universality results for the largest eigenvalues of some sample covariance matrix ensembles for sample covariance matrices with i i d entries with sub gaussian tails when both the number of samples and the number of variables become large and the ratio approaches one it is a well known result of soshnikov that the limiting distribution of the largest eigenvalue is same that of gaussian samples in this paper we extend this result to two cases the first case is when the ratio approaches an arbitrary finite value the second case is when the ratio becomes infinite or arbitrarily small

fuzzy theoretic approach to signals and systems static systems abstract fuzzy theoretic approach to signals and systems assumes all system variables and parameters as uncertain i e being characterized by membership functions develops a mathematical theory for analytically determining the membership functions on system variables and parameters derives algorithms for estimating the parameters of membership functions and establishes robustness and convergence properties of the estimation algorithms the membership functions are analytically determined via solving a variational optimization problem that maximizes the over uncertainties averaged log membership of the observed data around an initial guess this paper develops the analytical fuzzy theory for the particular case of a multi input single output static system affected by noises the theory facilitates designing an adaptive filtering algorithm the robustness of the adaptive filtering algorithm is proved theoretically via a mathematical analysis numerical experiments further demonstrate the robustness of the filtering algorithm a comparison of the algorithm with the state of art methods is made using the practical biomedical applications related to the modeling and analysis of heart rate signals for assessing the physiological state of an individual

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

tests for high dimensional covariance matrices we propose tests for sphericity and identity of high dimensional covariance matrices the tests are nonparametric without assuming a specific parametric distribution for the data they can accommodate situations where the data dimension is much larger than the sample size namely the large p small n situations we demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes we applied the proposed test on a microarray dataset on yorkshire gilts and tested for the covariance structure for the expression levels for sets of genes

outlier robust pca the high dimensional case principal component analysis plays a central role in statistics engineering and science because of the prevalence of corrupted data in real world applications much research has focused on developing robust algorithms perhaps surprisingly these algorithms are unequipped indeed unable to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation and the dataset contains some arbitrarily corrupted observations we propose a high dimensional robust principal component analysis algorithm that is efficient robust to contaminated points and easily kernelizable in particular our algorithm achieves maximal robustness it has a breakdown point of 50 the best possible while all existing algorithms have a breakdown point of zero moreover our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sublinearly in the dimension

stationary fuzzy fokker planck learning and stochastic fuzzy filtering the application of nonlinear optimization to the estimation of fuzzy model parameters is well known to do the reverse of this the concept of stationary fuzzy fokker planck learning sffpl is introduced i e sffpl applies the fuzzy modeling technique in nonlinear optimization problems sffpl is based on the fuzzy approximation of the stationary cumulative distribution function of a stochastic search process associated with the nonlinear optimization problem a carefully designed algorithm is suggested for sffpl to locate the optimum point this paper also considers the variational bayes vb based inference of a stochastic fuzzy filter whose consequents as well as antecedents are random variables the problem of vb inference of stochastic antecedents because of the nonlinearity of the likelihood function is analytically intractable the sffpl algorithm for high dimensional nonlinear optimization that does not require the derivative of the objective function can be used to numerically solve the stochastic fuzzy filtering problem

genetic training instance selection in multiobjective evolutionary fuzzy systems a coevolutionary approach when dealing with datasets that are characterized by a large number of instances multiobjective evolutionary learning moel of fuzzy rule based systems frbss suffers from high computational costs mainly because of the fitness evaluation the use of a reduced set of representative instances in place of the overall training set ts would considerably lessen the computational effort even though a large number of papers have proposed instance selection approaches mainly in classification problems how this selection should be performed especially in the context of regression is still an open issue in this paper we tackle the instance selection problem in the framework of moel of frbss through a coevolutionary approach in the execution of the moel periodically a single objective genetic algorithm soga evolves a population of reduced tss the soga aims to maximize a purposely defined index which measures how much the pareto fronts computed by using respectively the reduced ts and the overall ts are close to each other the closer the fronts the more the reduced ts is representative of the overall ts during the execution of the moel the rule base and the membership function parameters of the fuzzy sets are concurrently learned by maximizing the accuracy and minimizing the complexity we tested our approach on 12 large datasets we adopted reduced tss composed of 5 10 and 20 of the overall ts using nonparametric statistical tests we verified that with 10 and 20 of the overall ts the pareto front approximations that are generated by our coevolutionary approach are comparable with the ones generated by applying the moel with the overall ts although the coevolution allows us to save up to 86 36 of the execution time in addition the analysis of the behavior of three representative solutions on the test set highlights that the use of the reduced tss does not affect the generalization capabilities of the generated frbss

