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

adaptive fuzzy rule based classification systems this paper proposes an adaptive method to construct a fuzzy rule based classification system with high performance for pattern classification problems the proposed method consists of two procedures an error correction based learning procedure and an additional learning procedure the error correction based learning procedure adjusts the grade of certainty of each fuzzy rule by its classification performance that is when a pattern is misclassified by a particular fuzzy rule the grade of certainty of that rule is decreased on the contrary when a pattern is correctly classified the grade of certainty is increased because the error correction based learning procedure is not meaningful after all the given patterns are correctly classified we cannot adjust a classification boundary in such a case to acquire a more intuitively acceptable boundary we propose an additional learning procedure we also propose a method for selecting significant fuzzy rules by pruning unnecessary fuzzy rules which consists of the error correction based learning procedure and the concept of forgetting we can construct a compact fuzzy rule based classification system with high performance

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

a continuation problem for computing solutions of discretised evolution problems with application to plane quasi static contact problems with friction abstract a continuation problem for finding successive solutions of discretised abstract first order evolution problems is proposed and a general piecewise c 1 continuation problem is studied a condition ensuring local existence and uniqueness of its solution curves is given an analogy of the first order system of smooth problems is derived and results of existence and uniqueness of its solutions are stated possibility of continuation of a solution curve along directions solving the first order system is discussed a technique for numerical continuation of the solution curves is developed furthermore an application of the abstract continuation problem is presented for plane quasi static contact problems with friction various formulations of the first order system are derived for this case so that the analysis from the abstract frame can be developed and supplemented finally the proposed numerical continuation is tested on model examples

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

the integration and application of fuzzy and grey modeling methods abstract an integrated fuzzy and grey model and its applications to the prediction control problems are presented the basic grey model gm 1 1 is accompanied with the adaptive fuzzy method to improve its prediction capability the gradient descent scheme is applied to the fuzzy rules to determine whether the predicted results from the grey model should be adjusted the quantity of adjustment is judged from the degrees of the correlation between the past data and the current input we select a few most correlated patterns to decide the direction of adjustment due to the simplicity of the structure and its fast learning characteristics this model is good as a real time controller under the proposed methodology the simulation results are shown to be superior to those systems which exploit complicated control variables and rules a well known difference equation and the weather forecast prediction problems are depicted to verify the superiority of the proposed method

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

genetic training instance selection in multiobjective evolutionary fuzzy systems a coevolutionary approach when dealing with datasets that are characterized by a large number of instances multiobjective evolutionary learning moel of fuzzy rule based systems frbss suffers from high computational costs mainly because of the fitness evaluation the use of a reduced set of representative instances in place of the overall training set ts would considerably lessen the computational effort even though a large number of papers have proposed instance selection approaches mainly in classification problems how this selection should be performed especially in the context of regression is still an open issue in this paper we tackle the instance selection problem in the framework of moel of frbss through a coevolutionary approach in the execution of the moel periodically a single objective genetic algorithm soga evolves a population of reduced tss the soga aims to maximize a purposely defined index which measures how much the pareto fronts computed by using respectively the reduced ts and the overall ts are close to each other the closer the fronts the more the reduced ts is representative of the overall ts during the execution of the moel the rule base and the membership function parameters of the fuzzy sets are concurrently learned by maximizing the accuracy and minimizing the complexity we tested our approach on 12 large datasets we adopted reduced tss composed of 5 10 and 20 of the overall ts using nonparametric statistical tests we verified that with 10 and 20 of the overall ts the pareto front approximations that are generated by our coevolutionary approach are comparable with the ones generated by applying the moel with the overall ts although the coevolution allows us to save up to 86 36 of the execution time in addition the analysis of the behavior of three representative solutions on the test set highlights that the use of the reduced tss does not affect the generalization capabilities of the generated frbss

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

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

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

analytic network process for pattern classification problems using genetic algorithms the analytic network process anp is a useful technique for multi attribute decision analysis mcda that employs a network representation to describe interrelationships between diverse attributes owing to effectiveness of the anp in allowing for complex interrelationships between attributes this paper develops an anp based classifier for pattern classification problems with interdependence or independence among attributes to deal with interdependence this study employs genetic algorithms gas to automatically determine elements in the supermatrix that are not easily user specified to find degrees of importance of respective attributes then with the relative importance for each attribute in the limiting supermatrix the current work determines the class label of a pattern by its synthetic evaluation experimental results obtained by the proposed anp based classifier are comparable to those obtained by other fuzzy or non fuzzy classification methods

