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

information transfer through disordered media by diffuse waves we consider the information content h of a scalar multiple scattered diffuse wave field psi r and the information capacity c of a communication channel that employs diffuse waves to transfer the information through a disordered medium both h and c are shown to be directly related to the mesoscopic correlations between the values of psi r at different positions r in space arising due to the coherent nature of the wave for the particular case of a communication channel between two identical linear arrays of n 1 equally spaced transmitters or receivers receiver spacing a we show that the average capacity proportional variant n and obtain explicit analytic expressions for n in the limit of n infinity and kl infinity where k 2pi lambda lambda is the wavelength and l is the mean free path modification of the above results in the case of finite but large n and kl is discussed as well if the signal to noise ratio s n exceeds some critical value s n c n is a nonmonotonic function of a exhibiting maxima at ka mpi m 1 2 em leader for smaller s n ka mpi correspond to local minima while the absolute maximum of n is reached at some ka approximately s n 1 2 max as maximized over the receiver spacing a and the optimal normalized receiver spacing ka opt as the spacing maximizing both max n and ka opt scale as s n 1 2 for s n max n approximately log s n for s n s n c

density results for proper efficiencies several density results are established for different notions of proper efficiency in vector optimization without the requirement of ordering cones being boundedly based for a compact set the set of henig proper efficient points is shown to be dense in the set of all efficient points density theorems for borwein s proper efficiency and for positive scalarizable efficiency follow immediately with appropriate assumptions

systematic analysis of group identification in stock markets we propose improved methods to identify stock groups using the correlation matrix of stock price changes by filtering out the marketwide effect and the random noise we construct the correlation matrix of stock groups in which nontrivial high correlations between stocks are found using the filtered correlation matrix we successfully identify the multiple stock groups without any extra knowledge of the stocks by the optimization of the matrix representation and the percolation approach to the correlation based network of stocks these methods drastically reduce the ambiguities while finding stock groups using the eigenvectors of the correlation matrix

spectral analysis of time dependent market adjusted return correlation matrix abstract we present an adjusted method for calculating the eigenvalues of a time dependent return correlation matrix in a moving window first we compare the normalized maximum eigenvalue time series of the market adjusted return correlation matrix to that of the logarithmic return correlation matrix on an 18 year dataset of 310 s p 500 listed stocks for small and large window or memory sizes we observe that the resulting new eigenvalue time series is more stationary than the time series obtained without the adjustment second we perform this analysis while sweeping the window size 5 100 500 in order to examine the dependence on the choice of window size this approach demonstrates the multi modality of the eigenvalue distributions we find that the three dimensional distribution of the eigenvalue time series for our market adjusted return is significantly more stationary than that produced by classic method finally our model offers an approximate polarization domain characterized by a smooth l shaped strip the polarization with large amplitude is revealed while there is persistence in agreement of individual stock returns movement with market with small amplitude most of the time

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

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

statistical properties of random density matrices statistical properties of ensembles of random density matrices are investigated we compute traces and von neumann entropies averaged over ensembles of random density matrices distributed according to the bures measure the eigenvalues of the random density matrices are analysed we derive the eigenvalue distribution for the bures ensemble which is shown to be broader then the quarter circle distribution characteristic of the hilbert schmidt ensemble for measures induced by partial tracing over the environment we compute exactly the two point eigenvalue correlation function

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

a new method to estimate the noise in financial correlation matrices companies belonging to the same industrial branch are subject to similar economical influences hence the time series of their stocks can show similar trends implying a correlation financial correlation matrices measure the unsystematic correlations between time series of stocks such information is important for risk management it has been found by laloux et al that the correlation matrices are noise dressed a major reason being the finiteness of the time series we present a new and alternative method to estimate this noise we introduce a power mapping of the elements in the correlation matrix which suppresses the noise and thereby effectively prolongs the time series neither further data processing nor additional input is needed to develop and test our method we use a model suggested by noh which can be viewed as a special case of a factor model in economics we perform numerical simulations for the time series and obtain correlation matrices we support the numerics by a qualitative analytical discussion with our approach different correlation structures buried under this noise can be detected our method is general and can be applied to all systems in which time series are measured

