Transfer Information Energy: A Quantitative Indicator of Information Transfer between Time Series

We introduce an information-theoretical approach for analyzing information transfer between time series. Rather than using the Transfer Entropy (TE), we define and apply the Transfer Information Energy (TIE), which is based on Onicescu's Information Energy. Whereas the TE can be used as a measure of the reduction in uncertainty about one time series given another, the TIE may be viewed as a measure of the increase in certainty about one time series given another. We compare the TIE and the TE in two known time series prediction applications. First, we analyze stock market indexes from the Americas, Asia/Pacific and Europe, with the goal to infer the information transfer between them (i.e., how they influence each other). In the second application, we take a bivariate time series of the breath rate and instantaneous heart rate of a sleeping human suffering from sleep apnea, with the goal to determine the information transfer heart → breath vs. breath → heart. In both applications, the computed TE and TIE values are strongly correlated, meaning that the TIE can substitute the TE for such applications, even if they measure symmetric phenomena. The advantage of using the TIE is computational: we can obtain similar results, but faster.