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

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

improving the interpretability of tsk fuzzy models by combining global learning and local learning the fuzzy inference system proposed by takagi sugeno and kang known as the tsk model in fuzzy system literature provides a powerful tool for modeling complex nonlinear systems unlike conventional modeling where a single model is used to describe the global behavior of a system tsk modeling is essentially a multimodel approach in which simple submodels typically linear models are combined to describe the global behavior of the system most existing learning algorithms for identifying the tsk model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model although these algorithms can generate a tsk model with good global performance i e the model is capable of approximating the given system with arbitrary accuracy provided that sufficient rules are used and sufficient training data are available they cannot guarantee the resulting model to have a good local performance often the submodels in the tsk model may exhibit an erratic local behavior which is difficult to interpret since one of the important motivations of using the tsk model also other fuzzy models is to gain insights into the model it is important to investigate the interpretability issue of the tsk model we propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework this algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics but remains the component of global fitting in the existing learning algorithms the algorithm is capable of adjusting its parameters based on the user s preference generating models with good tradeoff in terms of global fitting and local interpretation we illustrate the performance of the proposed algorithm using a motorcycle crash modeling example

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

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

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

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

linguistic summarization using if then rules and interval type 2 fuzzy sets linguistic summarization ls is a data mining or knowledge discovery approach to extract patterns from databases many authors have used this technique to generate summaries like most senior workers have high salary which can be used to better understand and communicate about data however few of them have used it to generate if then rules like if x is large and y is medium then z is small which not only facilitate understanding and communication of data but can also be used in decision making in this paper an ls approach to generate if then rules for causal databases is proposed both type 1 and interval type 2 fuzzy sets are considered five quality measures the degrees of truth sufficient coverage reliability outlier and simplicity are defined among them the degree of reliability is especially valuable for finding the most reliable and representative rules and the degree of outlier can be used to identify outlier rules and data for close up investigation an improved parallel coordinates approach for visualizing the if then rules is also proposed experiments on two datasets demonstrate our ls and rule visualization approaches finally the relationships between our ls approach and the wang mendel wm method perceptual reasoning and granular computing are pointed out

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

distributed representation of fuzzy rules and its application to pattern classification abstract this paper introduces the concept of distributed representation of fuzzy rules and applies it to classification problems distributed representation is implemented by superimposing many fuzzy rules corresponding to different fuzzy partitions of a pattern space this means that we simulatenously employ many fuzzy rule tables corresponding to different fuzzy partitions in fuzzy inference in order to apply distributed representation of fuzzy rules to pattern classification problems we first propose an algorithm to generate fuzzy rules from numerical data next we propose a fuzzy inference method using the generated fuzzy rules the classification power of distributed representation is compared with that of ordinary fuzzy rules which can be viewed as local representation

a new approach of multi stage fuzzy logic inference abstract in general multi stage fuzzy logic inference there are three kinds of linguistic variables input output and intermediate variables when intermediate variables are present in a set of fuzzy rules to form multi stage fuzzy rules the inferring time involved is understandably lengthened to achieve fast reasoning of multi stage fuzzy logic the number of fuzzy set operations of input output and intermediate variables must be decreased as much as possible without information loss for the intermediate variables the fuzzy logic operations of overlapped fuzzy value pairs on several multi stage fuzzy rules may be pre computed at compiling time furthermore if the pre computations have been done the numbers of intersection and union scalar operations must also decrease in order to speed up reasoning at execution time moreover whether fuzzy logic operations of multi stage fuzzy rules are pre computed or not the types of fuzzy logic operators must be considered failing to do so may lead to information error by using the truck trailer backer parking system improvements over related methods are attained

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

a fast and efficient multi objective evolutionary learning scheme for fuzzy rule based classifiers during the last years multi objective evolutionary algorithms moeas have been extensively used to generate fuzzy rule based systems characterized by different trade offs between accuracy and complexity in this paper we propose an moea based approach to learn concurrently the rule and data bases of fuzzy rule based classifiers frbcs in particular the rule bases are generated by exploiting a rule and condition selection rcs strategy which selects a reduced number of rules from a heuristically generated set of candidate rules and a reduced number of conditions for each selected rule during the evolutionary process rcs can be considered as a rule learning in a constrained search space as regards the data base learning the membership function parameters of each linguistic term used in the rules are learned concurrently to the application of rcs we tested our approach on twenty four classification benchmarks and compared our results with the ones obtained by two similar state of the art moea based approaches and by two well known non evolutionary classification algorithms namely furia and c4 5 using non parametric statistical tests we show that our approach generates frbcs with accuracy and complexity statistically comparable to and sometimes better than the ones generated by the two moea based approaches exploiting however only the 5 of the number of fitness evaluations used by these approaches further the classifiers generated by our approach result to be more interpretable than the ones generated by the furia and c4 5 algorithms while achieving the same accuracy level