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

a comparative study of tests for homogeneity of variances with applications to the outer continental shelf bidding data many of the existing parametric and nonparametric tests for homogeneity of variances and some variations of these tests are examined in this paper comparisons are made under the null hypothesis for robustness and under the alternative for power monte carlo simulations of various symmetric and asymmetric distributions for various sample sizes reveal a few tests that are robust and have good power these tests are further compared using data from outer continental shelf bidding on oil and gas leases

model approximation for discrete time state delay systems in the t s fuzzy framework this paper is concerned with the problem of h model approximation for discrete time takagi sugeno t s fuzzy time delay systems for a given stable t s fuzzy system our attention is focused on the construction of a reduced order model which not only approximates the original system well in an h performance but is also translated into a linear lower dimensional system by applying the delay partitioning approach a delay dependent sufficient condition is proposed for the asymptotic stability with an h error performance for the error system then the h model approximation problem is solved by using the projection approach which casts the model approximation into a sequential minimization problem subject to linear matrix inequality lmi constraints by employing the cone complementary linearization algorithm moreover by further extending the results h model approximation with special structures is obtained i e delay free model and zero order model finally two numerical examples are provided to illustrate the effectiveness of the proposed methods

spectral statistics of large dimensional spearman s rank correlation matrix and its application between two independent random variables in this paper we will establish a clt for the linear spectral statistics of this non parametric random matrix model in the scenario of high dimension supposing that p p n and p n c2 0 1 as n 1 we propose a novel evaluation scheme to estimate the core quantity in anderson and zeitouni s cumulant method in 1 to bypass the so called joint cumulant summability in addition we raise a two step comparison approach to obtain the explicit formulae for the mean and covariance functions in the clt relying on this clt we then construct a distribution free statistic to test complete independence for components of random vectors owing to the non parametric property we can use this test on generally distributed random variables including the heavy tailed ones

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

pca consistency in high dimension low sample size context principal component analysis pca is an important tool of dimension reduction especially when the dimension or the number of variables is very high asymptotic studies where the sample size is fixed and the dimension grows i e high dimension low sample size hdlss are becoming increasingly relevant we investigate the asymptotic behavior of the principal component pc directions hdlss asymptotics are used to study consistency strong inconsistency and subspace consistency we show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others then the corresponding estimated pc directions are consistent or converge to the appropriate subspace subspace consistency and most other pc directions are strongly inconsistent broad sets of sucient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations in preparation for these results we show that the geometric representation of hdlss data holds under general conditions which includes a mixing condition and a broad range of sphericity measures of the covariance matrix

threshold selection for covariance estimation thresholding is a regularization method commonly used for covariance estimation which provides consistent estimators if the population covariance satisfies certain sparsity condition bickel and levina 2008a cai and liu 2011 however the performance of the thresholding estimators heavily depends on the threshold level by minimizing the frobenius risk of the adaptive thresholding estimator for covariances we conduct a theoretical study for the optimal threshold level and obtain its analytical expression a consistent estimator based on this expression is proposed for the optimal threshold level which is easy to implement in practice and efficient in computation numerical simulations and a case study on gene expression data are conducted to illustrate the proposed method

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

integration of an index to preserve the semantic interpretability in the multiobjective evolutionary rule selection and tuning of linguistic fuzzy systems in this paper we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions mfs is performed the proposed index is the aggregation of three metrics that preserve the original meanings of the mfs as much as possible while a tuning of their definition parameters is performed additionally rule selection mechanisms can be used to reduce the model complexity which involves another important interpretability aspect to this end we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy rule based systems with three objectives accuracy semantic interpretability maximization and complexity minimization we tested our approach on nine real world regression datasets in order to analyze the interaction between the fuzzy rule selection approach and the tuning approach these are also individually proved in a multiobjective framework and compared with their respective single objective counterparts we compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective based approaches results confirm the effectiveness of our approach and a wide range of solutions is obtained which are not only more interpretable but are also more accurate

central limit theorems for classical likelihood ratio tests for high dimensional normal distributions for random samples of size n obtained from p variate normal distributions we consider the classical likelihood ratio tests lrt for their means and covariance matrices in the high dimensional setting these test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chi square distributions as n goes to infinity and p remains fixed in this paper we consider the highdimensional case where both p and n go to infinity with p n y 0 1 we prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances we perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing high dimensional data

