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.

Figure 1

Nearest-neighbor distribution obtained by individual unfolding for $N_f$-tori where two from 20 time series are correlated due to Gaussian distributed partial phases with mean ${\mu }_{\delta }=\pi ∕2$ and standard deviation ${\sigma }_{\delta }=\pi ∕2$. Shown are the results for (a) $s_{19}$, (b) $s_{18}$, (c) $s_2$, and (d) $s_1$. The theoretical result for the GOE is drawn as a solid line. The histograms drawn by a black and gray line has been calculated for an ensemble of 100 000 and 1000 matrices, respectively.

Figure 2

Nearest-neighbor distribution obtained by individual unfolding for the Rössler system where two from 20 time series are correlated via diffusive coupling with strength $\eta =0.049$. Shown are the results for distance $s_{19}$ considering an ensemble of (a) 1000 and (b) 100 000 matrices and (c) the nearest-neighbor distribution for distance $s_1$ for 1000 (gray histogram) and 100 000 (black histogram). The theoretical result for the GOE is drawn as a solid line.

Figure 3

Nearest-neighbor distribution obtained by individual unfolding for $N_f$-tori where eight from 20 time series are correlated with ${\sigma }_{\delta }=\pi ∕2$. Shown are the results for (a) $s_8$, (b) $s_7$, and (c) $s_6$. The theoretical result for the GOE is drawn as a solid line. The histograms drawn by a black and gray line has been calculated for an ensemble of 100 000 and 1000 matrices, respectively.

Figure 4

Spectral averaged nearest-neighbor distribution for $N_f$-tori where eight from 20 time series are correlated. The theoretical result for the GOE is drawn as a solid line. The histograms drawn by a black and gray line has been calculated for an ensemble of 100 000 and 1000 matrices, respectively.

Figure 5

(a) A segment of $125\mathrm{s}$ from an EEG recording at electrode F3. From about $499\text{to}512\mathrm{s}$ the large amplitude activity of a generalized epileptic seizure can be seen. At approximately 405 and $485\mathrm{s}$ artifacts caused by muscular movement occur. (b) KS probability for the coincidence of the nearest-neighbor distributions of the first seven distances with the null hypothesis and (c) the same as in (b) for the distances $s_8$ to $s_{18}$. The null hypothesis was calculated as an average over an artifact-free period between $200\text{to}250\mathrm{s}$ of the same recording. The solid vertical line marks the onset of the seizure. The dashed vertical line indicates when the moving window starts to overlap with the seizure period.

Figure 6

Nearest-neighbor distributions obtained by individual unfolding for the EEG recording shown in Fig. 5 for distance $s_{18}$ [panels (a), (b), and (c)] and distance $s_1$ [panels (d), (e), and (f)]. The null hypothesis is displayed as a gray line. The histograms drawn with a black line present an average over the period $395\text{to}415\mathrm{s}$ [panels (a) and (d)], which includes a muscle artifact, an artifact-free period between 440 and $460\mathrm{s}$ [panels (b) and (e)], and during the seizure period betwen 499 and $512\mathrm{s}$ [panels (c) and (f)].