Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach

In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations.

Figure 1

(a) Permutation entropy ${\mathcal{H}}_S$ and (b) permutation statistical complexity ${\mathcal{C}}_{JS}$ as a function of the embedding delay $\tau$ with embedding dimension $D=6$ for the Mackey-Glass oscillator operating in a chaotic regime. The effect of different levels of observational noise can be contrasted with the results obtained for the original noise-free chaotic dynamics. Mean and standard deviation for ${\tau }_{H_{\min }}$ and ${\tau }_{C_{\max }}$ values corresponding to 20 independent numerical realizations of length $N={10}^5$ data points are depicted in (c) for different noise levels ($\mathrm{NL}=0.05,0.1,\cdots ,0.95,1.0$). Curves described by the permutation quantifiers in the CECP are illustrated in (d). The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 2

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delay $\tau$ with embedding dimension $D=6$ for fBm with Hurst exponent $H=0.3$ (upper plot), $H=0.5$ (central plot), and $H=0.7$ (lower plot). Mean values corresponding to 20 independent numerical realizations of length $N={10}^5$ data points are depicted for the noise-free case and for different noise levels (NL $=$ 0.01, NL $=$ 0.05 and NL $=$ 0.10). Curves described by the permutation quantifiers in the CECP are displayed in the insets. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 3

Permutation quantifiers as a function of the embedding delay $\tau$ with embedding dimension $D=6$ for the van der Pol's oscillator. (a) Noise-free case, (b) $D_1=D_2=0.01$, (c) $D_1=D_2=0.03$, and (d) $D_1=D_2=0.05$.

Figure 4

Curves described by the permutation quantifiers in the CECP for the van der Pol's oscillator ($D=6$). The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 5

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with different embedding dimensions ($D=4,\cdots ,7$ from top to bottom) for the chaotic laser data. Curves described by the symbolic quantifiers in the CECP are shown in the insets. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 6

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with embedding dimensions $D=6$ for the Grand River flow dynamics. Qualitative similar results were also obtained for $D=4,5$, and 7. Curve described by the symbolic quantifiers in the CECP is shown in the inset. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 7

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with embedding dimensions $D=6$ for the Madden-Julian oscillation. Qualitative similar results were also obtained for $D=4,5$, and 7. Curve described by the symbolic quantifiers in the CECP is shown in the inset. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 8

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with embedding dimension $D=5$ for the monthly mean North Atlantic oscillation index. Estimated values for both quantifiers are practically constant whatever the time scale used for the analysis. Qualitative similar results were also obtained for $D=4$.

Figure 9

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with embedding dimension $D=6$ for the daily price data of the crude oil and gold. In both cases ${\mathcal{H}}_S$ decreases and ${\mathcal{C}}_{JS}$ increases for increasing time scales. Qualitative similar results were also obtained for $D=4$ and $D=5$. Curves described by the symbolic quantifiers in the CECP are depicted in the inset. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.

Figure 10

Permutation quantifiers (${\mathcal{H}}_S$ and ${\mathcal{C}}_{JS}$) as a function of the embedding delays with embedding dimensions $D=5$ for postural sway measurements. Mean and standard deviation of the permutation quantifiers for ten independent trials associated to the same volunteer are depicted. Curve described by the symbolic quantifiers in the CECP is shown in the inset. The dashed lines represent the maximum and minimum complexity values for a fixed value of the entropy.