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

weighting fuzzy classification rules using receiver operating characteristics roc analysis in fuzzy rule based classification systems rule weight has often been used to improve the classification accuracy in past research a number of heuristic methods for rule weight specification have been proposed in this paper a method of fuzzy rule weight specification using receiver operating characteristic roc analysis is proposed in order to specify the weight of a fuzzy rule using 2 class roc analysis the threshold that the rule achieves its maximum accuracy is found this threshold is used as the weight of the rule the proposed method is compared with existing ones through computer simulations on some well known classification problems with continuous attributes simulation results show that the proposed method performs better than existing methods of fuzzy rule weight specification

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

fuzzy self organizing controller and its application for dynamic processes abstract this paper deals with a self organizing controller used in real time and its application for dynamic processes the control policy of the controller is able to develop and improve automatically a new self organizing control algorithm which modifies the fuzzy control decision table is presented in this paper the control algorithm needs much less computation time and memory of the computer this paper provides two examples of the fuzzy self organizing control a set of experiment results is described it shows that the controller can be applied to some processes such as processes with time lag and non linearity or one whose parameter or running conditions are varied which were usually not controlled very well with a conventional controller

a new method for design and reduction of neuro fuzzy classification systems in this paper we propose a new class of neuro fuzzy systems moreover we develop a novel method for reduction of such systems without the deterioration of their accuracy the reduction algorithm gradually eliminates inputs rules antecedents and the number of discretization points of integrals in the center of area defuzzification method it then automatically detects and merges similar input and output fuzzy sets through computer simulations it is shown that accuracy of the system after reduction and merging has not deteriorated despite the fact that in some cases up to 54 of various parameters and 74 of inputs were eliminated the reduction algorithm has been tested using well known classification benchmarks

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

power spectrum reduction by optimal hankel norm approximation of the phase of the outer spectral factor a new approach to stochastic processes power spectrum reduction based on akaike s canonical correlation between the process past and future is developed as a first result the correlation between the canonical past and the canonical future is given its precise functional analytic interpretation as the hankel operator generated lay the all pass function characterizing the phase of the process outer spectral factor the canonical correlation coefficients turn out to he nothing else but the singular values of this hankel operator next the all pass function characterizing the phase of the outer spectral factor is reduced using the optimal hankel norm procedure of adamjan arov and krein the reduction of the all pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through this involves bode s relations between the gain and the phase of an outer function nehari s extension and the winding number of the all pass phase function which is itself shown to be related to the definition of the past and the future finally the net result is a reduced order outer spectral factor that matches the high order spectral factor fairly well

sugeno type controllers are universal controllers abstract we show that given any continuous real valued function on a compact set there is a sugeno type controller that will approximate this function to any degree of accuracy an application of this result to fuzzy control is presented which shows that this type of sugeno controller is a universal controller

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

hierarchical fuzzy systems for function approximation on discrete input spaces with application this paper investigates the capabilities of hierarchical fuzzy systems to approximate functions on discrete input spaces first it is shown that any function on a discrete space has an arbitrary separable hierarchical structure and can be naturally approximated by hierarchical fuzzy systems as a by product of this result a discrete version of kolmogorov s theorem is obtained second it is proven that any function on a discrete space can be approximated to any degree of accuracy by hierarchical fuzzy systems with any desired separable hierarchical structure that is functions on discrete spaces can be approximated more simply and flexibly than those on continuous spaces third a hierarchical fuzzy system identification method is proposed in which human knowledge and numerical data are combined for system construction and identification finally the proposed method is applied to the market condition performance modeling problem in site selection decision support and shows the better performance in both accuracy and interpretability than the regression and neural network approaches in additions the reason and mechanism why hierarchical fuzzy systems outperform regression and neural networks in this type of application are analyzed

universal approximation by hierarchical fuzzy system with constraints on the fuzzy rule this paper presents a special hierarchical fuzzy system where the outputs of the previous layer are not used in the if parts but used only in the then parts of the fuzzy rules of the current layer the proposed scheme can be shown to be a universal approximator to any 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 simulation of ball and beam control system it is demonstrated that the proposed scheme approximates with good accuracy the model nonlinear controller with fewer fuzzy rules than the centralized fuzzy system and its control performance is comparable to that of the nonlinear controller

