a new gradient descent approach for local learning of fuzzy neural models the majority of reported learning methods for takagi sugeno kang tsk fuzzy neural models to date mainly focus on improvement of their accuracy however one of the key design requirements in building an interpretable fuzzy model is that each obtained rule consequent must match well with the system local behavior when all the rules are aggregated to produce the overall system output this is one of the distinctive characteristics from black box models such as neural networks therefore how to find a desirable set of fuzzy partitions and hence identify the corresponding consequent models which can be directly explained in terms of system behavior presents a critical step in fuzzy neural modeling in this paper a new learning approach considering both nonlinear parameters in the rule premises and linear parameters in the rule consequents is proposed unlike the conventional two stage optimization procedure widely practiced in the field where the two sets of parameters are optimized separately the consequent parameters are transformed into a dependent set on the premise parameters thereby enabling the introduction of a new integrated gradient descent learning approach thus a new jacobian matrix is proposed and efficiently computed to achieve a more accurate approximation of the cost function by using the second order levenberg marquardt optimization method several other interpretability issues regarding the fuzzy neural model are also discussed and integrated into this new learning approach numerical examples are presented to illustrate the resultant structure of the fuzzy neural models and the effectiveness of the proposed new algorithm and compared with the results from some well known methods

fuzzy wavelet networks for function learning inspired by the theory of multiresolution analysis mra of wavelet transforms and fuzzy concepts a fuzzy wavelet network fwn is proposed for approximating arbitrary nonlinear functions the fwn consists of a set of fuzzy rules each rule corresponding to a sub wavelet neural network wnn consists of single scaling wavelets through efficient bases selection the dimension of the approximated function does not cause the bottleneck for constructing fwn especially by learning the translation parameters of the wavelets and adjusting the shape of membership functions the model accuracy and the generalization capability of the fwn can be remarkably improved furthermore an algorithm for constructing and training the fuzzy wavelet networks is proposed simulation examples are also given to illustrate the effectiveness of the method

positive solutions to a multi point higher order boundary value problem the authors consider a higher order multi point boundary value problem some existence and nonexistence results for positive solutions of the problem are obtained by using krasnosel skii s fixed point theorem examples are included to illustrate the results

parametrization of approximate algebraic curves by lines it is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines in this paper given a tolerance e 0 and an e irreducible algebraic plane curve l of degree d having an e singularity of multiplicity d 1 we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines furthermore the error analysis shows that under certain initial conditions that ensures that points are projectively well defined the output curve lies within the offset region of l at distance at most 2 2e1 2d exp 2

numerical proper reparametrization of parametric plane curves we present an algorithm for reparametrizing algebraic plane curves from a numerical point of view more precisely given a tolerance 0 and a rational parametrization p of a plane curve c with perturbed float coefficients we present an algorithm that computes a parametrization q of a new plane curve d such that q is an proper reparametrization of d in addition the error bound is carefully discussed and we present a formula that measures the closeness between the input curve c and the output curve d we introduce the problem of numerical improper parametrization of plane curves the approximate improper index is firstly defined in this paper a numerical algorithm is designed to find the approximate proper reparametrization the relationship between the numerical curve and its reparametrization is well studied

distance bounds of e points on hypersurfaces e points were introduced by the authors see s perez diaz j r sendra j sendra parametrization of approximate algebraic curves by lines theoret comput sci 315 2 3 2004 627 650 special issue s perez diaz j r sendra j sendra parametrization of approximate algebraic surfaces by lines comput aided geom design 22 2 2005 147 181 s perez diaz j r sendra j sendra distance properties of e points on algebraic curves in series mathematics and visualization computational methods for algebraic spline surfaces springer berlin 2005 pp 45 61 as a generalization of the notion of approximate root of a univariate polynomial the notion of e point of an algebraic hypersurface is quite intuitive it essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module intuition says that an e point of a hypersurface is a point close to it in this paper we formally analyze this assertion giving bounds of the distance of the e point to the hypersurface for this purpose we introduce the notions of height depth and weight of an e point the height and the depth control when the distance bounds are valid while the weight is involved in the bounds

