Implementation of two causal methods based on predictions in reconstructed state spaces.

If deterministic dynamics is dominant in the data, then methods based on predictions in reconstructed state spaces can serve to detect causal relationships between and within the systems. Here we introduce two algorithms for such causal analysis. They are designed to detect causality from two time series but are potentially also applicable in a multivariate context. The first method is based on cross-predictions, and the second one on the so-called mixed predictions. In terms of performance, the cross-prediction method is considerably faster and less prone to false negatives. The predictability improvement method is slower, but in addition to causal detection, in a multivariate scenario, it also reveals which specific observables can help the most if we want to improve prediction. The study also highlights cases where our methods and state-space approaches generally seem to lose reliability. We propose a new perspective on these situations, namely that the variables under investigation have weak observability due to the complex nonlinear information flow in the system. Thus, in such cases, the failure of causality detection cannot be attributed to the methods themselves but to the use of data that do not allow reliable reconstruction of the underlying dynamics.