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

multipolar aggregation operators in reasoning methods for fuzzy rule based classification systems the reasoning methods in fuzzy rule based classification systems are studied and their relation to multipolar aggregation operators is discussed we describe the change in the output of the classification system based on the change of the fuzzy reasoning method we propose a new fuzzy reasoning method based on multipolar fusion owa operator which can control the transition between the maximum rule method and the maximum vote method by a weighting vector this new method uses principles of owa operators and assigns the rule weight according to the strength of activation of the rule we compare different fuzzy reasoning methods in case studies

qualitative analysis of solutions to discrete static contact problems with coulomb friction we analyze properties of solutions to discrete contact problems with coulomb friction which are parametrized by the coeffi cient of frictionf using a generalized variant of the implicit function theo rem we establish conditions under which there exists a local lipschitz continuous branch of solutions around a reference point finally a piecewise smooth continuation algorithm which allows to follow such branches of solutions is proposed y have certain knowledge on the qualitative behavior of their solutions which is not available in the continuous case indeed the discrete model has at least one solution for any friction coeffi cientf and this solution is unique iff is small enough on the other hand the bounds onf guaranteeing uniqueness of the solution are mesh dependent in other words the uniqueness property of the discrete problem depends on the size of the prob lem thus it may happen that a small change off leads to a large change of the solution but this phenomenon appears only when suffi ciently fine discretizations are used this catastrophic behavior can be observed in numerical experiments 8 9 the first step towards better understanding of the discrete model is to estab lish the existence of local lipschitz continuous branches of solutions parametrized byf the present paper ex tends results from 10 where f was assumed to be constant to the case when f depends on the spatial variable such results are important for the development of continuation algorithms which is another goal of this paper

a genetic algorithm for generating fuzzy classification rules abstract fuzzy classification systems based on fuzzy logic are capable of dealing with cognitive uncertainties such as vagueness and ambiguity involved in classification problems to build a fuzzy classification system the key is to find an optimal set of fuzzy rules machine learning methods such as fuzzy neural networks and fuzzy decision tree induction have been applied to learn fuzzy rules but they may be trapped into local optimal based on the principle of natural evolution and global searching a genetic algorithm is promising in obtaining better results but it is usually less efficient in this paper a fuzzy genetic algorithm fga is developed to generate fuzzy classification rules several techniques such as multi value logic coding composite fitness function viability check and rule extraction are used to improve the efficiency and the effectiveness of the algorithm two experiments are conducted to demonstrate the performance of fga and to compare fga with other machine learning algorithms

stochastic model reduction via hankel approximation abstract we consider the problem of approximating a multidimensional stationary process y t of mcmillan degree n with another stationary process yk t of mcmillan degree k n minimizing the l 2 norm of the difference in the predictions for y t and yk t we prove that one solution to this problem can be obtained by using the optimal hankel approximant of the phase function of yk t t z

a method of converting a fuzzy system to a two layered hierarchical fuzzy system and its run time efficiency in classical single layer fuzzy systems fss the number of rules and the run time computational requirements increase exponentially with the number of input domains in this paper we present a method for converting a multidimensional fs to a two layer hierarchical fs that reduces the number of rules and improves the run time efficiency the first layer of the two layer system consists of fss whose rule bases can be represented as linearly independent vectors the second layer constructs linear combinations of the rule base vectors the effectiveness of the hierarchical conversion in reducing the number of rules and improving efficiency is demonstrated on a classic control problem and a simulated higher dimensional fs

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

qualitative analysis of 3d elastostatic contact problems with orthotropic coulomb friction and solution dependent coefficients of friction this paper analyzes a discrete form of 3d contact problems with local orthotropic coulomb friction and coefficients of friction which may depend on the solution itself the analysis is based on the fixed point reformulation of the original problem conditions guaranteeing the existence and uniqueness of discrete solutions are established finally numerical results of a model example are presented

universal fuzzy controllers a collection b of fuzzy controllers is a universal fuzzy controller if and only if given any controllable process p it can also be controlled by some fuzzy controller in b this paper identifies a new type of universal fuzzy controller

state space approach to approximation by phase matching a state space realization of the stable part of the phase function of a proper spectral factor is provided the phase matching problem jonckheere and helton 1985 harshavardhana and jonckheere 1985 can now be solved completely in the state space framework using existing numerically reliable procedures further a connection is made between the stochastic balancing approach and the phase approximation procedure for the stochastic realization approximation

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

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

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

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

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

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

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 neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

efficient self evolving evolutionary learning for neurofuzzy inference systems this study proposes an efficient self evolving evolutionary learning algorithm seela for neurofuzzy inference systems nfiss the major feature of the proposed seela is that it is based on evolutionary algorithms that can determine the number of fuzzy rules and adjust the nfis parameters the seela consists of structure learning and parameter learning the structure learning attempts to determine the number of fuzzy rules a subgroup symbiotic evolution is adopted to yield several variable fuzzy systems and an elite based structure strategy is adopted to find a suitable number of fuzzy rules for solving a problem the parameter learning is to adjust parameters of the nfis it is a hybrid evolutionary algorithm of cooperative particle swarm optimization cpso and cultural algorithm called cultural cpso ccpso the ccpso which uses cooperative behavior among multiple swarms can increase the global search capacity using the belief space experimental results demonstrate that the proposed method performs well in predicting time series and solving nonlinear control problems

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

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