stability analysis and control of discrete type 1 and type 2 tsk fuzzy systems part ii control design this paper proposes a new control system design methodology for type 1 and type 2 takagi sugeno kang tsk fuzzy systems that are based on new stability conditions the stability conditions are discussed in a companion paper part i and are used in the proofs of our main results a major advantage of the new methodology is that it does not require a common lyapunov function and is therefore applicable to systems with nonstabilizable consequents our controllers include fuzzy type 1 proportional and proportional integral pi controllers as well as constant state feedback for the same systems the controller results in an exponentially stable system and the designer can specify the rate of exponential convergence the controller designs can be tested by the usage of linear matrix inequalities lmis the design methodology is demonstrated by the usage of simple examples where methods that are based on a common lyapunov function fail and physical systems where the new methodology provides better performance

a vectorization optimization method based type 2 fuzzy neural network for noisy data classification this paper proposes a vectorization optimization method vom based type 2 fuzzy neural network vom2fnn for noisy data classification in handling problems with uncertainties such as noisy data type 2 fuzzy systems usually outperform their type 1 counterparts hence type 2 fuzzy sets are adopted in the antecedent parts to model the uncertainty to consider the classification problems the discriminative capability is crucial to determine the performance therefore a vom is proposed in the consequent parts to increase the discriminability and reduce the parameters compared with other existing fuzzy neural networks the novelty of the proposed vom2fnn is its consideration of both uncertainty and discriminability the effectiveness of the proposed vom2fnn is demonstrated by three classification problems experimental results and theoretical analysis indicate that the proposed vom2fnn performs better than the other fuzzy neural networks

on line fuzzy modeling via clustering and support vector machines in this paper we propose a novel approach to identify unknown nonlinear systems with fuzzy rules and support vector machines our approach consists of four steps which are on line clustering structure identification parameter identification and local model combination the collected data are firstly clustered into several groups through an on line clustering technique then structure identification is performed on each group using support vector machines such that the fuzzy rules are automatically generated with the support vectors time varying learning rates are applied to update the membership functions of the fuzzy rules the modeling errors are proven to be robustly stable with bounded uncertainties by a lyapunov method and an input to state stability technique comparisons with other related works are made through a real application of crude oil blending process the results demonstrate that our approach has good accuracy and this method is suitable for on line fuzzy modeling

a fuzzy model with online incremental svm and margin selective gradient descent learning for classification problems this paper proposes a new incremental learning approach to endow a takagi sugeno type fuzzy classification model with high generalization ability the proposed fuzzy model is learned through incremental support vector machine svm and margin selected gradient descent learning and is called fm3 in this learning approach training samples are fed incrementally one by one instead of all in one batch the fm3 evolves from an empty rule set a one pass clustering algorithm is used to determine the number of rules and initial fuzzy sets in the rule antecedent part an online incremental linear svm is proposed to tune the rule consequent parameters to endow the fm3 with high generalization ability the use of incremental instead of batch svm enables the fm3 to handle online training problems with only one new sample available at a time for antecedent parameter learning a margin selected gradient descent algorithm is proposed to prevent overtraining simulation results and comparisons with svms and fuzzy classifiers with different learning algorithms demonstrate the advantage of the fm3

on solution of the periodic riccati equations results presented by mori in the article named above ibid vol ac 30 pp 162 164 feb 1985 which give some bounds for the trace and the determinant of the positive definite solution to the algebraic riccati matrix equation are shown to be erroneous and suggestions to remedy these errors are made

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

a self organizing ts type fuzzy network with support vector learning and its application to classification problems a self organizing takagi sugeno ts type fuzzy network with support vector learning sotfn sv is proposed in this paper the proposed sotfn sv is inspired by analysis of ts type fuzzy systems and composite kernel support vector machine svm sotfn sv is a fuzzy system constructed by the hybridization of fuzzy clustering and svm the antecedent part of sotfn sv is generated via fuzzy clustering of the input data and then svm is used to tune the consequent part parameters to give the network better generalization performance for demonstration sotfn sv is applied to several classification problems especially the skin color classification problem in the skin color classification application each color pixel is represented by hue and saturation hs color space to represent color information by histogram as accurately as possible a nonuniform partition of hs space is proposed for comparison svms and other fuzzy systems trained by svm or neural networks are applied to the same classification problems the advantages of sotfn sv are verified by comparisons with the results of these methods

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

a recurrent self evolving interval type 2 fuzzy neural network for dynamic system processing this paper proposes a recurrent self evolving interval type 2 fuzzy neural network rseit2fnn for dynamic system processing an rseit2fnn incorporates type 2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system the antecedent parts in each recurrent fuzzy rule in the rseit2fnn are interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type with interval weights the antecedent part of rseit2fnn forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself the tsk type consequent part is a linear model of exogenous inputs the rseit2fnn initially contains no rules all rules are learned online via structure and parameter learning the structure learning uses online type 2 fuzzy clustering for the parameter learning the consequent part parameters are tuned by a rule ordered kalman filter algorithm to improve learning performance the antecedent type 2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm the rseit2fnn is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise free and noisy conditions comparisons with type 1 recurrent fuzzy neural networks validate the performance of the rseit2fnn

an integer programming approach for fuzzy rule based classification systems abstract fuzzy rule based classification systems frbcss have been successfully employed as a data mining technique where the goal is to discover the hidden knowledge in a data set in the form of interpretable rules and develop an accurate classification model in this paper we propose an exact approach to learn fuzzy rules from a data set for a frbcs first we propose a mixed integer programming model that extracts optimal fuzzy rules from a data set the model s embedded feature selection allows absence of insignificant features in a fuzzy rule in order to enhance its accuracy and coverage in order to build a comprehensive rule base rb we use this model in an iterative procedure that finds multiple rules by converting the obtained optimal solutions into a set of taboo constraints that prevents the model from re finding the previously obtained rules furthermore it changes the search direction by temporarily removing the correctly predicted patterns from the training set aiming to find the optimal rules that predict uncovered patterns in the training set this procedure ensures that most of the patterns in the training set are covered by the rb next another mixed integer programming model is developed to maximize predictive accuracy of the classifier by pruning the rb and removing redundant rules the predictive accuracy of the proposed model is tested on the benchmark data sets and compared with the state of the art algorithms from the literature by non parametric statistical tests