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

the arithmetic of zero cycles on surfaces with geometric genus and irregularity zero let x be a smooth projective geometrically irreducible surface over a perfect field f throughout this paper it will be assumed that the geometric genus pg and the irregularity q of x both vanish denote the separable closure of f by f let x x be the surface obtained from x by base extension it will also be assumed that the group ao x of rational equivalence classes of zero cycles of degree zero on x vanishes this is a technical hypothesis which could presumably be eliminated at the expense of working with ker ao x ao x for want of a better name and for ease in stating various results any surface which satisfies these three hypotheses will be called a pseudo rational surface bloch 2 has conjectured that the vanishing of ao x should follow from the assumption that po q 0 this was proven by bloch kas and lieberman 5 for all such surfaces which are not of general type i e which have kodaira dimension less than 2 it has also been proven for particular surfaces of general type by inose and mizukami 23 barlow 1 and keum 25 consequently the class of pseudo rational surfaces includes rational surfaces enriques surfaces elliptic surfaces with q 0 the classical godeaux surface burniat inoue surfaces campedelti surfaces and the surfaces of barlow and of keum this paper will study ao x for pseudo rational surfaces defined over fields of number theoretic interest bloch 4 introduced k theoretic techniques into the study of zero cycles on rational surfaces his work was extended 7 12 15 27 28 32 36 to achieve a thorough understanding of such cycles colliot th616ne and raskind 10 developed this machinery further to study codimension two cycles on any variety the author 14 applied these techniques to enriques surfaces raskind 30 used them to study zero cycles on pseudo rational surfaces one of the main results of this paper is a new proof of the following theorem of raskind 31

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

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

structure of a financial cross correlation matrix under attack abstract we investigate the structure of a perturbed stock market in terms of correlation matrices for the purpose of perturbing a stock market two distinct methods are used namely local and global perturbation the former involves replacing a correlation coefficient of the cross correlation matrix with one calculated from two gaussian distributed time series while the latter reconstructs the cross correlation matrix just after replacing the original return series with gaussian distributed time series concerning the local case it is a technical study only and there is no attempt to model reality the term global means the overall effect of the replacement on other untouched returns through statistical analyses such as random matrix theory rmt network theory and the correlation coefficient distributions we show that the global structure of a stock market is vulnerable to perturbation however apart from in the analysis of inverse participation ratios iprs the vulnerability becomes dull under a small scale perturbation this means that these analysis tools are inappropriate for monitoring the whole stock market due to the low sensitivity of a stock market to a small scale perturbation in contrast when going down to the structure of business sectors we confirm that correlation based business sectors are regrouped in terms of iprs this result gives a clue about monitoring the effect of hidden intentions which are revealed via portfolios taken mostly by large investors

random but not so much a parameterization for the returns and correlation matrix of financial time series abstract a parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied this parameterization allows easy introduction of non stationarity and it shows several of the characteristics of the true observed realizations such as fat tails volatility clustering and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of random matrix theory results the predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of brazilian assets and it is shown that those predictions fit the data better than random matrix theory

construction of point vortex equilibria via brownian ratchets a theory capable of producing equilibrium configurations of point vortices in the plane along with a numerical scheme to compute them is described the theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation agz 0 where a is the 1 2 n nk1 n non normal configuration matrix obtained by requiring that all intervortical distances remain fixed and g2r n are the n vortex strengths for existence of an equilibrium a must have a non trivial nullspace we consider the singular values of a when this has one or more zero singular values the nullspace of a is non empty and an equilibrium exists for some choice of g new equilibrium configurations are found numerically by randomly depositing n points in the plane which generically gives rise to a configuration matrix a with empty nullspace using the sum of squares of the k smallest singular values of a as a ratchet we thermally fluctuate the configuration allowing each point to execute a random walk in the plane retaining only those configurations which reduce this quantity at the next step the configuration is thus driven to one with nullspace a zko0 these converged states are not necessarily nearby their initial configurations typically they are asymmetric and often we can drive the same initial state to several different equilibria a reverse ratchet method is also described which can produce initial conditions that would evolve to a specified equilibrium state once a converged final state is achieved the full singular value decomposition of a is used to calculate an optimal basis set for the nullspace of a and thus all allowable g the distribution of the singular values gives important information on the size of each equilibrium state as measured by frobenius norm their distance from each other spacing and density and how far a randomly chosen system of n points in the plane is from the nearest equilibrium configuration with a specified rank as well as its shannon entropy

data driven estimates of the number of clusters in multivariate time series an important problem in unsupervised data clustering is how to determine the number of clusters here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series two nonparametric and purely data driven algorithms are expounded and compared the first exploits the eigenvalue spectra of surrogate data while the second employs the eigenvector components of the interrelation matrix compared to the first algorithm the second approach is computationally faster and not limited to linear interrelation measures

