Change ΔS of the entropy in natural time under time reversal: Complexity measures upon change of scale

Department of Solid State Physics and Solid Earth Physics Institute, Faculty of Physics, School of Science, National and Kapodistrian University of Athens - Panepistimiopolis, Zografos 157 84, Athens, Greece, EU

Received: 28 May 2014
Accepted: 14 December 2014

Abstract

The entropy S in natural time as well as the entropy in natural time under time reversal $S_{-}$ have already found useful applications in the physics of complex systems, e.g., in the analysis of electrocardiograms (ECGs). Here, we focus on the complexity measures $\Lambda_l$ which result upon considering how the statistics of the time series $\Delta S\left[\equiv S- S_{-}\right] $ changes upon varying the scale l. These scale-specific measures are ratios of the standard deviations $\sigma(\Delta S_l)$ and hence independent of the mean value and the standard deviation of the data. They focus on the different dynamics that appear on different scales. For this reason, they can be considered complementary to other standard measures of heart rate variability in ECG, like SDNN, as well as other complexity measures already defined in natural time. An application to the analysis of ECG —when solely using NN intervals— is presented: We show how $\Lambda_l$ can be used to separate ECG of healthy individuals from those suffering from congestive heart failure and sudden cardiac death.