Statistics of return intervals in long-term correlated records

We consider long-term correlated data with several distribution densities (Gaussian, exponential, power law, and log normal) and various correlation exponents $\gamma$ $(0<\gamma <1)$, and study the statistics of the return intervals $r_j$ between events above some threshold $q$. We show that irrespective of the distribution, the return intervals are long-term correlated in the same way as the original record, but with additional uncorrelated noise. Due to this noise, the correlations are difficult to observe by the detrended fluctuation analysis (which exhibits a crossover behavior) but show up very clearly in the autocorrelation function. The distribution $P_q\left(r\right)$ of the return intervals is characterized at large scales by a stretched exponential with exponent $\gamma$, and at short scales by a power law with exponent $\gamma -1$. We discuss in detail the occurrence of finite-size effects for large threshold values for all considered distributions. We show that finite-size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to ${10}^6$ data points. Finally, in order to quantify the clustering of extreme events due to the long-term correlations in the return intervals, we study the conditional distribution function and the related moments. We find that they show pronounced memory effects, irrespective of the distribution of the original data.

Figure 1

(Color online) Return intervals: (a) Illustration of the definition of return intervals $r_j$ between events $x_i$ above given thresholds $q_1=1$ and $q_2=2$. The smallest possible $r$ value is $r=1$, which occurs if $x_i>q$ and $x_{i+1}>q$. (b) Sequence of return intervals $r_j$ for long-term correlated data $x_i$ with $\gamma =0.4$ for a fixed threshold $q=2.327$ so that the average $r$ value is $R_q=100$. Note the logarithmic scale of the $r$-axis. (c) Sequence of return intervals for uncorrelated data [shuffled $x_i$ of (b)]. In (b) more epochs with $r$ values of small and large size appear compared to (c).

Figure 2

Standard deviation of $r$ values divided by $R_q$ as a function of $R_q$ for (a) Gaussian and (b) exponential uncorrelated data (circles) and long-term correlated data (squares $\gamma =0.7$, triangles $\gamma =0.4$, and diamonds $\gamma =0.2$). The ${\sigma }_r\left(R_q\right)$ dependences are representative also for data with other distribution densities $D\left(x\right)$.

Figure 3

(Color online) Normalized distribution density function $R_q{P}_q\left(r\right)$ of $r$ values with $R_q=100$ as a function of $r∕R_q$ for long-term correlated data with $\gamma =0.4$ (open symbols) and $\gamma =0.2$ (filled symbols; we multiplied this data by a factor 100 to avoid overlapping curves). In (a) the original data was Gaussian distributed, in (b) exponentially distributed, in (c) power-law distributed with $\delta =5.5$, and in (d) log-normally distributed. All four figures follow quite well stretched exponential curves [solid lines, Eq. (7)] over several decades. For small $r∕R_q$ values a power-law regime seems to dominate, while on large scales deviations from the stretched exponential behavior are due to finite-size effects.

Figure 4

(Color online) Finite-size effects on $P_q\left(r\right)$. Deviations from the stretched exponential curve for large $r∕R_q$ and (a) small $R_q=10$ as well as (b) large $R_q=250$ in data with $\gamma =0.4$ for three lengths $N=2^{13}$ (triangles), $2^{17}$ (squares), and $2^{21}$ (circles) indicate finite-size effects. For large values of $q$ and small lengths $N$ the occurrence of very large $r$ is actually impossible $(r\le N-N∕R_q)$. The finite-size effects for Gaussian data are representative for all four distribution densities $D\left(x\right)$ we considered.

Figure 5

(Color online) Power-law decay of $P_q\left(r\right)$ for $r<R_q$ for $R_q$ = 150 (circles), 250 (squares), and 500 (triangles) for (a) Gaussian, (b) exponentially, (c) power-law, and (d) log normally distributed data with $\gamma =0.4$ (solid symbols) and $\gamma =0.2$ (open symbols, shifted upwards by a factor ten to avoid overlapping curves). The dashed line is the stretched exponential Eq. (7), the solid line is a power law with slope $\gamma -1$ as a guide to the eye. The power-law effect in (a) and (b) appears in the scale range $0.01<r∕R_q<1$ and in (c) and (d) in $0.1<r∕R_q<1$. For even smaller $r∕R_q$ values $(r=1,2,3,\dots )$ all four figures show an upward trend which is due to the discreteness of the $r$ values.

