on evolutionary designing and learning of flexible neuro fuzzy structures for nonlinear classification abstract in the paper the evolutionary strategy is applied for learning flexible neuro fuzzy systems in the process of evolution we determine fuzzy inference mamdani type or logical type described by an s implication specific fuzzy implication if the logical type system is found in the process of evolution or specific t norm connecting antecedents and consequences if the mamdani type system is found in the process of evolution specific t norm for aggregation of antecedents in each rule specific triangular norm describing aggregation operator shapes and parameters of fuzzy membership functions weights describing importance of antecedents of rules and weights describing importance of rules parameters of adjustable triangular norms parameters of soft triangular norms it should be noted that the crossover and mutation operators are chosen in a self adaptive way the method is tested using well known classification benchmarks

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

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

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

a new fuzzy set merging technique using inclusion based fuzzy clustering this paper proposes a new method of merging parameterized fuzzy sets based on clustering in the parameters space taking into account the degree of inclusion of each fuzzy set in the cluster prototypes the merger method is applied to fuzzy rule base simplification by automatically replacing the fuzzy sets corresponding to a given cluster with that pertaining to cluster prototype the feasibility and the performance of the proposed method are studied using an application in mobile robot navigation the results indicate that the proposed merging and rule base simplification approach leads to good navigation performance in the application considered and to fuzzy models that are interpretable by experts in this paper we concentrate mainly on fuzzy systems with gaussian membership functions but the general approach can also be applied to other parameterized fuzzy sets

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

the tase nf model for function approximation problems optimization of the local sub models when using neuro fuzzy systems to model input output data can be spoiled during the global optimization process therefore special care has to be taken to avoid this loss some approaches involving grid input space partitioning that optimize local sub models whilst still obtaining a good global system performance have appeared however this is a more complex task for radial basis function networks scatter partitioning fuzzy systems and neuro fuzzy models in general this work presents a modified neuro fuzzy model known as the tase nf model which is a special case of a scatter partitioned takagi sugeno kang fuzzy system or normalized radial basis function neural network the tase nf model maintains the optimization properties of the local sub models fuzzy rules or rbf nodes when the model is globally optimized thanks to the modified calculation of the final normalized activation of the rules which provides an appropriate input space partitioning to achieve these objectives to show the characteristics of the proposed model a learning methodology for this model which consists of a clustering algorithm especially suited to function approximation problems a local search technique and a membership function merging approach with the objective of improving the transparency i e ease of interpretability of the extracted fuzzy rule set is also proposed

theory and application of a novel fuzzy pid controller using a simplifier takagi sugeno rule scheme we first present a novel general purpose nonlinear pid controller realized via fuzzy pid control that uses our newly introduced simplified takagi sugeno ts rule scheme analytical structure of the fuzzy pid controller is derived and its structure is analyzed in relation to the linear pid controller the unique features of the fuzzy controller are as follows first the proportional integral and derivative gains constantly vary with the output of the system under control the gain variation leads to a shorter rise time a less overshoot and a smaller settling time as compared to a comparable linear pid controller second the characteristics of the gain variation are determined by the fuzzy rules and can intuitively be designed we have also investigated the local stability of the fuzzy pid control systems as an application demonstration we have developed a fuzzy pid control system to regulate in computer simulation blood pressure in postsurgical patients we have chosen this particular control problem because the studies in the literature have established that in order to achieve satisfactory control results using a nonlinear controller with variable gains is necessary the simulation results show that the fuzzy pid controller significantly outperforms its linear counterpart and is safer and more robust over a wide range of patient condition 2000 published by elsevier

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

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

design of interpretable fuzzy rule based classifiers using spectral analysis with structure and parameters optimization this paper presents a design method for fuzzy rule based systems that performs data modeling consistently according to the symbolic relations expressed by the rules the focus of the model is the interpretability of the rules and the model s accuracy such that it can be used as tool for data understanding the number of rules is defined by the eigenstructure analysis of the similarity matrix which is computed from data the rule induction algorithm runs a clustering algorithm on the dataset and associates one rule to each cluster each rule is selected among all possible combinations of one dimensional fuzzy sets as the one nearest to a cluster s center the rules are weighted in order to improve the classifier performance and the weights are computed by a bounded quadratic optimization problem the model complexity is minimized in a structure selection search performed by a genetic algorithm that selects simultaneously the most representative subset of variables and also the number of fuzzy sets in the fuzzy partition of the selected variables the resulting model is evaluated on a set of benchmark datasets for classification problems the results show that the proposed approach produces accurate and yet compact fuzzy classifiers the resulting model is also evaluated from an interpretability point of view showing how the rule weights provide additional information to help data understanding and model exploitation

self adaptive neuro fuzzy inference systems for classification applications this paper presents a self adaptive neuro fuzzy inference system sanfis that is capable of self adapting and self organizing its internal structure to acquire a parsimonious rule base for interpreting the embedded knowledge of a system from the given training data set a connectionist topology of fuzzy basis functions with their universal approximation capability is served as a fundamental sanfis architecture that provides an elasticity to be extended to all existing fuzzy models whose consequent could be fuzzy term sets fuzzy singletons or functions of linear combination of input variables without a priori knowledge of the distribution of the training data set a novel mapping constrained agglomerative clustering algorithm is devised to reveal the true cluster configuration in a single pass for an initial sanfis construction estimating the location and variance of each cluster subsequently a fast recursive linear nonlinear least squares algorithm is performed to further accelerate the learning convergence and improve the system performance good generalization capability fast learning convergence and compact comprehensible knowledge representation summarize the strength of sanfis computer simulations for the iris wisconsin breast cancer and wine classifications show that sanfis achieves significant improvements in terms of learning convergence higher accuracy in recognition and a parsimonious architecture