recurrent neuro fuzzy networks for nonlinear process modeling a type of recurrent neuro fuzzy network is proposed in this paper to build long term prediction models for nonlinear processes the process operation is partitioned into several fuzzy operating regions within each region a local linear model is used to model the process the global model output is obtained through the centre of gravity defuzzification which is essentially the interpolation of local model outputs this modeling strategy utilizes both process knowledge and process input output data process knowledge is used to initially divide the process operation into several fuzzy operating regions and to set up the initial fuzzification layer weights process i o data are used to train the network network weights are such trained so that the long term prediction errors are minimized through training membership functions of fuzzy operating regions are refined and local models are learnt based on the recurrent neuro fuzzy network model a novel type of nonlinear model based long range predictive controller can be developed and it consists of several local linear model based predictive controllers local controllers are constructed based on the corresponding local linear models and their outputs are combined to form a global control action by using their membership functions this control strategy has the advantage that control actions can be calculated analytically avoiding the time consuming nonlinear programming procedures required in conventional nonlinear model based predictive control the techniques have been successfully applied to the modeling and control of a neutralization process

bayesian takagi sugeno kang fuzzy classifier in this paper the takagi sugeno kang tsk fuzzy classifier is casted into the bayesian inference framework and a new fuzzy classifier called bayesian tsk fuzzy classifier b tsk fc is proposed accordingly the proposed classifier can be constructed by learning both the antecedent and consequent parameters of the involved fuzzy rules simultaneously as a result of the introduction of bayesian inference the proposed b tsk fc can be distinguished as follows 1 unlike most existing tsk fuzzy classifiers where the antecedent and consequent parameters of fuzzy rules are learnt in a decoupled manner and the antecedent parameters are learnt only in the input space the antecedent parameters of the fuzzy rules in b tsk fc are learnt by developing a fuzzy clustering method in the input output space and the consequent parameters of fuzzy rules are learnt in accordance with the maximum margin of separation principle thereby resulting to form an intrinsic link between the input and output spaces to achieve improved classification performance and better interpretability 2 with a dirichlet prior assumption about fuzzy memberships in fuzzy clustering a markov chain monte carlo technique is employed to estimate the parameters of the proposed classifier from a sampling perspective 3 rather than being rivals fuzziness and probability in b tsk fc are collaboratively modeled to enhance the performance of tsk fuzzy classifier in terms of classification and interpretability our experimental results in synthetic datasets as well as several real world datasets confirm such merits of the proposed fuzzy classifier

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

effects of increasing the footprints of uncertainty on analytical structure of the classes of interval type 2 mamdani and ts fuzzy controllers the fundamental difference between an interval type 2 it2 fuzzy controller and a type 1 fuzzy controller is the footprint of uncertainty fou of an it2 fuzzy set in this paper we study how fous affect the analytical structure i e the input output mathematical relationship of a broad class of it2 mamdani and takagi sugeno ts controllers the controllers employ arbitrary fuzzy rules the karnik mendel km or enhanced km type reducer the minimum and operator and the centroid defuzzifier the controllers utilize commonly used it2 fuzzy sets for their input variables and any kind of type 2 fuzzy sets for their output variable we prove that with increase of fous of the input fuzzy sets the mamdani controllers approach constant controllers and the ts controllers approach piecewise linear controllers the resemblance to the constant or pricewise linear controllers increases as the fous increase when all the fous are at their maximum to reflect the highest level of uncertainties the mamdani and ts controllers become the constant controllers and piecewise linear controllers respectively we investigate how change in the resemblance takes place progressively as fous increase we also show that an increase in the resemblance narrows control gain variations for part of the it2 controllers which can worsen control performance these findings implicit controller design too large fous are generally undesirable for the input fuzzy sets because they can make an it2 controller behave like a constant or piecewise linear controller real time control experiment results are provided to illustrate the theoretical analysis

robust stability analysis and systematic design of single input interval type 2 fuzzy logic controllers recent results on fuzzy control have shown that interval type 2 it2 fuzzy logic controllers flcs might achieve better control performance due to the additional degree of freedom provided by the footprint of uncertainty fou in their it2 fuzzy sets however the design and robust stability analysis of the it2 flcs are still challenging problems due to their relatively more complex internal structure in this paper we will derive the explicitly fuzzy mapping fm of a single input it2 flc sit2 flc to present design methods and investigate its robustness the analytical information of the it2 fm will give the opportunity to provide explanations on the roles of the fou parameters by taking advantage of the well developed framework of nonlinear control theory comparative theoretical explorations will be presented on the differences between the type 1 t1 fm and it2 fm to clearly show the role of the fou on the robust control system performance it will be proven that the robust stability of the it2 fuzzy system is guaranteed with the aids of the well known popov lyapunov method moreover analytical design methods are presented for sit2 flcs to generate commonly employed control curves by only tuning the size of the fous without a need of an optimization procedure it will be theoretically shown that the fou gives the opportunity to the sit2 flc to generate commonly employed nonlinear control curves while also providing a certain degree of robustness which cannot be accomplished by its t1 counterpart the presented results provide theoretical explanations on the role of the fou on the performance and robustness of the sit2 flc

fuzzy regression analysis by support vector learning approach support vector machines svms have been very successful in pattern classification and function approximation problems for crisp data in this paper we incorporate the concept of fuzzy set theory into the support vector regression machine the parameters to be estimated in the svm regression such as the components within the weight vector and the bias term are set to be the fuzzy numbers this integration preserves the benefits of svm regression model and fuzzy regression model and has been attempted to treat fuzzy nonlinear regression analysis in contrast to previous fuzzy nonlinear regression models the proposed algorithm is a model free method in the sense that we do not have to assume the underlying model function by using different kernel functions we can construct different learning machines with arbitrary types of nonlinear regression functions moreover the proposed method can achieve automatic accuracy control in the fuzzy regression analysis task the upper bound on number of errors is controlled by the user predefined parameters experimental results are then presented that indicate the performance of the proposed approach

the probabilistic constraints in the support vector machine in this paper a new support vector machine classifier with probabilistic constrains is proposed which presence probability of samples in each class is determined based on a distribution function noise is caused incorrect calculation of support vectors thereupon margin can not be maximized in the proposed method constraints boundaries and constraints occurrence have probability density functions which it help for achieving maximum margin experimental results show superiority of the probabilistic constraints support vector machine pc svm relative to standard svm

a study on reduced support vector machines recently the reduced support vector machine rsvm was proposed as an alternate of the standard svm motivated by resolving the difficulty on handling large data sets using svm with nonlinear kernels it preselects a subset of data as support vectors and solves a smaller optimization problem however several issues of its practical use have not been fully discussed yet for example we do not know if it possesses comparable generalization ability as the standard svm in addition we would like to see for how large problems rsvm outperforms svm on training time in this paper we show that the rsvm formulation is already in a form of linear svm and discuss four rsvm implementations experiments indicate that in general the test accuracy of rsvm are a little lower than that of the standard svm in addition for problems with up to tens of thousands of data if the percentage of support vectors is not high existing implementations for svm is quite competitive on the training time thus from this empirical study rsvm will be mainly useful for either larger problems or those with many support vectors experiments in this paper also serve as comparisons of 1 different implementations for linear svm and 2 standard svm using linear and quadratic cost functions

