design for self organizing fuzzy neural networks based on genetic algorithms a novel hybrid learning algorithm based on a genetic algorithm to design a growing fuzzy neural network named self organizing fuzzy neural network based on genetic algorithms sofnnga to implement takagi sugeno ts type fuzzy models is proposed in this paper a new adding method based on geometric growing criterion and the epsiv completeness of fuzzy rules is first used to generate the initial structure then a hybrid algorithm based on genetic algorithms backpropagation and recursive least squares estimation is used to adjust all parameters including the number of fuzzy rules this has two steps first the linear parameter matrix is adjusted and second the centers and widths of all membership functions are modified the ga is introduced to identify the least important neurons i e the least important fuzzy rules simulations are presented to illustrate the performance of the proposed algorithm

a new gradient descent approach for local learning of fuzzy neural models the majority of reported learning methods for takagi sugeno kang tsk fuzzy neural models to date mainly focus on improvement of their accuracy however one of the key design requirements in building an interpretable fuzzy model is that each obtained rule consequent must match well with the system local behavior when all the rules are aggregated to produce the overall system output this is one of the distinctive characteristics from black box models such as neural networks therefore how to find a desirable set of fuzzy partitions and hence identify the corresponding consequent models which can be directly explained in terms of system behavior presents a critical step in fuzzy neural modeling in this paper a new learning approach considering both nonlinear parameters in the rule premises and linear parameters in the rule consequents is proposed unlike the conventional two stage optimization procedure widely practiced in the field where the two sets of parameters are optimized separately the consequent parameters are transformed into a dependent set on the premise parameters thereby enabling the introduction of a new integrated gradient descent learning approach thus a new jacobian matrix is proposed and efficiently computed to achieve a more accurate approximation of the cost function by using the second order levenberg marquardt optimization method several other interpretability issues regarding the fuzzy neural model are also discussed and integrated into this new learning approach numerical examples are presented to illustrate the resultant structure of the fuzzy neural models and the effectiveness of the proposed new algorithm and compared with the results from some well known methods

cyclic self dual codes it is shown that if the automorphism group of a binary self dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 in particular no cyclic binary self dual code can have all its weights divisible by four the number of cyclic binary self dual codes of length n is determined and the shortest nontrivial code in this class is shown to have length 14

a ga based fuzzy modeling approach for generating tsk models this paper proposes a new genetic based modeling method for building simple and well defined tsk models with scatter type input partitions our approach manages all attributes characterizing the structure of a tsk model simultaneously particularly it determines the number of rules the input partition the participating inputs in each rule and the consequent parameters the model building process is divided into two phases in phase one the structure learning task is formulated as a multi objective optimization problem which is resolved using a novel genetic based structure learning gbsl scheme apart from the mean square error mse and the number of rules three additional criteria are introduced in the fitness function for measuring the quality of the partitions optimization of these measures leads to models with representative rules small overlapping and efficient data cover in order to obtain models with accurate data fitting and good local performance the consequent parameters are determined using a local mse function while the overall model is evaluated on the basis of a global mse function the search capabilities of the suggested structure learning scheme are significantly enhanced by including a highly effective local search operator implemented by a micro genetic algorithm and four problem specific operators finally a genetic based parameter learning gbpl scheme is suggested in phase two which performs fine tuning of the initial models obtained after structure learning the performance of the proposed modeling approach is evaluated using a static example and a well known dynamic benchmark problem simulation results demonstrate that our models outperform those suggested by other methods with regard to simplicity model structure and accuracy