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

testing conditional independence using maximal nonlinear conditional correlation in this paper the maximal nonlinear conditional correlation of two random vectors x and y given another random vector z denoted by 1 x y z is defined as a measure of conditional association which satisfies certain desirable properties when z is continuous a test for testing the conditional independence of x and y given z is constructed based on the estimator of a weighted average of the form nz k 1 f z z k 2 1 x y z z k where f z is the probability density function of z and the z k s are some points in the range of z under some conditions it is shown that the test statistic is asymptotically normal under conditional independence and the test is consistent

nonparametric detection of signals by information theoretic criteria performance analysis and an improved estimator determining the number of sources from observed data is a fundamental problem in many scientific fields in this paper we consider the nonparametric setting and focus on the detection performance of two popular estimators based on information theoretic criteria the akaike information criterion aic and minimum description length mdl we present three contributions on this subject first we derive a new expression for the detection performance of the mdl estimator which exhibits a much closer fit to simulations in comparison to previous formulas second we present a random matrix theory viewpoint of the performance of the aic estimator including approximate analytical formulas for its overestimation probability finally we show that a small increase in the penalty term of aic leads to an estimator with a very good detection performance and a negligible overestimation probability

a range test for the equivalency of means under unequal variances in this article we present a range test using a two stage sampling procedure for testing the hypothesis that the normal means are falling into a practical indifference zone both the level and the power associated with the proposed test are controllable and are completely independent of the unknown variances tables needed for implementation are given

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

random matrix ensembles with an effective extensive external charge recent theoretical studies of chaotic scattering have encountered ensembles of random matrices in which the eigenvalue probability density function contains a one body factor with an exponent proportional to the number of eigenvalues two such ensembles have been encountered an ensemble of unitary matrices specified by the so called poisson kernel and the laguerre ensemble of positive definite matrices here we consider various properties of these ensembles jack polynomial theory is used to prove a reproducing property of the poisson kernel and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the dyson circular ensemble for the laguerre ensemble the scaled global density is calculated exactly for all even values of the parameter while for random matrices with unitary symmetry the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class

two sample tests for high dimensional covariance matrices we propose two tests for the equality of covariance matrices between two high dimensional populations one test is on the whole variance covariance matrices and the other is on offdiagonal sub matrices which define the covariance between two non overlapping segments of the high dimensional random vectors the tests are applicable i when the data dimension is much larger than the sample sizes namely the large p small n situations and ii without assuming parametric distributions for the two populations these two aspects surpass the capability of the conventional likelihood ratio test the proposed tests can be used to test on covariances associated with gene ontology terms

a simple approximation for the percentiles of the t distribution data analytic techniques are used in the construction of a simple approximation for the percentiles of the t distribution the approximation is a direct function of the degrees of freedom and the tail probability it is easily evaluated with a pocket calculator and should provide sufficient accuracy for most applications

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

cramer type large deviations for simple linear rank statistics a cramer type large deviation theorem for simple linear rank statistics is obtained range 0 x 0 n 1 mathord left vphantom 1 6 right kern nulldelimiterspace 6 the method of proof consists in approximating the simple linear rank statistic by a sum of independent uniformly bounded random variables and then applying a cramer type large deviation theorem on this sum

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

optimization of rational powered membership functions using extended kalman filter this paper presents an extension to the application of the extended kalman filter ekf in membership function mf optimization since the ekf is a derivative based method and derivative calculations of various mf shapes are too complicated so far only triangular type has been optimized via this approach in this article the ekf is applied in optimizing the rational powered mfs which are extended form of triangular trapezoidal functions and hence can offer better performance in almost all applications in addition because of the efficiency of the singleton mfs which makes them a common and desirable choice in hardware software realization of fuzzy systems these mfs are also optimized using the ekf the corresponding analytical equations and derivative formulas are given and finally the efficiency of the ekf in mf optimization is discussed

a diagnostic criterion for approximate factor structure abstract we build a simple diagnostic criterion for approximate factor structure in large panel datasets given observable factors the criterion checks whether the errors are weakly cross sectionally correlated or share at least one unobservable common factor interactive effects a general version allows to determine the number of omitted common factors also for time varying structures the empirical analysis runs on ten thousand us stocks from january 1968 to december 2011 for monthly returns we select time invariant specifications with at least four financial factors and a scaled three factor specification for quarterly returns we cannot select macroeconomic models without the market factor

