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.

[1]  L. Bunimovich.  On the ergodic properties of nowhere dispersing billiards , 1979 .

[2]  S. Simon,et al.  Eigenvalue density of correlated complex random Wishart matrices. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Peter J. Smith,et al.  A determinant representation for the distribution of quadratic forms in complex normal vectors , 2000 .

[4]  P. Seba,et al.  Random matrix analysis of human EEG data. , 2003, Physical review letters.

[5]  S. Sarkar,et al.  Universality in the vibrational spectra of single-component amorphous clusters. , 2004, Physical review letters.

[6]  S. Péché,et al.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices , 2004, math/0403022.

[7]  A. Lakshminarayan,et al.  Fluctuations of finite-time stability exponents in the standard map and the detection of small islands. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  N. Deo,et al.  Correlation and volatility in an Indian stock market: A random matrix approach , 2007 .

[9]  P. A. Mello,et al.  Random matrix physics: Spectrum and strength fluctuations , 1981 .

[10]  C. Beenakker.  Random-matrix theory of quantum transport , 1996, cond-mat/9612179.

[11]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[12]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[13]  Peter J. Forrester,et al.  Complex Wishart matrices and conductance in mesoscopic systems: Exact results , 1994 .

[14]  A. Pandey.  Statistical properties of many-particle spectra. IV. New ensembles by Stieltjes transform methods☆ , 1981 .

[15]  S. Majumdar,et al.  Extreme statistics of complex random and quantum chaotic states. , 2007, Physical review letters.

[16]  H. Stöckmann,et al.  Quantum Chaos: An Introduction , 1999 .

[17]  P. Forrester,et al.  Asymptotic correlations at the spectrum edge of random matrices , 1995 .

[18]  Jizhong Zhou,et al.  Application of random matrix theory to microarray data for discovering functional gene modules. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Binary correlations in random matrix spectra , 1979 .

[20]  Erwin Kreyszig,et al.  Introductory Mathematical Statistics. , 1970 .

[21]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[22]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[23]  Akhilesh Pandey,et al.  Skew-Orthogonal Polynomials and Universality of Energy-Level Correlations , 2001 .

[24]  Yannick Malevergne,et al.  Collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices , 2002, cond-mat/0210115.

[25]  M. S. Santhanam,et al.  Statistics of atmospheric correlations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Romuald A. Janik,et al.  Effective matter superpotentials from Wishart random matrices , 2003 .

[27]  Proof of universality of the Bessel kernel for chiral matrix models in the microscopic limit , 1996, hep-th/9606099.

[28]  Ralf R. Muller.  A random matrix model of communication via antenna arrays , 2002 .

[29]  Z. Burda,et al.  Spectral moments of correlated Wishart matrices. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Akhilesh Pandey,et al.  Skew-orthogonal polynomials and random-matrix ensembles. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[32]  Anirvan M. Sengupta,et al.  Distributions of singular values for some random matrices. , 1997, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  J. Verbaarschot.  The spectrum of the Dirac operator near zero virtuality for Nc = 2 and chiral random matrix theory , 1994 .