Detecting determinism with improved sensitivity in time series: Rank-based nonlinear predictability score

The rank-based nonlinear predictability score was recently introduced as a test for determinism in point processes. We here adapt this measure to time series sampled from time-continuous flows. We use noisy Lorenz signals to compare this approach against a classical amplitude-based nonlinear prediction error. Both measures show an almost identical robustness against Gaussian white noise. In contrast, when the amplitude distribution of the noise has a narrower central peak and heavier tails than the normal distribution, the rank-based nonlinear predictability score outperforms the amplitude-based nonlinear prediction error. For this type of noise, the nonlinear predictability score has a higher sensitivity for deterministic structure in noisy signals. It also yields a higher statistical power in a surrogate test of the null hypothesis of linear stochastic correlated signals. We show the high relevance of this improved performance in an application to electroencephalographic (EEG) recordings from epilepsy patients. Here the nonlinear predictability score again appears of higher sensitivity to nonrandomness. Importantly, it yields an improved contrast between signals recorded from brain areas where the first ictal EEG signal changes were detected (focal EEG signals) versus signals recorded from brain areas that were not involved at seizure onset (nonfocal EEG signals).

Figure 1

Relative frequencies of the noise amplitudes ${\xi }_{l,n}$ for different $l$ determined in bins of $0.05$. We use a logarithmic scale for the ordinate to better resolve characteristics of the distribution tails. The abscissa is truncated, and nonzero relative frequencies are obtained for amplitudes towards $\pm \infty$. $l=1$. Black dashed line: $l=1$; Green (light gray) solid line: $l=3$; Black solid line: $l=5$.

Figure 2

Results for the noisy Lorenz signals in dependence on the signal-to-noise ratio $q$ and amplitude distribution parameter $l$. Error bars show the distribution means, $\langle \mathcal{E}\rangle$ and $\langle \mathcal{S}\rangle ,\pm$ two standard deviations obtained for the 250 independent realizations of the signals. All simulations are independent across different $q$ and $l$ values. The leftmost and rightmost distributions correspond to the noise-only $\left(q=0\right)$ and noise-free $\left(q\to \infty \right)$ case, respectively. The vertical red lines mark the minimal $q$ value for which the distributions' mean values pass the midpoint between the mean values obtained for $q=0$ and $q\to \infty$. Black dots indicate those $q$ for which the distributions significantly differ from the ones obtained for $q=0$ (Wilcoxon rank-sum test rejected with $p<{10}^{-4}$). Green (light gray) dots indicate those $q$ for which the distributions significantly differ from the ones obtained for next lower $q$ value (Wilcoxon rank-sum test rejected with $p<{10}^{-4}$).

Figure 3

Results for the noisy Lorenz signals and corresponding surrogates in dependence on the signal-noise-ratio $q$ and the amplitude distribution parameter $l$. Black: Distributions of $\mathcal{E}$ and $\mathcal{S}$ obtained for 250 independent realizations of the noisy Lorenz signals (already shown in Fig. 2). Green (light gray): distributions of $\mathcal{E}$ and $\mathcal{S}$ obtained for 250 surrogates. Here one surrogate was generated for each of the 250 independent noisy Lorenz signals. Error bars again correspond to the means $\pm$ 2 standard deviations. In contrast to Fig. 2, black dots now indicate those $q$ for which the distribution obtained for the original signals significantly differ from the one obtained for the surrogates (Wilcoxon rank-sum test rejected with $p<{10}^{-4}$).

Figure 4

Comparison of $\mathcal{E}$ and $\mathcal{S}$ values for focal and nonfocal EEG signals. Histograms across the 3750 focal (black) and 3750 nonfocal [green (light gray)] EEG signals. The bin size is 0.015.

Figure 5

Comparison of rejection fractions $\rho$ of the surrogate null hypothesis test obtained by $\mathcal{E}$ and $\mathcal{S}$ for focal and nonfocal EEG signals. The percentages are with regard to the 3750 focal and 3750 nonfocal EEG signals. The dashed line indicates the chance level. The error bars illustrate $95\%$ confidence intervals determined by $[\rho -1.96\sqrt{\rho (1-\rho )/3750},\rho +1.96\sqrt{\rho (1-\rho )/3750}]$, see Ref. [29]. (Note that the $\mathcal{E}$ results are a reproduction of the corresponding nonrandomness test results shown as part of Fig. 2(a)of Ref. [29]. There are slight differences, smaller than the confidence intervals, between the rejection percentages depicted here and those of Ref. [29]. They are caused by the random component of statistical type-I errors of the surrogate test across all signals.)