building confidence interval based fuzzy random regression models in real world regression analysis statistical data may be linguistically imprecise or vague given the co existence of stochastic and fuzzy uncertainty real data cannot be characterized by using only the formalism of random variables in order to address regression problems in the presence of such hybrid uncertain data fuzzy random variables are introduced in this study to serve as an integral component of regression models a new class of fuzzy regression models that is based on fuzzy random data is built and is called the confidence interval based fuzzy random regression model ci frrm first a general fuzzy regression model for fuzzy random data is introduced then using expectations and variances of fuzzy random variables sigma confidence intervals are constructed for fuzzy random input output data the ci frrm is established based on the sigma confidence intervals the proposed regression model gives rise to a nonlinear programming problem that consists of fuzzy numbers or interval numbers since sign changes in the fuzzy coefficients modify the entire programming structure of the solution process the inherent dynamic nonlinearity of this optimization makes it difficult to exploit the techniques of linear programming or classical nonlinear programming therefore we resort to some heuristics finally an illustrative example is provided

fuzzy rule based models with interactive rules and their granular generalization abstract processing realized in fuzzy rule based models is associated with matching individual rules with the existing input data and aggregation of the levels of matching and conclusions of the rules rules are processed as individual entities in this study we introduce an augmentation of fuzzy models by facilitating interaction among the rules leading to more flexible type membership functions of fuzzy sets forming conditions of the rules thus resulting in substantially advanced topology of the partition of the input space different ways of realizing interaction among the rules are studied in the sequel we develop a granular fuzzy model implied by the rule based model showing how granular parameters of the original rule based model enhance its quality expressed in terms of coverage of experimental data the two evaluation criteria of the constructed granular model namely coverage and specificity are studied experimental results are reported for a series of publicly available data

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

stochastic choice of basis functions in adaptive function approximation and the functional link net a theoretical justification for the random vector version of the functional link rvfl net is presented in this paper based on a general approach to adaptive function approximation the approach consists of formulating a limit integral representation of the function to be approximated and subsequently evaluating that integral with the monte carlo method two main results are 1 the rvfl is a universal approximator for continuous functions on bounded finite dimensional sets and 2 the rvfl is an efficient universal approximator with the rate of approximation error convergence to zero of order o c spl radic n where n is number of basis functions and with c independent of n similar results are also obtained for neural nets with hidden nodes implemented as products of univariate functions or radial basis functions some possible ways of enhancing the accuracy of multivariate function approximations are discussed

fuzzy rule based models with randomized development mechanisms abstract fuzzy rule based models have attracted attention because of their modular architectures well developed design methodologies and practices as well as interpretability aspects methods exploiting factors of randomness offer significant efficiency and implementation simplicity that are essential in numerous application areas in this study we propose an original development of fuzzy rule based models established with the aid of concepts of randomization algorithms several design strategies involving different random prototypes generation and basis functions approximation are studied we investigate performance aspects of randomized rule base and look at the performance versus the key components of the models such as the number of rules and the use of the randomized algorithms in the development furthermore a comparative study is offered to quantify the efficiency of randomized algorithms experimental studies are reported for a series of publicly available data sets to illustrate the effectiveness of the proposed method and discuss its main features

eigenvalue intervals for higher order problems with singular nonlinearities abstract we study eigenvalue intervals for higher order problems with multipoint boundary conditions we consider the case when the nonlinearity may be singular at t 0 or t 1 our approach is based on topological methods and cover both sublinear and superlinear cases we also study the continuous dependence of solutions on functional parameters

t s fuzzy model identification with a gravitational search based hyperplane clustering algorithm in order to improve the performance of the fuzzy clustering algorithm in fuzzy space partition in the identification of the takagi sugeno t s fuzzy model a hyperplane prototype fuzzy clustering model is proposed to solve the clustering objective function which could not be handled by the gradient method as the traditional clustering method fuzzy c means does a newly developed excellent global search method which is the gravitational search algorithm gsa is employed then the gsa based hyperplane clustering algorithm gshpc is proposed and illuminated gshpc is used to partition the fuzzy space and identify premise parameters of the t s fuzzy model and orthogonal least squares is exploited to identify the consequent parameters comparative experiments are designed to verify the validity of the proposed clustering algorithm and the t s fuzzy model identification method and the results show that the new method is effective in describing a complicated nonlinear system with significantly high accuracies compared with approaches in the literature

