Multiscale transfer entropy: Measuring information transfer on multiple time scales

In this paper, we propose a novel multiscale transfer entropy (MTE) approach to quantify the information transfer of time series on multiple time scales. The MTE combines the advantages of both the multiscale analysis and the transfer entropy, which is able to identify directional, dynamical and scale-dependent information flows. To minimize finite size effects and to avoid spurious detection of causality, we resort to a refined time-delayed multiscale transfer entropy (TMTE) estimator given on overlapping coarse-graining. We also suggest several extensions of the TMTE, including an effective TMTE and a net TMTE. Synthetic simulations including the linear vector autoregressive (VAR) models, the long-range correlated ARFIMA processes, and the nonlinear Rössler systems are analyzed, and an application of the TMTE to daily closing prices and trading volumes of S&P 500 is studied.