inclusion theorems for non explosive and strongly exposed cones in normed spaces abstract this paper is composed of five sections in the first one we classify cones into pseudonuclear intermediate and explosive the notion of strongly exposed cone and the equivalence between pseudonuclearity and strong exposition for cones in a normed space are recalled in the second one it is shown that several properties for a cone in a normed space exist each one being equivalent to strong exposition of the cone furthermore 2 5 an estimate by chung is improved in the third one through projections of rank one some cones that are an immediate extension of elliptic cones in elementary geometry are defined and it is established 3 3 that a cone is strongly exposed if and only if it can be contained in one of that type in the fourth one it is shown 4 2 that also the non explosivity of a cone is expressible in terms of metric properties more precisely a cone is non explosive if and only if it can be contained in a balanced cone of elliptic kind but in which the axis becomes a finite dimensional subspace it may be interesting to observe that the result does not depend on particular hypotheses e g reflexivity on the space in the fifth one it is put in evidence a property of a cone in a reflexive normed space which excludes that the cone is intermediate an orientative essential bibliography concludes the paper

geometry of cones and an application in the theory of pareto efficient points in this article we give a new criterion for the existence of a bounded base for a cone p of a normed space x also if p is closed we give a partial answer to the problem is 0 a point of continuity of p if and only if 0 is a denting point of p the above problems have applications in the theory of pareto efficient points

one factor model for the cross correlation matrix in the vietnamese stock market abstract random matrix theory rmt has been applied to the analysis of the cross correlation matrix of a financial time series the most important findings of previous studies using this method are that the eigenvalue spectrum largely follows that of random matrices but the largest eigenvalue is at least one order of magnitude higher than the maximum eigenvalue predicted by rmt in this work we investigate the cross correlation matrix in the vietnamese stock market using rmt and find similar results to those of studies realized in developed markets us europe japan 9 18 as well as in other emerging markets 20 21 19 22 importantly we found that the largest eigenvalue could be approximated by the product of the average cross correlation coefficient and the number of stocks studied we demonstrate this dependence using a simple one factor model the model could be extended to describe other characteristics of the realistic data

measures of globalization based on cross correlations of world financial indices abstract the cross correlation matrix of daily returns of stock market indices in a diverse set of 37 countries worldwide was analyzed comparison of the spectrum of this matrix with predictions of random matrix theory provides an empirical evidence of strong interactions between individual economies as manifested by three largest eigenvalues and the corresponding set of stable non random eigenvectors the observed correlation structure is robust with respect to changes in the time horizon of returns ranging from 1 to 10 trading days and to replacing individual returns with just their signs this last observation confirms that it is correlations between signs and not absolute values of fluctuations which are mostly responsible for the observed effect negative changes in the index are somewhat more correlated than the positive ones also in our data set the reaction of asian stock indices to changes in european and american ones persists for about 3 days

decomposing the stock market intraday dynamics abstract the correlation matrix formalism is used to study temporal aspects of the stock market evolution this formalism allows to decompose the financial dynamics into noise as well as into some coherent repeatable intraday structures the present study is based on the high frequency deutsche aktienindex dax data over the time period between november 1997 and september 1999 and makes use of both the corresponding returns as well as volatility variations one principal conclusion is that a bulk of the stock market dynamics is governed by the uncorrelated noise like processes there exists however a small number of components of coherent short term repeatable structures in fluctuations that may generate some memory effects seen in the standard autocorrelation function analysis laws that govern fluctuations associated with those various components are different which indicates an extremely complex character of the financial fluctuations

effect of changing data size on eigenvalues in the korean and japanese stock markets abstract in this study we attempted to determine how eigenvalues change according to random matrix theory rmt in stock market data as the number of stocks comprising the correlation matrix changes specifically we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correlation matrix we determined that the value of the eigenvalue increases in proportion with the number of stocks furthermore we noted that the largest eigenvalue maintains its identical properties regardless of the number and type whereas other eigenvalues evidence different features

