Range entropy: A bridge between signal complexity and self-similarity
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Sample entropy (SampEn) has been accepted as an alternate, and sometimes a replacement, measure to approximate entropy (ApEn) for characterizing temporal complexity of time series. However, it still suffers from issues such as inconsistency over short-length signals and its tolerance parameter r, susceptibility to signal amplitude changes and insensitivity to self-similarity of time series. We propose modifications to the ApEn and SampEn measures which are defined for 0<r<1, are more robust to signal amplitude changes and sensitive to self-similarity property of time series. We modified ApEn and SampEn by redefining the distance function used originally in their definitions. We then evaluated the new entropy measures, called range entropies (RangeEn) using different random processes and nonlinear deterministic signals. We further applied the proposed entropies to normal and epileptic electroencephalographic (EEG) signals under different states. Our results suggest that, unlike ApEn and SampEn, RangeEn measures are robust to stationary and nonstationary signal amplitude variations and that their trajectories in the tolerance r-plane are constrained between 0 (maximum entropy) and 1 (minimum entropy). We also showed that RangeEn have direct relationships with the Hurst exponent; suggesting that the new definitions are sensitive to self-similarity structures of signals. RangeEn analysis of epileptic EEG data showed distinct behaviours in the r-domain for extracranial versus intracranial recordings as well as different states of epileptic EEG data. The constrained trajectory of RangeEn in the r-plane makes them a good candidate for studying complex biological signals such as EEG during seizure and non-seizure states. The Python package used to generate the results shown in this paper is publicly available at: https://github.com/omidvarnia/RangeEn.
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