Complexity Analysis of Physiological Time Series Using a Novel Permutation-Ratio Entropy

To date, various information entropy methods have been employed to evaluate complexity within physiological time series. However, such methods cannot discern different levels of nonlinear chaotic properties within time series, indicating that incorrect results are yielded due to noise. Herein, a novel permutation-ratio entropy (PRE) method was proposed and compared with the classical permutation entropy (PE) method, multiscale PE with scale factors 4 and 8 (MPE_S4 and MPE_S8). Simulations with clean logistic mapping series and the logistic mapping series plus noise with a signal-to-noise ratio of 20 dB showed that only PRE monotonically declined with complexity reduction within time series for all 12 combinations of parameters (time delay <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> and embedded dimension <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>). By contrast, PE only monotonically decreased at three parameter combinations for the clean logistic series and failed at all 12 parameter combinations for the logistic series plus noise, and moreover, MPE_S4 and MPE_S8 failed to monotonically decline for the clean logistic series and the logistic series plus noise at all parameter combinations. Results of surrogate data analysis indicated that PRE could more effectively measure the deterministic components of nonlinear within time series than PE, MPE_S4 and MPE_S8. In addition, the parameter <inline-formula> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> could enable PE, MPE_S4, and MPE_S8 to yield incorrect results, but it could not do so for PRE. Both PRE and PE were relatively stable on various parameters of <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula>. Interictal and ictal electroencephalography (EEG) recordings from the Bonn database and the CHB-MIT scalp EEG database were also observed, and the results indicated that the PRE could accurately measure the complexity of EEG recordings, as shown by higher entropy values yielded from interictal intracranial EEG recordings versus those yielded from ictal ones (<inline-formula> <tex-math notation="LaTeX">$p < 0.01$ </tex-math></inline-formula>).