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

checking the coherence and redundancy of fuzzy knowledge bases checking the coherence of a set of rules is an important step in knowledge base validation coherence is also needed in the field of fuzzy systems indeed rules are often used regardless of their semantics and it sometimes leads to sets of rules that make no sense avoiding redundancy is also of interest in real time systems for which the inference engine is time consuming a knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base we more particularly consider knowledge bases composed of parallel fuzzy rules then coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted i e any value for these variables is possible or at least not more restrictive than integrity constraints fuzzy rule representations can be implication based or conjunction based we show that only implication based models may lead to coherence problems however unlike conjunction based models they allow to design coherence checking processes some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules the problem of redundancy which is also of interest for fuzzy knowledge bases validation is addressed for these two kinds of rules

on the optimization of fuzzy decision trees the induction of fuzzy decision trees is an important way of acquiring imprecise knowledge automatically fuzzy id3 and its variants are popular and efficient methods of making fuzzy decision trees from a group of training examples this paper points out the inherent defect of the likes of fuzzy id3 presents two optimization principles of fuzzy decision trees proves that the algorithm complexity of constructing a kind of minimum fuzzy decision tree is np hard and gives a new algorithm which is applied to three practical problems the experimental results show that with regard to the size of trees and the classification accuracy for unknown cases the new algorithm is superior to the likes of fuzzy id3

extracting fuzzy rules for system modeling using a hybrid of genetic algorithms and kalman filter abstract this paper proposes a hybrid algorithm for extracting important fuzzy rules from a given rule base to construct a parsimonious fuzzy model with a high generalization ability this algorithm combines the advantages of genetic algorithms strong search capacity and kalman filter s fast convergence merit each random combination of the rules in the rule base is coded into a binary string and treated as a chromosome in genetic algorithms the binary string indicates the structure of a fuzzy model the parameters of the model are then estimated using the kalman filter in order to achieve a trade off between the accuracy and the complexity of a fuzzy model the schwarz rissanen criterion is used as an evaluation function in the hybrid algorithm the practical applicability of the proposed algorithm is examined by computer simulations on a human operator modeling problem and a nonlinear system modeling problem

compact and transparent fuzzy models and classifiers through iterative complexity reduction in our previous work 2000 we showed that genetic algorithms gas provide a powerful tool to increase the accuracy of fuzzy models for both systems modeling and classification in addition to these results we explore the ga to find redundancy in the fuzzy model for the purpose of model reduction an aggregated similarity measure is applied to search for redundancy in the rule base description as a result we propose an iterative fuzzy identification technique starting with data based fuzzy clustering with an overestimated number of local models the ga is then applied to find redundancy among the local models with a criterion based on maximal accuracy and maximal set similarity after the reduction steps the ga is applied with another criterion searching for minimal set similarity and maximal accuracy this results in an automatic identification scheme with fuzzy clustering rule base simplification and constrained genetic optimization with low human intervention the proposed modeling approach is then demonstrated for a system identification and a classification problem

ga fuzzy modeling and classification complexity and performance the use of genetic algorithms gas and other evolutionary optimization methods to design fuzzy rules for systems modeling and data classification have received much attention in recent literature authors have focused on various aspects of these randomized techniques and a whole scale of algorithms have been proposed we comment on some recent work and describe a new and efficient two step approach that leads to good results for function approximation dynamic systems modeling and data classification problems first fuzzy clustering is applied to obtain a compact initial rule based model then this model is optimized by a real coded ga subjected to constraints that maintain the semantic properties of the rules we consider four examples from the literature a synthetic nonlinear dynamic systems model the iris data classification problem the wine data classification problem and the dynamic modeling of a diesel engine turbocharger the obtained results are compared to other recently proposed methods

a tabu search algorithm for computing an operational timetable abstract the many constraints of different types that must be taken into account and the volume of data make the timetabling problem very hard to solve various specific requirements are considered in this paper after having described precisely the problem to be solved we present a general technique based on tabu search for finding an acceptable timetable the resulting computerized algorithm is used for constructing different real course schedules the results we obtain are very satisfactory

