Recurrence measure of conditional dependence and applications

Identifying causal relations from observational data sets has posed great challenges in data-driven causality inference studies. One of the successful approaches to detect direct coupling in the information theory framework is transfer entropy. However, the core of entropy-based tools lies on the probability estimation of the underlying variables. Here we propose a data-driven approach for causality inference that incorporates recurrence plot features into the framework of information theory. We define it as the recurrence measure of conditional dependence (RMCD), and we present some applications. The RMCD quantifies the causal dependence between two processes based on joint recurrence patterns between the past of the possible driver and present of the potentially driven, excepting the contribution of the contemporaneous past of the driven variable. Finally, it can unveil the time scale of the influence of the sea-surface temperature of the Pacific Ocean on the precipitation in the Amazonia during recent major droughts.

Figure 1

Recurrence matrix ${\mathbf{R}}^X$ definition. Consider the system at instant $i$, when the trajectory at instant $j$ recurs to the vicinity of ${\mathbf{x}}_i$ we write 1 in the element $i,j$ of ${\mathbf{R}}^X$. In contradistinction, at instant $j+1$ the trajectory is outside the vicinity of ${\mathbf{x}}_i$ therefore we set 0 in $i,j+1$ of ${\mathbf{R}}^X$. Notice that the recurrence matrix is symmetrical.

Figure 2

RMCD value (solid black) and the respective confidence interval (dashed red) versus the lag for the $\mathrm{ARMA}[1,1]$ model, Eq. (17). Cross marks pinpoint the lags that pass the multiple comparison analysis.

Figure 3

Analysis of indirect influence between $X$ and $Z$, Eqs. (17) and (18), using: (a) RMD and (b) RMCD, versus the lag. The solid-black line is the RMD or RMCD value and the dashed red line is the respective confidence interval.

Figure 4

RMCD value (solid black) and the respective confidence interval (dashed red) versus the lag for two coupled logistic maps whose coupling is lagged by $\tau =10$ from one another, Eq. (19). Cross marks pinpoint the lags that pass the multiple comparison analysis.

Figure 5

RMCD value (solid black) and the respective confidence interval (dashed red) versus the lag for two coupled Lorenz system, Eq. (20). Cross mark pinpoints the lags that pass the multiple comparison analysis.

Figure 6

Time lag influence of temperature anomaly of the Pacific Ocean (ENSO30) over the precipitation anomaly in the Southwest Amazon during (a) 2005, and (b) 2010, respectively.