computing derivatives in interval type 2 fuzzy logic systems this paper makes type 2 fuzzy logic systems much more accessible to fuzzy logic system designers because it provides mathematical formulas and computational flowcharts for computing the derivatives that are needed to implement steepest descent parameter tuning algorithms for such systems it explains why computing such derivatives is much more challenging than it is for a type 1 fuzzy logic system it provides derivative calculations that are applicable to any kind of type 2 membership functions since the calculations are performed without prespecifying the nature of those membership functions some calculations are then illustrated for specific type 2 membership functions

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

corrections to an automatic approach for learning and tuning gaussian interval type 2 fuzzy membership functions applied to lung cad classification system the potential of type 2 fuzzy sets to manage high levels of uncertainty in the subjective knowledge of experts or of numerical information has focused on control and pattern classification systems in recent years one of the main challenges in designing a type 2 fuzzy logic system fls is how to estimate the parameters of the type 2 fuzzy membership function t2mf and the footprint of uncertainty fou from imperfect and noisy datasets this paper presents an automatic approach to learn and tune gaussian interval type 2 membership functions it2mfs with application to multidimensional pattern classification problems t2mfs and their fous are tuned according to the uncertainties in the training dataset by a combination of genetic algorithm ga and cross validation techniques in our ga based approach the structure of the chromosome has fewer genes than other ga methods and chromosome initialization is more precise the proposed approach addresses the application of the interval type 2 fuzzy logic system it2fls for the problem of nodule classification in a lung computer aided detection system the designed it2fls is compared with its type 1 fuzzy logic system t1fls counterpart the results demonstrate that the it2fls outperforms the t1fls by more than 30 in terms of classification accuracy

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

granular neural networks with evolutionary interval learning to deal with different membership functions of the same linguistic term a new interval reasoning method using new granular sets is proposed based on yin yang methodology to make interval valued granular reasoning efficiently and optimize interval membership functions based on training data effectively a granular neural network gnn with a new high speed evolutionary interval learning is designed simulation results in nonlinear function approximation and bioinformatics have shown that the gnn with the evolutionary interval learning is able to extract interval valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm

adaptive hybrid type 2 intelligent sliding mode control for uncertain nonlinear multivariable dynamical systems a new adaptive hybrid interval type 2 fuzzy neural network fnn controller incorporating sliding mode and lyapunov synthesis approaches is proposed in this paper to handle the training data corrupted by noise or rule uncertainties for a class of uncertain nonlinear multivariable dynamic systems the hybrid adaptive fnn controller the free parameters of which can be tuned on line by an output feedback control law and adaptive laws is a combination of interval type 2 indirect and direct adaptive fnn controllers to meet the requirement of sufficient reach for the sliding mode control a weighting factor which can be adjusted based on the trade off between plant knowledge and control knowledge is included when combining the control efforts of the indirect adaptive fnn controller and the direct adaptive fnn controller the overall adaptive control scheme guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded the mass spring damper nonlinear system is fully illustrated to track sinusoidal signals the resulting adaptive hybrid interval type 2 fnn control system shows better performance than the adaptive hybrid type 1 fnn control system it reduces both the tracking error and the control effort and it is more flexible in the design process

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

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

characterisation of lp norms via holder s inequality abstract we characterise l p norms on the space of integrable step functions defined on a probabilistic space via a holder type of inequality with an optimality condition

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

quasi monotonicity subadditive bijections of r and characterization of lp norm abstract real functions with a regular behaviour of the one sided upper and lower limits are considered the main result says that every subadditive bijection of r right continuous at 0 must be a homeomorphism of r an application to a characterization of lp norm is also given

the pexider type generalization of the minkowski inequality abstract let be a measure space such that 0 a 1 b for some a b the following converse minkowski inequality theorem is proved in matkowski 2008 4 if 0 0 are bijective is increasing and 1 x y x y d 1 x x d 1 y y d for all nonnegative integrable simple functions x y r where x stands for the support of x then there exists a real p 1 such that t 1 t 1 t 1 t p in the present paper we show that if in the basic measure space there is no a such that either 1 a or 0 a 1 then there are some broad classes of non power functions which satisfy the above minkowski type inequality moreover we prove that in the converse of the minkowski inequality theorem the assumption of the increasing monotonicity of is essential