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

an eigenvector based test for local stationarity applied to array processing in sonar array processing a challenging problem is the estimation of the data covariance matrix in the presence of moving targets in the water column since the time interval of data local stationarity is limited this work describes an eigenvector based method for proper data segmentation into intervals that exhibit local stationarity providing data driven higher bounds for the number of snapshots available for computation of time varying sample covariance matrices application of the test is illustrated with simulated data in a horizontal array for the detection of a quiet source in the presence of a loud interferer

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix let xk i i 1 k 1 be a double array of nondegenerate i i d random variables and let pn n 1 be a sequence of positive integers such that n pn is bounded away from 0 and in this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry l n rm max 1 leq i 0 we show that the following six statements are equivalent bf i quad lim n to infty n 2 int limits n log n 1 4 infty left f n 1 x f n 1 left frac sqrt n log n x right right df x 0 bf ii quad n mathbb p left max 1 leq i j leq n x 1 i x 1 j geq sqrt n log n right to 0 quad rm as n to infty bf iii quad frac w mu n sqrt n log n stackrel mathbb p rightarrow 2 sigma 2 bf iv quad left frac n log n right 1 2 l n stackrel mathbb p rightarrow 2 bf v quad lim n rightarrow infty mathbb p left frac w mu n 2 n sigma 4 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty bf vi quad lim n rightarrow infty mathbb p left n l n 2 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty where mu mathbb e x 1 1 sigma 2 mathbb e x 1 1 mu 2 and an 4 log pn log log pn the equivalences between i ii iii and v assume that only mathbb e x 1 1 2 1 2 and n 1 alpha l n stackrel mathbb p rightarrow 0 1 2 alpha leq 1 respectively the current work thus provides weak limit analogues of the strong limit theorems of li and rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of ln obtained by jiang some open problems are also posed

limits of spiked random matrices ii given a large high dimensional sample from a spiked population the top sample covariance eigenvalue is known to exhibit a phase transition we show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real wishart setting and its general analogue proving a conjecture of baik et al ann probab 33 1643 1697 2005 we also treat shifted mean gaussian orthogonal and ensembles such results are entirely new in the real case in the complex case we strengthen existing results by providing optimal scaling assumptions one obtains the known limiting random schrodinger operator on the half line but the boundary condition now depends on the perturbation we derive several characterizations of the limit laws in which appears as a parameter including a simple linear boundary value problem this pde description recovers known explicit formulas at 2 4 yielding in particular a new and simple proof of the painleve representations for these tracy widom distributions

a new coefficient of correlation abstract is it possible to define a coefficient of correlation which is a as simple as the classical coefficients like pearson s correlation or spearman s correlation and yet b consistently es

testing mutual independence in high dimension via distance covariance summary we introduce an l2 type test for testing mutual independence and banded dependence structure for high dimensional data the test is constructed on the basis of the pairwise distance covariance and it accounts for the non linear and non monotone dependences among the data which cannot be fully captured by the existing tests based on either pearson correlation or rank correlation our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal it exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non linear dependence in empirical data analysis successfully on the theory side asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size as a demonstration of good power properties for our distance covariance based test we further show that an infeasible version of our test statistic has the rate optimality in the class of gaussian distributions with equal correlation

an energy gain bounding approach to robust fuzzy identification a novel method for the robust identification of interpretable fuzzy models based on the criterion that identification errors are least sensitive to data uncertainties and modelling errors is suggested the robustness of identification errors towards unknown disturbances data uncertainties modelling errors etc is achieved by bounding i e minimizing the maximum possible value of energy gain from disturbances to the identification errors the solution of energy gain bounding problem being robust shows an improved performance of the identification method the flexibility of the proposed framework is shown by designing the variable learning rate identification algorithms in both deterministic and stochastic frameworks

a min max approach to fuzzy clustering estimation and identification this study for any unknown physical process y f x1 xn is concerned with the 1 fuzzy partition of n dimensional input space x x1timesmiddotmiddotmiddottimesxn into k different clusters 2 estimating the process behavior ycirc f xcirc for a given input xcirc xcirc1 middotmiddotmiddot xcircn isinx and 3 fuzzy approximation of the process with uncertain input output identification data x k plusmndeltaxk y k plusmnvk k 1 using a sugeno type fuzzy inference system a unified min max approach that attempts to minimize the worst case effect of data uncertainties and modeling errors on estimation performance is suggested to provide robustness against data uncertainties and modeling errors the proposed method of min max fuzzy parameters estimation does not make any assumption and does not require a priori knowledge of upper bounds statistics and distribution of data uncertainties and modeling errors to show the feasibility of the approach simulation studies and a real world application of physical fitness classification based on the fuzzy interpretation of physiological parameters have been provided

