzified neural networks asymmetric fuzzy numbers are used as connection weights the fuzzy connection weights of the fuzzified neural networks correspond to the fuzzy coefficients of the linear fuzzy models nonlinear fuzzy regression based on the fuzzified neural networks is illustrated by computer simulations where type i and type ii membership functions are determined from numerical data

generalization of adaptive neuro fuzzy inference systems the adaptive network based fuzzy inference systems anfis of jang 1993 is extended to the generalized anfis ganfis by proposing a generalized fuzzy model gfm and considering a generalized radial basis function grbf network the gfm encompasses both the takagi sugeno ts model and the compositional rule of inference cri model the conditions by which the proposed gfm converts to ts model or the cri model are presented the basis function in grbf is a generalized gaussian function of three parameters the architecture of the grbf network is devised to learn the parameters of gfm where the grbf network and gfm have been proved to be functionally equivalent it is shown that grbf network can be reduced to either the standard rbf or the hunt s rbf network the issue of the normalized versus the non normalized grbf networks is investigated in the context of ganfis an interesting property of symmetry on the error surface of grbf network is investigated the proposed ganfis is applied to the modeling of a multivariable system like stock market

spectral expansion of a non self adjoint differential operator on the whole axis abstract in this article we consider an operator l defined by the differential expression l y y q x y x r in l2 r where q is a complex valued function under the condition sup x exp x q x 0 we have proved a spectral expansion of l in terms of the principal functions taking into account the spectral singularities we have also investigated the convergence of the spectral expansion of l

principal functions of non selfadjoint matrix sturm liouville equations in this paper we investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem y q x y l 2y x r 0 y 0 0 where q is a non selfadjoint matrix valued function

spectral properties of a schrodinger equation with a class of complex potentials and a general boundary condition in this paper we investigate the spectrum and the spectral singularities of an operator l generalized in l2 r by the differential expression l y y k 0n 1 kqk x y x r 0 and the boundary condition 0 k x f x dx f 0 f 0 0 where is a complex parameter qk k 0 1 n 1 are complex valued functions q0 q1 qn 1 are differentiable on 0 k l2 r and c with 0 discussing the spectrum we obtain that l has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions limx qk x 0 supx r eex k 0n 1q k x k x hold where k 0 1 n 1 and e 0

extended fuzzy regression models using regularization method regularization approach to regression can be easily found in statistics and information science literatures the technique of regularization was introduced as a way of controlling the smoothness properties of regression function the main purpose of this paper is to extend the models of diamond inform sci 46 1988 141 to fuzzy multiple linear regression models and to extend the model f3 with numerical input of diamond inform sci 46 1988 141 to fuzzy nonlinear model here all extensions are based on regularization technique

spectral properties of non selfadjoint difference operators abstract let l denote the operator generated in l2 z by the difference expression ly n an 1yn 1 bnyn anyn 1 n z 0 1 2 where an n z and bn n z are complex sequences in this paper we investigated the spectrum the spectral singularities and the properties of the principal vectors corresponding to the spectral singularities of l we also studied similar problems for the discrete dirac operator m generated in l2 z c 2 by the system of difference expression y n 1 y n 2 y n y 2 n p n y 1 n y 1 n 1 q n y 2 n n z where y y 1 n y 2 n n z is the forward difference operator i e y i n y i n 1 y i n i 1 2 and pn n z qn n z are complex sequences

calculating insurance claim reserves with fuzzy regression claim provisions are crucial for the financial stability of insurance companies this is why actuarial literature has proposed numerous claim reserving methods which are usually based on statistical concepts however the mutant and uncertain behaviour of insurance environments does not make advisable to use a wide data base when calculating claim reserves on the one hand this involves a considerable loss in reliability of statistical methods but on the other it makes the use of fuzzy set theory very attractive in this paper then we propose a claim reserving method that combines the extension by ishibuchi and nii of tanaka s classical fuzzy regression method with sherman s claim reserving scheme

estimating a term structure of interest rates for fuzzy financial pricing by using fuzzy regression methods abstract several papers in the financial literature propose using fuzzy numbers to model interest rate uncertainty however in our opinion the first problem to be solved is how to estimate these rates with fuzzy numbers in this paper we attempt to provide a solution to this question with a method for adjusting the temporal structure of interest rates tsir that is based on the extension proposed by ishibuchi and nii fuzzy sets and systems 119 2001 273 of the fuzzy regression methods exposed in savic and predrycz fuzzy regression analysis physica verlag heidelberg 1992 pp 91 100 tanaka fuzzy sets and systems 24 1987 363 375 and tanaka and ishibuchi fuzzy regression analysis physica verlag heidelberg 1992 pp 47 this method will enable to quantify the anticipated rates in the fixed income markets for the future with fuzzy numbers in particular we discuss how to estimate the tsir with triangular fuzzy numbers because of the desirable properties of this kind of fuzzy numbers

