General simulation algorithm for autocorrelated binary processes

The apparent ubiquity of binary random processes in physics and many other fields has attracted considerable attention from the modeling community. However, generation of binary sequences with prescribed autocorrelation is a challenging task owing to the discrete nature of the marginal distributions, which makes the application of classical spectral techniques problematic. We show that such methods can effectively be used if we focus on the parent continuous process of beta distributed transition probabilities rather than on the target binary process. This change of paradigm results in a simulation procedure effectively embedding a spectrum-based iterative amplitude-adjusted Fourier transform method devised for continuous processes. The proposed algorithm is fully general, requires minimal assumptions, and can easily simulate binary signals with power-law and exponentially decaying autocorrelation functions corresponding, for instance, to Hurst-Kolmogorov and Markov processes. An application to rainfall intermittency shows that the proposed algorithm can also simulate surrogate data preserving the empirical autocorrelation.

Figure 1

ACFs (a, b) and sample signals (c–h) corresponding to HK processes with three different values of the characteristic parameter $H\in \left\{0.7,0.8,0.9\right\}$ and $p\in \left\{0.01,0.1\right\}$. Panels (a, b) show the analytical ACFs (—) along with the empirical ACFs of the simulated sequences of transition probabilities $\left\{q\right\}(\circ )$ and binary signal $\left\{x\right\}(•)$ for each of the parameter values reported in the legends. Panels (c–e) and (f–h) depict synthetic sequences corresponding to ACFs reported in (a) and (b), respectively. The simulated sequences exhibit an increasing clustering effect related to the increasing strength of the autocorrelation (viz., parameter values).

Figure 2

As in Fig. 1 but for Markov processes with parameter ${\rho }_1\in \left\{0.5,0.8,0.95\right\}$; the same caption and interpretation apply. Empirical ACF values exhibit strong fluctuations and lose their agreement with the theoretical curves for ${\rho }_X\left(\tau \right)\approx 0.005$ because of finite size effects.

Figure 3

Box plots showing the variability of $p$ computed on sequences of size ${10}^3$ to ${10}^4$ by steps of ${10}^3$ for HK processes with $H\in \left\{0.7,0.8,0.9\right\}$ and $p\in \left\{0.01,0.05,0.1\right\}$. Finite size does not introduce any bias, while the variance of $p$ decreases as the sample size increases.

Figure 4

Box plots summarizing the variability of ACF mean error computed on sequences of size ${10}^3$ to ${10}^4$ by steps of ${10}^3$ for HK processes with $H\in \left\{0.7,0.8,0.9\right\}$ and $p\in \left\{0.01,0.05,0.1\right\}$. Finite size introduces some bias that tends to decrease as the sample size increases. Notice that such a bias is not a limit of the proposed algorithm, but it is due to the estimation of the autocorrelation function from finite size sequences.

Figure 5

As in Fig. 4 but for Markov processes with ${\rho }_1\in \left\{0.5,0.8,0.95\right\}$; the same caption and interpretation apply. Notice that the convergence to theoretical ACF is faster than in the case of HK processes (Fig. 4), as the estimation bias of the second order moments due to finite size effects is larger for processes characterized by long-term persistence.

Figure 6

Box plots showing the sampling variability of the number of IAAFT iterations required to achieve convergence for binary sequences with HK ACF, $H\in \left\{0.7,0.8,0.9\right\}$, and $p\in \left\{0.01,0.05,0.1\right\}$. The expected number of IAAFT iterations increases with the sample size and $p$, while it is almost independent of $H$. Similar results hold for the Markov process (not shown).

Figure 7

(a) Time series of rainfall depth (mm) obtained by merging October data recorded in Casigliano (central Italy) at 30-min temporal resolution from 1995 to 2001. (b) Observed occurrence process corresponding to the rainfall records in panel (a). (c–f) Typical synthetic time series, of equal length, generated by the proposed algorithm respectively with HK and Markov dependence structures, Bernoulli pure randomness, and empirical ACF (surrogate sequence). Comparison with the observed occurrence process in panel (b) shows that HK and surrogate can reproduce the typical clustering behavior (also known as overdispersion) of rainfall events, whereas Markov and Bernoulli occurrences cover the time axis more homogeneously (indicating so-called equi- or underdispersion). (g–j) Comparison of the ACF of the observed occurrence process and the mean ACF obtained by averaging the ACFs of 100 synthetic signals with the same size as that of the observed sequence. The 90% confidence bands are also reported. Panels (g–i) show that the average ACF of HK sequences fits very well the observed ACF, and sampling fluctuations fall within the confidence bands; on the other hand, Markov and Bernoulli dependence structures clearly underestimate the observed ACF. The average ACF of surrogate series in panel (j) closely follows the empirical ACF with limited fluctuations, as expected from the definition of surrogates.