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

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 maybe useful in the analysis of multifactor non-linear physical systems where complex causal relationships may exist.