rule based modeling of nonlinear relationships we discuss a problem of rule based fuzzy modeling of multiple input single output nonlinear relationships f r sub n spl rarr r the model under investigation is viewed as a collection of conditional statements if state spl omega then y g sub i x at i 1 2 n with spl omega sub i being a fuzzy relation defined in the space of the input variables in contrast to the commonly encountered identification approach based exclusively upon discrete experimental data the one proposed in this study is concerned with the rule based modeling exploiting the available nonlinear input output relationship the main thrust is in the development of a relevant fuzzy partition of the input variables we introduce and study criteria of separability and variability as the key means guiding a distribution and granularity of the linguistic labels forming the condition part of the local models

effect of rule weights in fuzzy rule based classification systems this paper examines the effect of rule weights in fuzzy rule based classification systems each fuzzy if then rule in our classification system has antecedent linguistic values and a single consequent class we use a fuzzy reasoning method based on a single winner rule in the classification phase the winner rule for a new pattern is the fuzzy if then rule that has the maximum compatibility grade with the new pattern when we use fuzzy if then rules with certainty grades the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade in this paper the effect of rule weights is illustrated by drawing classification boundaries using fuzzy if then rules with without certainty grades it is also shown that certainty grades play an important role when a fuzzy rule based classification system is a mixture of general rules and specific rules through computer simulations we show that comprehensible fuzzy rule based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy if then rules with certainty grades

efficient least angle regression for identification of linear in the parameters models least angle regression as a promising model selection method differentiates itself from conventional stepwise and stagewise methods in that it is neither too greedy nor too slow it is closely related to l1 norm optimization which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability in this paper we propose an efficient least angle regression algorithm for model selection for a large class of linear in the parameters models with the purpose of accelerating the model selection process the entire algorithm works completely in a recursive manner where the correlations between model terms and residuals the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step the model coefficients are only computed when the algorithm finishes the direct involvement of matrix inversions is thereby relieved a detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency compared with the original approach where the well known efficient cholesky decomposition is involved in solving least angle regression three artificial and real world examples are employed to demonstrate the effectiveness efficiency and numerical stability of the proposed algorithm

a maximizing discriminability based self organizing fuzzy network for classification problems a maximizing discriminability based self organizing fuzzy network mdsofn that can classify highly confusable patterns is proposed in this paper the underlying notion of the proposed mdsofn is to split the generation of fuzzy rules into linear discriminant analysis lda and gaussian mixture model gmm in lda the weights are updated by seeking directions that are efficient for discrimination in gmm parameter learning adopts the gradient descent method to reduce the cost function since lda derived fuzzy rules increase the discriminative capability among different classes the proposed mdsofn can classify highly confusable patterns the effectiveness of the proposed mdsofn is demonstrated by two classification problems a detailed comparative performance analysis for the fuzzy networks using lda principal component analysis pca and the support vector machine svm with various noise types is presented experimental results and theoretical analysis indicate that the lda derived fuzzy network performs better than the pca based fuzzy network and the svm based fuzzy network

ts fuzzy system based support vector regression this paper proposes a new regression model the takagi sugeno fuzzy system based support vector regression tsfs svr the tsfs svr is motivated by ts type fuzzy rules and its parameters are learned by a combination of fuzzy clustering and linear svr in contrast to a kernel based svr the tsfs svr has a smaller number of parameters while retaining the svr s good generalization ability in tsfs svr a one pass clustering algorithm clusters the input training data a new ts kernel which corresponds to a ts type fuzzy rule is then constructed by the product of a cluster output and a linear combination of input variables the tsfs svr output is a linear weighted sum of the ts kernels to achieve high generalization ability tsfs svr weights are learned through linear svr this paper demonstrates the capabilities of tsfs svr by conducting simulations in clean and noisy function approximations and signal prediction this paper also compares simulation results from the tsfs svr with gaussian kernel based svr and other learning models

identification and control of dynamic systems using recurrent fuzzy neural networks proposes a recurrent fuzzy neural network rfnn structure for identifying and controlling nonlinear dynamic systems the rfnn is inherently a recurrent multilayered connectionist network for realizing fuzzy inference using dynamic fuzzy rules temporal relations are embedded in the network by adding feedback connections in the second layer of the fuzzy neural network fnn the rfnn expands the basic ability of the fnn to cope with temporal problems in addition results for the fnn fuzzy inference engine universal approximation and convergence analysis are extended to the rfnn for the control problem we present the direct and indirect adaptive control approaches using the rfnn based on the lyapunov stability approach rigorous proofs are presented to guarantee the convergence of the rfnn by choosing appropriate learning rates finally the rfnn is applied in several simulations time series prediction identification and control of nonlinear systems the results confirm the effectiveness of the rfnn

fr3 a fuzzy rule learner for inducing reliable classifiers this paper introduces a fuzzy rule based classification method called fuzzy round robin repeated incremental pruning to produce error reduction ripper fr3 as the name suggests fr3 builds upon the ripper algorithm a state of the art rule learner more specifically in the context of polychotomous classification it uses a fuzzy extension of ripper as a base learner within a round robin scheme and thus can be seen as a fuzzy variant of the r3 learner that has recently been introduced in the literature a key feature of fr3 in comparison with its nonfuzzy counterpart is its ability to represent different facets of uncertainty involved in a classification decision in a more faithful way fr3 thus provides the basis for implementing ldquoreliable classifiersrdquo that may abstain from a decision when not being sure enough or at least indicate that a classification is not fully supported by the empirical evidence at hand besides our experimental results show that fr3 outperforms r3 in terms of classification accuracy and therefore suggest that it produces predictions that are not only more reliable but also more accurate the superb classification performance of fr3 is furthermore confirmed by comparing it to other state of the art fuzzy rule learners

learning control using fuzzified self organizing radial basis function network this note describes an approach to integrating fuzzy reasoning systems with radial basis function rbf networks and shows how the integrated network can be employed as a multivariable self organizing and self learning fuzzy controller in particular by drawing some equivalence between a simplified fuzzy control algorithm sfca and a rbf network we conclude that the rbf network can be interpreted in the context of fuzzy systems and can be naturally fuzzified into a class of more general networks referred to as fbfn with a variety of basis functions not necessarily globally radial synthesized from each dimension by fuzzy logical operators on the other hand as a result of natural generalization from rbf to sfca we claim that the fuzzy system like rbf is capable of universal approximation next the fbfn is used as a multivariable rule based controller but with an assumption that no rule base exists leading to a challenging problem of how to construct such a rule base directly from the control environment we propose a simple and systematic approach to performing this task by using a fuzzified competitive self organizing scheme and incorporating an iterative learning control algorithm into the system we have applied the approach to a problem of multivariable blood pressure control with a fbfn based controller having six inputs and two outputs representing a complicated control structure

periodic solutions of riccati equations applied to multirate sampling from some basic relations in estimation theory a comprehensive set of formulae are obtained for the periodic solution of riccati equations to obtain numerically stable algorithms square root formulations are introduced both the discrete time and continuous time cases are considered closed loop properties for periodic systems are also investigated the results are applied to multirate measurement sampling the control of a continuous stirred tank reactor serves as an illustrative example the temperature measurements are then complemented by concentration analysis available at a slower rate than the basic sampling and control rate the results give some guidelines about the benefits of multirate sampling