distribution free tests of independence in high dimensions we consider the testing of mutual independence among all entries in a formula see text dimensional random vector based on formula see text independent observations we study two families of distribution free test statistics which include kendall s tau and spearman s rho as important examples we show that under the null hypothesis the test statistics of these two families converge weakly to gumbel distributions and we propose tests that control the type i error in the high dimensional setting where formula see text we further show that the two tests are rate optimal in terms of power against sparse alternatives and that they outperform competitors in simulations especially when formula see text is large

asymptotic behavior of support vector machine for spiked population model for spiked population model we investigate the large dimension n and large sample size m asymptotic behavior of the support vector machine svm classification method in the limit of n m at fixed m n we focus on the generalization performance by analytically evaluating the angle between the normal direction vectors of svm separating hyperplane and corresponding bayes optimal separating hyperplane this is an analogous result to the one shown in paul 2007 and nadler 2008 for the angle between the sample eigenvector and the population eigenvector in random matrix theorem we provide not just bound but sharp prediction of the asymptotic behavior of svm that can be determined by a set of nonlinear equations based on the analytical results we propose a new method of selecting tuning parameter which significantly reduces the computational cost a surprising finding is that svm achieves its best performance at small value of the tuning parameter under spiked population model these results are confirmed to be correct by comparing with those of numerical simulations on finite size systems we also apply our formulas to an actual dataset of breast cancer and find agreement between analytical derivations and numerical computations based on cross validation

edge universality of correlation matrices xij 0 and e e 2 1 m n m these entries have a sub exponential decay at the tails we will be working in the regime n m dn limn 1 dn 6 0 1 in this paper we prove the edge universality of correlation matrices x x where the rectangular matrix x called the standardized matrix is obtained by normalizing each column of the data matrix e x by its euclidean norm our main result states that asymptotically the k point k 1 correlation functions of the extreme eigenvalues at both edges of the spectrum of the correlation matrix x x converge to those of the gaussian correlation matrix i e tracy widom law and thus in particular the largest and the smallest eigenvalues of x x after appropriate centering and rescaling converge to the tracy widom distribution the asymptotic distribution of extreme eigenvalues of the gaussian correlation matrix has been worked out only recently as a corollary of the main result in this paper we also obtain that the extreme eigenvalues of gaussian correlation matrices are asymptotically distributed according to the tracywidom law the proof is based on the comparison of green functions but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix we achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

tests for covariance matrix with fixed or divergent dimension testing covariance structure is of importance in many areas of statistical analysis such as microarray analysis and signal processing conventional tests for finite dimensional covariance cannot be applied to high dimensional data in general and tests for high dimensional covariance in the literature usually depend on some special structure of the matrix in this paper we propose some empirical likelihood ratio tests for testing whether a covariance matrix equals a given one or has a banded structure the asymptotic distributions of the new tests are independent of the dimension

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

testing for complete independence in high dimensions a simple statistic is proposed for testing the complete independence of random variables having a multivariate normal distribution the asymptotic null distribution of this statistic as both the sample size and the number of variables go to infinity is shown to be normal consequently this test can be used when the number of variables is not small relative to the sample size and in particular even when the number of variables exceeds the sample size the finite sample size performance of the normal approximation is evaluated in a simulation study and the results are compared to those of the likelihood ratio test

on the estimation of parameters of takagi sugeno fuzzy filters this study derives a class of filtering algorithms for takagi sugeno fuzzy models via solving a nonlinear parameters estimation problem the considered estimation problem is related to the problem of minimizing the expected value of the exponential of filtering errors energy under some stochastic assumptions the filtering criteria which involve an expectation operator are replaced by the deterministic quadratic optimization problems whose solutions provide a class of fuzzy filtering algorithms from a viewpoint of errors in the estimation of linear parameters of the fuzzy filter the derived filtering algorithms were analyzed with emphasis on stability robustness and steady state error issues the stability and robustness analyses have been made deterministically without making any assumption