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

fuzzy modelling and identification with genetic algorithm based learning abstract a ga based learning algorithm is proposed in this paper for the identification of tsk models the algorithm consists of four blocks partition block ga block tuning block and termination block the partition block is to determine an estimated partition of input variables the ga block is to optimise the structure of a tsk model the tuning block is to fine tune the parameters of the tsk model using the gradient descent based approach and the termination block checks that the resultant tsk model is satisfactory the proposed gabl algorithm has the advantage of simplicity flexibility high accuracy and automation the presented numerical examples indicate that the gabl algorithm is effective in constructing a good tsk model for complex nonlinear systems

analysis and design of hierarchical fuzzy systems in this letter the hierarchical fuzzy systems are analyzed and designed in the analysis part we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs in the design part we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input output pairs the algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy

gp coach genetic programming based learning of compact and accurate fuzzy rule based classification systems for high dimensional problems in this paper we propose gp coach a genetic programming based method for the learning of compact and accurate fuzzy rule based classification systems for high dimensional problems gp coach learns disjunctive normal form rules generated by means of a context free grammar coded as one rule per tree the population constitutes the rule base so it is a genetic cooperative competitive learning approach gp coach uses a token competition mechanism to maintain the diversity of the population and this obliges the rules to compete and cooperate among themselves and allows the obtaining of a compact set of fuzzy rules the results obtained have been validated by the use of non parametric statistical tests showing a good performance in terms of accuracy and interpretability

adaptive tabu search for course timetabling this paper presents an adaptive tabu search algorithm denoted by ats for solving a problem of curriculum based course timetabling the proposed algorithm follows a general framework composed of three phases initialization intensification and diversification the initialization phase constructs a feasible initial timetable using a fast greedy heuristic then an adaptively combined intensification and diversification phase is used to reduce the number of soft constraint violations while maintaining the satisfaction of hard constraints the proposed ats algorithm integrates several distinguished features such as an original double kempe chains neighborhood structure a penalty guided perturbation operator and an adaptive search mechanism computational results show the high effectiveness of the proposed ats algorithm compared with five reference algorithms as well as the current best known results this paper also shows an analysis to explain which are the essential ingredients of the ats algorithm

cooperative evolutionary learning of linguistic fuzzy rules and parametric aggregation connectors for mamdani fuzzy systems there are two tasks in the design of linguistic fuzzy models for a concrete application the derivation of the linguistic rule base and the setup of the inference system and the defuzzification method traditionally the derivation of the linguistic rule base has been considered the most important task but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior in this paper we take in consideration this idea we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step the aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors improving the accuracy of the linguistic mamdani fuzzy systems our proposal has shown good results solving three different applications we introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study we must remark that we present an experimental study with a double intention a to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors and b to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules changes on labels that produce a better behavior of the linguistic model than the classic ones

school timetabling a case in large binary integer linear programming a timetabling problem is formulated as a large integer linear programming problem in 0 1 variables a solution method based on lagrangean relaxation coupled with subgradient optimization is presented the solution method also incorporates a branch and bound procedure which takes advantage of special ordered sets of variables we present computational results for a large timetabling problem involving 900 subjects in a one year graduate program

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

self tuning fuzzy modeling with adaptive membership function rules and hierarchical structure based on genetic algorithm abstract recently fuzzy systems have been used in many fields and places in order to apply the fuzzy system to the various fields the tuning and optimizing method of the fuzzy system is the key issue some self tuning methods have been proposed so far however these conventional self tuning methods do not have sufficient capability of learning in this paper we propose a new supervised self tuning fuzzy modeling which consist of some membership function expressed by the radial basis function with insensitive region learning is carried out by the genetic algorithms the descent method is also utilized for tuning the shapes of the membership function and consequent parts the effectiveness of the proposed methods is shown by some numerical examples

approximation capabilities of hierarchical fuzzy systems derived from practical application in location analysis and pricing and based on the approach of hierarchical structure analysis of continuous functions this paper investigates the approximation capabilities of hierarchical fuzzy systems by first introducing the concept of the natural hierarchical structure it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters then based on kolmogorov s theorem it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property further the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed

optifel a convergent heterogeneous particle swarm optimization algorithm for takagi sugeno fuzzy modeling data driven design of accurate and reliable takagi sugeno t s fuzzy systems has attracted a lot of attention where the model structures and parameters are important and often solved in an optimization framework the particle swarm optimization pso algorithm is widely applied in the field however the classical pso suffers from premature convergence and it is trapped easily into local optima which will significantly affect the model accuracy to overcome these drawbacks we have developed a new t s fuzzy system parameters searching strategy called optifel with a heterogeneous multiswarm pso mspso to enhance the searching performance mspso groups the whole population into multiple cooperative subswarms which perform different search behaviors for the potential solutions we have found that the multiple subswarms strategy proposed in this paper is greatly helpful for finding the optimal parameters suitable for the subspaces of the t s fuzzy model our theoretical proof has also demonstrated that the cooperation among the subswarms can maintain a balance between exploration and exploitation to ensure the particles converge to stable points experimental results show that mspso performs significantly better than traditional pso algorithms on six benchmark functions with the improved mspso optifel can generate a good fuzzy system model with high accuracy and strong generalization ability

