Eigenvalue density of cross-correlations in Sri Lankan financial market

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.     

Supersymmetric matrix models and branched polymers

We solve a supersymmetric matrix model with a general potential. While matrix models usually describe surfaces, supersymmetry enforces a cancellation of bosonic and fermionic loops and only diagrams corresponding to so-called branched polymers survive. The eigenvalue distribution of the random matrices near the critical point is of a new kind.     

Eigenvalue spectra of random matrices for neural networks

The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions, making results from random matrix theory highly relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances.     

Universal correlations and power-law tails in financial covariance matrices

We investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power-law tails. In order to generate individual eigenvalue distributions a chopping procedure is devised, which produces a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections. Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay.     

Normal random matrix ensemble as a growth problem

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples.     

Random Incidence Matrices: Moments of the Spectral Density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of small eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.     

基于随机矩阵理论的城市人群移动行为分析

移动通信应用为人类移动规律的研究提供了独特的数据来源. 本文通过城市手机用户的分布数据,研究城市移动人群的整体动力学行为. 借助随机矩阵理论的方法,通过比较移动人群数据与随机数据在互相关矩阵谱分布上的差异,发现移动人群数据互相关矩阵的相关系数均值、最大本征值及其对应的本征向量明显偏离于随机互相关矩阵的分布,指出这种差异体现了城市移动人群的整体行为特性,且这种差异在不同时间段也会有所不同. 研究结果体现出相关系数的均值和最大本征值的波动趋势,并指出本征向量成员权重的时空模式与城市移动人群整体行为特征的波动过 关键词： 随机矩阵理论 移动人群 宏观行为     

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

We consider the ensemble of adjacency matrices of Erd?s-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption \({p N \gg N^{2/3}}\), we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erd?s-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erd?s-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.     

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of the master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.     

Statistical properties of cross-correlation in the Korean stock market

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original cross-correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The b₄₇₃\beta_{473} coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function E(s)E(\sigma) with the portfolio risk σ for the original and filtered cross-correlation matrices are consistent with a power-law function, E(σ) ~ s^-g\sigma^{-\gamma}, with the exponent γ ~ 2.92 and those for Asian currency crisis decreases significantly.     

The smallest eigenvalue distribution at the spectrum edge of random matrices

Pfaffian expressions are derived for the smallest eigenvalue distributions of Laguerre orthogonal and symplectic ensembles of random matrices. Asymptotic forms of the smallest eigenvalue distributions are evaluated in the limit of large matrix dimension.     

Eigenvalue density of linear stochastic dynamical systems: A random matrix approach

Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large. The mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach. The applicability of a simple eigenvalue density function, known as the Marenko–Pastur (MP) density, is investigated. The analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties. The new results are validated using an experiment on a vibrating plate with randomly attached spring–mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.     

Rigidity and normal modes in random matrix spectra

We consider the Gaussian ensembles of random matrices and describe the normal modes of the eigenvalue spectrum, i.e., the correlated fluctuations of eigenvalues about their most probable values. The associated normal mode spectrum is linear, and for large matrices, the normal modes are found to be Chebyshev polynomials of the second kind. We contrast this with the behaviour of a sequence of uncorrelated levels, which has a quadratic normal mode spectrum. The difference in the rigidity of random matrix spectra and sequences of uncorrelated levels can be attributed to this difference in the normal mode spectra. We illustrate this by calculating the number variance in the two cases.     

Random matrix theory analysis of cross-correlations in the US stock market: Evidence from Pearson’s correlation coefficient and detrended cross-correlation coefficient

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 Mar?enko–Pastur distribution.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

One-factor model for the cross-correlation matrix in the Vietnamese stock market

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) , , , , , , , ,  and  as well as in other emerging markets, ,  and . 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.     

Large deviations of extreme eigenvalues of random matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.     

Universal distribution functions in two-dimensional localized systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.     

A Real Quaternion Spherical Ensemble of Random Matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form Y=A ^?1 B, where A and B are independent N×N matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue joint probability density function and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β=1,2,4. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure Y=A ^?1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.