Inference of couplings between variables of a given system using causal wavelets, causal information, equations reconstruction, and other techniques

To infer directional couplings from variables is a difficult problem in dynamical systems, especially when its variables are taken from the real world. Many approaches have been developed to infer such couplings directly from observational time series. The objective of the present study is to investigate the capabilities of a set of techniques in test situations where the dynamics are governed by either fully deterministic (ordinary differential) equations or partially deterministic equations (the same ones with a stochastic perturbation added, the deterministic part remaining dominant). The studied system is based on two three-dimensional chaotic subsystems with very different dynamics, but similar structure, considering various couplings between them (none, unidirectional, bidirectional). One system is dissipative, and the other one is conservative. The time evolution produced by their variables is clearly correlated with one system, almost totally decorrelated with the other one. The following techniques, some of which are introduced in this study, are considered: simple/causal correlation, mutual/causal information, Granger causality index, cross/causal wavelet coherence, bivariate global modeling, and equation reconstruction techniques. All the techniques are evaluated based on their ability to detect direct and indirect causal relationships. Most of them prove poorly capable of detecting direct couplings and are not really robust in the contexts with low variable correlation, external weak couplings, and stochastic perturbations. Applied to the current problems, the bivariate modeling and the equation reconstruction techniques, both based on a global modeling technique, appear to be the most effective approaches to infer causality. The detection of weak bidirectional couplings appears particularly challenging under noisy conditions. Causal detection is tested on a set of groundwater level observational time series, revealing deterministic but complex couplings between three sub-basins of the Se San River basin (Central Highlands, Vietnam).