enhanced fuzzy system models with improved fuzzy clustering algorithm although traditional fuzzy models have proven to have high capacity of approximating the real world systems they have some challenges such as computational complexity optimization problems subjectivity etc in order to solve some of these problems this paper proposes a new fuzzy system modeling approach based on improved fuzzy functions to model systems with continuous output variable the new modeling approach introduces three features i an improved fuzzy clustering ifc algorithm ii a new structure identification algorithm and iii a nonparametric inference engine the ifc algorithm yields simultaneous estimates of parameters of c regression models together with fuzzy c partitioning of the data to calculate improved membership values with a new membership function the structure identification of the new approach utilizes ifc instead of standard fuzzy c means clustering algorithm to fuzzy partition the data and it uses improved membership values as additional input variables along with the original scalar input variables for two different choices of regression methods least squares estimation or support vector regression to determine ldquofuzzy functionsrdquo for each cluster with novel ifc one could learn the system behavior more accurately compared to other fsm models the nonparametric inference engine is a new approach which uses the alike nearest neighbor method for reasoning empirical comparisons indicate that the proposed approach yields comparable or better accuracy than fuzzy or neuro fuzzy models based on fuzzy rules bases as well as other soft computing methods

a fuzzy c means variant for the generation of fuzzy term sets a fuzzy c means fcm variant is proposed for the generation of fuzzy term sets with overlap the proposed variant differs from the original mainly in two areas the first modification ensures that two end terms take the maximum and minimum domain values as their centers the second modification prevents the generation of non convex fuzzy terms that often occurs with the original algorithm the optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion the exponential weight m used in the algorithm is found to greatly affect the shape of the membership function the effect of data size used for the generation of fuzzy terms is also discussed a generalized shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the fcm variant using various m values

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

variational bayes for a mixed stochastic deterministic fuzzy filter this study under the variational bayes vb framework infers the parameters of a takagi sugeno fuzzy filter having deterministic antecedents and stochastic consequents the aim of this study is to take advantages of the vb framework to design fuzzy filtering algorithms which include an automated regularization incorporation of statistical noise models and model comparison capability the vb method can be easily applied to the linear in parameters models this paper applies the vb method to the nonlinear fuzzy filters without using taylor expansion for a linear approximation of some nonlinear function it is assumed that the nonlinear parameters i e antecedents of the fuzzy filter are deterministic while linear parameters are stochastic the vb algorithm by maximizing a strict lower bound on the data evidence makes the approximate posterior of linear parameters as close to the true posterior as possible the nonlinear deterministic parameters are tuned in a way to further increase the lower bound on data evidence the vb paradigm can be used to design an algorithm that automatically selects the most suitable fuzzy filter out of the considered finite set of fuzzy filters this is done by fitting the observed data as a stochastic combination of the different takagi sugeno fuzzy filters such that the individual filters compete with one another to model the data

painleve formulas of the limiting distributions for nonnull complex sample covariance matrices in a recent study of large non null sample covariance matrices a new sequence of functions gener alizing the gue tracy widom distribution of random matrix theory was obtained this paper derives painleve formulas of these functions and use them to prove that they are indeed distribution functions applications of these new distribution functions to last passage percolation queues in tandem and to tally asymmetric simple exclusion process are also discussed as a part of the proof a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

selecting the best population provided it is better than a standard the unequal variance case abstract bechhofer and turnbull 1978 proposed two procedures for selecting the best treatment that is better than a standard that is the procedures guarantee that a with probability at least p 0 specified no population is selected when the largest population mean is sufficiently less than the standard and b with probability at least p 1 specified the population with the largest mean is selected when that mean is sufficiently greater than the second largest mean and the standard their first procedure solved the problem for the case of known variances and the second was for the case of unknown but equal variances they suggested that the unequal variance case might be handled by using the method in rinott 1978 but no details were given this article provides the appropriate probability equations and the necessary table

multivariate sign based high dimensional tests for sphericity this article concerns tests for sphericity in cases where the data dimension is larger than the sample size the existing multivariate sign based procedure hallin paindaveine 2006 for sphericity is not robust with respect to high dimensionality producing tests with type i error rates that are much larger than the nominal levels this is mainly due to bias from estimating the location parameter we develop a correction that makes the existing test statistic robust with respect to high dimensionality and show that the proposed test statistic is asymptotically normal under elliptical distributions the proposed method allows the dimensionality to increase as the square of the sample size simulations demonstrate that it has good size and power in a wide range of settings

