Random matrix approach to multivariate correlations: Some limiting cases

Recent advances have shown that the empirical correlation matrices of dynamical systems can be modeled as random matrices, for most part, chosen from an appropriate ensemble of the random matrix theory (RMT). In this work, we study certain limiting cases where this approach could potentially break down. Using a combination of analytical and numerical tools, we especially study the eigenvalue density and its spacing distribution. We show that the correlation matrices obtained from multivariate spatiotemporal timeseries, in a regime of spatiotemporal chaos, lead to strong deviations from RMT. We illustrate the results with time-series data drawn from coupled map lattices. We also explore the transition to the RMT regime from the limiting cases.

Figure 1

Numerically determined eigenvalues of the correlation matrix for CML time-series data with $a=2.00$ and $ϵ=0.08$ (dashed curve). The solid curve is the analytical result of the tridiagonal model, Eq. (5) with $b$ taken as the average of the first off-diagonal elements of the CML matrix; $b=0.022$.

Figure 2

(a) Numerically obtained eigenvalues of a correlation matrix for a CML with $a=1.98$ and $ϵ=0.08$ (dashed line, almost invisible). Solid curve: analytical eigenvalues in Eq. (10) with $b=0.127,{ϵ}_1=0.025,{ϵ}_2=0.0023$ of the matrix ${\mathbf{X}}^p$, where the numerical values of the parameters were chosen according to the CML matrix. (b) Percentage error, $\Delta {\lambda }_j=\delta {\lambda }_j\times 100∕{\lambda }_j$ between numerical and analytical eigenvalues.

Figure 3

The same as Fig. 2 for different CML parameters, $a=1.95$ and $ϵ=0.2$.

Figure 4

Level density of spectra from the full correlation matrix for CML data with $a=2.0$ and $ϵ=0.05$. The dotted curve is the numerical result, while the solid curve represents Eq. (13).

Figure 5

Level density of the correlation matrix for CML with (a) $a=1.95$ and $ϵ=0.2$ (b) $a=1.98$ and $ϵ=0.08$. The solid curve in (b) is the analytical level density obtained from Eq. (18).

Figure 6

Spacing distribution for the truncated correlation matrix spectra for CML. The solid curve is the Gaussian orthogonal ensemble (GOE) spacing distribution.

Figure 7

Spacing distribution of the full correlation matrix spectra for CML. The solid curve is the GOE spacing distribution.

Figure 8

Spacing distribution of the full correlation matrix spectra for CML. The solid curve is the GOE spacing distribution.

Figure 9

Spacing distribution of the full correlation matrix spectra from multivariate random variables sampled (a) $6\times {10}^5$ times and (b) 4000 times. See text for details. The solid curve is the GOE spacing distribution.