optimization of type 2 fuzzy systems based on bio inspired methods a concise review a review of the optimization methods used in the design of type 2 fuzzy systems which are relatively novel models of imprecision has been considered in this work the fundamental focus of the work has been based on the basic reasons of the need for optimizing type 2 fuzzy systems for different areas of application recently bio inspired methods have emerged as powerful optimization algorithms for solving complex problems in the case of designing type 2 fuzzy systems for particular applications the use of bio inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems in this review we consider the application of genetic algorithms particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type 2 fuzzy systems we also provide a comparison of the different optimization methods for the case of designing type 2 fuzzy systems

type 2 interval fuzzy rule based systems in spatial analysis abstract in this paper we present a rule based system by using interval type 2 fuzzy sets it2 fs for solving problems of spatial analysis this system is encapsulated in a geographical information system gis by performing numerous tests for evaluating their reliability

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

global convergence of karnik mendel algorithms karnik mendel km algorithms are the most commonly used iterative type reduction methods in interval type 2 fuzzy sets and systems as well as new techniques for computing the fuzzy weighted average fwa various extensions and improvements have been proposed however no proof has been provided for the convergence of these extensions it is necessary to provide the proof because many of the iterative algorithms may have divergence cases in the present study we provide a theoretical proof that km algorithms exhibit global convergence different initialization methods and iteration formats can always obtain the same unique optimal solution thus there are no concerns about the possibility of divergence in extensions of km algorithms our proof provides theoretical support for the applications of km algorithms especially the type reduction designs used in type 2 fuzzy systems and fwa computations because of the important roles of km algorithms in these methods

hybrid intelligent systems for time series prediction using neural networks fuzzy logic and fractal theory in this paper we describe a new method for the estimation of the fractal dimension of a geometrical object using fuzzy logic techniques the fractal dimension is a mathematical concept which measures the geometrical complexity of an object the algorithms for estimating the fractal dimension calculate a numerical value using as data a time series for the specific problem this numerical crisp value gives an idea of the complexity of the geometrical object or time series however there is an underlying uncertainty in the estimation of the fractal dimension because we use only a sample of points of the object and also because the numerical algorithms for the fractal dimension are not completely accurate for this reason we have proposed a new definition of the fractal dimension that incorporates the concept of a fuzzy set this new definition can be considered a weaker definition but more realistic of the fractal dimension and we have named this the fuzzy fractal dimension we can apply this new definition of the fractal dimension in conjunction with soft computing techniques for the problem of time series prediction we have developed hybrid intelligent systems combining neural networks fuzzy logic and the fractal dimension for the problem of time series prediction and we have achieved very good results

scaling properties of the eigenvalue spacing distribution for band random matrices the authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2 n where b is the band half width and n is the size of the matrices it is also shown that the eigenvalue s density obeys the semicircle law the found scaling behaviour suggests that the fluctuation properties in the intermediate regime between wigner dyson and poisson are universal

adaptive inverse control of cable driven parallel system based on type 2 fuzzy logic systems this paper is concerned with the problem of type 2 fuzzy adaptive inverse control for a cable driven parallel system based on the heuristics and prior knowledge of the system the system is divided into six subsystems the proposed control scheme for each subsystem contains a forward model and a fuzzy adaptive inverse controller faic which are expressed by an interval type 2 fuzzy nonlinear autoregressive exogenous narx model respectively to construct the antecedents of the interval type 2 fuzzy narx forward models and faics the monotonic property of the fuzzy narx model is first proved and then their antecedent parameters can be determined by this property furthermore the consequent parameters of the forward models are computed offline via a constrained least squares algorithm and the consequent parameters of the faic are adjusted online via a recursive least squares algorithm experiment results are provided to show that the proposed type 2 fuzzy control scheme can realize the control objectives and achieve a good control performance

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

on level spacing distributions for 2d non normal gaussian random matrices the level spacing distribution of general non normal gaussian random 2d matrices is derived in particular tridiagonal matrices have no level repulsion and show a halfsided gaussian distribution general non normal matrices show strong level repulsion the repulsion exponent is 2 0log