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

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

uncorrelated fuzzy neighborhood preserving analysis based feature projection for driver drowsiness recognition driver drowsiness is reported as one of the main causal factors in many traffic accidents as it progressively impairs the driver s awareness about external events drowsiness detection can be approached through monitoring physiological signals while driving to correlate drowsiness with the change in the corresponding patterns of the electroencephalogram eeg electrooculogram eog and electrocardiogram ecg signals the main challenge in such an approach is to extract a set of features that can highly discriminate between the different drowsiness levels this paper proposes a new fuzzy neighborhood preserving analysis fnpa feature projection method that is used to extract the discriminant information relevant to the loss of attention caused by drowsiness unlike existing methods fnpa considers the fuzzy memberships of the input measurements into the different classes while constructing the graph laplacian thus it is able to identify both the discriminant and the geometrical structure of the input data while accounting for the overlapping nature of the drowsiness patterns furthermore in order to address the singularity problem that occurs in many real world problems the singular value decomposition svd and later the qr decomposition are utilized to extract a set of statistically uncorrelated features presenting the uncorrelated fnpa ufnpa in the current preliminary study with datasets collected from 31 subjects only while performing a driving simulation task the proposed method is capable of accurately classifying the drowsiness levels using a small number of features with an average accuracy of 93 on the other hand the possibility of developing a subject independent drowsiness recognition system is also investigated when the problem is converted into a binary classification task as imposed by the number of drowsiness levels exhibited by the drivers with accuracies ranging from 82 to 84

a method for deriving the analytical structure of a broad class of typical interval type 2 mamdani fuzzy controllers type 2 t2 fuzzy controllers are emerging as the related t2 fuzzy logic and algorithms have recently been advancing rapidly at present a t2 fuzzy controller is viewed and used as a black box function generator that produces a desired nonlinear mapping between the input and output of the controller we call the mapping analytical structure the mathematical expression of the analytical structure however is not explicitly known to the controller designer this is in sharp contrast with the analytical structure of a conventional controller which is not only always explicitly known but serves as a starting point for system analysis and design obviously the knowledge of a t2 fuzzy controller s analytical structure can have significant benefits they include 1 understanding more precisely how the controller works in the same sense as we understand how a conventional controller e g the pid controller functions 2 making t2 fuzzy control more acceptable to safety critical fields such as biomedicine 3 taking advantage of the well developed nonlinear control theory to develop better analysis and design methods for t2 control systems e g less conservative system stability criteria and 4 permitting rigorous comparative exploration on differences between the t2 and type 1 t1 fuzzy controllers and their relative merits and pitfalls e g performance and structural complexity in this paper we develop an innovative technique which is capable of deriving the analytical structure for a wide class of interval t2 mamdani fuzzy controllers the configuration of the controllers is typical and is substantially more general than the related efforts in the literature it uses any number and types of interval t2 input fuzzy sets any number and types of general or interval t2 output fuzzy sets arbitrary fuzzy rules zadeh and operator karnik mendel center of sets type reducer and the centroid defuzzifier we show in detail how the derivation method works in a general setting and provide the analytical structure of an example t2 controller as well in addition we utilize the method to prove that a subset of the t2 fuzzy controllers are the sum of two nonlinear pi or pd controllers each of which has a variable proportional gain and a variable integral gain or derivative gain plus a variable offset if and only if the input fuzzy sets are piecewise linear e g triangular and or trapezoidal the sum of the two nonlinear pi or pd controllers is a new discovery relative to the current literature

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

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

a sampling expansion for nonbandlimited signals in chromatic derivatives in this paper we construct infinite band filterbanks for perfect reconstruction pr using hermite polynomials and hermite functions the analysis filters are linear combinations of derivative operators based on these polynomials the so called chromatic derivative filters together with the synthesis filterbanks they give pr for a large class of signals that may have infinite bandwidth several other related filterbanks are discussed as well the error in reconstruction for finite channel chromatic derivative filterbanks is calculated examples are given to demonstrate the use of these chromatic filterbanks

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

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

an interval type 2 neural fuzzy classifier learned through soft margin minimization and its human posture classification application this paper proposes an interval type 2 neural fuzzy classifier learned through soft margin minimization it2nfc smm and applies it to human body posture classification the it2nfc smm consists of interval type 2 zero order takagi sugeno t s fuzzy rules established through online structure learning the antecedent part of the it2nfc smm uses interval type 2 fuzzy sets to decrease the number of rules and manage noisy data for parameter learning the consequent parameters are learned through a linear support vector machine svm for soft margin minimization to improve the generalization ability the proposed svm based learning addresses the problem that the orders of the fuzzy rules in computing the outputs of an interval type 2 fuzzy system depend on the consequent values that are unknown in advance to address this problem the it2nfc smm uses weighted bound set boundaries to simplify the type reduction operation and a novel crisp to interval linear svm learning algorithm based on the soft margin minimization the antecedent parameters are tuned using the gradient descent algorithm the it2nfc smm is applied to a vision based human body posture classification system the system uses two cameras and novel classification features extracted from a silhouette of the human body to classify the four postures of standing bending sitting and lying the classification performance of the it2nfc smm is verified through results in clean and noisy classification examples and through the posture classification problem as well as through comparisons with various type 1 and type 2 fuzzy classifiers the overall result shows that the it2nfc smm achieves higher classification rates with a smaller or similar model size than the classifiers used for comparison especially for noisy classification problems

fuzzy min max neural networks i classification a supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described each fuzzy set is an aggregate union of fuzzy set hyperboxes a fuzzy set hyperbox is an n dimensional box defined by a min point and a max point with a corresponding membership function the min max points are determined using the fuzzy min max learning algorithm an expansion contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining the use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher level decision making the relationship between fuzzy sets and pattern classification is described the fuzzy min max classifier neural network implementation is explained the learning and recall algorithms are outlined and several examples of operation demonstrate the strong qualities of this new neural network classifier