random matrix theory analysis of cross correlations in the us stock market evidence from pearson s correlation coefficient and detrended cross correlation coefficient abstract in this study we first build two empirical cross correlation matrices in the us stock market by two different methods namely the pearson s correlation coefficient and the detrended cross correlation coefficient dcca coefficient then combining the two matrices with the method of random matrix theory rmt we mainly investigate the statistical properties of cross correlations in the us stock market we choose the daily closing prices of 462 constituent stocks of s p 500 index as the research objects and select the sample data from january 3 2005 to august 31 2012 in the empirical analysis we examine the statistical properties of cross correlation coefficients the distribution of eigenvalues the distribution of eigenvector components and the inverse participation ratio from the two methods we find some new results of the cross correlations in the us stock market in our study which are different from the conclusions reached by previous studies the empirical cross correlation matrices constructed by the dcca coefficient show several interesting properties at different time scales in the us stock market which are useful to the risk management and optimal portfolio selection especially to the diversity of the asset portfolio it will be an interesting and meaningful work to find the theoretical eigenvalue distribution of a completely random matrix r for the dcca coefficient because it does not obey the marcenko pastur distribution

eigenvalue density in hermitian matrix models by the lax pair method in this paper a new method is discussed to derive the eigenvalue density in a hermitian matrix model with a general potential the density is considered on one interval or multiple disjoint intervals the method is based on lax pair theory and the cayley hamilton theorem by studying the orthogonal polynomials associated with the hermitian matrix model it is obtained that the restriction conditions for the parameters in the density are connected to the discrete painleve i equation and the results are related to the scalar riemann hilbert problem some special density functions are also discussed in association with the known results in this subject

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

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

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

a generalization of a theorem of arrow barankin and blackwell this paper presents a generalization of a known density theorem of arrow barankin and blackwell admissible points of convex sets in contributions to the theory of games h w kuhn and a w tucker eds princeton university press princeton nj 1953 this result is shown to hold even in a real normed space partially ordered by a bishop phelps cone

quantifying the dynamics of financial correlations abstract a novel application of the correlation matrix formalism to study dynamics of the financial evolution is presented this formalism allows to quantify the memory effects as well as some potential repeatable intraday structures in the financial time series the present study is based on the high frequency deutsche aktienindex dax data over the time period between november 1997 and december 1999 and demonstrates a power of the method in this way two significant new aspects of the dax evolution are identified i the memory effects turn out to be sizably shorter than what the standard autocorrelation function analysis seems to indicate and ii there exist short term repeatable structures in fluctuations that are governed by a distinct dynamics the former of these results may provide an argument in favour of the market efficiency while the latter one may indicate origin of the difficulty in reaching a gaussian limit expected from the central limit theorem in the distribution of returns on longer time horizons

denoising of diffusion mri using random matrix theory we introduce and evaluate a post processing technique for fast denoising of diffusion weighted mr images by exploiting the intrinsic redundancy in diffusion mri using universal properties of the eigenspectrum of random covariance matrices we remove noise only principal components thereby enabling signal to noise ratio enhancements this yields parameter maps of improved quality for visual quantitative and statistical interpretation by studying statistics of residuals we demonstrate that the technique suppresses local signal fluctuations that solely originate from thermal noise rather than from other sources such as anatomical detail furthermore we achieve improved precision in the estimation of diffusion parameters and fiber orientations in the human brain without compromising the accuracy and spatial resolution

on the analysis of cross correlations in south african market data we report briefly on an application of random matrix theory to the analysis of sa financial market data an analysis of cross correlations in south african financial market data e print cond mat 0402389 correlation matrices c are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange from january 1993 to december 2002 spectral properties of c are tested against random matrix predictions we highlight some quantitative differences which arise when treating prices as existing only when measured as opposed to interpolating missing or illiquid trading days with a zero order hold

accuracy of pseudo inverse covariance learning a random matrix theory analysis for many learning problems estimates of the inverse population covariance are required and often obtained by inverting the sample covariance matrix increasingly for modern scientific data sets the number of sample points is less than the number of features and so the sample covariance is not invertible in such circumstances the moore penrose pseudo inverse sample covariance matrix constructed from the eigenvectors corresponding to nonzero sample covariance eigenvalues is often used as an approximation to the inverse population covariance matrix the reconstruction error of the pseudo inverse sample covariance matrix in estimating the true inverse covariance can be quantified via the frobenius norm of the difference between the two the reconstruction error is dominated by the smallest nonzero sample covariance eigenvalues and diverges as the sample size becomes comparable to the number of features for high dimensional data we use random matrix theory techniques and results to study the reconstruction error for a wide class of population covariance matrices we also show how bagging and random subspace methods can result in a reduction in the reconstruction error and can be combined to improve the accuracy of classifiers that utilize the pseudo inverse sample covariance matrix we test our analysis on both simulated and benchmark data sets