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

evolving fuzzy classifiers using different model architectures in this paper we present two novel approaches for on line evolving fuzzy classifiers called eclass and flexfis class both methods can be applied with different model architectures including single model sm with class labels as consequents classification hyper planes as consequents and multi model mm architecture additionally eclass can have a multi input multi output mimo architecture with multiple hyper planes as consequents of each fuzzy rule the difference between mm and mimo architectures is that the former one applies one separate and independent fuzzy rule based frb classifier for each class and is using an indicator labelling scheme while the latter one applies a single frb where the rules are mimo rather than miso both eclass and flexfis class methods are designed to work on a per sample basis and are thus one pass incremental additionally their structure frb is evolving rather than fixed it adapts their parameters in antecedent and consequent parts with any newly loaded sample a special emphasis is placed on advanced issues for improving accuracy and robustness including a thorough comparison between global and local learning of consequent functions a novel approach for detecting of and reacting on drifts in the data streams and an enhanced outlier treatment strategy the methods and their extensions according to the advanced issues are evaluated on one benchmark problem of handwritten images recognition as well as on a real life problem of image classification framework where images should be classified into good and bad ones during an on line and interactive production process

a strengthening of the assmus mattson theorem let w sub 1 d w sub 2 w sub s be the weights of the nonzero codewords in a binary linear n k d code c and let w sub 1 w sub 2 w sub 3 be the nonzero weights in the dual code c1 let t be an integer in the range 0 or d 4 then either the words of any nonzero weight w sub i form a t 1 design or else the codewords of minimal weight d form a 1 2 t t 2 design if in addition c is self dual with all weights divisible by 4 then the codewords of any given weight w sub i form either a t 1 design or a 1 2 t t 2 design the proof avoids the use of modular forms

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

using an adaptive neuro fuzzy inference system based interpolant for impulsive noise suppression from highly distorted images a new impulsive noise in suppression filter entitled adaptive neuro fuzzy inference system anfis based impulsive noise suppression filter which shows a high performance at the restoration of images distorted by in is proposed in this paper the extensive simulation results show that the proposed filter achieves a superior performance to the other filters mentioned in this paper in the cases of being effective in noise suppression and detail preservation especially when the noise density is very high

neuro fuzzy system with learning tolerant to imprecision in this paper a new learning method tolerant to imprecision is introduced and used in neuro fuzzy modeling this method can be called e insensitive learning where in order to fit the fuzzy model to real data a weighted e insensitive loss function is used the proposed method makes it possible to exclude an intrinsic inconsistency of neuro fuzzy modeling where zero tolerance learning is used to obtain a fuzzy model tolerant to imprecision the e insensitive learning leads to a model with the minimal vapnik chervonenkis dimension complexity which results in improving generalization ability of this system and its robustness to outliers finally numerical examples are given to demonstrate the validity of the introduced method

duadic codes a new family of binary cyclic n n 1 2 and n n 1 2 codes are introduced which include quadratic residue qr codes when n is prime these codes are defined in terms of their idempotent generators and they exist for all odd n p 1 a 1 p 2 a 2 cdots p r a r where each p i is a prime equiv pm 1 pmod 8 dual codes are identified the minimum odd weight of a duadic n n 1 2 code satisfies a square root bound when equality holds in the sharper form of this bound vectors of minimum weight hold a projective plane the unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent 73 37 9 duadic codes all duadic codes of length less than 127 are identified and the minimum weights of their extensions are given one of the duadic codes of length 113 has greater minimum weight than the qr code of that length

transposition networks as a class of fault tolerant robust networks the paper proposes designs of interconnection networks graphs which can tolerate link failures the networks under study belong to a subclass of cayley graphs whose generators are subsets of all possible transpositions we specifically focus on star and bubble sort networks our approach is to augment existing dimensions or generators with one or more dimensions if the added dimension is capable of replacing any arbitrary failed dimension it is called a wildcard dimension it is shown that up to isomorphism among digits used in labeling the vertices the generators of the star graph are unique the minimum number of extra dimensions needed to acquire i wildcard dimensions is derived for the star and bubble sort networks interestingly the optimally augmented star network coincides with the transposition network t sub n transposition networks are studied rigorously these networks are shown to be optimally fault tolerant t sub n is also shown to possess wide containers with short length fault diameter of t sub n is shown to be n while the t can efficiently embed star and bubble sort graphs it can also lend itself to an efficient embedding of meshes and hypercubes

