Copula-based algorithm for generating bursty time series

Dynamical processes in various natural and social phenomena have been described by a series of events or event sequences showing non-Poissonian, bursty temporal patterns. Temporal correlations in such bursty time series can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. Modeling and simulating various dynamical processes requires us to generate event sequences with a heavy-tailed IET distribution and memory effects between IETs. For this, we propose a Farlie-Gumbel-Morgenstern copula-based algorithm for generating event sequences with correlated IETs when the IET distribution and the memory coefficient between two consecutive IETs are given. We successfully apply our algorithm to the cases with heavy-tailed IET distributions. We also compare our algorithm to the existing shuffling method to find that our algorithm outperforms the shuffling method for some cases. Our copula-based algorithm is expected to be used for more realistic modeling of various dynamical processes.

Figure 1

Schematic diagram of the copula-based algorithm using a given interevent time (IET) distribution $P\left(\tau \right)$ and a conditional probability distribution $P\left({\tau }_{i+1}\right|{\tau }_i)$ for two consecutive IETs in Eq. (7).

Figure 2

Numerical results of the copula-based algorithm: Conditional probability distributions $P\left({\tau }_{i+1}\right|{\tau }_i)$ for several values of ${\tau }_i$ (left) and the resultant IET distribution (right) for the exponential case in Eq. (14) with $\mu =100$ and $M=0.1$ (top), for the power-law case in Eq. (20) with $\alpha =3.5$ and $M=0.07$ (middle), and for the case of power law with exponential cutoff in Eq. (26) with $\alpha =2.1,{\tau }_c={10}^3$, and $M=0.015$ (bottom) are compared to analytical curves in Eqs. (17), (14), (23), (20), (28), and (26) in panels (a–f), respectively. In panel (a), the solid and dashed curves correspond to ${\tau }_i=2$ and 128, respectively. In panels (c, e) the solid and dashed curves correspond to ${\tau }_i=2$ and 16, respectively. In panels (a, b) the curves are linearly binned, while the curves in panels (c–f) are all log-binned. Numerical results are averaged over 100 event sequences of size $n={10}^5$.

Figure 3

Average computation times in seconds for generating an event sequence using the copula-based algorithm (square) and the shuffling method (triangle) for the exponential case (a, d), the power-law case (b, e), and the case of power law with exponential cutoff (“$\text{power}+\text{cutoff}$” in short) (c, f). In panels (a–c), the computation times for various values of parameters (see the text) are plotted as a function of $M$ for $n={10}^5$. In panels (d–f), we show the $n$ dependence of computation times when the event sequences are generated using the same parameter values as in Fig. 2, where the dashed line of $y=x$ is for guiding the eye. Each point is averaged over 100 event sequences for exponential and power-law cases and ten event sequences for the $\text{power}+\text{cutoff}$ case.

Figure 4

Numerical results of the shuffling method with the same parameter values as in Fig. 2, with $\epsilon ={10}^{-3}$.