multiobjective programming for type 2 hierarchical fuzzy inference trees this paper proposes a design of hierarchical fuzzy inference tree hfit an hfit produces an optimum tree like structure i e a natural hierarchical structure that accommodates simplicity by combining several low dimensional fuzzy inference systems fiss such a natural hierarchical structure provides a high degree of approximation accuracy the construction of the hfit takes place in two phases first a nondominated sorting based multiobjective genetic programming mogp is applied to obtain a simple tree structure a low complexity model with a high accuracy second the differential evolution algorithm is applied to optimize the obtained tree s parameters in the derived tree each node acquires a different input s combination where the evolutionary process governs the input s combination hence hfit nodes are heterogeneous in nature which leads to a high diversity among the rules generated by the hfit additionally the hfit provides an automatic feature selection because it uses mogp for the tree s structural optimization that accepts inputs only relevant to the knowledge contained in data the hfit was studied in the context of both type 1 and type 2 fiss and its performance was evaluated through six application problems moreover the proposed multiobjective hfit was compared both theoretically and empirically with recently proposed fiss methods from the literature such as mcit2fis tscit2fnn sit2fnn rit2fns wb et2fis mrit2nfs it2fnn svr etc from the obtained results it was found that the hfit provided less complex and highly accurate models compared to the models produced by the most of other methods hence the proposed hfit is an efficient and competitive alternative to the other fiss for function approximation and feature selection

designing a fuzzy model by adaptive macroevolution genetic algorithms abstract in this paper the adaptive macroevolution genetic algorithms are proposed to identify three different types of fuzzy models several newly established techniques such as adaptive choice function and macroevolution are adopted into the simple genetic algorithms to improve the optimization capability the genetic algorithms used here are controlled to retain the best solution in the population until a better one is found in the next generation the procedures of applying the proposed genetic algorithms to optimize the fuzzy models are also investigated we also compare whether the instantaneous or batch model is more suitable to adjust the fuzzy models the refined fuzzy models are then used to remedy the prediction output from a grey system the superiority of the adaptive macroevolution genetic algorithms to the simple ones is discussed and an example is given to verify our viewpoints simulation results from difference equation and prediction filter are presented to illustrate the effectiveness of the proposed genetic based fuzzy grey model

a new credibilistic clustering algorithm abstract this paper focuses on credibilistic clustering approach a data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition there are different models for objective function based fuzzy clustering such as fuzzy c means fcm possibilistic c mean pcm and their combinations credibilistic clustering is a new approach in this field in this paper a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering also in objective function the separation of clusters is considered in addition to the compactness within clusters the steps of clustering are designed based on this approach finally the main issues about model are discussed and the results of computational experiments are presented to show the efficiency of the proposed model

a hybrid coevolutionary algorithm for designing fuzzy classifiers rule learning is one of the most common tasks in knowledge discovery in this paper we investigate the induction of fuzzy classification rules for data mining purposes and propose a hybrid genetic algorithm for learning approximate fuzzy rules a novel niching method is employed to promote coevolution within the population which enables the algorithm to discover multiple rules by means of a coevolutionary scheme in a single run in order to improve the quality of the learned rules a local search method was devised to perform fine tuning on the offspring generated by genetic operators in each generation after the ga terminates a fuzzy classifier is built by extracting a rule set from the final population the proposed algorithm was tested on datasets from the uci repository and the experimental results verify its validity in learning rule sets and comparative advantage over conventional methods

self organized fuzzy system generation from training examples in the synthesis of a fuzzy system two steps are generally employed the identification of a structure and the optimization of the parameters defining it the paper presents a methodology to automatically perform these two steps in conjunction using a three phase approach to construct a fuzzy system from numerical data phase 1 outlines the membership functions and system rules for a specific structure starting from a very simple initial topology phase 2 decides a new and more suitable topology with the information received from the previous step it determines for which variable the number of fuzzy sets used to discretize the domain must be increased and where these new fuzzy sets should be located this in turn decides in a dynamic way in which part of the input space the number of fuzzy rules should be increased phase 3 selects from the different structures obtained to construct a fuzzy system the one providing the best compromise between the accuracy of the approximation and the complexity of the rule set the accuracy and complexity of the fuzzy system derived by the proposed self organized fuzzy rule generation procedure sofrg are studied for the problem of function approximation simulation results are compared with other methodologies such as artificial neural networks neuro fuzzy systems and genetic algorithms

designing and learning of adjustable quasi triangular norms with applications to neuro fuzzy systems we introduce a new class of operators called quasi triangular norms they are denoted by h and parameterized by a parameter spl nu h a sub 1 a sub 2 a sub n spl nu from the construction of function h it follows that it becomes a t norm for spl nu 0 and a dual t conorm for spl nu 1 for spl nu close to 0 function h resembles a t norm and for spl nu close to 1 it resembles a t conorm in the paper we also propose adjustable quasi implications and a new class of neuro fuzzy systems most neuro fuzzy systems proposed in the past decade employ engineering implications defined by a t norm as the minimum or product in our proposition a quasi implication i a b spl nu varies from an engineering implication t a b to corresponding s implication as spl nu goes from 0 to 1 consequently the structure of neuro fuzzy systems presented in this work is determined in the process of learning learning procedures are derived and simulation examples are presented

a self evolving interval type 2 fuzzy neural network with online structure and parameter learning this paper proposes a self evolving interval type 2 fuzzy neural network seit2fnn with online structure and parameter learning the antecedent parts in each fuzzy rule of the seit2fnn are interval type 2 fuzzy sets and the fuzzy rules are of the takagi sugeno kang tsk type the initial rule base in the seit2fnn is empty and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space to avoid generating highly overlapping fuzzy sets in each input variable an efficient fuzzy set reduction method is also proposed this method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated for parameter learning the consequent part parameters are tuned by the rule ordered kalman filter algorithm for high accuracy learning performance detailed learning equations on applying the rule ordered kalman filter algorithm to the seit2fnn consequent part learning with rules being generated online are derived the antecedent part parameters are learned by gradient descent algorithms the seit2fnn is applied to simulations on nonlinear plant modeling adaptive noise cancellation and chaotic signal prediction comparisons with other type 1 and type 2 fuzzy systems in these examples verify the performance of the seit2fnn