an enhanced type reduction algorithm for type 2 fuzzy sets karnik and mendel proposed an algorithm to compute the centroid of an interval type 2 fuzzy set efficiently based on this algorithm liu developed a centroid type reduction strategy to carry out type reduction for type 2 fuzzy sets a type 2 fuzzy set is decomposed into a collection of interval type 2 fuzzy sets by cuts then the karnik mendel algorithm is called for each interval type 2 fuzzy set iteratively however the initialization of the switch point in each application of the karnik mendel algorithm is not a good one in this paper we present an improvement to liu s algorithm we employ the previously obtained result to construct the starting values in the current application of the karnik mendel algorithm convergence in each iteration except the first one can then speed up and type reduction for type 2 fuzzy sets can be carried out faster the efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results

analysis of data generated from multidimensional type 1 and type 2 fuzzy membership functions due to the numerous applications that utilize different types of fuzzy membership functions mfs it may sometimes be difficult to choose an appropriate mf for a particular application in this paper we establish preliminary guidelines to direct this selection by proposing a three stage method in the forward stage different mfs such as crisp mfs type 1 t1 fuzzy mfs and type 2 t2 fuzzy mfs are generated from multidimensional data sets next in the reverse stage data is generated back from these mfs by considering different bin sizes in doing so various data sets may be generated for different applications which require fuzzy data finally for the similarity analysis stage we propose an iterative algorithm that makes use of the results of wilcoxon signed rank wsr and wilcoxon rank sum wrs tests to compare the original data and the generated data from the results of these tests recommendations concerning the suitability of mfs for a specific application may be suggested by observing the accuracy of representation and the requirements of the application with this analysis the objective is to gain insight on when t2 fuzzy sets may be considered to outperform t1 fuzzy sets and vice versa several examples are provided using synthetic and real data to validate the iterative algorithm for data sets in various dimensions

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

interaction measures for control configuration selection based on interval type 2 takagi sugeno fuzzy model interaction measure determines decentralized and sparse control configurations for a multivariable process control this paper investigates interval type 2 takagi sugeno fuzzy it2tsf model based interaction measures using two different criteria one is controllability and observability gramians the other is relative normalized gain array rnga the main contributions are first a data driven it2tsf modeling method is introduced second explicit formulas to execute the two measures based on it2tsf models are given third two interaction indexes are defined from rnga to select sparse control configuration fourth the calculations to derive sensitivities of the two measures with respect to parametric variations in the it2tsf models are developed and fifth the discussion to compare the two measures is presented three multivariable processes are used as examples to show that the results calculated from it2tsf models are more accurate than that from their type 1 counterparts and compared to gramian based measure rnga selects more reasonable control configurations and is more robust to the parametric uncertainties

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

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

numerical integration as a finite matrix approximation to multiplication operator abstract in this article numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite dimensional basis the idea is to approximate the continuous spectral representation of a multiplication operator on a hilbert space with a discrete spectral representation of a hermitian matrix the gaussian quadrature is shown to be a special case of the new method the placement of the nodes of numerical integration and convergence of the new method are studied

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

multiobjective optimization and comparison of nonsingleton type 1 and singleton interval type 2 fuzzy logic systems singleton interval type 2 fuzzy logic systems flss have been widely applied in several real world applications where it was shown that the singleton interval type 2 flss outperform their singleton type 1 counterparts in applications with high uncertainty levels however one of the main criticisms of singleton interval type 2 flss is the fact that they outperform singleton type 1 flss solely based on their use of extra degrees of freedom extra parameters and that type 1 flss with a sufficiently large number of parameters may provide the same performance as interval type 2 flss in addition most works on type 2 flss only compare their results with singleton type 1 flss but fail to consider nonsingleton type 1 systems in this paper we aim to directly address and investigate this criticism in order to do so we will perform a comparative study between optimized singleton type 1 nonsingleton type 1 and singleton interval type 2 flss under the presence of noise we will also present a multiobjective evolutionary algorithm moea for the optimization of singleton type 1 nonsingleton type 1 and singleton interval type 2 fuzzy systems for function approximation problems the moea will aim to satisfy two objectives to maximize the accuracy of the fls and minimize the number of rules in the fls thus improving its interpretability furthermore we will present a methodology to obtain optimal consequents for the flss hence this paper has two main contributions first it provides a common methodology to learn the three types of flss i e singleton type 1 nonsingleton type 1 and singleton interval type 2 flss from data samples the second contribution is the creation of a common framework for the comparison of type 1 and type 2 flss that allows us to address the aforementioned criticism we provide details of a series of experiments and include statistical analysis showing that the type 2 fls is able to handle higher levels of noise than its nonsingleton and singleton type 1 counterparts