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

some self dual codes invariant under the hall janko group abstract we construct self dual codes of length 100 invariant under the hall janko group j 2 we also construct a 2 100 22 168 design with the automorphism group j 2 2 which is the extension of j 2 by its outer automorphism

on the application of orthogonal transformation for the design and analysis of feedforward networks orthogonal transformation which can lead to compaction of information has been used in two ways to optimize on the size of feedforward networks 1 through the selection of optimum set of time domain inputs and the optimum set of links and nodes within a neural network nn and 2 through the orthogonalization of the data to be used in nn s in case of processes with periodicity the proposed methods are efficient and are also extremely robust numerically the singular value decomposition svd and qr with column pivoting factorization qrcp are the transformations used svd mainly serves as the null space detector qrcp coupled with svd is used for subset selection which is one of the main operations on which the design of the optimal network is based svd has also been used to devise a new approach for the assessment of the convergence of the nn s which is an alternative to the conventional output error analysis

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

learning fuzzy rules from iterative execution of games this paper discusses the linguistic knowledge extraction from the iterative execution of a multiplayer non cooperative repeated game linguistic knowledge is automatically extracted in the form of fuzzy if then rules our knowledge extraction is mainly based on the on line incremental learning of fuzzy rule based systems in this sense our linguistic knowledge extraction is the learning of fuzzy rules we first briefly describe a market selection game which is formulated as a non cooperative repeated game with many players and several alternative actions we also describe some simple strategies for our market selection game in our market selection game the payoff of each player depends on the actions of all players when a particular action is chosen by many players those players receive low payoff high payoff is obtained from actions chosen by only a small number of players this means that minority players with respect to their actions receive high payoff next we show how our market selection game can be handled as a pattern classification problem where a single training pattern is successively generated from every round of our game a fuzzy rule based classification system is used as a decision making system by each player for choosing an action in every round an on line incremental learning algorithm is proposed for adjusting the fuzzy rule based classification system then we show how our market selection game can be handled as a function approximation problem a fuzzy rule based approximation system is used as a value function for approximating the expected payoff from each action finally simulation results show that comprehensible linguistic knowledge is extracted by the learning of fuzzy rule based systems

how to determine the minimum number of fuzzy rules to achieve given accuracy a computational geometric approach to siso case how to construct fuzzy systems using as less as possible rules withguaranteed performance is a difficult but important problem in this paper a computational geometry approach is introduced to determine the minimum number of rules required in building a fuzzy model to achieve a given approximation accuracy from the input output data of an unknown nonlinear system withsingle input and single output the basic idea is to partition system input domain in a non uniform manner according to the sampling data distribution and the approximation error tolerance by borrowing concepts and tools from computational geometry the problem is formulated and transformed into an edge visibility problem and a tunnel algorithm is used to find the minimum rule number numerical examples are given to illustrate the ideas difficulties and potentials are discussed in extending to the multi input case 2004 elsevier b v all rights reserved

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

flexfis a robust incremental learning approach for evolving takagi sugeno fuzzy models in this paper we introduce a new algorithm for incremental learning of a specific form of takagi sugeno fuzzy systems proposed by wang and mendel in 1992 the new data driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents but also the incremental learning of premise parameters appearing in the membership functions fuzzy sets together with a rule learning strategy in sample mode a modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules premise parts the modifications include an automatic generation of new clusters based on the nature distribution and quality of new data and an alternative strategy for selecting the winning cluster rule in each incremental learning step antecedent and consequent learning are connected in a stable manner meaning that a convergence toward the optimal parameter set in the least squares sense can be achieved an evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high dimensional data recorded at engine test benches and at rolling mills for the latter the obtained data driven fuzzy models are even compared with an analytical physical model furthermore a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three input nonlinear function approximation example