two faced property of a market factor in asset pricing and diversification effect abstract this study empirically investigates the test hypothesis that a market factor acting as a representative common factor in the pricing models has a negative influence on constructing a well diversified portfolio from the markowitz mean variance optimization function mvof we use the comparative correlation matrix c cm method to control a single eigenvalue among all eigenvalues included in the sample correlation matrix s cm through the random matrix theory rmt in particular this study observes the effect of the largest eigenvalue that has the property of the market factor according to the results the largest eigenvalue has the highest explanatory power on the stock return changes the c cm without the largest eigenvalue in the s cm constructs a more diversified portfolio capable of improving the practical applicability of the mvof moreover the more diversified portfolio constructed from this c cm has better out of sample performance in the future period these results support the test hypothesis for the two faced property of the market factor defined by the largest eigenvalue

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

random matrix approach to categorical data analysis correlation and similarity measures are widely used in all the areas of sciences and social sciences often the variables are not numbers but are instead qualitative descriptors called categorical data we define and study similarity matrix as a measure of similarity for the case of categorical data this is of interest due to a deluge of categorical data such as movie ratings top 10 rankings and data from social media in the public domain that require analysis we show that the statistical properties of the spectra of similarity matrices constructed from categorical data follow random matrix predictions with the dominant eigenvalue being an exception we demonstrate this approach by applying it to the data for indian general elections and sea level pressures in the north atlantic ocean

localized short range correlations in the spectrum of the equal time correlation matrix we suggest a procedure to identify those parts of the spectrum of the equal time correlation matrix mathbf c where relevant information about correlations of a multivariate time series is induced using an ensemble average over each of the distances between eigenvalues all nearest neighbor distributions can be calculated individually we present numerical examples where a information about cross correlations is found in the so called bulk of eigenvalues which generally is thought to contain only random correlations and where b the information extracted from the lower edge of the spectrum of mathbf c is statistically more significant than that extracted from the upper edge we apply the analysis to electroencephalographic recordings with epileptic events

the shannon entropy of sudoku matrices we study properties of an ensemble of sudoku matrices a special type of doubly stochastic matrix when normalized using their statistically averaged singular values the determinants are very near

patterns of a spatial exploration under time evolution of the attractiveness persistent nodes degree distribution and spectral properties abstract this work explores the features of a graph generated by agents that hop from one node to another node where the nodes have evolutionary attractiveness the jumps are governed by boltzmann like transition probabilities that depend both on the euclidean distance between the nodes and on the ratio of the attractiveness between them it is shown that persistent nodes i e nodes that never been reached by this special random walk are possible in the stationary limit differently from the case where the attractiveness is fixed and equal to one for all nodes 1 simultaneously one also investigates the spectral properties and statistics related to the attractiveness and degree distribution of the evolutionary network finally a study of the crossover between persistent phase and no persistent phase was performed and it was also observed the existence of a special type of transition probability which leads to a power law behaviour for the time evolution of the persistence

econophysics financial time series from a statistical physics point of view in recent years physicists have started applying concepts and methods of statistical physics to study economic problems the word econophysics is sometimes used to refer to this work much recent work is focused on understanding the statistical properties of nancial time series one reason for this interest is that nancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that nancial time series viewed from a dierent perspective might yield new results this article reviews the results of three recent phenomenological studies i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to drastic events such as market crashes the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as eight orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by three orders of magnitude from 1 min up to approximately 10 d ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law and persist for several months iii correlations among dierent companies the third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two dierent stocks from a study of the eigenvalue statistics of the cross correlation matrix constructed from price uctuations of the leading 1000 stocks we nd that the largest 1 of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix whereas the rest of the eigenvalues conform to random matrix behavior suggesting that these 1 of the eigenvalues contain system specic information about correlated time evolution of dierent companies c 2000 published by elsevier science b v all rights reserved

