exact closed form performance analysis of optimum combining with multiple cochannel interferers and rayleigh fading a new closed form expression is derived for the exact bit error probability bep for optimum combining with binary phase shift keying the exact bep expression is for multiple equal power cochannel interferers and multiple reception branches it is assumed that the aggregate interference and noise is gaussian and that both the desired signal and interference are subject to rayleigh fading the derivation starts by expressing the optimum combining decision statistic as a sum of quadratic forms of gaussian random variables and it proceeds to average over the fading interference the new bep expression has low complexity as it contains only finite sums and products

upper bounds on the bit error rate of optimum combining in wireless systems this paper presents upper bounds on the bit error rate ber of optimum combining in wireless systems with multiple cochannel interferers in a rayleigh fading environment we present closed form expressions for the upper bound on the bit error rate with optimum combining for any number of antennas and interferers with coherent detection of bpsk and qam signals and differential detection of dpsk we also present bounds on the performance gain of optimum combining over maximal ratio combining these bounds are asymptotically tight with decreasing ber and results show that the asymptotic gain is within 2 db of the gain as determined by computer simulation for a variety of cases at a 10 sup 3 ber the closed form expressions for the bound permit rapid calculation of the improvement with optimum combining for any number of interferers and antennas as compared with the cpu hours previously required by monte carlo simulation thus these bounds allow calculation of the performance of optimum combining under a variety of conditions where it was not possible previously including analysis of the outage probability with shadow fading and the combined effect of adaptive arrays and dynamic channel assignment in mobile radio systems

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

performance analysis of optimum combining with multiple interferers in flat rayleigh fading this paper is a performance analysis of optimum combining in the presence of multiple equal power interferers and noise when the number of interferers is less than the number of antenna elements desired signal and interferers are subject to flat rayleigh fading and the propagation channels are independent an approximate expression of the probability density function of the output signal to interference plus noise ratio sinr is derived analytically it is then applied to obtain the cumulative distribution function of the sinr and the bit error rate ber of some binary modulations including coherent binary phase shift keying in the case of a single interferer an exact analysis is performed to prove the validity of the approximation in the case of multiple interferers the accuracy of the approximation is assessed through simulations although limited to equipower interferers this analysis is a convenient way of assessing the performance of optimum combining in some typical situations and comparing it with that of maximal ratio combining the final results are remarkably simple and provide a useful complement to previous analyzes especially in the region of reasonably high bers which are of practical interest

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

fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

algebraic isomonodromic deformations of logarithmic connections on the riemann sphere and finite braid group orbits on character varieties we study algebraic isomonodromic deformations of flat logarithmic connections on the riemann sphere with n ge 4 poles for arbitrary rank we introduce a natural property of algebraizability for the germ of universal deformation of such a connection we relate this property to a peculiarity of the corresponding monodromy representation to yield a finite braid group orbit on the appropriate character variety under reasonable assumptions on the deformed connection we may actually establish an equivalence between both properties we apply this result in the rank two case to relate finite branching and algebraicity for solutions of garnier systems for general rank a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension

characteristic functions for optimum combining output sinr with awgn and correlated interference optimum combining oc is a powerful means for combatting co channel interference its performance however is difficult to analyze the prevailing methodology in current use is to neglect the influence of additive white gaussian noise awgn inclusion of awgn into the analysis usually leads to mathematical intractability except for some extreme cases in this letter we formulate the problem in the framework of hypergeometric functions so that their good properties can be exploited to enable the analysis to account for the influence of awgn and co channel interference simultaneously we obtain closed form solutions to the characteristic function chf of the signal to interference plus noise ratio at the oc output for three important operational scenarios in rayleigh fading although the general solution still needs further effort to simplify the assumptions made here are that both signal and interferers suffer from rayleigh fading and the number of interferers is not less than the number of antennas the use of resulting chf formulas is illustrated through their application to the bit error and outage performance evaluation of oc