context dependent fuzzy systems with application to time series prediction in this paper we introduce an implementation of a fuzzy system whose parameters are mutable according to the context the construction of the system is done via two steps first we build a based type 1 takagi sugeno kang tsk fuzzy system whose membership functions will be later adjusted to the situation by means of contextual transformation to reflect the influence of context in the interpretation of fuzzy sets second an iterative algorithm is performed to identify the transformation matrix which is used to scale the membership functions of the reference based fuzzy sets in each of the contexts the identification of the premise part of the based fuzzy system is performed via a combination of an island model parallel genetic algorithm and a space search memetic algorithm while the identification of the consequent parameters of the system is done via an improved qr householder least squares method the proposed system is evaluated using the well known mackey glass time series prediction benchmark dataset and has shown better accuracy than any other previous works concerning the same problem

an efficient ls svm based method for fuzzy system construction this paper proposes an efficient learning mechanism to build fuzzy rule based systems through the construction of sparse least squares support vector machines ls svms in addition to the significantly reduced computational complexity in model training the resultant ls svm based fuzzy system is sparser while offers satisfactory generalization capability over unseen data it is well known that the ls svms have their computational advantage over conventional svms in the model training process however the model sparseness is lost which is the main drawback of ls svms this is an open problem for the ls svms to tackle the nonsparseness issue a new regression alternative to the lagrangian solution for the ls svm is first presented a novel efficient learning mechanism is then proposed in this paper to extract a sparse set of support vectors for generating fuzzy if then rules this novel mechanism works in a stepwise subset selection manner including a forward expansion phase and a backward exclusion phase in each selection step the implementation of the algorithm is computationally very efficient due to the introduction of a few key techniques to avoid the matrix inverse operations to accelerate the training process the computational efficiency is also confirmed by detailed computational complexity analysis as a result the proposed approach is not only able to achieve the sparseness of the resultant ls svm based fuzzy systems but significantly reduces the amount of computational effort in model training as well three experimental examples are presented to demonstrate the effectiveness and efficiency of the proposed learning mechanism and the sparseness of the obtained ls svm based fuzzy systems in comparison with other svm based learning techniques

extraction of fuzzy rules from support vector machines the relationship between support vector machines svms and takagi sugeno kang tsk fuzzy systems is shown an exact representation of svms as tsk fuzzy systems is given for every used kernel function restricted methods to extract rules from svms have been previously published their limitations are surpassed with the presented extraction method the behavior of svms is explained by means of fuzzy logic and the interpretability of the system is improved by introducing the l fuzzy rule based system l frbs the l frbs exactly approximates the svm s decision boundary and its rules and membership functions are very simple aggregating the antecedents with uninorms as compensation operators the rules of the l frbs are limited to two and the number of fuzzy propositions in each rule only depends on the cardinality of the set of support vectors for that reason the l frbs overcomes the course of dimensionality and problems with high dimensional data sets are easily solved with the l frbs

fuzzy rule base simplification using multidimensional scaling and constrained optimization this paper proposes a novel approach for the development of highly accurate and interpretable fuzzy models with gaussian fuzzy sets under the framework of multidimensional scaling and non linear constrained optimization upon the assumption that an accurate initial fuzzy model has already been designed we introduce an effective methodology to approximate the similarity measure between fuzzy sets the resulting similarity degrees guide the quantification of the dissimilarity degrees between rule antecedents we then put the multidimensional scaling in place in order to transform the rule antecedents into points in a low dimensional euclidean space the elaboration on the distribution of these points is carried out by means of an objective function based fuzzy clustering using a cluster validity index rules that correspond to points belonging to the same cluster are similar and therefore are unified through a specialized merging process with respect to each dimension the aforementioned merging process acts to create a topology of fuzzy sets that enables us to elicit interpretability constraints which are used to minimize the model s performance index in terms of non linear constrained optimization the established fuzzy model appears to possess a simple and transparent i e interpretable structure while maintaining a highly accurate behavior the overall method is rigorously tested and evaluated through a number of simulation experiments that involve low and high dimensional data sets

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

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

sgerd a steady state genetic algorithm for extracting fuzzy classification rules from data this paper considers the automatic design of fuzzy rule based classification systems from labeled data the performance of classifiers and the interpretability of generated rules are of major importance in these systems in past research some genetic based algorithms have been used for the rule learning process these genetic fuzzy systems have utilized different approaches to encode rules in this paper we have proposed a novel steady state genetic algorithm to extract a compact set of good fuzzy rules from numerical data sgerd the selection mechanism of this algorithm is nonrandom and only the best individuals can survive our approach is very simple and fast and can be applied to high dimensional problems with numerical attributes to select the rules having high generalization capabilities our algorithm makes use of some rule and data dependent parameters we have also proposed an enhancing function that modifies the rule evaluation measures in order to assess the candidate rules more effectively before their selection experiments on some well known data sets are performed to show the performance of sgerd