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

dynamic spatial panel models networks common shocks and sequential exogeneity this paper considers a class of generalized methods of moments gmm estimators for general dynamic panel models allowing for weakly exogenous covariates and cross sectional dependence due to spatial lags unspecified common shocks and time varying interactive effects we significantly expand the scope of the existing literature by allowing for endogenous time varying spatial weight matrices without imposing explicit structural assumptions on how the weights are formed an important area of application is in social interaction and network models where our specification can accommodate data dependent network formation we consider an exemplary social interaction model and show how identification of the interaction parameters is achieved through a combination of linear and quadratic moment conditions for the general setup we develop an orthogonal forward differencing transformation to aid in the estimation of factor components while maintaining orthogonality of moment conditions this is an important ingredient to a tractable asymptotic distribution of our estimators in general the asymptotic distribution of our estimators is found to be mixed normal due to random norming however the asymptotic distribution of our test statistics is still chi square

simple and reliable estimators of coefficients of interest in a model with high dimensional confounding effects abstract often an investigator is interested in a single parameter or low dimensional parameter vector e g a treatment effect in a regression model rather than in the full set of regression coefficients there may also be a relatively high dimensional set of potential explanatory variables other than the effect of interest which if omitted could bias the estimate of the parameter of interest however there may be too many such variables to include all of them without substantial efficiency loss we suggest a simple easily computed estimator for this case using principal components to compute auxiliary regressors from the set of potential controls and establish the limit properties of the estimator allowing for dependence and heterogeneity as well as increasing dimension of the set of controls we also provide finite sample evidence on the performance of the estimator where principal components are selected in a one dimensional search using an appropriate information criterion as stopping rule for the number of components the results suggest that the estimator has practical usefulness in small samples

generalized linear time series regression we consider a cross section model that contains an individual component a deterministic time trend and an unobserved latent common time series component we show the following oracle property the parameters of the latent time series and the parameters of the deterministic time trend can be estimated with the same asymptotic accuracy as if the parameters of the individual component were known we consider this model in two settings least squares fits of linear specifications of the individual component and the parameters of the deterministic time trend and more generally quasilikelihood estimation in a generalized linear time series model

testing for structural stability of factor augmented forecasting models abstract mild factor loading instability particularly if sufficiently independent across the different constituent variables does not affect the estimation of the number of factors nor subsequent estimation of the factors themselves see e g stock and watson 2009 this result does not hold in the presence of large common breaks in the factor loadings however in this case information criteria overestimate the number of breaks additionally estimated factors are no longer consistent estimators of true factors hence various recent research papers in the diffusion index literature focus on testing the constancy of factor loadings however forecast failure of factor augmented models can be due to either factor loading instability regression coefficient instability or both to address this issue we develop a test for the joint hypothesis of structural stability of both factor loadings and factor augmented forecasting model regression coefficients our proposed test statistic has a chi squared limiting distribution and we are able to establish the first order validity of block bootstrap critical values empirical evidence is also presented for 11 us macroeconomic indicators

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

consistent estimation of time varying loadings in high dimensional factor models in this paper we develop a two step maximum likelihood estimator of time varying loadings in high dimensional factor models we specify the loadings to evolve as stationary vector autoregressions var and show that consistent estimates of the loadings parameters can be obtained in the first step principal components are extracted from the data to formfactor estimates in the second step the parameters of the loadings vars are estimated as a set of linear regression models with time varying coefficients we document the finite sample properties of the maximum likelihood estimator through an extensive simulation study and illustrate the empirical relevance of the time varying loadings structure using a large quarterly dataset for the us economy

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

ten years of genetic fuzzy systems current framework and new trends fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains currently there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities two of the most successful approaches to hybridise fuzzy systems with learning and adaptation methods have been made in the realm of soft computing neural fuzzy systems and genetic fuzzy systems hybridise the approximate reasoning method of fuzzy systems with the learning capabilities of neural networks and evolutionary algorithms the objective of this paper is to provide an account of genetic fuzzy systems with special attention to genetic fuzzy rule based systems after a brief introduction to models and applications of genetic fuzzy systems the field is overviewed new trends are identified a critical evaluation of genetic fuzzy systems for fuzzy knowledge extraction is elaborated and open questions that remain to be addressed in the future are raised the paper also includes some of the key references required to quickly access implementation details of genetic fuzzy systems

a new indirect approach to the type 2 fuzzy systems modeling and design this paper proposes a new method for designing interval type 2 fuzzy logic systems it2 flss considering two issues first quality of clustering the output space and secondly approximating the output of the it2 fls based on a new output processing method based on these two issues we present a new cluster validity index capable of being used for type 1 fuzzy c means fcms interval type 2 fcm it2 fcm and possibilistic c means pcms clustering algorithms this validity index is highly efficient in determining clusters with the least similarity between them and the highest similarity between data vectors in each cluster then a new definition for uncertainty bounds is presented in order to eliminate the type reduction process in it2 flss and to increase accuracy of the existing uncertainty bounds in the literature finally effectiveness of the proposed approaches compared to several well known existing methods has been investigated computational results have verified accuracy and effectiveness of the proposed method