rspop rough set based pseudo outer product fuzzy rule identification algorithm system modeling with neuro fuzzy systems involves two contradictory requirements interpretability verses accuracy the pseudo outer product pop rule identification algorithm used in the family of pseudo outer product based fuzzy neural networks popfnn suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high dimensional data this decreases the interpretability of the popfnn in linguistic fuzzy modeling this article proposes a novel rough set based pseudo outer product rspop algorithm that integrates the sound concept of knowledge reduction from rough set theory with the pop algorithm the proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes as many possible reducts exist in a given rule set an objective measure is developed for popfnn to correctly identify the reducts that improve the inferred consequence experimental results are presented using published data sets and real world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the popfnn using compositional rule of inference and singleton fuzzifier popfnn cri s architecture results showed that the proposed rough set based pseudo outer product algorithm reduces computational complexity improves the interpretability of neuro fuzzy systems by identifying significantly fewer fuzzy rules and improves the accuracy of the popfnn

optimal self dual codes over f sub 2 spl times f sub 2 with respect to the hamming weight in this paper we study optimal self dual codes and type iv self dual codes over the ring f sub 2 spl times f sub 2 of order 4 we give improved upper bounds on minimum hamming and lee weights for such codes using the bounds we determine the highest minimum hamming and lee weights for such codes of lengths up to 30 we also construct optimal self dual codes and type iv self dual codes

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

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

using uncorrupted neighborhoods of the pixels for impulsive noise suppression with anfis in this paper a novel adaptive network based fuzzy inference system anfis based filter abf is presented for the restoration of images corrupted by impulsive noise in the abf is performed in two steps in the first step impulse detection is realized by using statistical tools in the second step a nonlinear filtering scheme based on anfis is performed for only the corrupted pixels detected in the first step to demonstrate the effectivity of abf at the removal of high level in extensive simulations were realized for abf and nine different comparison filters empirical results indicate that the proposed filter achieves a better performance than the comparison filters in terms of noise suppression and detail preservation even when the images are highly corrupted by in

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

shadow codes and weight enumerators the technique of using shadow codes to build larger self dual codes is extended to codes over arbitrary fields it is shown how to build the codes and how to determine the new weight enumerator as well for codes over fields equipped with a square root of 1 and not of characteristic 2 a self dual code of length n 2 can be built from a self dual code of length n for codes over a field without a square root of 1 and not of characteristic 2 a self dual code of length n 4 is built from a self dual code of length n and for codes over fields of characteristic 2 the length of the new self dual code depends on the presence of the all one vector in the subcode chosen in certain cases using the subcode of vectors orthogonal to the all one vector the new weight enumerator can be calculated directly from the original weight enumerator specific examples of the technique are illustrated for codes over f sub 3 f sub 4 and f sub 5

new extremal self dual codes of lengths 36 38 and 58 we construct new extremal self dual binary codes of lengths 36 38 and 58 we show that there are at least 14 inequivalent extremal self dual 38 18 8 codes and that there are at least 368 inequivalent extremal self dual 38 19 8 codes for length 58 we construct 11 extremal self dual 58 29 10 codes whose weight enumerators were previously unknown

constructing l2 svm based fuzzy classifiers in high dimensional space with automatic model selection and fuzzy rule ranking in this paper a new scheme for constructing parsimonious fuzzy classifiers is proposed based on the l2 support vector machine l2 svm technique with model selection and feature ranking performed simultaneously in an integrated manner in which fuzzy rules are optimally generated from data by l2 svm learning in order to identify the most influential fuzzy rules induced from the svm learning two novel indexes for fuzzy rule ranking are proposed and named as alpha values and omega values of fuzzy rules in this paper the alpha values are defined as the lagrangian multipliers of the l2 svm and adopted to evaluate the output contribution of fuzzy rules while the omega values are developed by considering both the rule base structure and the output contribution of fuzzy rules as a prototype based classifier the l2 svm based fuzzy classifier evades the curse of dimensionality in high dimensional space in the sense that the number of support vectors which equals the number of induced fuzzy rules is not related to the dimensionality experimental results on high dimensional benchmark problems have shown that by using the proposed scheme the most influential fuzzy rules can be effectively induced and selected and at the same time feature ranking results can also be obtained to construct parsimonious fuzzy classifiers with better generalization performance than the well known algorithms in literature