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

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

stability analysis and control of discrete type 1 and type 2 tsk fuzzy systems part i stability analysis this paper introduces sufficient conditions for the exponential stability of type 1 and type 2 takagi sugeno kang tsk fuzzy systems a major advantage of the new conditions is that they do not require the existence of a common lyapunov function and are therefore applicable to systems with unstable consequents in addition our results include two classes of type 2 tsk systems with type 1 consequents for which no stability tests are available the use of the conditions in stability testing is demonstrated using simple numerical examples that include cases where methods that are based on a common lyapunov function fail the application of the stability test to develop new controller design methodologies is presented in a separate paper i e part ii

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

a dynamically constrained multiobjective genetic fuzzy system for regression problems in this paper a multiobjective genetic fuzzy system gfs to learn the granularities of fuzzy partitions tuning the membership functions mfs and learning the fuzzy rules is presented it uses dynamic constraints which enable three parameter mf tuning to improve the accuracy while guaranteeing the transparency of fuzzy partitions the fuzzy models fms are initialized by a method that combines the benefits of wang mendel wm and decision tree algorithms thus the initial fms have less rules rule conditions and input variables than if wm initialization were to be used moreover the fuzzy partitions of initial fms are always transparent our approach is tested against recent multiobjective and monoobjective gfss on six benchmark problems it is concluded that the accuracy and interpretability of our fms are always comparable or better than those in the comparative studies furthermore on some benchmark problems our approach clearly outperforms some comparative approaches suitability of our approach for higher dimensional problems is shown by studying three benchmark problems that have up to 21 input variables

fuzzy modeling of high dimensional systems complexity reduction and interpretability improvement fuzzy modeling of high dimensional systems is a challenging topic this paper proposes an effective approach to data based fuzzy modeling of high dimensional systems an initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema redundant rules are removed based on a fuzzy similarity measure then the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method during optimization rules that have a very low firing strength are deleted finally interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization the resulting fuzzy system generated by this method has the following distinct features 1 the fuzzy system is quite simplified 2 the fuzzy system is interpretable and 3 the dependencies between the inputs and the output are clearly shown this method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data

on the solution of a second order linear homogeneous difference equation with variable coefficients abstract a closed form solution of a second order linear homogeneous difference equation with variable coefficients is presented as an application of this solution we obtain expressions for cos n and sin n 1 sin as polynomials in cos

asymptotical stability of partial difference equations with variable coefficients the dynamical systems considered have scalar state are multivariate linear time discrete and time variable and are described by an initial value problem for a class of evolutionary partial difference equations the time set is the nonnegative part of the integer lattice in several dimensions parts of the asymptotical stability set in the parameter space spanned by the time variable coefficients are explicitly found to assess the quality of the sufficient stability criteria a comparison with the exact stability set is made in an example

enhanced interval approach for encoding words into interval type 2 fuzzy sets and its convergence analysis construction of interval type 2 fuzzy set models is the first step in the perceptual computer which is an implementation of computing with words the interval approach ia has so far been the only systematic method to construct such models from data intervals that are collected from a survey however as pointed out in this paper it has some limitations and its performance can be further improved this paper proposes an enhanced interval approach eia and demonstrates its performance on data that are collected from a web survey the data part of the eia has more strict and reasonable tests than the ia and the fuzzy set part of the eia has an improved procedure to compute the lower membership function we also perform a convergence analysis to answer two important questions 1 does the output interval type 2 fuzzy set from the eia converge to a stable model as increasingly more data intervals are collected and 2 if it converges then how many data intervals are needed before the resulting interval type 2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals we show that the eia converges in a mean square sense and generally 30 data intervals seem to be a good compromise between cost and accuracy