identification of fuzzy models using a successive tuning method with a variant identification ratio the paper is concerned with the identification of fuzzy inference systems fuzzy models realized with the aid of the successive tuning method using a variant identification ratio that is based on hierarchical fair competition based parallel genetic algorithms hfc pga and information granulation the hfc pga is a certain multi population version of parallel genetic algorithms pga which is suitable for a simultaneous optimization of both the structure of the fuzzy model as well as its parameters the granulation of information is realized with the aid of the c means clustering algorithm information granules formed in this way become essential at further stages of the construction of the fuzzy models by forming the centers modal values of the fuzzy sets constituting individual rules of the inference schemes further optimization of the fuzzy model deals with an adjustment of a suite of parameters such as the number of input variables to be used in the model a collection of specific subsets of the input variables and the number of membership functions being used by these variables and the order and parameters of the polynomial occurring in the conclusions of the corresponding rules an iterative development of the fuzzy model deals with its structural as well as parametric optimization via hfc pga the c means algorithm and a standard least square estimation method to evaluate the performance of the proposed model we exploit well known and commonly used data sets such as gas furnace mackey glass time series as well as medical imaging system a comparative analysis demonstrates that the proposed model leads to the superior performance when compared with other fuzzy models reported in the literature

function approximation with decomposed fuzzy systems if the number of input signals increases a fuzzy system gets increasingly intractable for two reasons on the one hand the knowledge acquisition suffers increasingly from the knowledge engineering bottleneck on the other hand computational and memory demands of fuzzy systems increase strongly thus suffering from the curse of dimensionality the first problem is classically addressed by learning techniques the second by decomposing the fuzzy system the netfan approach network of fuzzy adaptive nodes combines decomposition with learning in order to apply fuzzy techniques to more complex applications different ways to decompose a fuzzy system are discussed in this paper for the favorite decomposition principle it is shown to be an universal function approximator despite decomposition the real world example of controlling a pneumatic positioning system finally demonstrates the applicability and benefits of this type of hierarchical decomposition namely a significantly reduced number of rules

hierarchical cluster based multispecies particle swarm optimization for fuzzy system optimization this paper proposes a hierarchical cluster based multispecies particle swarm optimization hcmspso algorithm for fuzzy system optimization the objective of this paper is to learn takagi sugeno kang tsk type fuzzy rules with high accuracy in the hcmspso designed fuzzy system fs each rule defines its own fuzzy sets which implies that the number of fuzzy sets for each input variable is equal to the number of fuzzy rules a swarm in hcmspso is clustered into multiple species at an upper hierarchical level and each species is further clustered into multiple subspecies at a lower hierarchical level for an fs consisting of r rules r species swarms are formed in the upper level where one species optimizes a single fuzzy rule initially there are no species in hcmspso an online cluster based algorithm is proposed to generate new species fuzzy rules automatically in the lower layer subspecies within the same species are formed adaptively in each iteration during the particle update several simulations are conducted to verify hcmspso performance comparisons with other neural learning genetic and pso algorithms demonstrate the superiority of hcmspso performance

mathematical programming models and algorithms for a class faculty assignment problem this paper presents mathematical programming models for assigning faculty members to classes including among typical academic class scheduling issues certain specialized central policies at kuwait university the time slots for classes are initially assumed to be given and an integer programming model cfam is constructed to solve the resulting problem which aims to minimize the individual and collective dissatisfaction of faculty members in a fair fashion where dissatisfaction is measured by a function of the assignment of faculty members to time slots and specific classes in order to enhance the quality of results obtained in practice the model is modified ecfam so that the time slots for the classes can be changed however with restrictions related to efficient facility utilization and permitting an administratively regulated maximum number of changes gender based modeling considerations are also introduced in order to maintain desirable class offering patterns computational results are presented based on solving the models directly by the cplex mip version 7 5 package and also using a specialized lp based heuristic the faculty schedules generated via the proposed approach based on a number of case studies related to the department of mathematics and computer science at kuwait university reveal that this approach yields improved schedules in terms of fairness and enhanced satisfaction levels among faculty members

implementation of evolutionary fuzzy systems evolutionary fuzzy systems are discussed in which the membership function shapes and types and the fuzzy rule set including the number of rules inside it are evolved using a genetic evolutionary algorithm in addition the genetic parameters operators of the evolutionary algorithm are adapted via a fuzzy system benefits of the methodology are illustrated in the process of classifying the iris data set possible extensions of the methods are summarized

