Detecting directional couplings from multivariate flows by the joint distance distribution

The identification of directional couplings (or drive-response relationships) in the analysis of interacting nonlinear systems is an important piece of information to understand their dynamics. This task is especially challenging when the analyst's knowledge of the systems reduces virtually to time series of observations. Spurred by the success of Granger causality in econometrics, the study of cause-effect relationships (not to be confounded with statistical correlations) was extended to other fields, thus favoring the introduction of further tools such as transfer entropy. Currently, the research on old and new causality tools along with their pitfalls and applications in ever more general situations is going through a time of much activity. In this paper, we re-examine the method of the joint distance distribution to detect directional couplings between two multivariate flows. This method is based on the forced Takens theorem, and, more specifically, it exploits the existence of a continuous mapping from the reconstructed attractor of the response system to the reconstructed attractor of the driving system, an approach that is increasingly drawing the attention of the data analysts. The numerical results with Lorenz and Rössler oscillators in three different interaction networks (including hidden common drivers) are quite satisfactory, except when phase synchronization sets in. They also show that the method of the joint distance distribution outperforms the lowest dimensional transfer entropy in the cases considered. The robustness of the results to the sampling interval, time series length, observational noise, and metric is analyzed too.