maximum likelihood estimation and inference for high dimensional generalized factor models with application to factor augmented regressions abstract this paper reestablishes the main results in bai 2003 and bai and ng 2006 for generalized factor models with slightly stronger conditions on the relative magnitude of n number of subjects and t number of time periods convergence rates of the estimated factor space and loading space and asymptotic normality of the estimated factors and loadings are established under mild conditions that allow for linear logit probit tobit poisson and some other single index nonlinear models the probability density mass function is allowed to vary across subjects and time thus mixed models are also allowed for for factor augmented regressions this paper establishes the limit distributions of the parameter estimates the conditional mean and the forecast when factors estimated from nonlinear mixed data are used as proxies for the true factors

on sampled data fuzzy control design approach for t s model based fuzzy systems by using discretization approach in this paper a sampled data fuzzy control design approach for takagi sugeno t s model based fuzzy systems is proposed firstly the t s model based fuzzy systems are approximated by t s fuzzy model based discrete systems with parameter uncertainties then a sufficient condition on the existence of a sampled data fuzzy controller for the fuzzy systems is formulated in the form of linear matrix inequalities which are proposed in some symmetrical form so the proposed fuzzy control design approach will be less conservative and some problems in the existing literature can be solved finally a simulation example is discussed to show the effectiveness of the obtained fuzzy control design approach in this paper

forecasting using random subspace methods textabstractrandom subspace methods are a novel approach to obtain accurate forecasts in high dimensional regression settings we provide a theoretical justification of the use of random subspace methods and show their usefulness when forecasting monthly macroeconomic variables we focus on two approaches the first is random subset regression where random subsets of predictors are used to construct a forecast second we discuss random projection regression where artificial predictors are formed by randomly weighting the original predictors using recent results from random matrix theory we obtain a tight bound on the mean squared forecast error for both randomized methods we identify settings in which one randomized method results in more precise forecasts than the other and than alternative regularization strategies such as principal component regression partial least squares lasso and ridge regression the predictive accuracy on the high dimensional macroeconomic fred md data set increases substantially when using the randomized methods with random subset regression outperforming any one of the above mentioned competing methods for at least 66 of the series

bootstrapping factor augmented regression models abstract this paper proposes and theoretically justifies bootstrap methods for regressions where some of the regressors are factors estimated from a large panel of data we derive our results under the assumption that t n c where 0 c n and t are the cross sectional and the time series dimensions respectively thus allowing for the possibility that the factor estimation error enters the limiting distribution of the ols estimator as an asymptotic bias term as was recently discussed by ludvigson and ng 2011 we consider general residual based bootstrap methods and provide a set of high level conditions on the bootstrap residuals and on the idiosyncratic errors such that the bootstrap distribution of a rotated ols estimator is consistent we subsequently verify these conditions for a simple wild bootstrap residual based procedure

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

consistent factor estimation in dynamic factor models with structural instability abstract this paper considers the estimation of approximate dynamic factor models when there is temporal instability in the factor loadings we characterize the type and magnitude of instabilities under which the principal components estimator of the factors is consistent and find that these instabilities can be larger than earlier theoretical calculations suggest we also discuss implications of our results for the robustness of regressions based on the estimated factors and of estimates of the number of factors in the presence of parameter instability simulations calibrated to an empirical application indicate that instability in the factor loadings has a limited impact on estimation of the factor space and diffusion index forecasting whereas estimation of the number of factors is more substantially affected

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

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

type 2 fuzzy neural networks with fuzzy clustering and differential evolution optimization in many real world problems involving pattern recognition system identification and modeling control decision making and forecasting of time series available data are quite often of uncertain nature an interesting alternative is to employ type 2 fuzzy sets which augment fuzzy models with expressive power to develop models which efficiently capture the factor of uncertainty the three dimensional membership functions of type 2 fuzzy sets offer additional degrees of freedom that make it possible to directly and more effectively account for model s uncertainties type 2 fuzzy logic systems developed with the aid of evolutionary optimization forms a useful modeling tool subsequently resulting in a collection of efficient if then rules the type 2 fuzzy neural networks take advantage of capabilities of fuzzy clustering by generating type 2 fuzzy rule base resulting in a small number of rules and then optimizing membership functions of type 2 fuzzy sets present in the antecedent and consequent parts of the rules the clustering itself is realized with the aid of differential evolution several examples including a benchmark problem of identification of nonlinear system are considered the reported comparative analysis of experimental results is used to quantify the performance of the developed networks

asymptotic distribution of factor augmented estimators for panel regression abstract in this paper we derive an asymptotic theory for linear panel regression augmented with estimated common factors we give conditions under which the estimated factors can be used in place of the latent factors in the regression equation for the principal components estimate of the factor space it is shown that these conditions are satisfied when t n 0 and n t 3 0 under regularity monte carlo studies verify the asymptotic theory

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

semiparametric ultra high dimensional model averaging of nonlinear dynamic time series we propose two semiparametric model averaging schemes for nonlinear dynamic time series regression models with a very large number of covariates including exogenous regressors and auto regressive l

