Permutation-information-theory approach to unveil delay dynamics from time-series analysis

In this paper an approach to identify delay phenomena from time series is developed. We show that it is possible to perform a reliable time delay identification by using quantifiers derived from information theory, more precisely, permutation entropy and permutation statistical complexity. These quantifiers show clear extrema when the embedding delay $\tau$ of the symbolic reconstruction matches the characteristic time delay ${\tau }_S$ of the system. Numerical data originating from a time delay system based on the well-known Mackey-Glass equations operating in the chaotic regime were used as test beds. We show that our method is straightforward to apply and robust to additive observational and dynamical noise. Moreover, we find that the identification of the time delay is even more efficient in a noise environment. Our permutation approach is also able to recover the time delay in systems with low feedback rate or high nonlinearity.

Figure 1

(Color online) (a) Permutation entropy ${\mathcal{H}}_S$ and permutation statistical complexity ${\mathcal{C}}_{JS}$ as a function of the embedding delay $\tau$ for embedding dimensions $4\le D\le 8$ ($N={10}^6$ data points). Enlargement near the time delay ${\tau }_S$ of the system in order to see more clearly the effect of the embedding dimension on the (b) ${\mathcal{H}}_S$ and (c) ${\mathcal{C}}_{JS}$ estimations. $D$ increases from top to bottom for ${\mathcal{H}}_S$ and from bottom to top for ${\mathcal{C}}_{JS}$.

Figure 2

(Color online) Permutation statistical complexity ${\mathcal{C}}_{JS}$ as a function of the embedding delay $\tau$ for embedding dimensions $D=8$ ($N={10}^6$ data points) in the case of a Mackey-Glass system with time delays ${\tau }_{S,1}=60$ and ${\tau }_{S,2}=96$.

Figure 3

(Color online) Semilogarithmic plot of the permutation statistical complexity ${\mathcal{C}}_{JS}$ as a function of the embedding delay $\tau$ for different levels of observational noise. The noise level associated with the different curves $\left(\text{NL}=0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.5,0.6,0.7,0.8,0.9,1.0\right)$ increases from top to bottom. The embedding dimension $D=8$ and $N={10}^6$ data points. Ten independent realizations for each noise level are plotted. Since the dispersion is very small, the differences between these ten lines are hardly distinguishable.

Figure 4

(Color online) Ratio $\rho ={\mathcal{C}}_{JS}^{\text{peak}}/{\mathcal{C}}_{JS}^{\text{back}}$ as a function of the observational noise level. The embedding dimension was fixed equal to 8 and $N={10}^6$ data points. Notice that the maximum ratio is obtained for a value of NL close to 0.2 and that more reliable time delay identifications are obtained with added observational noise in the range $0<\text{NL}<0.4$. Error bars indicate standard deviations from ten independent realizations.

Figure 5

(Color online) Ratio $\rho ={\mathcal{C}}_{JS}^{\text{peak}}/{\mathcal{C}}_{JS}^{\text{back}}$ as a function of the dynamical noise level. The embedding dimension was fixed equal to 8 and $N={10}^6$ data points. Observe the maximum obtained for a Langevin force $\mathcal{D}$ of intensity near 0.15. Error bars indicate standard deviations from ten independent realizations.

Figure 6

(Color online) Signal-to-noise ratio $\left(\rho \right)$ for the delayed mutual information (DMI) as a function of (a) the observational noise level (NL) and (b) the dynamical noise level $\left(\mathcal{D}\right)$. Error bars indicate standard deviations from ten independent realizations. The probability distributions were estimated with the usual histogram by using $2^8$ bins.

Figure 7

(Color online) Comparison between the permutation statistical complexity $\left({\mathcal{C}}_{JS}\right)$ and the DMI for the Mackey-Glass oscillator with low feedback strength $\left(a=1.2\right)$ and different dynamical noise levels: (a) and (b) $\mathcal{D}=0.15$, and (c) and (d) $\mathcal{D}=0.45$. The embedding dimension was fixed equal to 3 and $N=5\times {10}^3$ data points. DMI was estimated by using a histogram with $2^8$ bins.

Figure 8

(Color online) Time delay identification for a ring of four unidirectionally delay-coupled Mackey-Glass oscillators in the presence of dynamical noise $\left(\mathcal{D}=0.1\right)$. (a) Permutation statistical complexity $\left({\mathcal{C}}_{JS}\right)$ with embedding dimension $D=3$ and (b) DMI estimated by using a histogram with $2^8$ bins. For this realization both quantifiers are able to identify the presence of the feedback delay at 300 time units.

Figure 9

(Color online) Time delay identification for a ring of four unidirectionally delay-coupled Mackey-Glass oscillators in the presence of dynamical noise $\left(\mathcal{D}=0.1\right)$. (a) Permutation statistical complexity $\left({\mathcal{C}}_{JS}\right)$ with embedding dimension $D=3$ and (b) DMI estimated by using a histogram with $2^8$ bins. For this realization only the permutation quantifier is able to identify the presence of the feedback delay at 300 time units.