reduced interval type 2 neural fuzzy system using weighted bound set boundary operation for computation speedup and chip implementation this paper proposes a reduced interval type 2 neural fuzzy system using weighted bound set boundaries rit2nfs wb for the simplification of type reduction operations the objective of this simplification is to reduce the system training time in software implementation and chip size in hardware implementation especially when the number of rules is large the antecedent part in the rit2nfs wb uses interval type 2 fuzzy sets it2fss and the consequent part can be of the takagi sugeno kang tsk or mamdani type the rit2nfs wb is built through an online structure and parameter learning to improve model accuracy in addition the interpretability of the rit2nfs wb is improved by considering distributions of the it2fss in input variables a distinguishability oriented cost function is used in parameter learning to generate distinguishable it2fss and improve semantics based interpretability for highly overlapped it2fss they are merged to reduce the number of it2fss and improve complexity based interpretability the software implemented tsk type rit2nfs wb is hardware implemented on a field programmable gate array chip to accelerate the chip execution speed the chip utilizes not only the parallel execution properties of fuzzy rules and bound set boundaries but the pipeline technique as well in particular the flexibility of the chip is considered so that no redesign of the circuits is required when the rit2nfs wb is applied to different problems the characteristics of the software and hardware implemented rit2nfs wb are verified through various examples and comparisons with various type 1 and interval type 2 fuzzy models

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

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

products of 2 2 matrices related to non autonomous fibonacci difference equations a technique to compute arbitrary products of a class of fibonacci 2x2 square matrices is proved in this work general explicit solutions for non autonomous fibonacci difference equations are obtained from these products in the periodic non autonomous fibonacci difference equations the monodromy matrix the floquet multipliers and the binet s formulas are obtained in the periodic case explicit solutions are obtained and the solutions are analyzed

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

on ordinary difference equations with variable coefficients abstract a solution representation in matrix form is achieved for all orders of the title equation sufficient criteria for the asymptotical stability and the bounded input bounded output stability of the solution sequence are presented our solution representation is confronted with a fully explicit representation the latter is derived in a simple way and also generalized to a class of partial difference equation with variable coefficients

fuzzy interpolation and extrapolation a practical approach fuzzy interpolation does not only help to reduce the complexity of fuzzy models but also makes inference in sparse rule based systems possible it has been successfully applied to systems control but limited work exists for its applications to tasks like prediction and classification almost all fuzzy interpolation techniques in the literature make strong assumptions that there are two closest adjacent rules available to the observation and that such rules must flank the observation for each attribute also some interpolation approaches cannot handle fuzzy sets whose membership functions involve vertical slopes to avoid such limitations and develop a more practical approach this paper extends the work of huang and shen the result enables both interpolation and extrapolation which involve multiple fuzzy rules with each rule consisting of multiple antecedents two realistic applications namely truck backer upper control and computer activity prediction are provided in this paper to demonstrate the utility of the extended approach experiment based comparisons to the most commonly used mamdani fuzzy reasoning mechanism and to other existing fuzzy interpolation techniques are given to show the significance and potential of this research

on constructing parsimonious type 2 fuzzy logic systems via influential rule selection type 2 fuzzy systems are increasing in popularity and there are many examples of successful applications while many techniques have been proposed for creating parsimonious type 1 fuzzy systems there is a lack of such techniques for type 2 systems the essential problem is to reduce the number of rules while maintaining the system s approximation performance in this paper four novel indexes for ranking the relative contribution of type 2 fuzzy rules are proposed which are termed r values c values omega1 values and omega2 values the r values of type 2 fuzzy rules are obtained by applying a qr decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model the c values rank rules based on the effects of rule consequents while the omega1 values and omega2 values consider both the rule base structure via firing strength matrices and the output contribution of fuzzy rule consequents two procedures for utilizing these indexes in fuzzy rule selection termed ldquoforward selectionrdquo and ldquobackward eliminationrdquo are described experiments are presented which demonstrate that by using the proposed methodology the most influential type 2 fuzzy rules can be effectively retained in order to construct parsimonious type 2 fuzzy models

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems we derive inner and outer bound sets for the type reduced set of an interval type 2 fuzzy logic system fls based on a new mathematical interpretation of the karnik mendel iterative procedure for computing the type reduced set the bound sets can not only provide estimates about the uncertainty contained in the output of an interval type 2 fls but can also be used to design an interval type 2 fls we demonstrate by means of a simulation experiment that the resulting system can operate without type reduction and can achieve similar performance to one that uses type reduction therefore our new design method based on the bound sets can relieve the computation burden of an interval type 2 fls during its operation which makes an interval type 2 fls useful for real time applications

on product of companion matrices this paper describes an explicit combinatorial formula for the product of companion matrices the method relies on the connections between matrix algebra and associated combinatorial structures to enumerate the paths in an unweighted digraph as an application we obtain bases for the solution space of the linear difference equation with variable coefficients

