correlated wishart ensembles and chaotic time series we study the correlated wishart ensembles in the context of time series analysis we are interested in the statistics of eigenlevels viz variables associated with independent eigenmodes in the system the motivation of this work is to study the effect of time series correlations on the wishart ensembles in this connection we derive the level density and the two point function for the correlated wishart ensembles by using the binary correlation method using our analytic results we analyze spectra of autocovariance matrices derived from single variable stationary time series we consider the stochastic time series of gaussian variables with exponentially decaying correlations and time series of chaotic maps viz the arnold map the standard map and the stadium billiard map in both cases correlated time series are encountered and analyzed under the framework of random matrix theory it is shown that the eigenlevel statistics for the chaotic maps follow closely those of correlated wishart ensembles it is indicated that the presence of collective modes in the spectra of autocovariance matrices is related to the integrability of the system

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

skew orthogonal polynomials and random matrix ensembles there is considerable interest in understanding the relation between random matrix ensembles and quantum chaotic systems in the context of the universality of energy level correlations in this connection while gaussian ensembles of random matrices have been studied extensively not much is known about ensembles with non gaussian weight functions dyson has shown that the n level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew orthogonal polynomials orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances we have obtained the following results 1 skew orthogonal polynomials of both types are derived for the jacobi class of weight functions including the limiting cases of associated laguerre and hermite or gaussian 2 matrix integral representations are given for the general weight functions 3 asymptotic forms of the polynomials are obtained rigorously for the jacobi class and in the form of an ansatz for the general case 4 for the three types of ensembles the asymptotic n level correlation functions with appropriate scaling are shown to be universal being independent of the weight function and location in the spectrum and identical with the well known gaussian results this provides a rigorous justification for the universality of the gaussian ensemble results observed in quantum chaotic systems as expected the level density is not universal

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

quantifying and modeling long range cross correlations in multiple time series with applications to world stock indices we propose a modified time lag random matrix theory in order to study time lag cross correlations in multiple time series we apply the method to 48 world indices one for each of 48 different countries we find long range power law cross correlations in the absolute values of returns that quantify risk and find that they decay much more slowly than cross correlations between the returns the magnitude of the cross correlations constitutes bad news for international investment managers who may believe that risk is reduced by diversifying across countries we find that when a market shock is transmitted around the world the risk decays very slowly we explain these time lag cross correlations by introducing a global factor model gfm in which all index returns fluctuate in response to a single global factor for each pair of individual time series of returns the cross correlations between returns or magnitudes can be modeled with the autocorrelations of the global factor returns or magnitudes we estimate the global factor using principal component analysis which minimizes the variance of the residuals after removing the global trend using random matrix theory a significant fraction of the world index cross correlations can be explained by the global factor which supports the utility of the gfm we demonstrate applications of the gfm in forecasting risks at the world level and infinding uncorrelated individual indices we find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices which is relevant information for world managers in reducing their portfolio risk finally we argue that this general method can be applied to a wide range of phenomena in which time series are measured ranging from seismology and physiology to atmospheric geophysics

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

emerging spectra of singular correlation matrices under small power map deformations correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular yet most analysis assume stationarity of the underlying time series this tends to be an assumption of varying and often dubious validity the validity of the assumption improves as shorter time series are used if many time series are used this implies an analysis of highly singular correlation matrices we attack this problem by using the so called power map which was introduced to reduce noise its nonlinearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so emerging spectra to correlations this sensitivity will be demonstrated for uncorrelated and correlated wishart ensembles

random matrix analysis of human eeg data we use random matrix theory to demonstrate the existence of generic and subject independent features of the ensemble of correlation matrices extracted from human eeg data in particular the spectral density as well as the level spacings was analyzed and shown to be generic and subject independent we also investigate number variance distributions in this case we show that when the measured subject is visually stimulated the number variance displays deviations from the random matrix prediction

