Kendall transfer entropy: a novel measure for estimating information transfer in complex systems.

Transfer entropy (TE) has been widely used to infer causal relationships among dynamical systems, especially in neuroscience. Kendall transformation provides a novel quantization method for estimating information-theoretic measures and shows potential advantages for small-sample neural signals. But it has yet to be introduced into the framework of TE estimation, which commonly suffers from the limitation of small sample sizes. This paper aims to introduce the idea of Kendall correlation into TE estimation and verify its effect.We proposed the Kendall TE (KTE) which combines the improved Kendall transformation and the TE estimation. To confirm its effectiveness, we compared KTE with two common TE estimation techniques: the adaptive partitioning algorithm (D-V partitioning) and the symbolic TE. Their performances were estimated by simulation experiments which included linear, nonlinear, linear + nonlinear models and neural mass models. Moreover, the KTE was also applied to real electroencephalography (EEG) recordings to quantify the directional connectivity between frontal and parietal regions with propofol-induced general anesthesia.The simulation results showed that the KTE outperformed the other two methods by many measures: (1) identifying the coupling direction under a small sample size; (2) the sensitivity to coupling strength; (3) noise resistance; and (4) the sensitivity to time-dependent coupling changes. For real EEG recordings, the KTE clearly detected the disrupted frontal-to-parietal connectivity in propofol-induced unconsciousness, which is in agreement with previous findings.We reveal that the proposed KTE method is a robust and powerful tool for estimating TE, and is particularly suitable for small sample sizes. The KTE also provides an innovative form of quantizing continuous time series for information-theoretic measures.