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

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

noninteracting adaptive control of pmsm using interval type 2 fuzzy logic systems based on the type 2 fuzzy logic systems and adaptive control theory this paper develops a decentralized adaptive fuzzy controller for a permanent magnet synchronous motor pmsm the adaptive law is designed to compensate for the interconnection effects and the reconstruction errors using a sliding mode term the type 2 fuzzy logic systems are used to model the unknown functions based on lyapunov s stability theorem the stability of closed loop systems can be verified moreover the proposed overall control schemes guarantee that all the signals that are involved are bounded to demonstrate the effectiveness of the proposed methods simulation results are illustrated in this paper

a generalized concept for fuzzy rule interpolation the concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993 it has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction the first implemented technique of fuzzy rule interpolation was termed as spl alpha cut distance based fuzzy rule base interpolation despite its advantageous properties in various approximation aspects and in complexity reduction it was shown that it has some essential deficiencies for instance it does not always result in immediately interpretable fuzzy membership functions this fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies this paper is an attempt into this direction it proposes an interpolation methodology whose key idea is based on the interpolation of relations instead of interpolating spl alpha cut distances and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques the proposed concept of interpolating relations is elaborated here using fuzzy and semantic relations this paper presents numerical examples in comparison with former approaches to show the effectiveness of the proposed interpolation methodology

type 2 fuzzy hidden markov models and their application to speech recognition this paper presents an extension of hidden markov models hmms based on the type 2 t2 fuzzy set fs referred to as type 2 fuzzy hmms t2 fhmms membership functions mfs of t2 fss are three dimensional and this new third dimension offers additional degrees of freedom to evaluate the hmms fuzziness therefore t2 fhmms are able to handle both random and fuzzy uncertainties existing universally in the sequential data we derive the t2 fuzzy forward backward algorithm and viterbi algorithm using t2 fs operations in order to investigate the effectiveness of t2 fhmms we apply them to phoneme classification and recognition on the timit speech database experimental results show that t2 fhmms can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical hmms

sofmls online self organizing fuzzy modified least squares network in this paper an online self organizing fuzzy modified least square sofmls network is proposed the algorithm has the ability to reorganize the model and adapt itself to a changing environment where both the structure and learning parameters are performed simultaneously the network generates a new rule if the smallest distance between the new data and all the existing rules the winner rule is more than a prespecified radius the major contributions of this paper are as follows 1 a new network is proposed in which unidimensional membership functions are used and only two parameters for each rule are employed thus reducing the number of parameters the network avoids the singularity produced by the widths in the antecedent part for online learning 2 a new pruning algorithm based on the density is proposed where the density is the number of times each rule is used in the algorithm the rule that has the smallest density the looser rule in a selected number of iterations is pruned if the value of its density is smaller than a prespecified threshold and 3 the stability of the proposed algorithm is proven and the bound for the average of the identification error is found the condition that led the algorithm to avoid the local minimum is found and it is proven that the parameter error is bounded by the initial parameter error three simulations give the effectiveness of the suggested algorithm

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

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

solutions of linear difference equations with variable coefficients the explicit solution of a linear difference equation of unbounded order with variable coefficients is presented as special cases the solutions of nonhomogeneous and homogeneous linear difference equations of order n with variable coefficients are obtained from these solutions we also get expressions for the product of companion matrices and the power of a companion matrix

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

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

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

symbol error probability and bit error probability for optimum combining with mpsk modulation new expressions are derived for the exact symbol error probability and bit error probability for optimum combining with multiple phase shift keying the expressions are for any numbers of equal power cochannel interferers and receive branches it is assumed that the aggregate interference and noise is gaussian and that both the desired signal and interference are subject to flat rayleigh fading the new expressions have low computational complexity as they contain only a single integral form with finite limits and finite integrand