an interval type 2 neural fuzzy system for online system identification and feature elimination we propose an integrated mechanism for discarding derogatory features and extraction of fuzzy rules based on an interval type 2 neural fuzzy system nfs in fact it is a more general scheme that can discard bad features irrelevant antecedent clauses and even irrelevant rules high dimensional input variable and a large number of rules not only enhance the computational complexity of nfss but also reduce their interpretability therefore a mechanism for simultaneous extraction of fuzzy rules and reducing the impact of or eliminating the inferior features is necessary the proposed approach namely an interval type 2 neural fuzzy system for online system identification and feature elimination it2nfs sife uses type 2 fuzzy sets to model uncertainties associated with information and data in designing the knowledge base the consequent part of the it2nfs sife is of takagi sugeno kang type with interval weights the it2nfs sife possesses a self evolving property that can automatically generate fuzzy rules the poor features can be discarded through the concept of a membership modulator the antecedent and modulator weights are learned using a gradient descent algorithm the consequent part weights are tuned via the rule ordered kalman filter algorithm to enhance learning effectiveness simulation results show that it2nfs sife not only simplifies the system architecture by eliminating derogatory irrelevant antecedent clauses rules and features but also maintains excellent performance

second order sliding fuzzy interval type 2 control for an uncertain system with real application a new second order sliding mode type 2 fuzzy controller for nonlinear uncertain perturbed systems is developed in this paper to overcome the constraint on the knowledge of the system model we have used local models that are related to some operating points to synthesize a type 2 nominal fuzzy global model the control is based on the super twisting algorithm which is among second order sliding mode controls smcs moreover two adaptive fuzzy type 2 systems have been introduced to generate the two super twisting signals to avoid both the chattering and the constraint on the knowledge of disturbances and uncertainties upper bounds these adaptive fuzzy type 2 systems are adjusted online by two adaptation laws that are deduced from the stability analysis in the lyapunov sense it has only one input i e the sliding surface and one output i e the optimal values of the gains control which are hard to compute with the original algorithm many results of the one link manipulator are obtained first by the simulation in order to compare the performances of the proposed method with that given by levant and then in a real time application in order to confirm the efficiency of the proposed approach the experimentation and simulation are done for the tracking control problem

an extension to fuzzy cognitive maps for classification and prediction fuzzy cognitive maps fcms as an illustrative causative representation of modeling and manipulation of complex systems can be used to model the dynamic behavior of the investigated systems however due to defects in expression and architecture the traditional fcms and most of their relevant extensions are not applicable to classification problems to solve this problem this paper presents an approach that directly extends the model by translating the reasoning mechanism of traditional fcms to a set of fuzzy if then rules moreover the proposed approach fully considers the contribution of the inputs to the activation of the fuzzy rules and quantifies the causalities using mutual subsethood which works in conjunction with volume defuzziflcation in a gradient descent learning framework in this manner our approach enhances the capability of the conventional fcms to automatically identify membership functions and quantify causalities despite the increase in the number of tunable parameters experimental results show that the proposed approach efficiently extends the application of the traditional fcms into classification problems while keeping the ability for prediction and approximation

genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction tuning fuzzy rule based systems for linguistic fuzzy modeling is an interesting and widely developed task it involves adjusting some of the components of the knowledge base without completely redefining it this contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions to adjust the former component we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability to alter the latter component two different approaches changing their basic parameters and using nonlinear scaling factors are proposed as the accomplished experimental study shows the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance the paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process a good interpretability accuracy tradeoff is obtained combining both processes with a sequential scheme first reducing the rule set and subsequently tuning the model

interval type 2 mutual subsethood fuzzy neural inference system it2msfunis this paper presents an interval type 2 mutual subsethood fuzzy neural inference system it2msfunis a mutual subsethood measure between two interval type 2 fuzzy sets it2 fs has been derived and has been used in determining the similarity between the it2 fs inputs and it2 fs antecedents the consequent weights are taken to be interval sets the inputs to the system are fuzzified into it2 fss with gaussian primary membership function having fixed center and uncertain variance aggregation of type 2 mutual subsethood based activation spreads is performed using product operator the output is obtained using simplified type reduction followed by defuzzification the system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space the mathematical modeling and empirical studies of it2msfunis bring forth its efficacy in problems pertaining to function approximation time series prediction control and classification comparisons with other type 1 and type 2 neuro fuzzy systems verify that it2msfunis compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy empirical studies indicate the intelligent decision making capability of the proposed model the main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between it2 fss and to find out its applicability in diverse application domains the improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of it2 fss the integration of mutual subsethood with interval type 2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well known problems reported in the literature