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

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

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

the three pass regression filter a new approach to forecasting using many predictors we forecast a single time series using many predictor variables with a new estimator called the three pass regression lter 3prf it is calculated in closed form and conveniently represented as a set of ordinary least squares regressions 3prf forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large this requires only specifying the number of relevant factors driving the forecast target regardless of the total number of common including potentially irrelevant factors driving the cross section of predictors we derive inferential theory including limiting distributions for estimated relevant factors predictive coefcients and forecasts and provide consistent standard error estimators we explore two empirical applications forecasting macroeconomic aggregates with a large panel of economic indices and forecasting stock market aggregates with many individual assets price dividend ratios these combined with a range of monte carlo experiments demonstrate the 3prf s forecasting power

an interval type 2 fuzzy model of computing with words as a methodology computing with words allows the use of words instead of numbers or symbols in the process of computing and reasoning and thus conforms more to humans inference when it is used to describe real world problems motivated by zadeh s paradigm of computing with words the literature has proposed a kind of type 1 fuzzy automata as a formal model of computing with words which takes type 1 fuzzy subsets of symbols as input however type 1 fuzzy representation provides a limited platform for approximating the meaning of words since it is not able to capture linguistic uncertainty in this paper we develop a formal interval type 2 fuzzy model of computing with words by generalizing the existing type 1 fuzzy sets based model to an interval type 2 fuzzy environment concretely we take interval type 2 fuzzy automata i e interval type 2 fuzzy finite automata and interval type 2 fuzzy pushdown automata which combine interval type 2 fuzzy set theory and automaton theory as a computational model of computing with words furthermore we develop the extension principles to extend from computing with values to computing with words and show that computing with words can be implemented with computing with values in the interval type 2 fuzzy setting at an extra computational cost

indirect adaptive type 2 fuzzy impulsive control of nonlinear systems in this paper a novel indirect adaptive interval type 2 fuzzy tracking control is proposed strategy with impulsive and lyapunov synthesis approach it handles the training data corrupted by noise or rule uncertainties for linear and nonlinear systems involving external disturbances in comparison with conventional methods the advocated approach guarantees closed loop stability the output tracking error of the overall system will converge to zero asymptotically by the generalized barbalat s lemma numerical simulations are given to illustrate the effectiveness of the proposed techniques

reduction methods of type 2 uncertain variables and their applications to solid transportation problem uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity changeability and non decidability of practical environments an uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution this paper proposes the definition of type 2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties some uncertainty reduction methods associated with type 2 uncertain variables are also proposed for convenience of applicability including reduction of optimistic value pessimistic value and expected value moreover four classes of type 2 uncertain variables are reduced to type 1 uncertain variables with specific uncertainty distributions type 2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type 2 uncertain parameters where the solution methods are also provided for the proposed models finally numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches

forecasting economic time series using targeted predictors abstract this paper studies two refinements to the method of factor forecasting first we consider the method of quadratic principal components that allows the link function between the predictors and the factors to be non linear second the factors used in the forecasting equation are estimated in a way to take into account that the goal is to forecast a specific series this is accomplished by applying the method of principal components to targeted predictors selected using hard and soft thresholding rules our three main findings can be summarized as follows first we find improvements at all forecast horizons over the current diffusion index forecasts by estimating the factors using fewer but informative predictors allowing for non linearity often leads to additional gains second forecasting the volatile one month ahead inflation warrants a high degree of targeting to screen out the noisy predictors a handful of variables notably relating to housing starts and interest rates are found to have systematic predictive power for inflation at all horizons third the targeted predictors selected by both soft and hard thresholding changes with the forecast horizon and the sample period holding the set of predictors fixed as is the current practice of factor forecasting is unnecessarily restrictive

estimating cross section common stochastic trends in nonstationary panel data this paper studies large dimension factor models with nonstationary dynamic factors also referred to as cross section common stochastic trends we consider the problem of estimating the dimension of the common stochastic trends and the stochastic trends themselves we derive the rates of convergence and the limiting distributions for the estimated common trends and for the estimated loading coefficients generalized dynamic factor models with nonstationary factors are also considered cointegration among the factors is permitted the method is applied to the study of employment fluctuations across 60 industries for the u s we examine the hypothesis that these fluctuations can be explained by a small number of aggregate factors we also test whether some observable macroeconomic variables are the underlying factors

