Abstract
Since its original formulation in 2000, transfer entropy has become an invaluable tool in the toolbox of nonlinear dynamicists working with empirical data. Transfer entropy and its generalizations provide a precise definition of uncertainty and information transfer that are central to the coupled systems studied in nonlinear science. However, a canonical definition of state-dependent transfer entropy has yet to be introduced. We introduce a candidate measure, the specific transfer entropy, and compare its properties to both total and local transfer entropy. Specific transfer entropy makes possible both state- and time-resolved analysis of the predictive impact of a candidate input system on a candidate output system. We also present principled methods for estimating total, local, and specific transfer entropies from empirical data. We demonstrate the utility of specific transfer entropy and our proposed estimation procedures with two model systems, and find that specific transfer entropy provides more, and more easily interpretable, information about an input-output system compared to currently existing methods.
Highlights
One of the hallmarks of a complex system is that the interaction of relatively simple units gives rise to complex overall dynamics
Granger causality quantifies the predictive impact of a candidate input system on a candidate output system accounting for the past of the candidate output system
We have developed specific transfer entropy, and compared its theoretical properties to both total and local transfer entropies
Summary
One of the hallmarks of a complex system is that the interaction of relatively simple units gives rise to complex overall dynamics. We consider three approaches to estimating total, local, and specific transfer entropies from data: plug-in estimators via kernel density estimators with bandwidths based on a normal reference, plug-in estimators using kernel density estimators with bandwidths tuned by l-block cross validation, and plug-in estimators using kth-nearest-neighbor estimators. For model selection for the kth-nearest-neighbor-based estimator of the transfer entropies, we choose the model order p to minimize the mean squared error between the kreg-nearestneighbors prediction of the output future based on the output pasts of order p and the true output future. This is the self-predictively optimal (SPO) formulation of [41] using the. Note that unlike the kernel density-based estimators of specific entropy rate, the kth-nearest-neighbor-based estimator can result in negative specific transfer entropies
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