matrix methods for quadrature formulas on the unit circle a survey in this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle like nodes and weights of gauss quadrature formulas for the estimation of integrals with respect to measures on the real line can be computed from the eigenvalue decomposition of the jacobi matrix szeg quadrature formulas for the approximation of integrals with respect to measures on the unit circle can be obtained from certain unitary five diagonal or unitary hessenberg matrices that characterize the recurrence for an orthogonal laurent polynomial basis these quadratures are exact in a maximal space of laurent polynomials orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles more general szeg quadrature formulas can be obtained that are exact in certain spaces of rational functions in this context the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator mobius transform of the same five diagonal or hessenberg matrices

converse theorem for the minkowski inequality let a measure space such that 0 a 1 b for some a b under some natural conditions on the bijective functions 1 2 1 2 0 0 we prove that if x y x y d 1 x 1 xd 2 y 2 yd for all nonnegative integrable simple functions x y r where x stands for the support of x then there exists a real p 1 such that t 1 i t i 1 tp t 1 i t i 1 t1 p i 1 2 some generalizations and relevant results for the reversed inequality are also presented

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

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 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

design of a new adaptive neuro fuzzy inference system based on a solution for clustering in a data potential field in this study we propose a new method for building adaptive neuro fuzzy inference systems anfis via datasets in order to improve the performance of conventional anfis to handle noisy data we focus on ameliorating the cluster data space established from a given dataset to achieve this we propose a weighted clustering process in the joint input output data space thus during the clustering process the cluster with the smallest potential distance which is a combination of the euclidean distance and the size of the clusters has priority when obtaining the surveyed sample based on this principle we formulate a new algorithm for synthesizing an anfis via the proposed data potential field called anfis pf which has the following features it establishes a data potential field that covers the entire initial data space a cluster data space is built based on the generated data potential field and the anfis is synthesized using this cluster data space finally we performed experiments using datasets with and without noise to demonstrate the effectiveness of the proposed method in several applications including dynamic response noisy datasets obtained from a magnetorheological damper

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

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

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

study on enhanced karnik mendel algorithms initialization explanations and computation improvements computing the centroid of an interval type 2 fuzzy set is an important operation in a type 2 fuzzy logic system and is usually implemented by karnik mendel km iterative algorithms by connecting km algorithms and continuous km algorithms together this paper gives theoretical explanations on the initialization methods of km and enhanced karnik mendel ekm algorithms proposes exact methods for centroid computation of an interval type 2 fuzzy set and extends the enhanced karnik mendel ekm algorithms to three different forms of weighted ekm wekm algorithms it shows that ekm algorithms become a special case of the wekm algorithms when the weights of the latter are constant value it also shows that in general the weighted ekm algorithms have smaller absolute error and faster convergence speed than the ekm algorithms which make them very attractive for real time applications of fuzzy logic system four numerical examples are used to illustrate and analyze the performance of wekm algorithms

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

data based system modeling using a type 2 fuzzy neural network with a hybrid learning algorithm we propose a novel approach for building a type 2 neural fuzzy system from a given set of input output training data a self constructing fuzzy clustering method is used to partition the training dataset into clusters through input similarity and output similarity tests the membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster then a type 2 fuzzy takagi sugeno kang if then rule is derived from each cluster to form a fuzzy rule base a fuzzy neural network is constructed accordingly and the associated parameters are refined by a hybrid learning algorithm which incorporates particle swarm optimization and a least squares estimation for a new input a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type 2 fuzzy set which is then defuzzified by applying a refined type reduction algorithm experimental results are presented to demonstrate the effectiveness of our proposed approach

similarity measures for general type 2 fuzzy sets based on the α plane representation abstract similarity measures are very important concepts in fuzzy sets fss theory there are many different definitions of similarity measures for both type 1 t1 fss and interval type 2 it2 fss in this paper one similarity measure for it2 fss which is extended from the jaccard similarity measure for t1 fss is reviewed then based on the plane representation for a general type 2 gt2 fs this similarity measure is generalized to such t2 fss some examples that demonstrate how to compute the similarity measures for different t2 fss are given

orthogonal polynomials on the unit circle distribution of zeros departamento de matemritica aplicada escuela tpcnica superior de ingenieros industriales universidad de vigo apartado de correos no 62 36280 vigo spain received 30 august 1990 revised 29 january 1991 abstract marcellan f and e godoy orthogonal polynomials on the unit circle distribution of zeros journal of computational and applied mathematics 37 1991 195 208 in this paper we summarize some results concerning zeros of orthogonal polynomials with respect to an indefinite inner product we analyze the inverse problem i e a discrete representation for the functional in terms of the n th orthogonal polynomial finally some numerical examples are shown keywords toeplitz matrices orthogonal polynomials on the unit circle zeros quadrature formulas