forecasting with factor augmented regression a frequentist model averaging approach abstract this paper considers forecast combination with factor augmented regression in this framework a large number of forecasting models are available varying by the choice of factors and the number of lags we investigate forecast combination across models using weights that minimize the mallows and the leave h out cross validation criteria the unobserved factor regressors are estimated by principle components of a large panel with n predictors over t periods with these generated regressors we show that the mallows and leave h out cross validation criteria are asymptotically unbiased estimators of the one step ahead and multi step ahead mean squared forecast errors respectively provided that n t however the paper does not establish any optimality properties for the methods in contrast to well known results in the literature this result suggests that the generated regressor issue can be ignored for forecast combination without restrictions on the relation between n and t simulations show that the mallows model averaging and leave h out cross validation averaging methods yield lower mean squared forecast errors than alternative model selection and averaging methods such as aic bic cross validation and bayesian model averaging we apply the proposed methods to the us macroeconomic data set in stock and watson 2012 and find that they compare favorably to many popular shrinkage type forecasting methods

identification and estimation of a large factor model with structural instability this paper tackles the identification and estimation of a high dimensional factor model with unknown number of latent factors and a single break in the number of factors and or factor loadings occurring at unknown common date first we propose a least squares estimator of the change point based on the second moments of estimated pseudo factors and show that the estimation error of the proposed estimator is op 1 we also show that the proposed estimator has some degree of robustness to misspecification of the number of pseudo factors with the estimated change point plugged in consistency of the estimated number of pre and post break factors and convergence rate of the estimated pre and post break factor space are then established under fairly general assumptions the finite sample performance of our estimators is investigated using monte carlo experiments

limit theory for panel data models with cross sectional dependence and sequential exogeneity abstract the paper derives a general central limit theorem clt and asymptotic distributions for sample moments related to panel data models with large n the results allow for the data to be cross sectionally dependent while at the same time allowing the regressors to be only sequentially rather than strictly exogenous the setup is sufficiently general to accommodate situations where cross sectional dependence stems from spatial interactions and or from the presence of common factors the latter leads to the need for random norming the limit theorem for sample moments is derived by showing that the moment conditions can be recast such that a martingale difference array central limit theorem can be applied we prove such a central limit theorem by first extending results for stable convergence in hall and heyde 1980 to non nested martingale arrays relevant for our applications we illustrate our result by establishing a generalized estimation theory for gmm estimators of a fixed effect panel model without imposing i i d or strict exogeneity conditions we also discuss a class of maximum likelihood ml estimators that can be analyzed using our clt

structural analysis with multivariate autoregressive index models we address the issue of parameter dimensionality reduction in vector autoregressive models vars for many variables by imposing specific reduced rank restrictions on the coefficient matrices that simplify the vars into multivariate autoregressive index mai models we derive the wold representation implied by the mais and show that it is closely related to that associated with dynamic factor models then the theoretical analysis is extended to the case of general rank restrictions on the var coefficients next we describe classical and bayesian estimation of large mais and discuss methods for rank determination finally the performance of the mais is compared with that of large bayesian vars in the context of monte carlo simulations and two empirical applications on the transmission mechanism of monetary policy and on the propagation of demand and supply shocks

confidence intervals for diffusion index forecasts and inference for factor augmented regressions we consider the situation when there is a large number of series n e ach witht observations and each series has some predictive ability for some variable of interest a methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors in this paper we show that the least squares estimates obtained from these factor augmented regressions are t consistent and asymptotically normal if t n 0 the conditional mean predicted by the estimated factors is min t n consistent and asymptotically normal except when t n goes to zero inference should take into account the effect of estimated regressors on the estimated conditional mean we present analytical formulas for prediction intervals that are valid regardless of the magnitude of n t and that can also be used when the factors are nonstationary

tests of equal accuracy for nested models with estimated factors abstract in this paper we develop asymptotics for tests of equal predictive ability between nested models when factor augmented regressions are used to forecast we provide conditions under which the estimation of the factors does not affect the asymptotic distributions developed in clark and mccracken 2001 and mccracken 2007 this enables researchers to use the existing tabulated critical values when conducting inference despite the presence of estimated predictors as an intermediate result we derive the asymptotic properties of the principal components estimator over recursive windows we provide simulation evidence on the finite sample effects of factor estimation and apply the tests to the case of forecasting excess returns to the s p 500 composite index

