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 , RangeEnmeasures 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: URL
Approximate entropy ( ApEn) and sample entropy (SampEn) are widely used for temporal complexity analysis of real-world phenomena . However, their relationship with the Hurst exponent as a measure of self-similarity has not been deeply studied. Also, they still suffer from susceptibility to signal amplitude changes . A common practice for addressing this issue is to correct their input signal amplitude by its standard deviation. In this study, we first show using simulations that ApEn and SampEn are related to the Hurst exponent through their tolerance and embedding dimension parameters. We then propose a modification to ApEn and SampEn called Range entropy or RangeEn. RangeEn measures represent a better robustness to nonstationary changes in signals and a more linear link to the Hurst exponent compared to ApEn and SampEn. They are always bounded in the tolerance r-plane between 0 (maximum entropy) and 1 (minimum entropy) and need no signal amplitude correction . We also perform complexity and self-similarity analyses on epileptic EEG data as a real-world example. Python implementation of this study is publicly available at: URL
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Sample entropy (SampEn) has been accepted as an alternate, and sometimes a replacement, measure to approximate entropy (
Approximate entropy (
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for characterizing temporal complexity of time series
and sample entropy (SampEn) are widely used for temporal complexity analysis of real-world phenomena
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it still suffers from issues such as inconsistency over short-length signals and its tolerance parameter r,
their relationship with the Hurst exponent as a measure of self-similarity has not been deeply studied. Also, they still suffer from
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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
. A common practice for addressing this issue is to correct their input signal amplitude by its standard deviation. In this study, we first show using simulations that
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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
are related to the Hurst exponent through their tolerance and embedding dimension parameters. We then propose a modification to
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, RangeEnmeasures are robust to stationary and nonstationary signal amplitude variations and that their trajectories
called Range entropy or RangeEn. RangeEn measures represent a better robustness to nonstationary changes in signals and a more linear link to the Hurst exponent compared to ApEn and SampEn. They are always bounded
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are constrained
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and need no signal amplitude correction
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showed that RangeEn have direct relationships with the Hurst exponent; suggesting that the new definitions are sensitive to
perform complexity and
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structures of signals. RangeEn analysis of
analyses on
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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
as a real-world example. Python implementation of this study