fuzzy basis functions comparisons with other basis functions fuzzy basis functions fbf s which have the capability of combining both numerical data and linguistic information are compared with other basis functions because a fbf network is different from other networks in that it is the only one that can combine numerical and linguistic information comparisons are made when only numerical data is available in particular a fbf network is compared with a radial basis function rbf network from the viewpoint of function approximation their architectural interrelationships are discussed additionally a rbf network which is implemented using a regularization technique is compared with a fbf network from the viewpoint of overcoming ill posed problems a fbf network is also compared with specht s probabilistic neural network and his general regression neural network grnn from an architectural point of view a fbf network is also compared with a gaussian sum approximation in which gaussian functions play a central role finally we summarize the architectural relationships between all the networks discussed in this paper

design of natural classification kernels using prior knowledge a new class of kernels has been designed to enhance the usability of prior knowledge prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems the prior knowledge is expressed in natural language via fuzzy rules first the concepts of a fuzzy rule base and prior confidence region are proposed to formulate the prior knowledge then the new kernels which are referred to as natural classification kernels ncks are represented by fuzzy equivalence relations based on the formulation of prior knowledge an nck is interpreted as a measure of similarities between samples it is proven that ncks have two desired properties 1 transitivity with respect to the triangular norms and 2 the ability to provide higher similarities to spatially closer samples from the same class using transitivity a large number of ncks may be directly obtained by means of triangular norms additionally the theoretical results show that the second property makes it possible for the support vector machine svm and convex hull separation algorithm to generalize from training samples to test samples in the prior confidence region finally some synthetic datasets and a benchmark dataset are employed to validate the proposed approach

analytical structure and characteristics of symmetric karnik mendel type reduced interval type 2 fuzzy pi and pd controllers this paper presents the analytical structure of a class of interval type 2 it2 fuzzy proportional derivative pd and proportional integral pi controllers that have symmetrical rule base and symmetrical consequent sets two assumptions are made 1 the zadeh and operator is employed as the t norm operator 2 type reduction is performed by the karnik mendel km type reduction method the main contributions are the methodology that identifies the input conditions where the km algorithm uses a new switch point to compute the bounds of the type reduced set the closed form expressions that relate the inputs and output of an it2 fuzzy controller and insights into the potential performance improvement because of the inclusion of the footprint of uncertainty fou compared with its t1 counterpart two additional fou parameters generate 31 extra local regions each providing a unique relationship between the inputs and output signals the generation of a relatively large number of local regions at the cost of two extra design parameters indicates that an it2 fuzzy controller may be able to provide better performance furthermore by comparing the analytical structure with the corresponding t1 counterpart the potential advantages to use the it2 over the t1 fuzzy controller are studied four interesting characteristics are identified and they provide insights into why the it2 fuzzy controller may better balance the conflicting aims of fast rise time and small overshoot

a locally recurrent fuzzy neural network with support vector regression for dynamic system modeling this paper proposes a new recurrent model known as the locally recurrent fuzzy neural network with support vector regression lrfnn svr that handles problems with temporal properties structurally an lrfnn svr is a five layered recurrent network the recurrent structure in an lrfnn svr comes from locally feeding the firing strength of each fuzzy rule back to itself the consequent layer in an lrfnn svr is a takagi sugeno kang t s k type consequent which is a linear function of current states regardless of system input and output delays for the structure learning a one pass clustering algorithm clusters the input training data and determines the number of network nodes in hidden layers for the parameter learning an iterative linear svr algorithm is proposed to tune free parameters in the rule consequent part and feedback loops the motivation for using svr for parameter learning is to improve the lrfnn svr generalization ability this paper demonstrates lrfnn svr capabilities by conducting simulations in dynamic system prediction and identification problems with noiseless and noisy data in addition this paper compares simulation results from the lrfnn svr with other recurrent fuzzy models

fuzzy control theory a nonlinear case abstract sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs fuzzy control rules type of fuzzy logic used for evaluating the fuzzy control rules and the defuzzification algorithm used for the controller output we analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms we prove theoretically that such a fuzzy controller the smallest possible with two inputs error and rate change of error and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional integral pi controller with proportional gain and integral gain changing with error and rate change of error about a setpoint furthermore this fuzzy controller is precisely equivalent to a conventional linear pi controller if a linear defuzzification algorithm is employed computer simulation showed that the performance of the fuzzy controller was almost the same as that of the pi controller when first order and second order linear processes were used furthermore the fuzzy controller was significantly better when a first order with a time delay model was used more importantly the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled but the pi controller was unstable

boundary problems and periodic riccati equations we consider the resolution problem of a periodic riccati equation from the point of view of a generalized boundary problem that lets us reduce the differential problem to an algebraic equation of riccati type sufficient conditions for the existence and calculus of special types of solutions are given

a formula for the general solution of riccati type matrix differential equations abstract in this paper a closed formula for the general solution of a class of non symmetric riccati matrix differential equations in given in terms of exponentials of matrices related to the data

deriving analytical input output relationship for fuzzy controllers using arbitrary input fuzzy sets and zadeh fuzzy and operator a fuzzy controller uses either zadeh or product fuzzy and operator with the former being more frequently used than the latter we have recently published a novel technique for deriving analytical input output relation for the fuzzy controllers that use zadeh and operator and arbitrary trapezoidal input fuzzy sets including triangular ones as special cases in this paper we have developed a general technique based on that technique to cover arbitrary types of input fuzzy sets moreover we have established some necessary and sufficient conditions to characterize general relationship between shape of input fuzzy sets and shape of input space divisions an important and integral issue because analytical relationship differs in different regions of input space the new technique and the shape relations are applicable to any type of fuzzy controllers e g mamdani type or takagi sugeno type the analytical structures that we have derived provide an unprecedented opportunity to insightfully and rigorously examine the advantages and shortcomings of different design choices available for various components of the fuzzy controllers we have focused on type selection for input fuzzy sets of mamdani fuzzy controllers our preliminary analysis indicates that the fuzzy controllers using trapezoidal fuzzy sets may be understood and possibly analyzed and designed more sensibly and easily in the context of conventional control theory than the fuzzy controllers using any other types of fuzzy sets our proposition is that trapezoidal fuzzy sets should be the first choice and used most of time possible implication for automatic learning of input fuzzy sets via neural networks or genetic algorithms is briefly discussed