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

hybrid identification in fuzzy neural networks this paper introduces an identification method for nonlinear models in the form of fuzzy neural networks fnn in this model we use two forms of the fuzzy inference methods a simplified and linear fuzzy inference and exploit a standard error back propagation learning algorithm the fnn modeling and identification environment realizes parameter identification through a synergistic usage of clustering techniques genetic optimization and a complex search method we use a hard c means hcm clustering algorithm to determine initial apexes of the membership functions of the information granules used in this fuzzy model the parameters such as apexes of membership functions learning rates and momentum coefficients are then adjusted using hybrid algorithm the proposed hybrid identification algorithm is carried out by combining both genetic optimization genetic algorithm ga and the improved complex method an aggregate objective function performance index with a weighting factor is introduced to achieve a sound balance between approximation and generalization of the model according to the selection and adjustment of the weighting factor of this objective function we reveal how to design a model with sound approximation and generalization abilities the proposed model is experimented with using several time series data gas furnace sewage treatment process and nox emission process data of gas turbine power plant

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

quadratic pencil of schrodinger operators with spectral singularities discrete spectrum and principal functions abstract in this article we investigated the spectrum of the quadratic pencil of schrodinger operatorsl generated inl2 r by the equation formula and the boundary condition formula whereu vare complex valued functions anduis absolutely continuous in each finite subinterval of r discussing the spectrum we proved thatl has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions formula and formula hold it is shown that the principal functions corresponding to eigenvalues ofl are in the spacel2 r and the principal functions corresponding to spectral singularities are in another hilbert space which containsl2 r some results about the spectrum ofl have also been applied to radial klein gordon and one dimensional schrodinger equations

fuzzy linear least squares for the identification of possibilistic regression models abstract the stochastic approach to regression e g the least squares method is a commonly used technique in a wide range of disciplines as it offers an intuitive and well founded solution to the problem of fitting a set of parameters to existing data specifically in the case where the parameters of meta models are identified the linear least squares formulation receives wide appreciation among engineers due to its simple yet powerful formulation and solution usually this approach provides a reasonable fit depending largely on the quality of the model that is employed however it has two main drawbacks from a statistical point of view the premises justifying its use cannot always be assumed in good faith and most importantly the resulting model will almost certainly fail at correctly predicting the output without falling victim to the effect of over fitting the presented fuzzy regression algorithm originates from a set of different axioms for the solution of the linear least squares problem it is able to provide the worst case optimal parameter variation necessary to cover all of the crisp data points while encoding meaningful information in the membership function this is accomplished by means of a sensible fuzzification procedure of the crisp data points and the application of an exact inverse fuzzy arithmetic for estimating the fuzzy valued parameters of the model in question

q measures an efficient extension of the sugeno spl lambda measure in this paper we introduce a new class of fuzzy measures referred to as q measures based on a modification of the definition of the well known sugeno spl lambda measure our proposed definition of the q measures not only includes the spl lambda measure as a special case but also preserves all desirable properties and avoids some of the limitations of the conventional spl lambda measure the q measure approach provides a more flexible and powerful method for constructing various fuzzy measures we provide an iterative algorithm for constructing an interesting sequence of q measures and analytically prove its convergence as a distinguishing characteristic of the proposed formulation

additive and nonadditive fuzzy hidden markov models we present a novel approach for the development of fuzzy hidden markov models fhmms by exploiting both additive and nonadditive properties of input fuzzy sets in the fuzzy rules of generalized fuzzy model gfm this development utilizes 1 gaussian mixture model gmm to manipulate the mixture parameters for the input fuzzy sets and 2 gfm rules for the inclusion of states in the consequent part to be able to use hmm taking the components of gaussian mixture density conditioned on the past system states and making use of equivalence of gmm with gfm parameters of the additive and nonadditive fhmms are estimated using the forward backward procedure of the baum welch algorithm the additive and nonadditive fhmms are validated on three benchmark applications involving time series prediction and the results are compared and found to be better than or equal to those of the existing recent fuzzy models

fuzzy neural networks a survey abstract in this paper a fuzzy neural network will be a layered feedforward neural net that has fuzzy signals and or fuzzy weights we survey recent results on learning algorithms and applications for fuzzy neural networks

estimating a fuzzy term structure of interest rates using fuzzy regression techniques abstract several papers in the financial literature propose using fuzzy numbers fns to model interest rate uncertainty however in our opinion the first problem to be solved is how to estimate these rates with fns in this paper we attempt to provide a solution to this question with a method for adjusting the temporal structure of interest rates tsir that is based on fuzzy regression techniques this method will enable to quantify the anticipated rates in the fixed income markets for the future with fns in particular we discuss how to estimate the tsir with triangular fuzzy numbers because of their desirable properties