on the monotonicity of hierarchical sum product fuzzy systems abstract motivated by the authors previous work on the control of queueing systems the assessment of sustainable development and the measurement of material recyclability this paper provides sufficient conditions on the parameters of hierarchical fuzzy systems under which the output of the system is monotonic with respect to its inputs this property could be useful in designing multistage fuzzy inference systems and fuzzy controllers

quasioptimal uniformly convergent finite element methods for the elliptic boundary layer problem we consider the bilinear finite element method for the singularly perturbed elliptic boundary value problem in two dimensions we construct our method on a shishkin mesh through the boundary layer correction technique and elaborate barrier functions the method is shown to be convergent uniformly in the perturbation parameter of order n 2 ln2 n in the l2 norm numerical results confirm our theoretical analysis

fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

on stability of fuzzy systems expressed by fuzzy rules with singleton consequents addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents it first presents two canonical forms of a fuzzy system a parametric expression and a state space expression a fuzzy system with singleton consequents is found to be a piecewise polytopic affine system then the paper gives stability theorems for a discrete time system as well as for a continuous time system it also gives a brief survey on the stability issues of fuzzy control systems

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary korteweg de vries hierarchy the isochronous variant is exhibited of the dynamical system corresponding to the mth ordinary differential equation of the stationary korteweg de vries kdv hierarchy new diophantine relations are thereby obtained in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbitrary degree having integer zeros generalizations of these formulas to relations among rational functions are also obtained the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

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

fuzzy modeling of high dimensional systems complexity reduction and interpretability improvement fuzzy modeling of high dimensional systems is a challenging topic this paper proposes an effective approach to data based fuzzy modeling of high dimensional systems an initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema redundant rules are removed based on a fuzzy similarity measure then the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method during optimization rules that have a very low firing strength are deleted finally interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization the resulting fuzzy system generated by this method has the following distinct features 1 the fuzzy system is quite simplified 2 the fuzzy system is interpretable and 3 the dependencies between the inputs and the output are clearly shown this method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data

a dynamically constrained multiobjective genetic fuzzy system for regression problems in this paper a multiobjective genetic fuzzy system gfs to learn the granularities of fuzzy partitions tuning the membership functions mfs and learning the fuzzy rules is presented it uses dynamic constraints which enable three parameter mf tuning to improve the accuracy while guaranteeing the transparency of fuzzy partitions the fuzzy models fms are initialized by a method that combines the benefits of wang mendel wm and decision tree algorithms thus the initial fms have less rules rule conditions and input variables than if wm initialization were to be used moreover the fuzzy partitions of initial fms are always transparent our approach is tested against recent multiobjective and monoobjective gfss on six benchmark problems it is concluded that the accuracy and interpretability of our fms are always comparable or better than those in the comparative studies furthermore on some benchmark problems our approach clearly outperforms some comparative approaches suitability of our approach for higher dimensional problems is shown by studying three benchmark problems that have up to 21 input variables

ten ways to generate the il in and related schemes abstract for one dimensional singularly perturbed boundary value problems there exist a remarkable variety of possibilities to generate uniformly convergent schemes we describe the basic ideas of ten different approaches and the advantages and disadvantages of several methods

necessary convergence conditions for upwind schemes in the two dimensional case this paper considers nine point difference schemes for a two dimensional boundary value singular perturbation problem without turning points and parabolic boundary layers necessary conditions are given for the uniform convergence in the sense of the maximum norm of a scheme using these conditions several widely used schemes are analysed it is shown that some common schemes are not uniformly convergent in and that in some cases we are able to compute uniquely free parameters in the scheme some remarks on the treatment of a problem with a parabolic boundary layer are given

integration of an index to preserve the semantic interpretability in the multiobjective evolutionary rule selection and tuning of linguistic fuzzy systems in this paper we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions mfs is performed the proposed index is the aggregation of three metrics that preserve the original meanings of the mfs as much as possible while a tuning of their definition parameters is performed additionally rule selection mechanisms can be used to reduce the model complexity which involves another important interpretability aspect to this end we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy rule based systems with three objectives accuracy semantic interpretability maximization and complexity minimization we tested our approach on nine real world regression datasets in order to analyze the interaction between the fuzzy rule selection approach and the tuning approach these are also individually proved in a multiobjective framework and compared with their respective single objective counterparts we compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective based approaches results confirm the effectiveness of our approach and a wide range of solutions is obtained which are not only more interpretable but are also more accurate

radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems this paper presents a means to approximate the dynamic and static equations of stochastic nonlinear systems and to estimate state variables based on radial basis function neural network rbfnn after a nonparametric approximate model of the system is constructed from a priori experiments or simulations a suboptimal filter is designed based on the upper bound error in approximating the original unknown plant with nonlinear state and output equations the procedures for both training and state estimation are described along with discussions on approximation error nonlinear systems with linear output equations are considered as a special case of the general formulation finally applications of the proposed rbfnn to the state estimation of highly nonlinear systems are presented to demonstrate the performance and effectiveness of the method

ga based fuzzy neural controller design for municipal incinerators the successful operation of mass burn municipal incinerators for solid waste management involves many uncertain factors not only the physical composition and chemical property of waste streams but also the complexity of combustion mechanism would significantly influence the performance of waste incineration due to the rising concerns of dioxin furan emissions from those incineration facilities in the metropolitan region the applications of fuzzy control technologies for reducing the operational risks have gradually received wide attention in the scientific community recent advances of intelligent combustion control icc technologies indicate that hybrid fuzzy control schemes integrating some ideas and paradigms existing in different soft computing approaches may provide more reliable control performance of combustion process the proposed integrated methodology using genetic algorithms and neural networks as tools to aid in fuzzy logic control therefore employs a three stage analysis it applies three soft computing approaches simultaneously for generating a representative state function searching for a set of multi objective control strategies and auto tuning the fuzzy control rule base for use in the case study the findings from this research clearly indicate that the control performance at least in three types of municipal incinerators i e patented stokers are takuma volund and martin is greatly enhanced via the use of icc technologies by using this integrated methodology the case study not only verifies the applicability and suitability for controlling municipal incinerators but also presents application potentials for controlling many other types of industrial incinerators such as modular rotary kiln and fluidized bed incinerators

fuzzy association rules general model and applications the theory of fuzzy sets has been recognized as a suitable tool to model several kinds of patterns that can hold in data in this paper we are concerned with the development of a general model to discover association rules among items in a crisp set of fuzzy transactions this general model can be particularized in several ways each particular instance corresponds to a certain kind of pattern and or repository of data we describe some applications of this scheme paying special attention to the discovery of fuzzy association rules in relational databases

a uniformly convergent galerkin method on a shishkin mesh for a convection diffusion problem abstract a galerkin finite element method that uses piecewise bilinears on a simple piecewise equidistant mesh is applied to a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of ordern 1 ln nin a global energy norm and of ordern 1 2ln3 2 npointwise near the outflow boundary where the total number of mesh points iso n2

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

only smooth rule bases can generate monotone mamdani assilian models under center of gravity defuzzification at first sight it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model however this is not the case in this study we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone input output behavior in the mamdani assilian models considered in this study the linguistic values of the input variables as well as the output variable are described by trapezoidal membership functions that form a fuzzy partition the rule base is monotone and the crisp output is obtained by the center of gravity cog defuzzification method it is verified that for each of the three basic t norms i e the minimum tm the product tp and the lukasiewicz t norm tl a monotone input output behavior is obtained for any monotone rule base or at least for any monotone smooth rule base the outcome of this study is a guideline for designers of monotone linguistic fuzzy models for the t norms tm and tl models with a single input variable show a monotone input output behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length the latter restriction can easily be bypassed by an auxiliary interpolation procedure for the t norm tp models with a single input variable show a monotone input output behavior for any monotone rule base and any fuzzy output partition when designing a monotone model with more than one input variable one should opt for the t norm tp and use a monotone smooth rule base it is shown that monotonicity of models with two input variables that apply tp is guaranteed for any monotone smooth rule base and any fuzzy output partition finally it is proved that a monotone input output behavior is always obtained for models with three input variables that apply tp and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape

necessary l2 uniform convergence conditions for difference schemes for two dimensional convection diffusion problems we derive necessary conditions for the uniform convergence with respect to the perturbation parameter in l2 of schemes for singularly perturbed convection diffusion problems applying these conditions we show that the convergence order of the streamline diffusion finite element method with piecewise linear trial functions cannot be better than 12 uniformly in the singular perturbation parameter

a finite element method with a large mesh width for a stiff two point boundary value problem abstract we describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two point boundary value problem on 1 1 u pu f u 1 u 1 0 0 d dx on a quasi uniform mesh we construct exponentially fitted trial spaces which consist of piece wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint we discretise the galerkin form for the boundary problem using such exponentially fitted trial spaces we derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh points the error depending on the mesh width and the degree of the piece wise polynomials