stable subspace tracking algorithm based on a signed urv decomposition subspace estimation and tracking are of fundamental importance in many signal processing algorithms the class of schur subspace estimators provides a complete parametrization of all principal subspace estimates defined as the column spans of corresponding low rank matrix approximants that lie within a specified 2 norm distance of a given matrix the parametrization is found in terms of a two sided hyperbolic decomposition hyperbolic urv or hurv which can be computed using hyperbolic rotations unfortunately such rotations are commonly associated with numerical instabilities in this paper we present a numerically stable non iterative algorithm to compute the hurv called the signed urv surv algorithm we show that this algorithm implicitly imposes certain constraints on the hurv such that important norm bounds that guarantee stability are satisfied the constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the svd which is a special case within this class the complexity of the algorithm is of the same order as that of a qr update updating and downdating are of the same complexity and are both numerically stable surv is proven to provide rank estimates consistent with the svd with the same rank threshold it can replace an svd where only subspace estimation is needed typical applications would e g be the detection of the number of signals in array signal processing and subspace estimation for source separation and interference mitigation such as the first step in music and esprit type algorithms simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the svd

adaptive fuzzy filtering in a deterministic setting many real world applications involve the filtering and estimation of process variables this study considers the use of interpretable sugeno type fuzzy models for adaptive filtering our aim in this study is to provide different adaptive fuzzy filtering algorithms in a deterministic setting the algorithms are derived and studied in a unified way without making any assumptions on the nature of signals i e process variables the study extends in a common framework the adaptive filtering algorithms usually studied in signal processing literature and p norm algorithms usually studied in machine learning literature to semilinear fuzzy models a mathematical framework is provided that allows the development and an analysis of the adaptive fuzzy filtering algorithms we study a class of nonlinear lms like algorithms for the online estimation of fuzzy model parameters a generalization of the algorithms to the p norm is provided using bregman divergences a standard tool for online machine learning algorithms

sample size determination in stratified trials to establish the equivalence of two treatments when designing a trial to establish that a new treatment is as effective as a standard one the conventional test procedure and sample size based on a null hypothesis of no difference between two treatments is inappropriate several authors have investigated test statistics and corresponding sample sizes based on the null hypothesis that the standard treatment is more effective than the new by at least some specific value for a single 2 x 2 table this paper considers a trial that involves several 2 x 2 tables and presents an approximate formula for the sample size required to obtain a given power of a one tailed score test for a null hypothesis of a specific common non zero difference between two treatments across strata i show that the sample size for a trial based on an unstratified test is always larger than that based on a stratified test when the design is balanced

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

a multiobjective evolutionary approach to concurrently learn rule and data bases of linguistic fuzzy rule based systems in this paper we propose the use of a multiobjective evolutionary approach to generate a set of linguistic fuzzy rule based systems with different tradeoffs between accuracy and interpretability in regression problems accuracy and interpretability are measured in terms of approximation error and rule base rb complexity respectively the proposed approach is based on concurrently learning rbs and parameters of the membership functions of the associated linguistic labels to manage the size of the search space we have integrated the linguistic two tuple representation model which allows the symbolic translation of a label by only considering one parameter with an efficient modification of the well known 2 2 pareto archived evolution strategy paes we tested our approach on nine real world datasets of different sizes and with different numbers of variables besides the 2 2 paes we have also used the well known nondominated sorting genetic algorithm nsga ii and an accuracy driven single objective evolutionary algorithm ea we employed these optimization techniques both to concurrently learn rules and parameters and to learn only rules we compared the different approaches by applying a nonparametric statistical test for pairwise comparisons thus taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective eas finally a data complexity measure which is typically used in pattern recognition to evaluate the data density in terms of average number of patterns per variable has been introduced to characterize regression problems results confirm the effectiveness of our approach particularly for possibly high dimensional datasets with high values of the complexity metric

an approach to theory of fuzzy signals basic definitions the paper presents a new approach to the description of fourier transform and correlation function taking into consideration the uncertainty of the transformed function a conventional approach to uncertainty employs probabilistic description here fuzzy logic theory is applied to describe this uncertainty new definitions namely fuzzy fourier transform fuzzy correlation and possibility correlation are proposed the definition of correlation function is based on the concept of fuzzy functional spaces some examples of applications and a comparison with the conventional approach are shown

geometric representation of high dimension low sample size data summary high dimension low sample size data are emerging in various areas of science we find a common structure underlying many such data sets by using a non standard type of asymptotics the dimension tends to while the sample size is fixed our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex essentially all the randomness in the data appears only as a random rotation of this simplex this geometric representation is used to obtain several new statistical insights

a table of percentage points of the range of independent t variables a table of selected percentiles of the range of independent student s t variables is provided these percentiles are required by several multiple comparison procedures notably those not requiring the assumption of equal variances an illustration is given

