Characterizing time series by extended complexity-entropy curves based on Tsallis, Rényi, and power spectral entropy

In this paper, we create three different entropy curves, Tsallis q-complexity-entropy curve, Rényi r-complexity-entropy curve, and Tsallis-Rényi entropy curve via extending the traditional complexity-entropy causality plane and replacing the permutation entropy into power spectral entropy. This kind of method is free of any parameters and some features that are obscure in the time domain can be extracted in the frequency domain. Results from numerical simulations verify that these three entropy curves can characterize time series efficiently. Chaotic and stochastic time series can be distinguished based on whether the q-complexity-entropy curves are opened or closed. The unrelated stochastic process has a negative curvature associated with the Rényi r-complexity-entropy curve, whereas there are positive curvatures for related cases. In addition, the Tsallis-Rényi entropy curve can display the relationship between two entropies. Finally, we apply this method to sleep electrocardiogram and electroencephalography signals. It is proved that these signals possess similar features with long-range correlated 1/f noise. It is robust enough to exhibit different characteristics for each sleep stage. By using surrogate data sets, the nonlinearity of simulated chaotic time series and sleep data can be identified.