Detection of time delays and directional interactions based on time series from complex dynamical systems

Data-based and model-free accurate identification of intrinsic time delays and directional interactions is an extremely challenging problem in complex dynamical systems and their networks reconstruction. A model-free method with new scores is proposed to be generally capable of detecting single, multiple, and distributed time delays. The method is applicable not only to mutually interacting dynamical variables but also to self-interacting variables in a time-delayed feedback loop. Validation of the method is carried out using physical, biological, and ecological models and real data sets. Especially, applying the method to air pollution data and hospital admission records of cardiovascular diseases in Hong Kong reveals the major air pollutants as a cause of the diseases and, more importantly, it uncovers a hidden time delay (about 30–40 days) in the causal influence that previous studies failed to detect. The proposed method is expected to be universally applicable to ascertaining and quantifying subtle interactions (e.g., causation) in complex systems arising from a broad range of disciplines.

Figure 1

Accurate detection of time delays and directional interactions in a logistic model of two species. For $n=2$ and $\delta t=1$: (a–c) single delay under unidirectional coupling; (d) multiple delays where the coupling term in the population of the second species is $K_2(X_{t-{\tau }_{21}}+X_{t-{\tau }_{22}})$; (e, f) multiple delays under bidirectional coupling.

Figure 2

Distributions of terms in the first equation (a–d) and the second equation (e–h) of Eq. (2). The horizontal intervals represents the domains of the relevant quantities and the vertical axis represents the frequencies of the values. The parameters are taken as $v=3,K_1=0.05$, and $K_2=0.08$, and the length of the time series is ${10}^3$.

Figure 3

Identification of diverse types of time delays. For $n=6$ and $\delta t=0.12$, identification of discrete time delays in two directions for different parameter sets (a) and (b) and distributed time delay for different kernel function $k$ (c) in different coupled LR systems. (d) For $n=4,\delta t=0.2$, and different ${\tau }_0$ inducing different dynamics, successful and failed detections of delays associated with self feedback in the MG system.

Figure 4

Identification of time delayed causal influence of air pollutants on cardiac disease from environmental and medical data collected in Hong Kong. The CME scores for disease occurrence against concentrations of pollutants ${\mathrm{NO}}_2$ (a), Rspar (b), and ${\mathrm{SO}}_2$ (c), respectively, are depicted in both directional interactions [i.e., the interaction from disease occurrence to specific pollutant concentration (square lines) and the interaction from pollutant concentration to disease occurrence (dot lines)]. The embedding parameters are $n=14$ for cardiac disease data, $n=7$ for the pollutant data, and $\delta t=1$ day.

Figure 5

Performance of alternative CME scores. For the logistic system of two species in Fig. 1 for $K_1=0,K_2=0.08,{\tau }_1=0$, and ${\tau }_2=3$, alternative definitions of the CME scores yield the same result of time delay detection as that based on Eq. (1). Shown are the mean value and the corresponding standard deviation for each candidate delay for 100 trails from random initial values.

Figure 6

Performance comparisons of our CME and the recent CCM method with increased embedding dimension. Here, the data is from the unidirectional coupled logistic model with the same parameters as those in Fig. 1 where the true time delay $\tau =3$. For each intrinsic time delay, the mean value and the corresponding standard deviation are computed based on 100 independent trails.

Figure 7

Performances of the CME method with various factors. (a, b) CME scores with the growth of the length of time series produced by a specific coupled LR system. Here, $X$ and $Y$ stand, respectively, for $x_1$ and $y_2$ in the system. (c, d) CME scores with the growth of the SNR for the noise-deteriorated time series of the above coupled Lorenz-Rössler system. (e, f) CME scores with the growth of the coupling strength for the unidirectionally coupled Lorenz systems. Here, $X$ and $Y$ stand, respectively, for the driving and the driven systems. The inset in (f) shows the correlation between $x(t-\tau )$ and $y\left(t\right)$ as the coupling strength increases.

Figure 8

Original data of the daily hospital admissions of cardiovascular disease and the pollutants data in Hong Kong from 1995 to 1997.