corrections to lrt on large dimensional covariance matrix by rmt in this paper we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from gaussian populations when the dimension p is large compared to the sample size n next using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random f matrices we propose necessary corrections for these lr tests to cope with high dimensional effects the asymptotic distributions of these corrected tests under the null are given simulations demonstrate that the corrected lr tests yield a realized size close to nominal level for both moderate p around 20 and high dimension while the traditional lr tests with 2 approximation fails another contribution from the paper is that for testing the equality between two covariance matrices the proposed correction applies equally for non gaussian populations yielding a valid pseudo likelihood ratio test

sparse principal component analysis and iterative thresholding principal component analysis pca is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix however it behaves poorly when the number of features p is comparable to or even much larger than the sample size n in this paper we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse under a spiked covariance model we find that the new approach recovers the principal subspace and leading eigenvectors consistently and even optimally in a range of high dimensional sparse settings simulated examples also demonstrate its competitive performance

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

deterministic parallel analysis an improved method for selecting factors and principal components factor analysis and principal component analysis are used in many application areas the first step choosing the number of components remains a serious challenge our work proposes improved methods for this important problem one of the most popular state of the art methods is parallel analysis pa which compares the observed factor strengths with simulated strengths under a noise only model the paper proposes improvements to pa we first derandomize it proposing deterministic pa which is faster and more reproducible than pa both pa and deterministic pa are prone to a shadowing phenomenon in which a strong factor makes it difficult to detect smaller but more interesting factors we propose deflation to counter shadowing we also propose to raise the decision threshold to improve estimation accuracy we prove several consistency results for our methods and test them in simulations we also illustrate our methods on data from the human genome diversity project where they significantly improve the accuracy

hybrid identification in fuzzy neural networks this paper introduces an identification method for nonlinear models in the form of fuzzy neural networks fnn in this model we use two forms of the fuzzy inference methods a simplified and linear fuzzy inference and exploit a standard error back propagation learning algorithm the fnn modeling and identification environment realizes parameter identification through a synergistic usage of clustering techniques genetic optimization and a complex search method we use a hard c means hcm clustering algorithm to determine initial apexes of the membership functions of the information granules used in this fuzzy model the parameters such as apexes of membership functions learning rates and momentum coefficients are then adjusted using hybrid algorithm the proposed hybrid identification algorithm is carried out by combining both genetic optimization genetic algorithm ga and the improved complex method an aggregate objective function performance index with a weighting factor is introduced to achieve a sound balance between approximation and generalization of the model according to the selection and adjustment of the weighting factor of this objective function we reveal how to design a model with sound approximation and generalization abilities the proposed model is experimented with using several time series data gas furnace sewage treatment process and nox emission process data of gas turbine power plant

two sample covariance matrix testing and support recovery in high dimensional and sparse settings in the high dimensional setting this article considers three interrelated problems a testing the equality of two covariance matrices and b recovering the support of and c testing the equality of and row by row we propose a new test for testing the hypothesis h 0 and investigate its theoretical and numerical properties the limiting null distribution of the test statistic is derived and the power of the test is studied the test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives the simulation results show that the test significantly outperforms the existing methods both in terms of size and power analysis of a prostate cancer dataset is carried out to demonstrate the application of the testing procedures when the null hypothesis of equal covariance matrices is rejected it is often of significant interest to further investigate how they differ from each other motivated by applications in genomics we also consider recovering the support of and

how to establish equivalence when data are censored a randomized trial of treatments for b non hodgkin lymphoma interest in equivalence trials has been increasing for many years though the methodology which has been developed for such trials is mainly for uncensored data in cancer research we are more often concerned with survival in an efficacy trial the null hypothesis specifies equality of the two survival distributions but in an equivalence trial a null hypothesis of inequivalence h0 has to be tested the usual logrank test has to be modified to test whether the true value r of the ratio of hazard rates in two treatment groups is at least equal to a limit value r0 if prognostic factors have to be taken into account the cox model provides tests of ho and a useful confidence interval for the adjusted relative derived from the regression parameter for the treatment indicator an equivalence trial of maintenance therapy was carried out in children with b non hodgkin lymphoma and serves as an illustration

a robust design criterion for interpretable fuzzy models with uncertain data we believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation h sup 2 and h sup spl infin filtering techniques because of the fact that not only linear parameters consequents but also the nonlinear parameters membership functions attempt to identify the uncertain behavior of the unknown system however the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system s behavior is optimal in some sense this study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data we consider online identification of an interpretable fuzzy model based on the robust solution of a regularized least squares fuzzy parameters estimation problem the aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties the study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors a time domain feedback analysis of the proposed identification approach is carried out with emphasis on stability robustness and steady state issues the simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods