Unraveling coupling delays through a transfer entropy analysis in stochastic processes and non-linear systems

Abstract In this work we propose a transfer entropy approach to discern time-delay couplings within non-linear and stochastic coupled systems. We introduce the concept of ‘time-wise transfer entropy’, which quantifies the reduction in future uncertainty for a process Y by considering the values of processes X and Y at a specific past moment. The key advantage of our approach is a reduction in the number of parameters required for estimation when compared to other transfer entropy methodologies. Our proposed time-wise transfer entropy not only lends itself to effective estimation from actual data but also enhances our understanding of the origins of seemingly ‘spurious’ couplings observed in some transfer entropy approaches. To validate our method, we apply it to determine coupling delays in minimal stochastic models where the time-wise transfer entropy can be precisely derived in terms of the Shannon entropy. We further assess the technique performance in coupled non-linear systems with delays, demonstrating its capacity to accurately reproduce the corresponding coupling delays. The developed technique may be useful in the analysis of multifactor non-linear physical systems where complex causal relationships may exist.