Reliable detection of directional couplings using cross-vector measures

Detecting directional couplings from time series is crucial in understanding complex dynamical systems. Various approaches based on reconstructed state-spaces have been developed for this purpose, including a cross-distance vector measure, which we introduced in our recent work. Here, we devise two new cross-vector measures that utilize ranks and time series estimates instead of distances. We analyze various deterministic and stochastic dynamics to compare our cross-vector approach against some established state-space-based approaches. We demonstrate that all three cross-vector measures can identify the correct coupling direction for a broader range of couplings for all considered dynamics. Among the three cross-vector measures, the rank-based variant performs the best. Comparing this novel measure to an established rank-based measure confirms that it is more noise-robust and less affected by linear cross-correlation. To extend this comparison to real-world signals, we combine both measures with the method of surrogates and analyze a database of electroencephalographic (EEG) recordings from epilepsy patients. This database contains signals from brain areas where the patients' seizures were detected first and signals from brain areas that were not involved in the seizure onset. A better discrimination between these signal classes is obtained by the cross-rank vector measure. Additionally, this measure proves to be robust to non-stationarity, as its results remain nearly unchanged when the analysis is repeated for the subset of EEG signals that were identified as stationary in previous work. These findings suggest that the cross-vector approach can serve as a valuable tool for researchers analyzing complex time series and for clinical applications.