automatic tuning of a fuzzy visual system using evolutionary algorithms single objective versus multiobjective approaches one of the main advantages of fuzzy systems is their ability to design comprehensible models of real world systems thanks to the use of a fuzzy rule structure easily interpretable by human beings this is especially useful for the design of fuzzy logic controllers where the knowledge base can be extracted from expert knowledge even more the availability of a readable structure allows the human expert to customize the fuzzy controller to different environments by manually tuning its components nevertheless this tuning task is usually a time consuming procedure when done manually especially when several measures are considered to evaluate the controller performance and thus the interest in the design of automatic tuning procedures for fuzzy systems has increased along the last few years in this paper we tackle the tuning of the fuzzy membership functions of a fuzzy visual system for autonomous robots this fuzzy visual system is based on a hierarchical structure of three different fuzzy classifiers whose combined action allows the robot to detect the presence of doors in the images captured by its camera although the global knowledge represented in the fuzzy system knowledge base makes it perform properly in the door detection task its adaptation to the specific conditions of the environment where the robot is operating can significantly improve the classification accuracy however the tuning procedure is complex as two different performance indexes are involved in the optimization process true positive and false positive detections thus becoming a multiobjective problem hence in order to automatically put the fuzzy system tuning into effect different single and multiobjective evolutionary algorithms are considered to optimize the two criteria and their behavior in problem solving is compared

a proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings however since the accuracy and the interpretability of the obtained model are contradictory properties the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled to solve this problem one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level in this paper a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability to do so we consider a new rule representation scheme base on the linguistic 2 tuples representation model which allows the lateral variation of the involved labels furthermore the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper presenting results on different real applications the obtained results show the good performance of the proposed approach in high dimensional problems and its ability to cooperate with methods to remove unnecessary rules

multi objective hierarchical genetic algorithm for interpretable fuzzy rule based knowledge extraction a new scheme based on multi objective hierarchical genetic algorithm mohga is proposed to extract interpretable rule based knowledge from data the approach is derived from the use of multiple objective genetic algorithm moga where the genes of the chromosome are arranged into control genes and parameter genes these genes are in a hierarchical form so that the control genes can manipulate the parameter genes in a more effective manner the effectiveness of this chromosome formulation enables the fuzzy sets and rules to be optimally reduced some important concepts about the interpretability are introduced and the fitness function in the moga will consider both the accuracy and interpretability of the fuzzy model in order to remove the redundancy of the rule base proactively we further apply an interpretability driven simplification method to newborn individuals in our approach we first apply the fuzzy clustering to generate an initial rule based model then the multi objective hierarchical genetic algorithm and the recursive least square method are used to obtain the optimized fuzzy models the accuracy and the interpretability of fuzzy models derived by this approach are studied and presented in this paper we compare our work with other methods reported in the literature on four examples a synthetic nonlinear dynamic system a nonlinear static system the lorenz system and the mackey glass system simulation results show that the proposed approach is effective and practical in knowledge extraction

an analysis of smoothing effects of upwinding strategies for the convection diffusion equation using a technique for constructing analytic expressions for discrete solutions to the convection diffusion equation we examine and characterize the effects of upwinding strategies on solution quality in particular for grid aligned flow and discretization based on bilinear finite elements with streamline upwinding we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when different types of boundary layers are present this analysis provides a basis for choosing a streamline upwinding parameter which also gives accurate solutions for problems with non grid aligned and variable speed flows in addition we show that the same analytic techniques provide insight into other discretizations such as a finite difference method that incorporates streamline diffusion and the isotropic artificial diffusion method

integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary burgers hierarchy an isochronous system is introduced by modifying the nth ode of the stationary burgers hierarchy and then by investigating its behaviour near its equilibria neat diophantine relations are identified involving well known polynomials of arbitrary degree having integer zeros or equivalently matrices the determinants of which yield such polynomials the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

fuzzy systems as universal approximators an additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy an additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input output state space and averaging patches that overlap the fuzzy system computes a conditional expectation e y x if we view the fuzzy sets as random sets each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state space geometry neural or statistical clustering systems can approximate the unknown fuzzy patches from training data these adaptive fuzzy systems approximate a function at two levels at the local level the neural system approximates and tunes the fuzzy rules at the global level the rules or patches approximate the function