improving the speed of center of sets type reduction in interval type 2 fuzzy systems by eliminating the need for sorting in the deployment of interval type 2 fuzzy systems one of the most important steps is the type reduction the commonly used center of sets type reducer requires the solution of two nonlinear constrained optimization problems frequently used approaches to solve them are the karnik mendel algorithms and their variants however these algorithms suffer from the need for sorting which is known to be computationally very expensive using the reformulations proposed in this paper for center of sets type reducer it is possible to eliminate the need for sorting this makes interval type 2 fuzzy systems more appropriate for cost sensitive real time applications extensive simulations are presented to illustrate the faster convergence speed of the proposed method over six other enhanced variants of the karnik mendel algorithm as applied to center of sets type reduction of interval type 2 fuzzy systems

an incremental support vector machine trained ts type fuzzy system for online classification problems this paper proposes an incremental support vector machine trained ts type fuzzy classifier isvm fc the isvm fc is a fuzzy system that consists of takagi sugeno ts type fuzzy rules structure and parameters in the isvm fc are trained incrementally from one subset of training data at a time this incremental training approach avoids the use of large amounts of memory required for storing training data in batch learning reduces training time and adapts the classifier to time dependent classification systems where training data are available sequentially initially there are no fuzzy rules for structure learning with the isvm fc it generates all rules according to the distribution of the training data an incremental linear support vector machine svm is used to tune the resulting rule parameters to give the classifier better generalization performance the use of incremental learning discards past training data adaptively according to its distance to the linear hyperplane thereby improving learning efficiency three simulations are conducted to verify the performance of the isvm fc comparisons with fuzzy classifiers and gaussian kernel svm with batch and incremental learning modes demonstrate that the isvm fc improves training and test times and reduces memory consumption for classifier storage without deteriorating the generalization ability

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

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

a recurrent self organizing neural fuzzy inference network a recurrent self organizing neural fuzzy inference network rsonfin is proposed the rsonfin is inherently a recurrent multilayered connectionist network for realizing the basic elements and functions of dynamic fuzzy inference and may be considered to be constructed from a series of dynamic fuzzy rules the temporal relations embedded in the network are built by adding some feedback connections representing the memory elements to a feedforward neural fuzzy network each weight as well as node in the rsonfin has its own meaning and represents a special element in a fuzzy rule there are no hidden nodes initially in the rsonfin they are created online via concurrent structure identification and parameter identification the structure learning together with the parameter learning forms a fast learning algorithm for building a small yet powerful dynamic neural fuzzy network two major characteristics of the rsonfin can thus be seen 1 the recurrent property of the rsonfin makes it suitable for dealing with temporal problems and 2 no predetermination like the number of hidden nodes must be given since the rsonfin can find its optimal structure and parameters automatically and quickly moreover to reduce the number of fuzzy rules generated a flexible input partition method the aligned clustering based algorithm is proposed various simulations on temporal problems are done and performance comparisons with some existing recurrent networks are also made efficiency of the rsonfin is verified from these results

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

an interval type 2 fuzzy neural network with support vector regression for noisy regression problems this paper proposes an interval type 2 fuzzy neural network with support vector regression it2fnn svr for noisy regression problems the antecedent part in each fuzzy rule of an it2fnn svr uses interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type the use of interval type 2 fuzzy sets helps improve the network s noise resistance the network inputs may be numerical values or type 1 fuzzy sets with the latter being used for further improvements in robustness it2fnn svr learning consists of both structure learning and parameter learning the structure learning algorithm is responsible for online rule generation the parameters are optimized for structural risk minimization using a two phase linear svr algorithm in order to endow the network with high generalization ability it2fnn svr performance is verified through comparisons with type 1 and type 2 fuzzy logic systems and other regression models on noisy regression problems

support vector machines for histogram based image classification traditional classification approaches generalize poorly on image classification tasks because of the high dimensionality of the feature space this paper shows that support vector machines svm can generalize well on difficult image classification problems where the only features are high dimensional histograms heavy tailed rbf kernels of the form k x y e sup spl rho spl sigma sub i sup xia yia b with a spl les 1 and b spl les 2 are evaluated on the classification of images extracted from the corel stock photo collection and shown to far outperform traditional polynomial or gaussian radial basis function rbf kernels moreover we observed that a simple remapping of the input x sub i spl rarr x sub i sup a improves the performance of linear svm to such an extend that it makes them for this problem a valid alternative to rbf kernels

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

an online self constructing neural fuzzy inference network and its applications a self constructing neural fuzzy inference network sonfin with online learning ability is proposed in this paper the sonfin is inherently a modified takagi sugeno kang tsk type fuzzy rule based model possessing neural network learning ability there are no rules initially in the sonfin they are created and adapted as online learning proceeds via simultaneous structure and parameter identification in the structure identification of the precondition part the input space is partitioned in a flexible way according to an aligned clustering based algorithm as to the structure identification of the consequent part only a singleton value selected by a clustering method is assigned to each rule initially afterwards some additional significant terms selected via a projection based correlation measure for each rule will be added to the consequent part incrementally as learning proceeds the combined precondition and consequent structure identification scheme can set up an economic and dynamically growing network a main feature of the sonfin in the parameter identification the consequent parameters are tuned optimally by either least mean squares or recursive least squares algorithms and the precondition parameters are tuned by a backpropagation algorithm to enhance the knowledge representation ability of the sonfin a linear transformation for each input variable can be incorporated into the network so that much fewer rules are needed or higher accuracy can be achieved

generalizations of chromatic derivatives and series expansions chromatic series expansions of bandlimited functions provide an alternative representation to the whittaker shannon kotel nikov sampling series chromatic series share similar properties with taylor series insofar as the coefficients of the expansions which are called chromatic derivatives are based on the ordinary derivatives of the function chromatic derivatives are linear combinations of ordinary derivatives in which the coefficients of the combinations are related to a system of orthonormal polynomials but unlike taylor series chromatic series have more useful applications in signal processing because they represent bandlimited functions more efficiently than taylor series the goal of this article is to generalize the notion of chromatic derivatives and then extend chromatic series expansions to a larger class of signals than the class of bandlimited signals in particular we extend chromatic series to signals given by integral transforms other than the fourier transform such as the laplace and hankel transforms

improved structure optimization for fuzzy neural networks fuzzy neural network based inference systems are well known universal approximators which can produce linguistically interpretable results unfortunately their dimensionality can be extremely high due to an excessive number of inputs and rules which raises the need for overall structure optimization in the literature various input selection methods are available but they are applied separately from rule selection often without considering the fuzzy structure this paper proposes an integrated framework to optimize the number of inputs and the number of rules simultaneously first a method is developed to select the most significant rules along with a refinement stage to remove unnecessary correlations an improved information criterion is then proposed to find an appropriate number of inputs and rules to include in the model leading to a balanced tradeoff between interpretability and accuracy simulation results confirm the efficacy of the proposed method