topological properties of star graphs abstract our purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n cubes or hypercubes these properties are interesting by themselves from a graph theory point of view as well as they have direct in generating vertex disjoint paths and minimal paths in a star graph they can also be readily utilized to design routing algorithms and to compute contention and traffic congestion in networks that use star graphs as the underlying topology

packing radius covering radius and dual distance tietaivainen 1991 derived an upper bound on the covering radius of codes as a function of the dual distance this was generalized to the minimum distance and to q polynomial association schemes by levenshtein and fazekas both proofs use a linear programming approach in particular levenshtein and fazekas 1990 use linear programming bounds for codes and designs in this article proofs relying solely on the orthogonality relations of krawtchouk 1929 lloyd and more generally krawtchouk adjacent orthogonal polynomials are derived as a by product upper bounds on the minimum distance of formally self dual binary codes are derived

fault diameter of interconnection networks abstract we introduce the concept of fault diameter of interconnection networks the motivation is to estimate the degradation of performance under maximally fault conditions we derive fault diameter of specific classes of commonly used interconnection networks and various types of product graphs

on the minimal weight of some singly even codes it is shown that the minimal distance d of a singly even self dual 24t 8 12t 4 code is at most 4t 2 if its shadow contains a weight 4 vector t is even and sub t sup 5t is odd it is proved particularly that there does not exist a 56 28 12 singly even self dual code with 4862 words of weight 12 this answers a question raised by conway and sloane see ibid vol 36 p 1319 33 1990

estimating the concentration of optically active constituents of sea water by takagi sugeno models with quadratic rule consequents determining the concentrations of chlorophyll suspended particulate matter and coloured dissolved organic matter in the sea water is basic to support the monitoring of upwelling phenomena algae blooms and changes in the marine ecosystem since these concentrations affect the spectral distribution of the solar light back scattered by the water body their estimation can be computed by using a set of remotely sensed multispectral measurements of the reflected sunlight in this paper the relation between the concentrations of interest and the average subsurface reflectances is modelled by means of a set of second order takagi sugeno ts fuzzy rules unlike first order ts rules which adopt linear functions as consequent second order ts rules exploit quadratic functions thus improving the modelling capability of the rule in the subspace determined by the antecedent first we show how we can build a second order ts model through a simple transformation which allows estimating the consequent parameters using standard linear least squares algorithms and by adopting one of the most used methods proposed in the literature to generate first order ts models then we compare first order and second order ts models against mean square error and interpretability of rules we highlight how second order ts models allow us to achieve better approximation than first order ts models though maintaining interpretability of the rules finally we show how second order ts models perform considerably better the mean square error is lower by two orders of magnitude than the specific implementations of radial basis function networks and multi layer perceptron networks used in previous papers for the same application domain

takagi sugeno fuzzy modeling incorporating input variables selection fuzzy models especially takagi sugeno t s fuzzy models have received particular attention in the area of nonlinear modeling due to their capability to approximate any nonlinear behavior based only on measured data without any prior knowledge there is no systematic way to obtain a t s fuzzy model with a simple structure and sufficient accuracy the main idea discussed in this paper is to reduce the complexity of t s fuzzy models by estimating an optimal number of fuzzy rules and selecting relevant inputs as antecedent variables independently of the selection of consequent regressors a systematic procedure is proposed here and illustrated on static and dynamical nonlinear systems

a new upper bound for the minimum of an integral lattice of determinant 1 let be an n dimensional integral lattice of determinant 1 we show that for all sufficiently large n the minimal nonzero squared length in does not exceed n 6 10 this bound is a consequence of some new conditions on the theta series of these lattices these conditions also enable us to find the greatest possible minimal squared length in all dimensions n 33 in particular we settle the no roots problem there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n 23 n 25 there are also analogues of all these results for codes