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a generalized class teacher model for some timetabling problems abstract we consider a theoretical model which extends the basic class teacher model of timetabling and which corresponds to some situations which occur frequently in the basic training programs of universities and schools we are given m teachers t 1 t m and n classes c 1 c n the set of classes is partitioned into p disjoint subsets g 1 g p in such a way that in addition to the lectures given by one teacher to one class there are some lectures given by one teacher to the students of all classes in group g l 1 l p such lectures will be called group lectures the number a lj of one hour group lectures which teacher t j must deliver to group g l and the number b ij of one hour class teaching which t j must give to class c i are given is there a timetable of t hours or length t so that each class c i and each group g l receive all their lectures but no student is scheduled to be taught by more than one teacher in each hour and no teacher must teach to more than one group or class in each hour we show that this problem is np complete and find some sufficient conditions for the existence of a timetable of length t we also describe an algorithm for constructing a timetable corresponding to the requirement matrices a a lj and b b ij and show that under a natural assumption on a and b this algorithm finds a timetable within 7 6 of the optimum length

universal approximation by hierarchical fuzzy systems abstract a serious problem limiting the applicability of standard fuzzy controllers is the rule explosion problem that is the number of rules increases exponentially with the number of input variables to the fuzzy controller a way to deal with this curse of dimensionality is to use the hierarchical fuzzy systems a hierarchical fuzzy system consists of a number of hierarchically connected low dimensional fuzzy systems it can be shown that the number of rules in the hierarchical fuzzy system increases linearly with the number of input variables in this paper we prove that the hierarchical fuzzy systems are universal approximators that is they can approximate any nonlinear function on a compact set to arbitrary accuracy our proof is constructive that is we first construct a hierarchical fuzzy system in a step by step manner then prove that the constructed fuzzy system satisfies an error bound and finally show that the error bound can be made arbitrarily small

evolving decision trees with beam search based initialization and lexicographic multi objective evaluation decision tree induction algorithms represent one of the most popular techniques for dealing with classification problems however traditional decision tree induction algorithms implement a greedy approach for node splitting that is inherently susceptible to local optima convergence evolutionary algorithms can avoid the problems associated with a greedy search and have been successfully employed to the induction of decision trees previously we proposed a lexicographic multi objective genetic algorithm for decision tree induction named legal tree in this work we propose extending this approach substantially particularly w r t two important evolutionary aspects the initialization of the population and the fitness function we carry out a comprehensive set of experiments to validate our extended algorithm the experimental results suggest that it is able to outperform both traditional algorithms for decision tree induction and another evolutionary algorithm in a variety of application domains

a multiobjective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy rule based systems in this paper we propose the use of a multiobjective evolutionary approach to generate a set of linguistic fuzzy rule based systems with different tradeoffs between accuracy and interpretability in regression problems accuracy and interpretability are measured in terms of approximation error and rule base rb complexity respectively the proposed approach is based on concurrently learning rbs and parameters of the membership functions of the associated linguistic labels to manage the size of the search space we have integrated the linguistic two tuple representation model which allows the symbolic translation of a label by only considering one parameter with an efficient modification of the well known 2 2 pareto archived evolution strategy paes we tested our approach on nine real world datasets of different sizes and with different numbers of variables besides the 2 2 paes we have also used the well known nondominated sorting genetic algorithm nsga ii and an accuracy driven single objective evolutionary algorithm ea we employed these optimization techniques both to concurrently learn rules and parameters and to learn only rules we compared the different approaches by applying a nonparametric statistical test for pairwise comparisons thus taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective eas finally a data complexity measure which is typically used in pattern recognition to evaluate the data density in terms of average number of patterns per variable has been introduced to characterize regression problems results confirm the effectiveness of our approach particularly for possibly high dimensional datasets with high values of the complexity metric

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

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

robust controllability of t s fuzzy model based control systems with parametric uncertainties the robust controllability problem for the takagi sugeno t s fuzzy model based control systems is studied in this paper under the assumption that the nominal t s fuzzy model based control systems are locally controllable i e each fuzzy rule of the nominal t s fuzzy model based control systems has a full row rank for its controllability matrix a sufficient condition is proposed to preserve the assumed property when the parameter uncertainties are added into the nominal t s fuzzy model based control systems the proposed sufficient condition can provide the explicit relationship of the bounds on parameter uncertainties to preserve the assumed property besides a robustly global controllability condition and the related robustly global stabilizability condition of the uncertain t s fuzzy model based control systems are also presented in this paper a nonlinear mass spring damper mechanical system with parameter uncertainties is given as an example to illustrate the application of the proposed sufficient conditions

application of statistical information criteria for optimal fuzzy model construction theoretical studies have shown that fuzzy models are capable of approximating any continuous function on a compact domain to any degree of accuracy however constructing a good fuzzy model requires finding a good tradeoff between fitting the training data and keeping the model simple a simpler model is not only easily understood but also less likely to overfit the training data even though heuristic approaches to explore such a tradeoff for fuzzy modeling have been developed few principled approaches exist in the literature due to the lack of a well defined optimality criterion in this paper we propose several information theoretic optimality criteria for fuzzy models construction by extending three statistical information criteria 1 the akaike information criterion aic 1974 2 the bhansali downham information criterion bdic 1977 and 3 the information criterion of schwarz 1978 and rissanen 1978 sric we then describe a principled approach to explore the fitness complexity tradeoff using these optimality criteria together with a fuzzy model reduction technique based on the singular value decomposition svd the role of these optimality criteria in fuzzy modeling is discussed and their practical applicability is illustrated using a nonlinear system modeling example

evolutionary design of fuzzy rule base for nonlinear system modeling and control in designing fuzzy models and controllers we encounter a major difficulty in the identification of an optimized fuzzy rule base which is traditionally achieved by a tedious trial and error process the paper presents an approach to the evolutionary design of an optimal fuzzy rule base for modeling and control evolutionary programming is used to simultaneously evolve the structure and the parameter of fuzzy rule base for a given task to check the effectiveness of the suggested approach four numerical examples are examined the performance of the identified fuzzy rule bases is demonstrated