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

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

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

weighted fuzzy interpolative reasoning systems based on interval type 2 fuzzy sets abstract in this paper we present a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on interval type 2 fuzzy sets we also apply the proposed weighted fuzzy interpolative reasoning method to deal with the truck backer upper control problem the proposed method satisfies the seven evaluation indices for fuzzy interpolative reasoning the experimental results show that the proposed method outperforms the existing methods it provides us with a useful way for dealing with fuzzy interpolative reasoning in sparse fuzzy rule based systems

bootstrapping factor models with cross sectional dependence abstract we consider bootstrap methods for factor augmented regressions with cross sectional dependence among idiosyncratic errors this is important to capture the bias of the ols estimator derived recently by goncalves and perron 2014 we first show that a common approach of resampling cross sectional vectors over time is invalid in this context because it induces a zero bias we then propose the cross sectional dependent csd bootstrap where bootstrap samples are obtained by taking a random vector and multiplying it by the square root of a consistent estimator of the covariance matrix of the idiosyncratic errors we show that if the covariance matrix estimator is consistent in the spectral norm then the csd bootstrap is consistent and we verify this condition for the thresholding estimator of bickel and levina 2008 finally we apply our new bootstrap procedure to forecasting inflation using convenience yields as recently explored by gospodinov and ng 2013

principal components estimation and identification of static factors a b s t r a c t it is known that the principal component estimates of the factors and the loadings are rotations of the un derlying latent factors and loadings we study conditions under which the latent factors can be estimated asymptotically without rotation we derive the limiting distributions for the estimated factors and factor loadings when n and t are large and make precise how identification of the factors affects inference based on factor augmented regressions we also consider factor models with additive individual and time effects the asymptotic analysis can be modified to analyze identification schemes not considered in this analysis

simulation of the bird age structured population growth based on an interval type 2 fuzzy cellular structure in this paper an age structured population growth model based on a fuzzy cellular structure is proposed an age structured population growth model enables a better description of population dynamics in this paper the dynamics of a particular bird species is considered the dynamics is governed by the variation of natality mortality and emigration rates which in this work are evaluated using an interval type 2 fuzzy logic system the use of type 2 fuzzy logic enables handling the effects caused by environment heterogeneity on the population a set of fuzzy rules about population growth are derived from the interpretation of the ecological laws and the bird life cycle the proposed model is formulated using discrete mathematics within the framework of a fuzzy cellular structure the fuzzy cellular structure allows us to visualize the evolution of the population s spatial dynamics the spatial distribution of the population has a deep effect on its dynamics moreover the model enables not only to estimate the percentage of occupation on the cellular space when the species reaches its stable equilibrium level but also to observe the occupation patterns

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

rank regularized estimation of approximate factor models abstract it is known that the common factors in a large panel of data can be consistently estimated by the method of principal components and principal components can be constructed by iterative least squares regressions replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with small singular values from the common component the method has been used in the machine learning literature to recover low rank matrices we study the procedure from the perspective of estimating an approximate factor model under the rank constraint the common component is estimated by the space spanned by factors whose singular values exceed a threshold the desire for minimum rank and parsimony lead to a data dependent penalty for selecting the number of factors the new criterion is more conservative than the existing deterministic penalties and is appropriate when the nominal number of factors is inflated by the presence of weak factors or large measurement noise we provide asymptotic results that can be used to test economic hypotheses

gmm estimation with cross sectional dependence this paper presents a spatial model of dependence among agents using a metric of economic distance measurements of this economic distance provide cross sectional data with a structure similar to that provided by the time index in time series data generalized method of moments estimators using such dependent data are shown to be consistent and asymptotically normal this paper presents a class of non parametric positive semi definite covariance matrix estimators that allow for general forms of dependence characterized by economic distance these covariance matrix estimators are shown to remain consistent when economic distances are not precisely observed

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

interval type 2 fuzzy adaptive tracking control design for pmdc motor with the sector dead zones we propose it2fac scheme to reduce the sector dead zone nonlinearities this scheme integrates variable structure to achieve the h tracking performance this scheme uses the lyapunov criterion to ensure the boundedness of all states this scheme exhibits faster tracking responses for the pmdc motor systems this paper deals with the permanent magnet dc motor system with sector dead zones and external disturbances via interval type 2 fuzzy adaptive tracking control scheme to achieve h tracking performance the type 2 fuzzy dynamic model is used to approximate the motor dynamics without constructing sector dead zone inverse where the parameters of the fuzzy model are obtained both from the fuzzy inference and online update laws based on the lyapunov criterion and riccati inequality the control scheme is derived to stabilize the closed loop system such that all states of the system are guaranteed to be bounded and h tracking performance is achieved due to uncertainties dead zone nonlinearities and external disturbances the advantage of the proposed control scheme is that it can better handle the vagueness or uncertainties inherent in linguistic words using fuzzy set membership functions with adaptation capability by linear analytical results instead of estimating non linear system functions as the system parameters are unknown finally the simulations of the pmdc motor system with sector dead zone nonlinearities are used to illustrate the effectiveness of the proposed control scheme and the performance comparisons with the three dimensional autonomous r o ssler system wang and chua 2008 are used to validate the h tracking performance of the pmdc motor system

