<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Recent work has paid close attention to the first principle of Granger causality, according to which cause precedes effect. In this context, the question may arise whether the detected direction of causality also reverses after the time reversal of unidirectionally coupled data. Recently, it has been shown that for unidirectionally causally connected autoregressive (AR) processes <italic>X</italic> → <italic>Y</italic>, after time reversal of data, the opposite causal direction <italic>Y</italic> → <italic>X</italic> is indeed detected, although typically as part of the bidirectional <italic>X ↔ Y</italic> link. As we argue here, the answer is different when the measured data are not from AR processes but from linked deterministic systems. When the goal is the usual forward data analysis, cross-mapping-like approaches correctly detect <italic>X</italic> → <italic>Y</italic>, while Granger causality-like approaches, which should not be used for deterministic time series, detect causal independence <italic>X</italic> ⫫ <italic>Y</italic> . The results of backward causal analysis depend on the predictability of the reversed data. Unlike AR processes, observables from deterministic dynamical systems, even complex nonlinear ones, can be predicted well forward, while backward predictions can be difficult (notably when the time reversal of a function leads to one-to-many relations). To address this problem, we propose an approach based on models that provide multiple candidate predictions for the target, combined with a loss function that consideres only the best candidate. The resulting good forward and backward predictability supports the view that unidirectionally causally linked deterministic dynamical systems <italic>X</italic> → <italic>Y</italic> can be expected to detect the same link both before and after time reversal.</p>
</abstract>

Recent work has paid close attention to the first principle of Granger causality, according to which cause precedes effect. In this context, the question may arise whether the detected direction of causality also reverses after the time reversal of unidirectionally coupled data. Recently, it has been shown that for unidirectionally causally connected autoregressive (AR) processes X → Y, after time reversal of data, the opposite causal direction Y → X is indeed detected, although typically as part of the bidirectional X ↔ Y link. As we argue here, the answer is different when the measured data are not from AR processes but from linked deterministic systems. When the goal is the usual forward data analysis, cross-mapping-like approaches correctly detect X → Y, while Granger causality-like approaches, which should not be used for deterministic time series, detect causal independence X ⫫ Y . The results of backward causal analysis depend on the predictability of the reversed data. Unlike AR processes, observables from deterministic dynamical systems, even complex nonlinear ones, can be predicted well forward, while backward predictions can be difficult (notably when the time reversal of a function leads to one-to-many relations). To address this problem, we propose an approach based on models that provide multiple candidate predictions for the target, combined with a loss function that consideres only the best candidate. The resulting good forward and backward predictability supports the view that unidirectionally causally linked deterministic dynamical systems X → Y can be expected to detect the same link both before and after time reversal.

Against the Flow of Time with Multi-Output Models

Measurement Science Review

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Against the Flow of Time with Multi-Output Models"}

Recent work has paid close attention to the first principle of Granger causality, according to which cause precedes effect. In this context, the question may...

Against the Flow of Time with Multi-Output Models

Published Online: Sep 21, 2023

Page range: 175 - 183

Received: Jan 16, 2023

Accepted: Jul 26, 2023

DOI: https://doi.org/10.2478/msr-2023-0023

Keywords
Causality, deterministic dynamical systems, reversibility, multi-output prediction

© 2023 Jozef Jakubík et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Against the Flow of Time with Multi-Output Models

Published Online: Sep 21, 2023

Page range: 175 - 183

Received: Jan 16, 2023

Accepted: Jul 26, 2023

DOI: https://doi.org/10.2478/msr-2023-0023

KeywordsCausality, deterministic dynamical systems, reversibility, multi-output prediction

© 2023 Jozef Jakubík et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Causality, deterministic dynamical systems, reversibility, multi-output prediction