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

neuro fuzzy rule generation survey in soft computing framework the present article is a novel attempt in providing an exhaustive survey of neuro fuzzy rule generation algorithms rule generation from artificial neural networks is gaining in popularity in recent times due to its capability of providing some insight to the user about the symbolic knowledge embedded within the network fuzzy sets are an aid in providing this information in a more human comprehensible or natural form and can handle uncertainties at various levels the neuro fuzzy approach symbiotically combining the merits of connectionist and fuzzy approaches constitutes a key component of soft computing at this stage to date there has been no detailed and integrated categorization of the various neuro fuzzy models used for rule generation we propose to bring these together under a unified soft computing framework moreover we include both rule extraction and rule refinement in the broader perspective of rule generation rules learned and generated for fuzzy reasoning and fuzzy control are also considered from this wider viewpoint models are grouped on the basis of their level of neuro fuzzy synthesis use of other soft computing tools like genetic algorithms and rough sets are emphasized rule generation from fuzzy knowledge based networks which initially encode some crude domain knowledge are found to result in more refined rules finally real life application to medical diagnosis is provided

competitive interaction reasoning in designing fuzzy rule based classification systems frbcss complex fuzzy rule extraction techniques and tuning membership functions are frequently used to enhance classification accuracy however these approaches not only decrease system s transparency which is the hallmark of fizzy design but also are oftentimes computationally expensive and require multiple parameters to be optimized using a large amount of training data in this paper inspired by the so called competitive behavior of mini columns in the brain neuronal circuitry we proposed a new reasoning method for fuzzy classifiers referred to as competitive interaction reasoning cir that employs the cumulative information provided by all fuzzy rules and adjusts the decision boundaries as if the membership functions are directly modified this mechanism is mathematically implemented by a linear transformation and resembles the competitive interaction observed in brain neuronal columns cross rule competition weights are optimized using hebbian reinforcement learning using a large number of simulations on benchmark data sets we show that the proposed cir significantly improves classification accuracy without compromising interpretability of the fuzzy classifier in addition cir can further facilitate formation of the fuzzy rules and incorporation of the expert knowledge by confining the destructive effects of noisy rules or expert inconsistencies experiments on 23 well known benchmark data sets confirm high performance of cir in comparison with a number of popular classifiers

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

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

automatic design of hierarchical takagi sugeno type fuzzy systems using evolutionary algorithms this paper presents an automatic way of evolving hierarchical takagi sugeno fuzzy systems ts fs the hierarchical structure is evolved using probabilistic incremental program evolution pipe with specific instructions the fine tuning of the if then rule s parameters encoded in the structure is accomplished using evolutionary programming ep the proposed method interleaves both pipe and ep optimizations starting with random structures and rules parameters it first tries to improve the hierarchical structure and then as soon as an improved structure is found it further fine tunes the rules parameters it then goes back to improve the structure and the rules parameters this loop continues until a satisfactory solution hierarchical ts fs model is found or a time limit is reached the proposed hierarchical ts fs is evaluated using some well known benchmark applications namely identification of nonlinear systems prediction of the mackey glass chaotic time series and some classification problems when compared to other neural networks and fuzzy systems the developed hierarchical ts fs exhibits competing results with high accuracy and smaller size of hierarchical architecture

a class of hierarchical fuzzy systems with constraints on the fuzzy rules this paper presents a class of hierarchical fuzzy systems where previous layer outputs are used not in if parts but only in then parts of the fuzzy rules of the current layer the proposed scheme is shown to be a universal approximator to any real continuous function on a compact set if complete fuzzy sets are used in the if parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier from the example of ball and beam control system simulation it is demonstrated that the proposed scheme approximates with high accuracy a model nonlinear controller with fewer fuzzy rules than a centralized fuzzy system and its control performance is comparable to that of a nonlinear controller

fuzzy system identification using an adaptive learning rule with terminal attractors a fuzzy model with an adaptive learning rule is proposed to approximate a class of nonlinear systems the proposed adaptive learning rule uses the concept of terminal attractors and guarantees that the identification is stable and fast convergence in finite time another subject of this paper is to determine the appropriate rule number of the fuzzy model required for a desired modeling error a learning algorithm is also introduced to identify the centers of membership functions of fuzzy outputs

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

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

look ahead based fuzzy decision tree induction decision tree induction is typically based on a top down greedy algorithm that makes locally optimal decisions at each node due to the greedy and local nature of the decisions made at each node there is considerable possibility of instances at the node being split along branches such that instances along some or all of the branches require a large number of additional nodes for classification in this paper we present a computationally efficient way of incorporating look ahead into fuzzy decision tree induction our algorithm is based on establishing the decision at each internal node by jointly optimizing the node splitting criterion information gain or gain ratio and the classifiability of instances along each branch of the node simulations results confirm that the use of the proposed look ahead method leads to smaller decision trees and as a consequence better test performance

a note on universal approximation by hierarchical fuzzy systems this paper proves in a constructive manner that the general n dimensional hierarchical fuzzy systems are universal approximators it is an extension of the results in l x wang fuzzy sets and systems 93 1998 223 230 an upper bound of approximation error is also given