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

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

monotonic smooth takagi sugeno fuzzy systems with fuzzy sets with compact support in this paper sufficient conditions on monotonicity of takagi sugeno t s fuzzy systems with linear consequent parts with smooth membership functions with compact support i e equal to zero outside a compact set are derived the presented conditions considerably reduce conservatism of those known for t s fuzzy systems utilizing membership functions with the support that is not compact besides two illustrative examples will demonstrate that the monotonicity requirement imposed on fuzzy systems may significantly improve their extrapolation capability

parametric conditions for a monotone tsk fuzzy inference system to be an n ary aggregation function despite the popularity and practical importance of the fuzzy inference system fis the use of an fis model as an n ary aggregation function which is characterized by both the monotonicity and boundary properties is yet to be established this is because research on ensuring that fis models satisfy the monotonicity property i e monotone fis is relatively new not to mention the additional requirement of satisfying the boundary property the aim of this paper therefore is to establish the parametric conditions for the takagi sugeno kang tsk fis model to operate as an n ary aggregation function hereafter denoted as n tsk fis via the specifications of fuzzy membership functions fmfs and fuzzy rules an absorption property with fuzzy rules interpretation is outlined and the use of n tsk fis as a uni norm is explained exploiting the established parametric conditions a framework for which an n tsk fis model can be constructed from data samples is formulated and analyzed along with a number of remarks synthetic data sets and a benchmark example on education assessment are presented and discussed to be best of the authors knowledge this study serves as the first use of the tsk fis model as an n ary aggregation function

a review of the application of multiobjective evolutionary fuzzy systems current status and further directions over the past few decades fuzzy systems have been widely used in several application fields thanks to their ability to model complex systems the design of fuzzy systems has been successfully performed by applying evolutionary and in particular genetic algorithms and recently this approach has been extended by using multiobjective evolutionary algorithms which can consider multiple conflicting objectives instead of a single one the hybridization between multiobjective evolutionary algorithms and fuzzy systems is currently known as multiobjective evolutionary fuzzy systems this paper presents an overview of multiobjective evolutionary fuzzy systems describing the main contributions on this field and providing a two level taxonomy of the existing proposals in order to outline a well established framework that could help researchers who work on significant further developments finally some considerations of recent trends and potential research directions are presented

hierarchical genetic fuzzy systems abstract this paper introduces a hierarchical evolutionary approach to optimize the parameters of takagi sugeno ts fuzzy systems the approach includes a least squares method to determine the parameters of nonlinear consequents a pruning procedure is developed to avoid redundancy in each rule consequent and to achieve proper representation flexibility the performance of the hierarchical evolutionary approach is evaluated using function approximation and classification problems they demonstrate that the evolutionary algorithm working together with optimization and pruning procedures provides structurally simple fuzzy systems whose performance seems to be better than the ones produced by alternative approaches

an upwind finite element method via numerical intergration upwind finite element schemes remove spurious oscillations that occur in the solution of diffusion convection equations up to now these schemes lose part of their accuracy when the peclet is large as an improvement it is proposed to move the integration nodes along the streamlines before evaluating the elementary convection matrices the displacements of the nodes along the streamlines which are one dimensional manifolds are calculated analogously to well known one dimensional formulae the last section of this paper illustrates this new method with the help of four examples which show its validity

diophantine properties of the zeros of certain laguerre and para jacobi polynomials matrices and of arbitrary rank n given by simple expressions in terms of the n zeros of certain laguerre or para jacobi polynomials of degree n feature a diophantine property in the laguerre case this property states that the 2n zeros of the polynomial are all integers indeed we conjecture that the results in the para jacobi case are somewhat analogous they refer to the zeros of the general solution of the ode generally characterizing jacobi polynomials in the special case in which this solution is a para jacobi polynomial featuring an additional arbitrary parameter

distinguishability quantification of fuzzy sets distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models distinguishability has been mathematically defined through different measures which are addressed in this paper special emphasis is given to similarity which exhibits sound theoretical properties but its calculation is usually computationally intensive and possibility whose calculation can be very efficient but it does not exhibit the same properties of similarity it is shown that under mild conditions usually met in interpretable fuzzy modeling possibility can be used as a valid measure for assessing distinguishability thus overcoming the computational inefficiencies of similarity measures moreover procedures that minimize possibility also minimize similarity and consequently improve distinguishability in this sense the use of possibility is fully justified in interpretable fuzzy modeling