approximation with random bases in this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a hilbert space of functions we consider and analyze published procedures both randomized and deterministic for selecting elements from these families that have been shown to ensure the rate of convergence in l2 norm of order o 1 n where n is the number of elements we show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided in the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and or extreme sensitivity of the outcome of the approximation to parameters implications of our analysis for applications of neural networks in modeling and control are illustrated with examples

eigenvalues of a general class of boundary value problem with derivative dependent nonlinearity we consider a general class of boundary value problem bvp comprising the differential equation y m t f t y t y t y t y q t t 0 1 where 1 q m 1 and 0 together with multi point boundary conditions y 0 y 0 y 0 y q 1 0 0 a i y q t j y q 1 t j y m 1 t j 0 j r 0 1 i m q where 0 t 0 t 1 t r 1 particular cases of this general bvp include the well known abel gontscharoff focal m p sturm liouville and complementary lidstone bvps it is noted that bvps with derivative dependent nonlinear terms are less investigated in the literature due to technical difficulty in this paper a new technique is developed to characterize the eigenvalues so that the bvp has a positive solution explicit eigenvalue intervals are also established we include several well known examples in the literature to illustrate the results obtained

linear programming with fuzzy random variable coefficients abstract linear programming with fuzzy random variable coefficients is introduced by discussing a practical engineering problem in order to study the solution of the linear programming with fuzzy random variable coefficients we discuss the simplex algorithm for linear programming with random variable coefficients furthermore the solution and distribution problem of this new fuzzy random programming are studied

enhanced fuzzy system models with improved fuzzy clustering algorithm although traditional fuzzy models have proven to have high capacity of approximating the real world systems they have some challenges such as computational complexity optimization problems subjectivity etc in order to solve some of these problems this paper proposes a new fuzzy system modeling approach based on improved fuzzy functions to model systems with continuous output variable the new modeling approach introduces three features i an improved fuzzy clustering ifc algorithm ii a new structure identification algorithm and iii a nonparametric inference engine the ifc algorithm yields simultaneous estimates of parameters of c regression models together with fuzzy c partitioning of the data to calculate improved membership values with a new membership function the structure identification of the new approach utilizes ifc instead of standard fuzzy c means clustering algorithm to fuzzy partition the data and it uses improved membership values as additional input variables along with the original scalar input variables for two different choices of regression methods least squares estimation or support vector regression to determine ldquofuzzy functionsrdquo for each cluster with novel ifc one could learn the system behavior more accurately compared to other fsm models the nonparametric inference engine is a new approach which uses the alike nearest neighbor method for reasoning empirical comparisons indicate that the proposed approach yields comparable or better accuracy than fuzzy or neuro fuzzy models based on fuzzy rules bases as well as other soft computing methods

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

support vector learning mechanism for fuzzy rule based modeling a new approach this paper describes a fuzzy modeling framework based on support vector machine a rule based framework that explicitly characterizes the representation in fuzzy inference procedure the support vector learning mechanism provides an architecture to extract support vectors for generating fuzzy if then rules from the training data set and a method to describe the fuzzy system in terms of kernel functions thus it has the inherent advantage that the model does not have to determine the number of rules in advance and the overall fuzzy inference system can be represented as series expansion of fuzzy basis functions the performance of the proposed approach is compared to other fuzzy rule based modeling methods using four data sets

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

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

rational hausdorff divisors a new approach to the approximate parametrization of curves in this paper we introduce the notion of rational hausdorff divisor we analyze the dimension and irreducibility of its associated linear system of curves and we prove that all irreducible real curves belonging to the linear system are rational and are at finite hausdorff distance among them as a consequence we provide a projective linear subspace where all irreducible elements are solutions of the approximate parametrization problem for a given algebraic plane curve furthermore we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization therefore we present a generic answer to the approximate parametrization problem in addition we introduce the notion of hausdorff curve and we prove that every irreducible hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve