Against the Flow of Time with Multi-Output Models

Recent work has paid close attention to the first principle of Granger causality about the cause preceding the effect. In this context, the question may arise as to whether the detected direction of causality will also be reversed after the time reversal of unidirectionally coupled data. It has been shown recently that in the case of unidirectionally causally connected autoregressive (AR) processes X -> Y , after time-reversal of data, the detected causality often changes to Y -> X but mostly into bidirectional X <-> Y link. As we argue here, the answer is different when the data examined does not come from AR processes, but rather from linked deterministic systems. If the goal is the usual forward data analysis, cross-mapping causal methods correctly detect X -> Y , while tests based on Granger, which should not be used for deterministic time series, detect causal independence X (sic) Y . The results of the backward causal analysis depend on the predictability of the reversed data. Unlike AR processes, observables from deterministic dynamical systems, even complex nonlinear ones, are well predicted forwards while backward predictions can sometimes seem difficult (notably when the time reversal of a function leads to one-to-many relations). To address this problem, here we propose an approach based on models that provide multiple candidate predictions for the target, combined with a loss function that takes into account only the best candidate. The resulting good forward and backward predictability supports the view that with unidirectionally causally linked deterministic dynamical systems X -> Y can be expected to detect the same link both before and after time reversal.