Recurrence time analysis, long-term correlations, and extreme events

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent $\gamma$, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different $\gamma$, which shows a lack of invariance under the change of observables.

Figure 1

Gaussian distributed time series with recurrence intervals with measure $\mu \left(I\right)=0.01$, which implies $⟨T⟩=100$. One extreme interval $I_{\mathit{ext}}(q=2.3263)=[2.3263,\infty [$ and two recurrence intervals $I_1(-2,0.0922)=[-1.9078,-2.0922]$, $I_2(0,0.0125)=[-0.0125,0.0125]$ illustrated in the (a) time series and (b) probability density function.

Figure 2

The stretched exponential distribution (10) for different values of the exponent $\gamma$.

Figure 3

(Color online) RTS of long-term correlated linear time series with $N=2^{25}\approx 3\times {10}^7$ points for different values of ${\gamma }_c$ (symbols). Lines are the stretched exponential distribution (10) with $\gamma ={\gamma }_c$. The recurrence interval is extreme with $\mu \left(I_{\mathit{ext}}\right)={10}^{-2}$.

Figure 4

(Color online) RTS of long-term correlated linear time series with ${\gamma }_c=0.1$ and different recurrence intervals [centered in $X_c$ with measure $\mu \left(I\right)={10}^{-3}$]. The lines are stretched exponential distributions and the symbols connected by lines are the numerical simulations. From (a) to (b) we use the values given by $a_{\gamma }$ of the best fitting of Eq. (10) in (a). In (c) we analyze the case $X_c=0$ for different values of $\mu \left(I\right)$, from bottom to top: $\mu \left(I\right)={10}^{-1}$ (shifted down by ${10}^2$), $\mu \left(I\right)={10}^{-2}$ (shifted by 10) and $\mu \left(I\right)={10}^{-3}$. The gray lines are the Poissonian distribution [$\gamma =1$ in Eq. (10)]. $a_{\gamma =1}=1$ was used for all three cases.

Figure 5

PDF of the series (a) of $x$ (Gaussian) and (b) of $y$ [Eq. (14) with $X_c=1$, $\delta =0.0207$]. The points inside the interval $I_1(X_c=1,\delta =0.0207)$ in $x$ become extreme events in $y$. The opposite happens for the extreme interval $I_2=I_{\mathit{ext}}(q=2.3263)$.

Figure 6

Generalized Hurst exponent of the time series of $x$ and $y$ ($N=2^{20}\approx {10}^6$ points) as a function of the scale $q_{DFA}$. The horizontal gray lines are the noncorrelated value $(h=0.5)$ and the expected value for ${\gamma }_c=0.1$ $(h=1-{\gamma }_c∕2=0.95)$. The difference between the original and the shuffled time series measures the effect of the correlation at each scale $q_{DFA}$.

Figure 7

Multifractal spectrum for the series of recurrence times $\left(\{T_1,T_1,\dots ,T_{Ne}\}\right)$ obtained for intervals with $\mu \left(I\right)=0.01$, and different values of $X_c$ in long-term correlated linear time series with ${\gamma }_c=0.1$ $(N_e=2^{18}\approx 2\times {10}^5)$.