an interval type 2 fuzzy linmap method with approximate ideal solutions for multiple criteria decision analysis the purpose of this paper is to develop a linear programming technique for multidimensional analysis of preference linmap to address multiple criteria decision analysis problems within the interval type 2 fuzzy environment based on interval type 2 trapezoidal fuzzy numbers considering the issue of anchor dependency we use multiple anchor points in the decision making process and employ approximate positive ideal and negative ideal solutions as the points of reference selected useful properties of the approximate ideal solutions are also investigated in contrast to the classical linmap methods this paper directly generates approximate ideal solutions from the characteristics of all alternatives next this work presents the concept of closeness based indices using minkowski distances with approximate ideal solutions to develop a new approach for determining measurements of consistency and inconsistency under incomplete preference information on paired comparisons of the alternatives this paper provides a novel method that uses the concept of comprehensive closeness based indices to measure the poorness of fit and the goodness of fit by applying the consistency indices and inconsistency indices this work formulates an optimization problem that can be solved for the optimal weights of the criteria and thus acquires the best compromise alternative additionally this paper explores the problem of supplier selection and conducts a comparative discussion to validate the effectiveness and applicability of the proposed interval type 2 fuzzy linmap method with approximate ideal solutions furthermore the proposed method is applied to address a marketplace decision difficulty mpdd prone decision making problem to provide additional contributions for practical implications

t2fela type 2 fuzzy extreme learning algorithm for fast training of interval type 2 tsk fuzzy logic system a challenge in modeling type 2 fuzzy logic systems is the development of efficient learning algorithms to cope with the ever increasing size of real world data sets in this paper the extreme learning strategy is introduced to develop a fast training algorithm for interval type 2 takagi sugeno kang fuzzy logic systems the proposed algorithm called type 2 fuzzy extreme learning algorithm t2fela has two distinctive characteristics first the parameters of the antecedents are randomly generated and parameters of the consequents are obtained by a fast learning method according to the extreme learning mechanism in addition because the obtained parameters are optimal in the sense of minimizing the norm the resulting fuzzy systems exhibit better generalization performance the experimental results clearly demonstrate that the training speed of the proposed t2fela algorithm is superior to that of the existing state of the art algorithms the proposed algorithm also shows competitive performance in generalization abilities

simplified interval type 2 fuzzy neural networks this paper describes a self evolving interval type 2 fuzzy neural network fnn for various applications as type 1 fuzzy systems cannot effectively handle uncertainties in information within the knowledge base we propose a simple interval type 2 fnn which uses interval type 2 fuzzy sets in the premise and the takagi sugeno kang tsk type in the consequent of the fuzzy rule the tsk type consequent of fuzzy rule is a linear combination of exogenous input variables given an initially empty the rule base all rules are generated with on line type 2 fuzzy clustering instead of the time consuming k m iterative procedure the design factors ql and qr are learned to adaptively adjust the upper and lower positions on the left and right limit outputs using the parameter update rule based on a gradient descent algorithm simulation results demonstrate that our approach yields fewer test errors and less computational complexity than other type 2 fnns

type 1 and type 2 fuzzy inference systems as integration methods in modular neural networks for multimodal biometry and its optimization with genetic algorithms we describe in this paper a comparative study between fuzzy inference systems as methods of integration in modular neural networks for multimodal biometry these methods of integration are based on techniques of type 1 fuzzy logic and type 2 fuzzy logic also the fuzzy systems are optimized with simple genetic algorithms with the goal of having optimized versions of both types of fuzzy systems first we considered the use of type 1 fuzzy logic and later the approach with type 2 fuzzy logic the fuzzy systems were developed using genetic algorithms to handle fuzzy inference systems with different membership functions like the triangular trapezoidal and gaussian since these algorithms can generate fuzzy systems automatically then the response integration of the modular neural network was tested with the optimized fuzzy systems of integration the comparative study of the type 1 and type 2 fuzzy inference systems was made to observe the behavior of the two different integration methods for modular neural networks for multimodal biometry

on bootstrapping panel factor series this paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series two main results are shown firstly a bootstrap invariance principle is derived pointwise in i obtaining an upper bound for the order of truncation of the ar polynomial that depends on n and t consistent estimation of the long run variances is also studied for n t secondly joint bootstrap asymptotics is also studied investigating the conditions under which the bootstrap is valid in particular the extent of cross sectional dependence which can be allowed for is investigated whilst we show that for general forms of cross dependence consistent estimation of the long run variance and therefore validity of the bootstrap is fraught with difficulties however we show that one cross sectional unit at a time resampling schemes yield valid bootstrap based inference under weak forms of cross sectional dependence