lmi based design of constrained fuzzy predictive control predictive control of nonlinear systems subject to output and input constraints is considered a fuzzy model is used to predict the future behavior two new ideas are proposed here first an added constraint on the applied control action is used to ensure the decrease of a quadratic lyapunov function and so guarantee lyapunov exponential stability of the closed loop system second the feasibility of the finite horizon optimization problem with the added constraints is ensured based on an off line solution of a set of lmis the novel stability method is compared to the existing methods such as the techniques based on the end point constraints terminal constraint set and the robust stability techniques based on the small gain theory the proposed method ensures lyapunov exponential stability does not need an auxiliary controller and can be used with any feasible controller parameters illustrative examples including the predictive control of a highly nonlinear chemical reactor cstr are discussed

cubic self dual binary codes we study binary self dual codes with a fixed point free automorphism of order three all binary codes of that type can be obtained by a cubic construction that generalizes turyn s we regard such cubic codes of length 3 spl lscr as codes of length spl lscr over the ring f sub 2 spl times f sub 4 classical notions of type ii codes shadow codes and weight enumerators are adapted to that ring two infinite families of cubic codes are introduced new extremal binary codes in lengths spl les 66 are constructed by a randomized algorithm necessary conditions for the existence of a cubic 72 36 16 type ii code are derived

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

on the weight enumerators of duadic and quadratic residue codes in this correspondence we compute the weight enumerators of various quadratic residue codes over f sub 2 and f sub 3 together with certain codes of related families like the duadic and the quadratic double circulant codes we use a parallel algorithm to find the number of codewords of a given not too high weight from which we deduce by usual classical methods for self dual and formally self dual codes over f sub 2 and f sub 3 their associated previously unknown weight enumerators we compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes

a hierarchical knowledge based environment for linguistic modeling models and iterative methodology although linguistic models are highly descriptive they suffer from inaccuracy in some complex problems this fact is due to problems related to the inflexibility of the linguistic rule structure that has been considered moreover methods often employed to design these models from data are also biased by the former structure and by their nature which is close to prototype identification algorithms in order to deal with these problems of linguistic modeling an extension of the knowledge base of linguistic fuzzy rule based systems was previously introduced i e the hierarchical knowledge base hkb ieee trans fuzzy systems 10 1 2002 2 hierarchical linguistic fuzzy models derived from this structure are viewed as a class of local modeling approaches they attempt to solve a complex modeling problem by decomposing it into a number of simpler linguistically interpretable subproblems from this perspective linguistic modeling using an hkb can be regarded as a search for a decomposition of a non linear system that gives a desired balance between the interpretability and the accuracy of the model using this approach we are able to effectively explore the fact that the complexity of the systems is usually not uniform we propose a well defined hierarchical environment adopting a more general treatment than the typical prototype oriented learning methods this iterative hierarchical methodology takes the hkb as a base and performs a wide variety of linguistic modeling more specifically from fully interpretable to fully accurate as well as intermediate trade offs hierarchical linguistic models with the aim of analyzing the behavior of the proposed methodology two real world electrical engineering distribution problems from spain have been selected successful results were obtained in comparison with other system modeling techniques

a method for constructing inequivalent self dual codes with applications to length 56 necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes as an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 sixteen such codes are inequivalent

classification of extremal and s extremal binary self dual codes of length 38 in this paper we classify all extremal and s extremal binary self dual codes of length 38 there are exactly 2744 extremal self dual codes two s extremal codes and 1730 s extremal codes we obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self dual codes of length 36 and from a generalization of this recursive algorithm for the shadow the classification of extremal codes permits to achieve the classification of all extremal codes with

multiobjective identification of takagi sugeno fuzzy models the problem of identifying the parameters of the constituent local linear models of takagi sugeno fuzzy models is considered in order to address the tradeoff between global model accuracy and interpretability of the local models as linearizations of a nonlinear system two multiobjective identification algorithms are studied particular attention is paid to the analysis of conflicts between objectives and we show that such information can be easily computed from the solution of the multiobjective optimization this information is useful to diagnose the model and tune the weighting priorities of the multiobjective optimization moreover the result of the conflict analysis can be used as a constructive tool to modify the fuzzy model structure including membership functions in order to meet the multiple objectives simple illustrative examples as well as experimental results show the usefulness of the method

