Long-Term Memory: A Natural Mechanism for the Clustering of Extreme Events and Anomalous Residual Times in Climate Records

We study the statistics of the return intervals between extreme events above a certain threshold in long-term persistent records. We find that the long-term memory leads (i) to a stretched exponential distribution of the return intervals, (ii) to a pronounced clustering of extreme events, and (iii) to an anomalous behavior of the mean residual time to the next event that depends on the history and increases with the elapsed time in a counterintuitive way. We present an analytical scaling approach and demonstrate that all these features can be seen in long climate records. The phenomena should also occur in heartbeat records, Internet traffic, and stock market volatility and have to be taken into account for an efficient risk evaluation.

Figure 1

(a) Illustration of the return intervals $r_q(l)$, $l=1,\dots ,N_q$ for three equidistant threshold values $q_1,q_2,q_3$ for the water levels of the Nile at Roda (near Cairo, Egypt). One return interval for each quantile $q$ is indicated by arrows. (b) DFA2 analysis for (i) the reconstructed Mann record of northern hemisphere temperatures (981 yr, circles) [12], (ii) the historical record of water levels of the Nile River [5] (663 yr, squares), (iii) the reconstructed precipitation record of New Mexico [13] (2129 yr, triangles down), (iv) the reconstructed temperature record of Baffin Island [14] (1241 yr, triangles up), and (v) the reconstructed river flow of the Sacramento River [15] (1109 yr, diamonds).

Figure 2

(a) Distributions $P_q(r)$ of the return intervals $r$ for the thresholds $q=1.5$ ($R_q\approx 15$, squares), 2.0 ($R_q\approx 44$, circles), and 2.5 ($R_q\approx 161$, triangles) for simulated long-term correlated records with $\gamma =0.5$ (top), 0.3 (middle), and 0.1 (bottom) (filled symbols) and for the corresponding shuffled data (open symbols). For the simulations, we used 1000 records of length $N=2\times {10}^6$ for each value of $\gamma$. (b) Distributions $P_q(r)$ of the return intervals $r$ for the five climate records considered in Fig. 1b with the same symbols, for both original data (filled symbols) and shuffled data (open symbols). The data have been averaged over all quantiles $q$ with $R_q>3\mathrm{y}\mathrm{r}$ and more than 50 return intervals. The lines are the theoretical curves following Eq. (1).

Figure 3

(a) Autocorrelation function $C_r(s)$ of the return intervals for $q=1.5$ (circles), 1.75 (squares), and 2.0 (triangles) for simulated long-term correlated records with $\gamma =0.4$. (b) Conditional distribution function $P_q(r|r_0)$ versus $r/R_q$ for $\gamma =0.4$ and $q=1.5$ (circles), 1.75 (squares), and 2.0 (triangles), for $r_0=R_q/4$ (open symbols) and $r_0=4R_q$ (full symbols). (c),(d) $R_q{P}_q^{+}(r)$ (open symbols) and $R_q{P}_q^{-}(r)$ (full symbols), averaged over all $r_0$ above and below the median return interval, respectively, versus $r/R_q$ for (c) artificial records and (d) the five climate records from Fig. 1b. The artificial data in (c) have the same $\gamma$ values and mean record lengths as the climate records; we studied 1000 records of size $N=1250$.

Figure 4

(a) Mean residual time to the next event ${\tau }_q(x|r_0)$ (in units of $R_q$) versus $r_0/R_q$ for four $q$ values ($q=1.0$, 1.25, 1.5, and 1.75, different symbols) and four values of the elapsed time $x$ since the last event ($x/R_q=0$, 0.25, 1, and 4, from bottom to top). (b) Mean residual time ${\tau }_q(x|r_0)$ as a function of $x/R_q$, for $r_0/R_q=1/8$ (lower curve) and $r_0/R_q=8$ (upper curve). The middle curve represents the mean residual time averaged over all $r_0$. (c),(d) Mean residual times ${\tau }_q^{-}(x)$ (full line) and ${\tau }_q^{+}(x)$ (dashed line), averaged over all $r_0$ below and above the median return interval, respectively, for (c) $x=0$ and (d) $x=R_q/4$. The symbols are for the five climate records from Fig. 1b, for both original data (filled symbols) and shuffled data (open symbols). For obtaining the curves in (a),(b) we used the same statistics as in Figs. 2a and 3a, 3b, and for the theoretical curves in (c),(d) and the error bars we used the same statistics as in Fig. 3c.