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.
Abstract. We study time evolution of a subsystem's density matrix under unitary evolution, generated by a sufficiently complex, say quantum chaotic, Hamiltonian, modeled by a random matrix. We exactly calculate all coherences, purity and fluctuations. We show numerically that the reduced density matrix can be described in terms of a noncentral correlated Wishart ensemble for which we are able to perform analytical calculations of the eigenvalue density. Our description accounts for a transition from an arbitrary initial state towards a random state at large times, enabling us to determine the convergence time after which random states are reached. We identify and describe a number of other interesting features, like a series of collisions between the largest eigenvalue and the bulk, accompanied by a phase transition in its distribution function.
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 non-linearity 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.
The Wishart model for real symmetric correlation matrices is defined as W = AA t , where matrix A is usually a rectangular Gaussian random matrix and A t is the transpose of A. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices A and B as AB t . The corresponding Wishart model, thus, is defined as C = AB t BA t . We study the spectral density of C for the case when A and 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 large C. We complement our analytic results with numerics.
We study spectral densities for systems on lattices, which, at a phase transition display, power-law spatial correlations. Constructing the spatial correlation matrix we prove that its eigenvalue density shows a power law that can be derived from the spatial correlations. In practice time series are short in the sense that they are either not stationary over long time intervals or that they are not available over long time intervals. Also we usually do not have time series for all variables available. We shall make numerical simulations on a 2-D Ising model with the usual Metropolis algorithm as time-evolution. Using all spins on a grid with periodic boundary conditions we find a power law, that is, for large grids, compatible with the analytic result. We still find a power law even if we choose a fairly small subset of grid points at random. The exponents of the power laws will be smaller under such circumstances. For very short time series leading to singular correlation matrices we use a recently developed technique to lift the degeneracy at zero in the spectrum and find a significant signature of critical behavior even in this case as compared to high temperature results which tend to those of random matrix models.
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