constructing accurate and parsimonious fuzzy models with distinguishable fuzzy sets based on an entropy measure parsimony is very important in system modeling as it is closely related to model interpretability in this paper a scheme for constructing accurate and parsimonious fuzzy models by generating distinguishable fuzzy sets is proposed in which the distinguishability of input space partitioning is measured by a so called local entropy by maximizing this entropy measure the optimal number of merged fuzzy sets with good distinguishability can be obtained which leads to a parsimonious input space partitioning while preserving the information of the original fuzzy sets as much as possible different from the existing merging algorithms the proposed scheme takes into account the information provided by input output samples to optimize input space partitioning furthermore this scheme possesses the ability to seek a balance between the global approximation ability and distinguishability of input space partitioning in constructing takagi sugeno ts fuzzy models experimental results have shown that this scheme is able to produce accurate and parsimonious fuzzy models with distinguishable fuzzy sets

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

multivariable fuzzy inference system for fingerprinting indoor localization the emergence of wireless sensor network has raised the need for cheap wireless indoor localization technique this paper considers the problem of fingerprinting indoor localization based on signal strength measurements rss a new approach based on fuzzy logic has been put forward the proposal makes use of k nearest neighbor classification in signal space the localization of target node is then determined as a weighted combination of nearest fingerprints the weights are determined using takagi sugeno fuzzy controller with two inputs a new enhancement to the knn is proposed to enhance the accuracy of location estimation this enhancement allows the knn to outlier some miss elected neighbors based on triangular area measurements the performance of the developed estimation algorithm has been evaluated using both monte carlo simulations and real testbed scenarios while compared to other alternative approaches

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

on ternary self dual codes of length 24 a partial classification is given of the self dual codes of length 24 over gf 3 the main results are as follows there are exactly two codes with minimum hamming distance d 9 most of the codes have d 6 and are indecomposable one code with d 6 has a trivial automorphism group this is the first such self dual code that has been found the codes generated by the 59 inequivalent 24 times 24 hadamard matrices have been investigated and there appear to be only nine inequivalent codes two with d 9 and seven with d 6 and in all there are 27 decomposable codes at least 96 indecomposable codes with d 6 and the total number of inequivalent codes is at least 140

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

the maximum partition matching problem with applications let cal s c1 c2 ck be a collection of pairwise disjoint subsets of u 1 2 n such that bigcup i 1 k ci u a partition matching of cal s consists of two subsets a1 am and b1 bm of u together with a sequence of distinct partitions of cal s cal a 1 cal b 1 ldots cal a m cal b m such that ai is contained in a subset in the collection cal a i and bi is contained in a subset in the collection cal b i for all i 1 m an efficient algorithm is developed that constructs a maximum partition matching for a given collection cal s the algorithm can be used to construct optimal parallel routing between two nodes in interconnection networks

a characterization of a 56 28 extremal self dual code corresp a 54 27 code is characterized in terms of a 27 times 27 matrix made up of circulant 3 times 3 matrices the minimum distance of this code is found to be 11 and its weight distribution is computed the 54 27 d 11 code is extended to a 56 28 d 12 external self dual code a 54 27 d 10 self dual code is also described in terms of the incidence matrix of a 27 13 6 group difference set defined on gf 3 3

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

a 72 36 16 doubly even code does not have an automorphism of order 11 corresp we prove that there does not exist a 72 36 16 doubly even code with an automorphism of order 11 if such a code exists it can be decomposed as a direct sum of two codes one can be viewed as a self dual 12 6 4 binary code and the other a self dual 6 3 4 code over f 1024 this is shown to be impossible