tests for overidentifying restrictions in factor augmented var models this paper develops tests for overidentifying restrictions in factor augmented vector autoregressive favar models the favar combines a high dimensional factor model and a conventional var for the latent factors the identification of structural shocks in favar can lead to restrictions on the factor loadings of many variables so it can involve infinitely many identifying restrictions as the number of cross sections goes to infinity our focus is to test the joint null hypothesis that all the restrictions are satisfied conventional tests cannot be used due to the large dimension we transform the infinite dimensional problem into a finite dimensional one by combining the individual statistics across the cross section dimension we find the limit distribution of our joint test statistic under the null hypothesis and prove that it is consistent against the alternative that a fraction of or all identifying restrictions are violated the monte carlo results show that the joint test statistic has good finite sample size and power we implement our tests to an updated version of stock and watson s 2005 data set the proposed test rejects the null hypotheses that the number of fast shocks is two or more but does not reject the null that there is only one fast shock which is the monetary policy shock this result is further confirmed by the impulse responses of major macroeconomic variables to the monetary policy shock the impulse responses based on one fast shock are generally more economically plausible than those based on two or more fast shocks and the price puzzle is either considerably reduced or entirely solved for all price indexes when only one fast shock is used

energy level statistics and localization in sparsed banded random matrix ensemble the matrix elements of the quantum hamiltonian which corresponds to the classical kam system h h0 epsilon v where h0 is integrable in an energy ordered eigenbasis of h0 are considered their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble sbrme in analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the brody distribution the numerical study of sbrme shows two interesting scaling laws i the connection between the scaling variable x alpha b3 2 alpha mean diagonal increment b bandwidth and the level repulsion parameter beta as deduced from the brody distribution and ii the connection between the same scaling variable x and an expression which contains two types of localization length entropy and geometric the universal aspects and the importance of sbrme for the hamiltonian systems in the transition region between integrability and chaos are discussed

new universal aspects of diffusion in strongly chaotic systems we study some new universal aspects of diffusion in chaotic systems especially those having very large lyapunov coefficients on the chaotic indecomposable topologically transitive component we do this by discretizing the chaotic component on the surface of section sos in a large number n of simplectically equally big cells in the sense of equal relative invariant ergodic measure normalized so that the total measure of the chaotic component is unity by iterating the transition of the chaotic orbit through the sos where j counts the number of iterations discrete time and assuming complete lack of correlation even between consecutive crossings which can be justified due to the very large lyapunov exponents we show the universal approach of the relative measure of the occupied cells denoted by to the asymptotic value of unity in the following way so that in the limit of big n we have for j n fixed the exponential law this analytic result is verified numerically in a variety of specific systems for a plane billiard robnik 1983 for a 3d billiard prosen 1997 for an ergodic logistic map tent map for a standard map k 400 and for the hydrogen atom in a strong magnetic field the agreement is almost perfect except in the latter two systems for some long time deviations on very small scales however for henon heiles system and for the standard map k 3 the deviations are noticeable though not very big only about 1 we have tested the random number generators press et al 1986 and confirmed that some are almost perfect ran0 and ran3 whilst two of them ran1 and ran2 exhibit big deviations

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

data driven elastic fuzzy logic system modeling constructing a concise system with human like inference mechanism the construction of fuzzy logic systems flss using data driven techniques has become the most popular modeling approach however this approach still faces critical challenges including the difficulty in obtaining concise models for high dimensional data and generating accurate fuzzy rules to simulate human inference mechanism to tackle these issues a new fls modeling framework called data driven elastic fls dd efls is proposed in this paper the dd efls has two key characteristics first the fuzzy rules in the rule base can use different feature subspaces that are extracted from the original high dimensional space to yield simple and accurate rules in feature spaces of lower dimensionality second fuzzy inferences from various views are implemented by embedding different rules in the corresponding subspaces to imitate human inference mechanism based on the dd efls framework an elastic takagi sugeno kang tsk fls modeling method etsk fls is proposed to train the elastic tsk fls using the concise rules and a more human like inference mechanism for modeling tasks based on high dimensional datasets the characteristics and advantages of the proposed framework and the etsk fls method are validated experimentally using both synthetic and real world datasets

intermediate statistics in the regime of mixed classical dynamics we explain the arguments in support of the berry robnik 1984 17 2413 picture of the energy level statistics in the asymptotic strict far semiclassical limit of sufficiently small where the entire energy spectrum can be represented as a statistically independent superposition of regular and irregular level sequences the regular ones obeying the poissonian statistics and the irregular ones the rmt statistics goe or gue we generalize the results to describe not only the level spacing distribution the number variance and the delta statistics but also arbitrary statistics the probability to find k levels inside an interval of length l after unfolding very useful and effective approximations for are described we demonstrate very clearly that this regime is excellently described by this picture for k as high as the outer energy scale even when taking into account the regular component and only one the dominant chaotic component this we show numerically for the compactified standard map and for the quartic 2d billiard

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

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