Analytically solvable autocorrelation function for weakly correlated interevent times

Long-term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions. Such temporal correlations can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. In contrast to the role of heterogeneous IETs on the autocorrelation function, little is known about the effects due to the correlations between IETs. To rigorously study these effects, we derive an analytical form of the autocorrelation function for the arbitrary IET distribution in the case with weakly correlated IETs, where the Farlie-Gumbel-Morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive IETs. Our analytical results are confirmed by numerical simulations for exponential and power-law IET distributions. For the power-law case, we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between IETs. Our analytical approach enables us to better understand long-term temporal correlations induced by the correlations between IETs.

Figure 1

Case with the exponential IET distribution in Eq. (24): Simulation results of the autocorrelation function $A_M\left(t_d\right)$ for various values of $\mu$ and $M$ (symbols) are collapsed when rescaled properly. They are in good agreement with the analytical result up to the first order of $M$ in Eq. (26) (black dotted curve). Each point and its standard error were obtained over ${10}^4$ event sequences of $n=5\times {10}^4$.

Figure 2

Case with the power-law IET distribution in Eq. (27) for the simulation [Eq. (28) for the analysis]: Simulation results of the autocorrelation function $A_0\left(t_d\right)$ (a,c) and the difference between $A_{0.002}\left(t_d\right)$ and $A_0\left(t_d\right)$ (b,d) for $\alpha =1.4$ (a,b) and 2.7 (c,d) (blue circles) are compared to the corresponding analytical result in Eqs. (33) or (34) (red solid curve). We also plot the curves of $A_0\left(t_d\right)$ and $A_{0.002}\left(t_d\right)-A_0\left(t_d\right)$ that are obtained by the numerical inverse Laplace transform of Eq. (23) with $\tilde{P}\left(s\right)$ in Eq. (29) and $\tilde{Q}\left(s\right)$ in Eq. (30) (black dotted curve). Each point and its standard error were obtained over up to ${10}^4$ event sequences of $n=5\times {10}^4$.

Figure 3

Case with the power-law IET distribution in Eq. (27): (a,b) Simulation results of the autocorrelation function $A_M\left(t_d\right)$ for various values of $M$ when $\alpha =1.4$ (a) and 2.7 (b), respectively. The standard errors for the points are smaller than the symbol size. (c) Estimated values of the apparent decaying exponent ${\gamma }_M$, defined by Eq. (35), for several values of $M$. Each point and its standard error were obtained from up to ${10}^4$ event sequences of $n=5\times {10}^4$. For each $\alpha$, the estimated value of ${\gamma }_0$ for $M=0$ is also plotted by a horizontal dotted line for comparison.