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

adaptive filter design using type 2 fuzzy cerebellar model articulation controller this paper aims to propose an efficient network and applies it as an adaptive filter for the signal processing problems an adaptive filter is proposed using a novel interval type 2 fuzzy cerebellar model articulation controller t2fcmac the t2fcmac realizes an interval type 2 fuzzy logic system based on the structure of the cmac due to the better ability of handling uncertainties type 2 fuzzy sets can solve some complicated problems with outstanding effectiveness than type 1 fuzzy sets in addition the lyapunov function is utilized to derive the conditions of the adaptive learning rates so that the convergence of the filtering error can be guaranteed in order to demonstrate the performance of the proposed adaptive t2fcmac filter it is tested in signal processing applications including a nonlinear channel equalization system a time varying channel equalization system and an adaptive noise cancellation system the advantages of the proposed filter over the other adaptive filters are verified through simulations

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

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

reduced dimension map turbo blast detection the bell labs layered space time blast architecture is a simple and efficient multiantenna coding structure that can achieve high spectral efficiency many blast detectors require more receiver antennas than transmitter antennas we propose two novel turbo processing blast detectors that can operate in systems with fewer receiver antennas than transmitter antennas both detectors are based on the group detection strategy the first proposed detector the reduced dimension maximum a posteriori rdmap detector uses a dynamically formed group for each bit decision while the second proposed detector the group maximum a posteriori gmap uses a static grouping for both detectors a maximum a posteriori map decision is made using a group of transmitted symbols and the remaining signal contribution is treated as interference the interference is characterized as nonzero mean colored noise source that is whitened before a decision is made both proposed detectors are generalizations of the map detector and the turbo processing minimum mean squared error mmse detector in sellathurai and haykin and abe and matsumoto an uncoded bit error rate analysis for an independent rayleigh fading environment is also presented simulated results are presented which show that both the rdmap and gmap detectors have a performance improvement over the mmse detector especially in systems having an excess number of transmitter antennas

non schlesinger deformations of ordinary differential equations with rational coefficients we consider deformations of 2 2 and 3 3 matrix linear odes with rational coefficients with respect to singular points of fuchsian type which do not satisfy the well known system of schlesinger equations or its natural generalization some general statements concerning the reducibility of such deformations for 2 2 odes are proved an explicit example of the general non schlesinger deformation of 2 2 matrix odes of the fuchsian type with four singular points is constructed and application of such deformations to the construction of special solutions of the corresponding schlesinger systems is discussed some examples of isomonodromic and non isomonodromic deformations of 3 3 matrix odes are considered the latter arise as the compatibility conditions with linear odes with non single valued coefficients

discrete interval type 2 fuzzy system models using uncertainty in learning parameters fuzzy system modeling fsm is one of the most prominent tools that can be used to identify the behavior of highly nonlinear systems with uncertainty conventional fsm techniques utilize type 1 fuzzy sets in order to capture the uncertainty in the system however since type 1 fuzzy sets express the belongingness of a crisp value x of a base variable x in a fuzzy set a by a crisp membership value mua x they cannot fully capture the uncertainties due to imprecision in identifying membership functions higher types of fuzzy sets can be a remedy to address this issue since the computational complexity of operations on fuzzy sets are increasing with the increasing type of the fuzzy set the use of type 2 fuzzy sets and linguistic logical connectives drew a considerable amount of attention in the realm of fuzzy system modeling in the last two decades in this paper we propose a black box methodology that can identify robust type 2 takagi sugeno mizumoto and linguistic fuzzy system models with high predictive power one of the essential problems of type 2 fuzzy system models is computational complexity in order to remedy this problem discrete interval valued type 2 fuzzy system models are proposed with type reduction in the proposed fuzzy system modeling methods fuzzy c means fcm clustering algorithm is used in order to identify the system structure the proposed discrete interval valued type 2 fuzzy system models are generated by a learning parameter of fcm known as the level of membership and its variation over a specific set of values which generate the uncertainty associated with the system structure

computation of highly ramified coverings an almost belyi covering is an algebraic covering of the projective line such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line in general there are 1 dimensional families of these coverings with a fixed ramification pattern that is hurwitz spaces for these coverings are curves in this paper three almost belyi coverings of degrees 11 12 and 20 are explicitly constructed we demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth painleve equation