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

long term learning for type 2 neural fuzzy systems abstract the development of a new long term learning framework for interval valued neural fuzzy systems is presented for the first time in this article the need for such a framework is twofold to address continuous batch learning of data sets and to take advantage the extra degree of freedom that type 2 fuzzy logic systems offer for better model predictive ability the presented long term learning framework uses principles of granular computing grc to capture information knowledge from raw data in the form of interval valued sets in order to build a computational mechanism that has the ability to adapt to new information in an additive and long term learning fashion the latter is to accommodate new input output mappings and new classes of data without significantly disturbing existing input output mappings therefore maintaining existing performance while creating and integrating new knowledge rules this is achieved via an iterative algorithmic process which involves a two step operation iterative rule base growth capturing new knowledge and iterative rule base pruning removing redundant knowledge for type 2 rules the two step operation helps create a growing but sustainable model structure the performance of the proposed system is demonstrated using a number of well known non linear benchmark functions as well as a highly nonlinear multivariate real industrial case study simulation results show that the performance of the original model structure is maintained and it is comparable to the updated model s performance following the incremental learning routine the study is concluded by evaluating the performance of the proposed framework in frequent and consecutive model updates where the balance between model accuracy and complexity is further assessed

spectral methods for orthogonal rational functions abstract we present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure two alternatives are discussed leading to representations which are linear fractional transformations with matrix coefficients acting on infinite hessenberg or five diagonal unitary matrices this approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions besides the zeros of the orthogonal and para orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite hessenberg or five diagonal matrices as an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions

holder minkowski type inequality for generalized sugeno integral abstract we give a necessary and sufficient condition for validity of the holder minkowski type inequality for the generalized sugeno integral any monotone measure and the class of associated functions we also show a connection between integral inequalities and the problem of finding binary operations satisfying the distributivity inequalities we characterize all semicopulas satisfying holder and minkowski inequalities for the seminormed fuzzy integral one result has been obtained for the recently introduced q integral we describe the relationships among the existing sufficient conditions and show that our result generalizes most known holder minkowski type inequalities for comonotone functions in the available literature we also provide two other methods to get the above mentioned inequality for the sugeno integral and the class of m subadditive functions as well as for the sugeno integral of 0 1 valued functions using a duality approach we point out some flaws and correct the corresponding false statements appearing in the literature at the end we pose several problems and questions for further research

an evolving interval type 2 neurofuzzy inference system and its metacognitive sequential learning algorithm in this paper we propose an evolving interval type 2 neurofuzzy inference system it2fis and its fully sequential learning algorithm it2fis employs interval type 2 fuzzy sets in the antecedent part of each rule and the consequent realizes takagi sugeno kang fuzzy inference mechanism in order to render the inference fast and accurate we propose a data driven interval reduction approach to convert interval type 1 fuzzy set in antecedent to type 1 fuzzy number in the consequent during learning the sequential algorithm learns a sample one by one and only once the it2fis structure evolves automatically and adapts its network parameters using metacognitive learning mechanism concurrently the metacognitive learning regulates the learning process by appropriate selection of learning strategies and helps the proposed it2fis to approximate the input output relationship efficiently an evolving it2fis employing a metacognitive learning algorithm is referred to as mcti2fis performance of metacognitive interval type 2 neurofuzzy inference system mcit2fis is evaluated using a set of benchmark time series problems and is compared with existing type 2 and type 1 fuzzy inference systems finally the performance of the proposed mcit2fis has been evaluated using a practical stock price tracking problem the results clearly highlight that mcit2fis performs better than other existing results in the literature

uncertain fuzzy clustering interval type 2 fuzzy approach to c means in many pattern recognition applications it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms therefore various types of uncertainty may be taken into account when performing several pattern recognition methods when one performs clustering with fuzzy sets fuzzy membership values express assignment availability of patterns for clusters however when one assigns fuzzy memberships to a pattern set imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment when one encounters fuzzy clustering fuzzy membership design includes various uncertainties e g distance measure fuzzifier prototypes etc in this paper we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final c partition in the fuzzy c means fcm algorithm to design and manage uncertainty for fuzzifier m we extend a pattern set to interval type 2 fuzzy sets using two fuzzifiers m1 and m2 which creates a footprint of uncertainty fou for the fuzzifier m then we incorporate this interval type 2 fuzzy set into fcm to observe the effect of managing uncertainty from the two fuzzifiers we also provide some solutions to type reduction and defuzzification i e cluster center updating and hard partitioning in fcm several experimental results are given to show the validity of our method