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
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 is not widely studied. Additionally, ApEn and SampEn are susceptible 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 in their tolerance r and embedding dimension m parameters. We then propose a modification to ApEn and SampEn called range entropy or RangeEn. We show that RangeEn is more robust to nonstationary signal changes, and it has a more linear relationship with the Hurst exponent , compared to ApEn and SampEn. RangeEn is bounded in the tolerance r-plane between 0 (maximum entropy) and 1 (minimum entropy) and it has no need for signal amplitude correction. Finally, we demonstrate the clinical usefulness of signal entropy measures for characterisation of epileptic EEG data as a real-world example. 
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Python implementation of this study is publicly available at: URL