a random matrix theory approach to financial cross correlations it is common knowledge that any two rms in the economy are correlated even rms belonging to dierent sectors of an industry may be correlated because of indirect correlations how can we analyze and understand these correlations this article reviews recent results regarding cross correlations between stocks specically we use methods of random matrix theory rmt which originated from the need to understand the interactions between the constituent elements of complex interacting systems to analyze the cross correlation matrix c of returns we analyze 30 min returns of the largest 1000 us stocks for the 2 year period 1994 1995 we nd that the statistics of approximately 20 of the largest eigenvalues 2 show deviations from the predictions of rmt to test that the rest of the eigenvalues are genuinely random we test for universal properties such as eigenvalue spacings and eigenvalue correlations and demonstrate that c shares universal properties with the gaussian orthogonal ensemble of random matrices the statistics of the eigenvectors of c conrm the deviations of the largest few eigenvalues from the rmt prediction we also nd that these deviating eigenvectors are stable in time in addition we quantify the number of rms that participate signicantly to an eigenvector using the concept of inverse participation ratio borrowed from localization theory c 2000 published by elsevier science b v all rights reserved

similarities and differences between physics and economics in this opening talk we discuss some of the similarities between work being done by economists and by physicists seeking to contribute to economics we also mention some of the dierences in the approaches taken and justify these dierent approaches by developing the argument that by approaching the same problem from dierent points of view new results might emerge in particular we review some recent results for example the 1nding that there are two new universal scaling models in economics i the 3uctuation of price changes of any stock market is characterized by a pdf which is a simple power law with exponent 4 that extends over 10 2 standard deviations a factor of 10 8 on the y axis ii for a wide range of economic organizations the histogram that shows how size of organization is inversely correlated to 3uc tuations in size with an exponent 1 6 neither of these two new laws has a 1rm theoretical foundation we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic 1eld leads to large 3uctuations c 2001 published by elsevier science b v

spectral distribution of random matrices from binary linear block codes let c be a binary linear block code of length n dimension k and minimum hamming distance d over gf 2 n let d denote the minimum hamming distance of the dual code of c over gf 2 n let e gf 2 n 1 1 n be the component wise mapping e vi 1 vi for v v1 v2 vn gf 2 n finally for p n let mmb c be a p n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of c under e it is shown that for d large enough and y p n 0 1 fixed as n the empirical spectral distribution of the gram matrix of 1 n mmb c resembles that of a random i i d rademacher matrix i e the marchenko pastur distribution moreover an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the gram matrix of 1 n mmb c to the marchenko pastur distribution as a function of y and d is presented

correlation and network analysis of global financial indices random matrix theory rmt and network methods are applied to investigate the correlation and network properties of 20 financial indices the results are compared before and during the financial crisis of 2008 in the rmt method the components of eigenvectors corresponding to the second largest eigenvalue form two clusters of indices in the positive and negative directions the components of these two clusters switch in opposite directions during the crisis the network analysis uses the fruchterman reingold layout to find clusters in the network of indices at different thresholds at a threshold of 0 6 before the crisis financial indices corresponding to the americas europe and asia pacific form separate clusters on the other hand during the crisis at the same threshold the american and european indices combine together to form a strongly linked cluster while the asia pacific indices form a separate weakly linked cluster if the value of the threshold is further increased to 0 9 then the european indices france germany and the united kingdom are found to be the most tightly linked indices the structure of the minimum spanning tree of financial indices is more starlike before the crisis and it changes to become more chainlike during the crisis the average linkage hierarchical clustering algorithm is used to find a clearer cluster structure in the network of financial indices the cophenetic correlation coefficients are calculated and found to increase significantly which indicates that the hierarchy increases during the financial crisis these results show that there is substantial change in the structure of the organization of financial indices during a financial crisis

collective behavior of cryptocurrency price changes abstract digital assets termed cryptocurrencies are correlated we analyze cross correlations between price changes of different cryptocurrencies using methods of random matrix theory and minimum spanning trees we find that the cross correlation matrix exhibits non trivial hierarchical structures and groupings of cryptocurrency pairs which are not present in the partial cross correlations in sharp contrast to the predictions for other financial markets we discover that most of the eigenvalues in the spectrum of the cross correlation matrix do not agree with the universal predictions of random matrix theory but the few of the largest eigenvalues deviate as expected the minimum spanning tree of cryptocurrency cross correlations reveals distinct community structures that are surprisingly stable collective behaviors that are present in the cryptocurrency market can be useful for the construction of portfolio of cryptocurrencies as well as for future research on the subject

