Analyzing long-term correlated stochastic processes by means of recurrence networks: Potentials and pitfalls

Long-range correlated processes are ubiquitous, ranging from climate variables to financial time series. One paradigmatic example for such processes is fractional Brownian motion (fBm). In this work, we highlight the potentials and conceptual as well as practical limitations when applying the recently proposed recurrence network (RN) approach to fBm and related stochastic processes. In particular, we demonstrate that the results of a previous application of RN analysis to fBm [Liu et al. Phys. Rev. E 89, 032814 (2014)] are mainly due to an inappropriate treatment disregarding the intrinsic nonstationarity of such processes. Complementarily, we analyze some RN properties of the closely related stationary fractional Gaussian noise (fGn) processes and find that the resulting network properties are well-defined and behave as one would expect from basic conceptual considerations. Our results demonstrate that RN analysis can indeed provide meaningful results for stationary stochastic processes, given a proper selection of its intrinsic methodological parameters, whereas it is prone to fail to uniquely retrieve RN properties for nonstationary stochastic processes like fBm.

Figure 1

(a) Three example trajectories of fBm with $H=0.7$ and (b) the corresponding ACFs. (c) Average ACFs taken over 200 independent realizations of fBm with the same Hurst exponent $H$. Different line colors correspond to different values of $H$ from $0.1$ to $0.9$ in steps of $0.1$ (from bottom to top at small $\tau$). In all cases, the time series length has been set to $N=2^{13}$. (d) As described for panel (c) for $N=2^{15}$.

Figure 2

Example trajectory of a fBm with $H=0.7$ in a two-dimensional reconstructed phase space with embedding delays (a) $\tau =100$ and (b) $\tau =750$ ($N=2^{13}$).

Figure 3

Fraction of false nearest-neighbors (FNN) for (a) fBm and (b) fGn with $H=0.7$ and different time series lengths $N$ (black (lower line): $N=1000$, blue (middle): $N=21000$, and red (upper line): $N=31000$).

Figure 4

As described in the legend of Fig. 1 for fGn obtained by differencing the previous fBms. In (b), the additional green (smooth) line represents the averaged ACF over 200 independent realizations.

Figure 5

Dependence of (a) RN transitivity $\mathcal{T}$ and (b) global clustering coefficient $\mathcal{C}$ for fGn on the Hurst exponent $H$ for different embedding dimensions ($m=3$: $\square$, $m=4$: $◃$, $m=5$: $*$, $m=6$: $•$), taken over 200 independent realizations and using a RN edge density of $\rho =0.03$. The embedding delay has been kept at the same value for all realizations with the same $H$ according to the decorrelation time ${\tau }_{0.1}$. In all cases, $N=2^{12}$.

Figure 6

Dependence of RN transitivity $\mathcal{T}$ and global clustering coefficient $\mathcal{C}$ on the embedding dimension $m$ for fGn with $H=0.7$ (averages over 200 realizations) for two different values of $\rho$ [(a) $\rho =0.01$, (b) $\rho =0.03$]. The dashed line corresponds to the expected analytical values $\mathcal{T}={(3/4)}^m$ for $m$-dimensional Gaussian processes. In all cases, $N=2^{12}$.