Figure 6

Dependence of $P_q\left(1\right)$ and $P_q\left(2\right)$ on $\gamma$ for various quantiles $q$. The distribution of the return intervals of size $r=1$ is plotted versus $R_q$ for (a) Gaussian and (b) exponentially distributed data with correlation exponents $\gamma =0.2$ (circles), 0.4 (diamonds), 0.6 (triangles), 0.8 (squares) as well as uncorrelated data (stars). The stronger the correlations, i.e., the smaller $\gamma$, the higher the probability that two consecutive events $x_i$ and $x_{i+1}$ exceed the given threshold $q$. The straight lines indicate power laws with exponents $-\gamma$, suggesting the discretization effects in Fig. 5 to be independent of $q$ or $R_q$. Note that the larger $R_q$, the stronger the correlation effect. Figures (c) and (d) show the same for return intervals of size $r=2$.

Figure 7

(Color online) Autocorrelation functions $C_x\left(s\right)$ of the original data $(\times )$ and $C_r\left(s\right)$ of the return interval series for (a) Gaussian, (b) exponentially, (c) power-law, and (d) log-normally distributed data with $\gamma =0.4$ for $R_q=$ 10 (circles), 20 (squares), 50 (triangles), and 100 (diamonds). All curves show a power-law behavior indicating long-term correlations in the sequence of return intervals with the same correlation exponent $\gamma$ (straight lines). Curves for large $R_q$ show weaker correlations at large scales $s$ due to finite-size effects. All results were averaged over 150 configurations of original data with length $N=2^{21}$.

Figure 8

(Color online) Fluctuation analysis (FA) and detrended fluctuation analysis (DFA) of the series of return intervals for (a) $R_q=10$, (b) 20, (c) 50, and (d) 100. The FA $(\times )$ and the DFA1 $(\circ )$ fluctuation functions $F_r\left(s\right)$ of the sequence of return intervals are shown for Gaussian long-term correlated data with $\gamma =0.4$. The asymptotic slope $\alpha$ is related to $\gamma$ via $\alpha =1-\gamma ∕2$. The solid lines have the slope $\alpha =0.8$ $(\gamma =0.4)$. On small scales a slope close to $\alpha =0.5$ is observed due to the large uncorrelated component in the return interval series, while the relative prefactor of the long-term correlated component is decreasing with increasing $R_q$.

Figure 9

(Color online) Conditional distribution density $P_q(r\mid r_0)$ divided by $P_q\left(r\right)$ for three different conditions $r_0∕R_q=1∕4$ (a), 1 (b), and 4 (c) for long-term correlated $(\gamma =0.4)$ Gaussian distributed original data. The maximum of the ratio moves from small $r∕R_q$ to large $r∕R_q$ with increasing the condition $r_0∕R_q$. The cutoff on the right-hand side is due to finite set length and therefore limited maximum $r$ values. The data collapse consists of six curves for quantiles with $R_q=20$ (circles), 50 (squares), 100 (triangles down), 150 (diamonds), 250 (crosses), and 500 (triangles up). Parts (d) to (f) show the same for exponentially distributed data. The collapse seems noisier because the curves are averaged over only 150 configurations compared to 1000 configurations in (a) to (c). For uncorrelated data, $P_q(r\mid r_0)=P_q\left(r\right)$ exactly, as indicated by the straight line at the ratio one.

Figure 10

(Color online) Conditional return periods $R_q\left(r_0\right)$ in units of $R_q$ versus the condition $r_0∕R_q$ for $R_q$ = 5 (circles), 10 (squares), 50 (triangles), and 250 (diamonds) for long-term correlated data with $\gamma =0.4$. All figures clearly display the memory effect in form of increasing $R_q\left(r_0\right)$ with increasing $r_0$, caused by the long-term correlations in the original data. In uncorrelated data, $R_q\left(r_0\right)=R_q$, as indicated by the horizontal lines at ratio one.

Figure 11

(Color online) Conditional return periods for long-term correlated data with four values of $\gamma$ and uncorrelated data (shown at $\gamma =1$); $R_q$ is fixed at $R_q=50$. Figure (a) shows the single-conditional and the double-conditional return periods ($R_q^{+∕-}$ and $R_q^{++∕--}$) in units of $R_q$ for Gaussian data. $R_q^{+}$ (filled symbols, upper curve) and $R_q^{-}$ (filled symbols, lower curve) are the single-conditional return periods, where the predecessor $r_0$ is larger or smaller than $R_q$. $R_q^{++}$ (open symbols, upper curve) and $R_q^{--}$ (open symbols, lower curve) are the double-conditional return periods where the preceding and the prepreceding return intervals are both larger or both smaller than $R_q$ [$R_q^{++}$ = $R_q(r\mid r_0>R_q\mid r_{-1}>R_q)$, $R_q^{--}$ = $R_q(r\mid r_0<R_q\mid r_{-1}<R_q)$]. The splitting of the curves is a consequence of the long-term correlations in the sequence of $r$ values. In the uncorrelated case all four conditional return periods degenerate to $R_q$. Figure (b) shows the same for exponentially distributed data, (c) for power-law distributed data, and (d) for log-normally distributed data. All points were averaged over 20 configurations with roughly $40000r$ values.