two generalizations of a theorem of arrow barankin and blackwell in 1953 arrow barankin and blackwell proved that if r n is equipped with its natural ordering and if a is a closed convex subset of r n then the set of points in a that can be supported by strictly positive linear functionals is dense in the set of all efficient maximal points of a in this note two generalizations of this result are given the first of these is in the setting of a dual system and requires relatively weak assumptions on the ordering cone but a rather strong compactness assumption on the set a the second generalization which is in the setting of a locally convex space relaxes the compactness assumption on the set a but demands more stringent assumptions on the ordering cone this second result was recently obtained by petschke for normed spaces m petschke on a theorem of arrow barankin and blackwell siam j control optim 28 1990 pp 395 401 the proof given here is substantially different from that given by petschke

the effect of a market factor on information flow between stocks using the minimal spanning tree abstract we empirically investigated the effects of market factors on the information flow created from n n 1 2 linkage relationships among stocks we also examined the possibility of employing the minimal spanning tree mst method which is capable of reducing the number of links to n 1 we determined that market factors carry important information value regarding information flow among stocks moreover the information flow among stocks showed time varying properties according to the changes in market status in particular we noted that the information flow increased dramatically during periods of market crises finally we confirmed via the mst method that the information flow among stocks could be assessed effectively with the reduced linkage relationships among all links among stocks from the perspective of the overall market

time scales involved in emergent market coherence abstract in addressing the question of the time scales characteristic for the market formation we analyze high frequency tick by tick data from the nyse and from the german market by using returns on various time scales ranging from seconds or minutes up to 2 days we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales for various sets of stocks of different capitalization and the average trading frequency we observe a significant elevation of the largest eigenvalue with increasing time scale our results from the correlation matrix study can be considered as a manifestation of the so called epps effect there is no unique explanation of this effect and it seems that many different factors play a role here one of such factors is randomness in transaction moments for different stocks another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history

some more density results for proper efficiencies in this paper we give sufficient conditions for a set and or a cone ensuring the density of henig s proper efficient points and positive proper efficient points in the set of efficient points in hausdorff locally convex spaces these conditions are based on the concept of the point of continuity an d generalize many existing results

detection and characterization of changes of the correlation structure in multivariate time series we propose a method based on the equal time correlation matrix as a sensitive detector for phase shape correlations in multivariate data sets the key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix consequently detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors the performance of the technique is demonstrated by application to n f tori autoregressive models and coupled chaotic systems the high sensitivity the comparatively small computational effort and the excellent time resolution of the method recommend it for application to the analysis of complex spatially extended nonstationary systems

collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices abstract through simple analytical calculations and numerical simulations we demonstrate the generic existence of a self organized macroscopic state in any large multivariate system possessing non vanishing average correlations between a finite fraction of all pairs of elements the coexistence of an eigenvalue spectrum predicted by random matrix theory rmt and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom up collective effect of the underlying time series rather than a top down impact of factors our results in excellent agreement with previous results obtained on large financial correlation matrices show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner

random matrix theory analysis of cross correlations in financial markets we confirm universal behaviors such as eigenvalue distribution and spacings predicted by random matrix theory rmt for the cross correlation matrix of the daily stock prices of tokyo stock exchange from 1993 to 2001 which have been reported for new york stock exchange in previous studies it is shown that the random part of the eigenvalue distribution of the cross correlation matrix is stable even when deterministic correlations are present some deviations in the small eigenvalue statistics outside the bounds of the universality class of rmt are not completely explained with the deterministic correlations as proposed in previous studies we study the effect of randomness on deterministic correlations and find that randomness causes a repulsion between deterministic eigenvalues and the random eigenvalues this is interpreted as a reminiscent of level repulsion in rmt and explains some deviations from the previous studies observed in the market data we also study correlated groups of issues in these markets and propose a refined method to identify correlated groups based on rmt some characteristic differences between properties of tokyo stock exchange and new york stock exchange are found

collective behavior of stock price movements a random matrix theory approach we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt developed in physics to describe the energy levels ofcomplex systems rmt makes predictions f universal properties that do not depend on the interactions between the elements comprising the system deviations from rmt provide clues regarding system speci c properties we compare the statistics ofthe cross correlation matrix c whose elements cij are the correlation coe cients ofprice uctuations ofstock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks c 2001 published by elsevier science b v