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

nmeef sd non dominated multiobjective evolutionary algorithm for extracting fuzzy rules in subgroup discovery a non dominated multiobjective evolutionary algorithm for extracting fuzzy rules in subgroup discovery nmeef sd is described and analyzed in this paper this algorithm which is based on the hybridization between fuzzy logic and genetic algorithms deals with subgroup discovery problems in order to extract novel and interpretable fuzzy rules of interest and the evolutionary fuzzy system nmeef sd is based on the well known non dominated sorting genetic algorithm ii nsga ii model but is oriented toward the subgroup discovery task using specific operators to promote the extraction of interpretable and high quality subgroup discovery rules the proposal includes different mechanisms to improve diversity in the population and permits the use of different combinations of quality measures in the evolutionary process an elaborate experimental study which was reinforced by the use of nonparametric tests was performed to verify the validity of the proposal and the proposal was compared with other subgroup discovery methods the results show that nmeef sd obtains the best results among several algorithms studied

toward necessity of parametric conditions for monotonic fuzzy systems input output monotonicity is an important constraint found in many application domains a monotonic fuzzy system mfs is defined as a takagi sugeno kang tsk system whose output is monotonically increasing or decreasing with respect to one or more inputs this paper reviews the authors previous work which derived the parametric conditions for the mfs and discusses the rationale lying behind the conditions an mfs is developed by creating a monotonic rule base while preserving the relative monotonicity among the membership functions corresponding to the fuzzy rules this paper also proves that the parametric conditions are necessary and sufficient to build a single input zeroth order mfs with two rules only the sufficiency of the conditions holds for a multi input first order or higher order tsk fuzzy system with three or more rules

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

three objective genetics based machine learning for linguisitc rule extraction abstract this paper shows how a small number of linguistically interpretable fuzzy rules can be extracted from numerical data for high dimensional pattern classification problems one difficulty in the handling of high dimensional problems by fuzzy rule based systems is the exponential increase in the number of fuzzy rules with the number of input variables another difficulty is the deterioration in the comprehensibility of fuzzy rules when they involve many antecedent conditions our task is to design comprehensible fuzzy rule based systems with high classification ability this task is formulated as a combinatorial optimization problem with three objectives to maximize the number of correctly classified training patterns to minimize the number of fuzzy rules and to minimize the total number of antecedent conditions we show two genetic algorithm based approaches one is rule selection where a small number of linguistically interpretable fuzzy rules are selected from a large number of prespecified candidate rules the other is fuzzy genetics based machine learning where rule sets are evolved by genetic operations these two approaches search for non dominated rule sets with respect to the three objectives

on the monotonicity of fuzzy inference methods related to t s inference method yubazaki et al have proposed a single input rule modules connected type fuzzy inference method sirms method whose final output is obtained by combining the products of the importance degrees and the inference results from single input fuzzy rule modules moreover seki et al have proposed a functional type sirms method whose consequent parts are generalized to functions from real numbers it is expected that inference results from the functional type sirms method are monotone if the antecedent parts and the consequent parts of fuzzy rules in the functional type sirms rule modules are monotone however this paper points out that even if consequent parts in the functional type sirms rule modules are monotone the inference results are not necessarily monotone when the antecedent parts are noncomparable fuzzy sets and it clarifies the conditions for the monotonicity of inference results from the functional type sirms method moreover for the takagi sugeno t s inference method the monotonicity condition is clarified in the case of two inputs by using the equivalence relation of fuzzy inference

the wm method completed a flexible fuzzy system approach to data mining in this paper the so called wang mendel wm method for generating fuzzy rules from data is enhanced to make it a comprehensive and flexible fuzzy system approach to data description and prediction in the description part the core ideas of the wm method are used to develop three methods to extract fuzzy if then rules from data the first method shows how to extract rules for the user specified cases the second method generates all the rules that can be generated directly from the data and the third method extrapolates the rules generated by the second method over the entire domain of interest in the prediction part two fuzzy predictive models are constructed based on the fuzzy if then rules extracted by the methods of the description part the first model gives a continuous output and is suitable for predicting continuous variables and the second model gives a piecewise constant output and is suitable for predicting categorical variables we show that by comparing the prediction accuracy of the fuzzy predictive models with different numbers of fuzzy sets covering the input variables we can rank the importance of the input variables we also propose an algorithm to optimize the fuzzy predictive models and show how to use the models to solve pattern recognition problems throughout this paper we use a set of real data from a steel rolling plant to demonstrate the ideas and test the models

modified streamline diffusion schemes for convection diffusion problems abstract we consider the design of robust and accurate finite element approximation methods for solving convection diffusion problems we develop some two parameter streamline diffusion schemes with piecewise bilinear or linear trial functions and show that these schemes satisfy the necessary conditions for l 2 uniform convergence of order greater than 1 2 introduced by stynes and tobiska m stynes and l tobiska necessary l 2 uniform convergence conditions for difference schemes for two dimensional convection diffusion problems comput math applic 29 1995 45 53 for smooth problems the schemes satisfy error bounds of the form o h u 2 in an energy norm in addition extensive numerical experiments show that they effectively reproduce boundary layers and internal layers caused by discontinuities on relatively coarse grids without any requirements on alignment of flow and grid

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100