a representation of the solution of the nth order linear difference equation with variable coefficients abstract the linear difference equation of the n th order with variable coefficients and a related difference equation are considered it is shown that for each problem the determinants of submatrices of a single solution matrix provides the solution sequence applications to sequences and series are shown results on toeplitz matrices involving the roots of associated polynomial equations are obtained

circular pentagons and real solutions of painleve vi equations we study real solutions of a class of painleve vi equations to each such solution we associate a geometric object a one parametric family of circular pentagons we describe an algorithm that permits to compute the numbers of zeros poles 1 points and fixed points of the solution on the interval 1 infty and their mutual position the monodromy of the associated linear equation and parameters of the painleve vi equation are easily recovered from the family of pentagons

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

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

weighted fuzzy rule interpolation based on ga based weight learning techniques in this paper we propose a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems it is based on a genetic algorithm ga based weight learning technique the proposed method can deal with fuzzy rule interpolation with weighted antecedent variables it also can deal with fuzzy rule interpolation based on polygonal membership functions and bell shaped membership functions we also propose a ga based weight learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules furthermore we apply the proposed weighted fuzzy interpolative reasoning method and the proposed ga based weight learning algorithm to deal with the truck backer upper control problem the computer activity prediction problem multivariate regression problems and time series prediction problems based on statistical analysis techniques the experimental results show that the proposed weighted fuzzy interpolative reasoning method by the use of the optimally learned weights that were obtained by the proposed ga based weight learning algorithm has statistically significantly smaller error rates than the existing methods

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

exact bit error probability for optimum combining with a rayleigh fading gaussian cochannel interferer we derive expressions for the exact bit error probability bep for the detection of coherent binary phase shift keying signals of the optimum combiner employing space diversity when both the desired signal and a gaussian cochannel interferer are subject to flat rayleigh fading two different methods are employed to reach two different but numerically identical expressions with the direct method the conditional bep is averaged over the fading of both signal and interference with the moment generating function based method expressions are derived from an alternative representation of the gaussian q function

a probabilistic framework for interval type 2 fuzzy linguistic summarization many studies on linguistic summarization have addressed the use of ordinary fuzzy set type 1 fuzzy set t1fs for modeling words however few of them have exploited interval type 2 fuzzy set it2fs although it2fs is better able to deal with the uncertainty associated with words the existing studies on linguistic summarization using it2fs have focused on scalar cardinality based degree of truth in this paper for the first time we propose a probabilistic framework based on the idea of interval mass assignment to evaluate interval type 2 fuzzy linguistic summaries based on the type ii quantified sentences and the semi fuzzy quantifiers as an alternative method to the scalar cardinality based methods we implement a real case study on linguistic summarization of time series data of europe brent spot price as well as a comparison of the results obtained with our approach and those of the existing approaches

a hybrid learning algorithm for a class of interval type 2 fuzzy neural networks in real life information about the world is uncertain and imprecise the cause of this uncertainty is due to deficiencies on given information the fuzzy nature of our perception of events and objects and on the limitations of the models we use to explain the world the development of new methods for dealing with information with uncertainty is crucial for solving real life problems in this paper three interval type 2 fuzzy neural network it2fnn architectures are proposed with hybrid learning algorithm techniques gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation at the antecedents layer a interval type 2 fuzzy neuron it2fn model is used and in case of the consequents layer an interval type 1 fuzzy neuron model it1fn in order to fuzzify the rule s antecedents and consequents of an interval type 2 takagi sugeno kang fuzzy inference system it2 tsk fis it2 tsk fis is integrated in an adaptive neural network in order to take advantage the best of both models this provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy if then rules in addition to handling uncertainty and imprecision on the other hand neural networks are highly adaptable with learning and generalization capabilities experimental results are divided in two kinds in the first one a non linear identification problem for control systems is simulated here a comparative analysis of learning architectures it2fnn and anfis is done for the second kind a non linear mackey glass chaotic time series prediction problem with uncertainty sources is studied finally it2fnn proved to be more efficient mechanism for modeling real world problems

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