quantifying economic fluctuations abstract this manuscript is a brief summary of a talk designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on interpreting empirical data on economic fluctuations using this conceptual framework as a guide we empirically quantify the relation between trading activity measured by the number of transactions n and the price change g t for a given stock over a time interval t t t we relate the time dependent standard deviation of price changes volatility to two microscopic quantities the number of transactions n t in t and the variance w2 t of the price changes for all transactions in t we find that the long ranged volatility correlations are largely due to those of n we then argue that the tail exponent of the distribution of n is insufficient to account for the tail exponent of p g x since n and w display only weak inter dependency our results show that the fat tails of the distribution p g x arises from w finally we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt rmt makes predictions for universal properties that do not depend on the interactions between the elements comprising the system and deviations from rmt provide clues regarding system specific properties we compare the statistics of the cross correlation matrix c whose elements cij are the correlation coefficients of price fluctuations of stock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks we also discuss results that are reminiscent of phase transitions in spin systems where the divergent behavior of the response function at the critical point zero magnetic field leads to large fluctuations and we discuss a curious symmetry breaking a feature qualitatively identical to the behavior of the probability density of the magnetization for fixed values of the inverse temperature

random matrix theory for portfolio optimization a stability approach we apply random matrix theory rmt to an empirically measured financial correlation matrix c and show that this matrix contains a large amount of noise in order to determine the sensitivity of the spectral properties of a random matrix to noise we simulate a set of data and add different volumes of random noise having ascertained that the eigenspectrum is independent of the standard deviation of added noise we use rmt to determine the noise percentage in a correlation matrix based on real data from s p500 eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart finally we attempt to separate the noisy part from the non noisy part of c we apply an existing technique to cleaning c and then discuss its associated problems we propose a technique of filtering c that has many advantages from the stability point of view over the existing method of cleaning

cross correlations in an emerging market financial data abstract we report the results of a study of the spectral properties of the correlation matrix of price fluctuations in the stock prices in an emerging market istanbul stock exchange market by using random matrix theory although properties such as the distributions of correlation matrix elements eigenvalue spacings and components of the eigenvectors show similar trends to those observed for the other markets the eigenvalue distribution is found to be markedly different compared to similar studies

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

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

an analysis of cross correlations in an emerging market abstract we apply random matrix theory to compare correlation matrix estimators c obtained from emerging market data the correlation matrices are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange jse from january 1993 to december 2002 we test the spectral properties of c against random matrix predictions and find some agreement between the distributions of eigenvalues nearest neighbour spacings distributions of eigenvector components and the inverse participation ratios for eigenvectors we show that interpolating both missing data and illiquid trading days with a zero order hold increases agreement with rmt predictions for the more realistic estimation of correlations in an emerging market we suggest a pairwise measured data correlation matrix for the data set used this approach suggests greater temporal stability for the leading eigenvectors an interpretation of eigenvectors in terms of trading strategies is given as opposed to classification by economic sectors

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

sparls the sparse rls algorithm we develop a recursive l1 regularized least squares sparls algorithm for the estimation of a sparse tap weight vector in the adaptive filtering setting the sparls algorithm exploits noisy observations of the tap weight vector output stream and produces its estimate using an expectation maximization type algorithm we prove the convergence of the sparls algorithm to a near optimal estimate in a stationary environment and present analytical results for the steady state error simulation studies in the context of channel estimation employing multipath wireless channels show that the sparls algorithm has significant improvement over the conventional widely used recursive least squares rls algorithm in terms of mean squared error mse moreover these simulation studies suggest that the sparls algorithm with slight modifications can operate with lower computational requirements than the rls algorithm when applied to tap weight vectors with fixed support

on a theorem of arrow barankin and blackwell a well known theorem of arrow barankin and blackwell states that if bf r n is equipped with the natural ordering then for every compact convex subset s of bf r n the set of properly minimal elements of s is dense in the set of minimal elements of s in this note a result of jahn is used to show a generalization of the density theorem of arrow barankin and blackwell it will be shown that this theorem holds in a real normed space that is partially ordered by a convex cone with a closed bounded base

maximal spanning trees asset graphs and random matrix denoising in the analysis of dynamics of financial networks abstract we study the time dependence of maximal spanning trees and asset graphs based on correlation matrices of stock returns in these networks the nodes represent companies and links are related to the correlation coefficients between them special emphasis is given to the comparison between ordinary and denoised correlation matrices the analysis of single and multi step survival ratios of the corresponding networks reveals that the ordinary correlation matrices are more stable in time than the denoised ones our study also shows that some information about the cluster structure of the companies is lost in the denoising procedure cluster structure that makes sense from an economic point of view exists and can easily be observed in networks based on denoised correlation matrices however this structure is somewhat clearer in the networks based on ordinary correlation matrices some technical aspects such as the random matrix denoising procedure are also presented