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

stability of finite delay difference systems the stability of finite delay difference systems has recently been discussed in l in this paper we consider the delay difference systems with infinite delay of the following form x n 1 g n x s s 1 1 1 n g n x 1 where 1 is an integer or w we assume g n 0 0 forn l l l so that 1 has the zero solution x n 0 clearly for any given integer n l 1 and a given bounded initial function 4 l 1 1 no rk there is a unique solution of l denoted by x n n 4 such that it satisfies 1 for all integers n 2 n and x n no 4 444 forn f l l no for any integer m 2 co let 2 m m 1 i and 2 z the integer set if m a for any n e z define ii411 tlnl sup l s 15 s 5 n and for any h 0 let c n 4 z rk i nl h in the sequel we always assume that the variables n r and s take integer values and the corresponding intervals are discrete intervals

boundedness of neutral delay difference systems abstract for neutral delay differences systems of the general form two kinds of boundedness criteria are established in this paper one is by means of the discrete liapunov functionals while the other is in terms of the discrete liapunov functions with razumikhin techniques the obtained results are new in the sense that their counterparts for neutral delay differential systems have not appeared so far

eigenvalue densities of real and complex wishart correlation matrices wishart correlation matrices are the standard model for the statistical analysis of time series the ensemble averaged eigenvalue density is of considerable practical and theoretical interest for complex time series and correlation matrices the eigenvalue density is known exactly in the real case a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results we use the supersymmetry method to fully circumvent this problem we present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums

new bounds and asymptotic expansions for the constants of landau and lebesgue abstract the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively in this paper we establish new bounds and asymptotic expansions for the constants of landau and lebesgue

impulsive control of discrete systems with time delay this note deals with impulsive control of discrete systems with time delay it s the first time that an impulsive controller was introduced for discrete systems with time delay firstly some stability criteria for delay discrete systems with impulse effects are obtained by using lyapunov functionals and razumikhin technique these stability criteria show that if the impulses satisfy some conditions then the original stable systems can keep their stability property even with impulse effects then by using lyapunov functionals or lyapunov functions together with razumikhin technique two stability criteria for the discrete systems with time delay under impulsive control are given these two results show that unstable systems can be stable under impulsive control some examples are also presented to illustrate the efficiency of the obtained results

stability of neutral delay difference systems abstract for neutral delay difference systems of the general form two kinds of stability criteria are established in this paper one is by means of the discrete liapunov functionals while the other is with the discrete liapunov functions both make use of razumikhin techniques

spectral density of the correlation matrix of factor models a random matrix theory approach we studied the eigenvalue spectral density of the correlation matrix of factor models of multivariate time series by making use of the random matrix theory we analytically quantified the effect of statistical uncertainty on the spectral density due to the finiteness of the sample we considered a broad range of models ranging from one factor models to hierarchical multifactor models

statistics of atmospheric correlations for a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions recent advances show that the scope of random matrix theory is much wider in this work we show that the random matrix approach can be beneficially applied to a completely different classical domain namely to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterize the state of atmosphere we show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription in particular the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution

spectral density of a wishart model for nonsymmetric correlation matrices the wishart model for real symmetric correlation matrices is defined as mathsf w mathsf aa t where matrix mathsf a is usually a rectangular gaussian random matrix and mathsf a t is the transpose of mathsf a analogously for nonsymmetric correlation matrices a model may be defined for two statistically equivalent but different matrices mathsf a and mathsf b as mathsf ab t the corresponding wishart model thus is defined as mathbf c mathsf ab t mathsf ba t we study the spectral density of mathbf c for the case when mathsf a and mathsf b are not statistically independent the ensemble average of such nonsymmetric matrices therefore does not simply vanishes to a null matrix in this paper we derive a pastur self consistent equation which describes spectral density of mathbf c at large matrix dimension we complement our analytic results with numerics

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

application of random matrix theory to microarray data for discovering functional gene modules we show that spectral fluctuation of coexpression correlation matrices of yeast gene microarray profiles follows the description of the gaussian orthogonal ensemble goe of the random matrix theory rmt and removal of small values of the correlation coefficients results in a transition from the goe statistics to the poisson statistics of the rmt this transition is directly related to the structural change of the gene expression network from a global network to a network of isolated modules