adaptive fuzzy wavelet network control design for nonlinear systems this paper presents a new adaptive fuzzy wavelet network controller a fwnc for control of nonlinear affine systems inspired by the theory of multiresolution analysis mra of wavelet transforms and fuzzy concepts the proposed adaptive gain controller which results from the direct adaptive approach has the ability to tune the adaptation parameter in the then part of each fuzzy rule during real time operation each fuzzy rule corresponds to a sub wavelet neural network sub wnn and one adaptation parameter each sub wnn consists of wavelets with a specified dilation value the degree of contribution of each sub wnn can be controlled flexibly orthogonal least square ols method is used to determine the number of fuzzy rules and to purify the wavelets for each sub wnn since the efficient procedure of selecting wavelets used in the ols method is not very sensitive to the input dimension the dimension of the approximated function does not cause the bottleneck for constructing fwn fwn is constructed based on the training data set of the nominal system and the constructed fuzzy rules can be adjusted by learning the translation parameters of the selected wavelets and also determining the shape of membership functions then the constructed adaptive fwn controller is employed such that the feedback linearization control input can be best approximated and the closed loop stability is guaranteed the performance of the proposed a fwnc is illustrated by applying a second order nonlinear inverted pendulum system and compared with previously published methods simulation results indicate the remarkable capabilities of the proposed control algorithm it is worth noting that the proposed controller significantly improves the transient response characteristics and the number of fuzzy rules and on line adjustable parameters are reduced

design of interval type 2 fuzzy sliding mode controller in this paper an interval type 2 fuzzy sliding mode controller it2fsmc is proposed for linear and nonlinear systems the proposed it2fsmc is a combination of the interval type 2 fuzzy logic control it2flc and the sliding mode control smc which inherits the benefits of these two methods the objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed loop system the lyapunov stability method is adopted to verify the stability of the interval type 2 fuzzy sliding mode controller system the design procedure of the it2fsmc is explored in detail a typical second order linear interval system with 50 parameter variations an inverted pendulum with variation of pole characteristics and a duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method the simulation results show that the it2fsmc achieves the best tracking performance in comparison with the type 1 fuzzy logic controller t1flc the it2flc and the type 1 fuzzy sliding mode controller t1fsmc

on support vector regression machines with linguistic interpretation of the kernel matrix initially the idea of approximate reasoning using generalized modus ponens and a fuzzy implication is recalled next a fuzzy system based on logical interpretation of if then rules and with parametric conclusions is presented then it is shown that global and local e insensitive learning of the above fuzzy system may be presented as the learning of a support vector regression machine with a special type of a kernel matrix obtained from clustering the kernel matrix may be interpreted in terms of linguistic values based on the premises of if then rules a new method of obtaining a fuzzy system by means of a support vector machine svm with a data dependent kernel matrix is introduced this paper contains examples of a svm used to design fuzzy models of real life data simulation results show an improvement in the generalization ability of a fuzzy system learned by the new method compared with traditional learning methods

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

dynamic non singleton fuzzy logic systems for nonlinear modeling we investigate dynamic versions of fuzzy logic systems flss and specifically their non singleton generalizations nsflss and derive a dynamic learning algorithm to train the system parameters the history sensitive output of the dynamic systems gives them a significant advantage over static systems in modeling processes of unknown order this is illustrated through an example in nonlinear dynamic system identification since dynamic nsfls s can be considered to belong to the family of general nonlinear autoregressive moving average narma models they are capable of parsimoniously modeling narma processes we study the performance of both dynamic and static flss in the predictive modeling of a narma process

experimental study of intelligent controllers under uncertainty using type 1 and type 2 fuzzy logic uncertainty is an inherent part in control systems used in real world applications the use of new methods for handling incomplete information is of fundamental importance type 1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in control systems type 2 fuzzy sets that are used in type 2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom this paper deals with the design of control systems using type 2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise etc the experimental results are divided in two classes in the first class simulations of a feedback control system for a non linear plant using type 1 and type 2 fuzzy logic controllers are presented a comparative analysis of the systems response in both cases was performed with and without the presence of uncertainty for the second class a non linear identification problem for time series prediction is presented based on the experimental results the conclusion is that the best results are obtained using type 2 fuzzy systems

support vector based fuzzy neural network for pattern classification fuzzy neural networks fnns for pattern classification usually use the backpropagation or c cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data however such kinds of learning algorithms usually cannot minimize the empirical risk training error and expected risk testing error simultaneously and thus cannot reach a good classification performance in the testing phase to tackle this drawback a support vector based fuzzy neural network svfnn is proposed for pattern classification in this paper the svfnn combines the superior classification power of support vector machine svm in high dimensional data spaces and the efficient human like reasoning of fnn in handling uncertainty information a learning algorithm consisting of three learning phases is developed to construct the svfnn and train its parameters in the first phase the fuzzy rules and membership functions are automatically determined by the clustering principle in the second phase the parameters of fnn are calculated by the svm with the proposed adaptive fuzzy kernel function in the third phase the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method to investigate the effectiveness of the proposed svfnn classification it is applied to the iris vehicle dna satimage ijcnn1 datasets from the uci repository statlog collection and ijcnn challenge 2001 respectively experimental results show that the proposed svfnn for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

a ts type maximizing discriminability based recurrent fuzzy network for classification problems this work proposes a takagi sugeno ts type maximizing discriminability based recurrent fuzzy network mdrfn that can classify highly confusable patterns the discriminative capability plays a significant role in determining classification performance to increase the discriminative capability the proposed mdrfn considers minimum classification error mce and minimum training error mte in mce the weights are updated by maximizing the discrimination among different classes in mte the parameter learning adopts the gradient descent method to reduce the cost function therefore the novelty of mdrfn is that it not only minimizes the cost function but maximizes the discriminative capability as well the effectiveness of the proposed mdrfn is demonstrated by three temporal classification problems in the experiments other rfns including the singleton type recurrent neural fuzzy network srnfn ts type rfn trfn and simple rfn srfn are compared analysis results indicate that the proposed mdrfn exhibits excellent classification performance

feature selection for monotonic classification monotonic classification is a kind of special task in machine learning and pattern recognition monotonicity constraints between features and decision should be taken into account in these tasks however most existing techniques are not able to discover and represent the ordinal structures in monotonic datasets thus they are inapplicable to monotonic classification feature selection has been proven effective in improving classification performance and avoiding overfitting to the best of our knowledge no technique has been specially designed to select features in monotonic classification until now in this paper we introduce a function which is called rank mutual information to evaluate monotonic consistency between features and decision in monotonic tasks this function combines the advantages of dominance rough sets in reflecting ordinal structures and mutual information in terms of robustness then rank mutual information is integrated with the search strategy of min redundancy and max relevance to compute optimal subsets of features a collection of numerical experiments are given to show the effectiveness of the proposed technique

a three domain fuzzy wavelet system for simultaneous processing of time frequency information and fuzziness traditional wavelet system is a two domain time and frequency domains wavelet system 2dws which works only in time and frequency domains the 2dws is not able to treat time frequency information and fuzziness simultaneously for this reason a three domain fuzzy time and frequency domains fuzzy wavelet system 3dfws is proposed where the three domain mechanism provides a solution to handle fuzzy uncertainties and time frequency information together the major advantage of 3dfws is able to use the prior knowledge via the novel fuzzy domain to analyze uncertain data and signals which will enhance the potentials of 2dws experimental and simulation studies show that the performance of the proposed 3dfws is superior to the traditional one for simultaneous processing of time frequency and fuzziness

a fuzzy system constructed by rule generation and iterative linear svr for antecedent and consequent parameter optimization this paper proposes a new fuzzy regression model i e the fuzzy system constructed by rule generation and iterative linear support vector regression fs rglsvr for structural risk minimization the fs rglsvr is composed of takagi sugeno ts type fuzzy if then rules these rules are automatically constructed by a self splitting rule generation algorithm that introduces the self splitting technique to the k means clustering algorithm this new algorithm regards a cluster as a fuzzy rule where no preassignment of the cluster rule number is necessary the cost function for parameter learning is defined based on structural risk instead of empirical risk minimization in order to achieve generalizability tuning all of the free parameters in the fs rglsvr using linear support vector regression svr is proposed to minimize the cost function each of the consequent and antecedent part parameters is expressed as a linear combination coefficient in a transformed input space so that the linear svr is applicable this paper introduces iterative linear svr to tune antecedent and consequent parameters this paper demonstrates the capabilities of fs rglsvr by two simulated and four practical regression examples comparisons with fuzzy systems with different types of learning algorithms verify the performance of the fs rglsvr

fuzzy approximation relations on fuzzy n cell number space and their applications in classification in this paper the problems of fuzzy binary relations on fuzzy n cell number space and their applications are investigated firstly we have defined some fuzzy approximation relations on fuzzy n cell number space and studied their properties secondly as application we have developed an algorithmic version of classification in an imprecise or uncertain environment by using the fuzzy approximation relations practical examples are provided to show the application and rationality of the proposed techniques

solving algebraic and differential riccati operator equations abstract by application of the riesz dunford functional calculus a method for solving generalized algebraic riccati operator equations in hilbert space is given results are applied to the problem of finding explicit solutions for cauchy problems and two point boundary value problems for generalized riccati operator differential equations

fuzzy one class support vector machines in one class classification the problem is to distinguish one class of data from the rest of the feature space it is important in many applications where one of the classes is characterized well while no measurements are available for the other class scholkopf et al first introduced a method of adapting the support vector machine svm methodology to the one class classification problem called one class svm in this paper we incorporate the concept of fuzzy set theory into the one class svm we apply a fuzzy membership to each input point and reformulate the one class svm such that different input points can make different contributions to the learning of decision surface besides the parameters to be identified in one class svm such as the components within the weight vector and the bias term are fuzzy numbers this integration preserves the benefits of svm learning theory and fuzzy set theory where the svm learning theory characterizes the properties of learning machines which enable them to effectively generalize the unseen data and the fuzzy set theory might be very useful for finding a fuzzy structure in an evaluation system

derivation and analysis of the analytical structures of the interval type 2 fuzzy pi and pd controllers research results on type 2 t2 fuzzy control have started to emerge in the literature over the past several years none of these results however are concerned with the explicit input output mathematical structure of a t2 fuzzy controller as the literature on type 1 t1 fuzzy control has demonstrated revealing such structure information is important as it will deepen our precise understanding of how and why t2 fuzzy controllers function in the context of control theory and lay a foundation for more rigorous system analysis and design in this paper we derive the mathematical structure of two mamdani interval t2 fuzzy proportional integral pi controllers that use the following identical elements two interval t2 triangular input fuzzy sets for each of the two input variables four singleton t1 output fuzzy sets a zadeh and operator and the center of sets type reducer one controller employs the popular centroid defuzzifier while the other employs a new defuzzifier that we propose which is called the average defuzzifier the advantages of using the latter defuzzifier are given which include the fact that the derivation method originally developed by us in previous papers for the t1 fuzzy controllers can be directly adopted for the t2 controller and the results are general with respect to the design parameters this is not the case for the other t2 controller for which we have developed a novel derivation approach partially depending on numerical computations our derivation results prove explicitly both controllers to be nonlinear pi controllers with variable gains i e the expressions are different we analyze the gain variation characteristics and extend these findings to the corresponding t2 fuzzy proportional derivative pd controllers our new results are consistent with the relevant structure results on the t1 fuzzy pi and pd controllers in the literature and contain them as special cases we discuss how the new structure information can be utilized to design and tune the t2 controllers even when the mathematical model of the system to be controlled is unknown neither derivation method is restrictive only to the t2 controllers in this paper they are directly applicable to other t2 controllers with more complex configurations

a general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and zadeh and operator deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding insightful analysis and more effective design of fuzzy control systems we previously developed a technique for deriving the analytical structure of the fuzzy controllers that use zadeh fuzzy and operator and the symmetric identical trapezoidal or triangular input fuzzy sets many fuzzy controllers use arbitrary trapezoidal triangular input fuzzy sets that are asymmetric at present there exists no technique capable of deriving the analytical structure of these fuzzy controllers extending our original technique we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller mamdani type or ts type that employs the arbitrary trapezoidal input fuzzy sets and zadeh fuzzy and operator the new technique contains our original technique as a special case given the importance of pid control we focus on mamdani fuzzy pi and pd controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy pi pd controllers the controllers use two arbitrary trapezoidal fuzzy sets for each input variable four arbitrary singleton output fuzzy sets four fuzzy rules zadeh fuzzy and operator and the centroid defuzzifier this configuration is more general and complicated than the mamdani fuzzy pi pd controllers in the current literature it actually contains them as special cases we call this configuration the generalized fuzzy pi pd controller

chromatic derivatives and local approximations we present a detailed motivation for the notions of chromatic derivatives and chromatic expansions chromatic derivatives are special numerically robust linear differential operators chromatic expansions are the associated local expansions which possess the best features of both the taylor and the nyquist expansions we give a simplified treatment of some of the basic properties of chromatic derivatives and chromatic expansions which are relevant for applications we also consider some signal spaces with a scalar product defined by a cesaro type sum of products of chromatic derivatives as well as an approximation of such a scalar product which is relevant for signal processing we also introduce a new kind of local approximations based on trigonometric functions