adaptive t s fuzzy neural modeling and control for general mimo unknown nonaffine nonlinear systems using projection update laws this paper describes a novel design of an on line takagi sugeno t s fuzzy neural controller for a class of general multiple input multiple output mimo systems with unknown nonlinear functions and external disturbances instead of modeling the unknown systems directly the t s fuzzy neural model approximates a virtual linearized system vls of a real system with modeling errors and external disturbances compared with previous approaches the main contribution of this paper is an investigation of more general mimo unknown systems using on line adaptive t s fuzzy neural controllers in this paper we also use projection update laws which generalize the projection algorithm to tune the adjustable parameters this prevents parameter drift and ensures that the parameter matrix is bounded away from singularity we prove that the closed loop system controlled by the proposed controller is robust stable and the effect of all the modeling errors and external disturbances on the tracking error can be attenuated finally two examples covering four cases are simulated in order to confirm the effectiveness and applicability of the proposed approach in this paper

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

tase a taylor series based fuzzy system model that combines interpretability and accuracy typically takagi sugeno kang tsk fuzzy rules have been used as a powerful tool for function approximation problems since they have the capability of explaining complex relations among variables using rule consequents that are functions of the input variables but they present the great drawback of the lack of interpretability which makes them not to be so suitable for a wide range of problems where interpretability of the obtained model is a fundamental key in this paper we present a novel approach that extends the work by bikdash ieee trans fuzzy systems 7 6 1999 686 696 in order to obtain an interpretable and accurate model for function approximation from a set of i o data samples which make use of the taylor series expansion of a function around a point to approximate the function using a low number of rules our approach also provides an automatic methodology for obtaining the optimum structure of our taylor series based tase fuzzy system as well as its pseudo optimal rule parameters both antecedents and consequents

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

the covering radius of extremal self dual code d11 and its application d11 constructed by conway and sloane is an extremal self dual 56 28 12 code with weight enumerator 1 8190y sup 12 622314y sup 16 11699688y sup 20 we find that the covering radius of d11 is 10 this shows that the inequality r c sub 56 spl les 10 cannot be improved any more where r c sub 56 is the covering radius of any extremal doubly even 56 28 12 code c sub 56 and we use it to get ten new 58 29 10 codes

tsk fuzzy modeling based on spl epsiv insensitive learning in this paper new learning methods tolerant to imprecision are introduced and applied to fuzzy modeling based on the takagi sugeno kang fuzzy system the fuzzy modeling has an intrinsic inconsistency it may perform thinking tolerant to imprecision but learning methods are zero tolerant to imprecision the proposed methods make it possible to exclude this intrinsic inconsistency of a fuzzy modeling where zero tolerance learning is used to obtain fuzzy model tolerant to imprecision these new methods can be called spl epsiv insensitive learning or spl epsiv learning where in order to fit the fuzzy model to real data the spl epsiv insensitive loss function is used this leads to a weighted or fuzzified version of vapnik s support vector regression machine this paper introduces two approaches to solving the spl epsiv insensitive learning problem the first approach leads to the quadratic programming problem with bound constraints and one linear equality constraint the second approach leads to a problem of solving a system of linear inequalities two computationally efficient numerical methods for the spl epsiv insensitive learning are proposed the spl epsiv insensitive learning leads to a model with the minimal vapnik chervonenkis dimension which results in an improved generalization ability of this model and its outliers robustness finally numerical examples are given to demonstrate the validity of the introduced methods

affine ts model based fuzzy regulating servo control design an affine t s fuzzy system is derived naturally from linearizing a nonlinear system or from some data driven identification techniques for example cluster based algorithms or soft computation learning structures little research is proposed for the intrinsic analysis of an affine type fuzzy system further the controllers to regulate or to achieve servo control of affine ts based nonlinear systems are also few both affine type fuzzy regulation and servo control design scheme are theoretically derived a simple lyapunov based stability criterion and some extra conditions are proposed to guarantee the global stability of the generated closed loop fuzzy systems the exponential stability of the feedback fuzzy system is ensured under some conditions the performance of the proposed fuzzy controllers and fuzzy servo controllers is examined by three case studies simulation results show that the proposed controllers can stabilize these affine fuzzy systems and the proposed servomechanism can adapt itself to various incoming signals in very short time spans

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen