Abstract
Entropy indicates irregularity or randomness of a dynamic system. Over the decades, entropy calculated at different scales of the system through subsampling or coarse graining has been used as a surrogate measure of system complexity. One popular multi-scale entropy analysis is the multi-scale sample entropy (MSE), which calculates entropy through the sample entropy (SampEn) formula at each time scale. SampEn is defined by the “logarithmic likelihood” that a small section (within a window of a length m) of the data “matches” with other sections will still “match” the others if the section window length increases by one. “Match” is defined by a threshold of r times standard deviation of the entire time series. A problem of current MSE algorithm is that SampEn calculations at different scales are based on the same matching threshold defined by the original time series but data standard deviation actually changes with the subsampling scales. Using a fixed threshold will automatically introduce systematic bias to the calculation results. The purpose of this paper is to mathematically present this systematic bias and to provide methods for correcting it. Our work will help the large MSE user community avoiding introducing the bias to their multi-scale SampEn calculation results.
Highlights
Complexity is an important property of a complex system such as the living organisms, Internet, traffic system etc
multi-scale entropy (MSE) is based on Sample entropy (SampEn) [2,3], which is an extension of the well-known Approximate entropy (ApEn) [3,4] after removing the self-matching induced bias
When applied to Gaussian noise and 1/f noise, it was observed that SampEn of Gaussian noise decreases with the signal subsampling scale while it stays at the same level for most of scales of a 1/f process
Summary
Complexity is an important property of a complex system such as the living organisms, Internet, traffic system etc. A popular one is the multi-scale entropy (MSE) proposed by Costa et al [1]. Because complex signal often presents self similarity when the signal is observed at different time scale, Costa et al first applied SampEn to the same signal but at different time scales after coarse graining. When applied to Gaussian noise and 1/f noise, it was observed that SampEn of Gaussian noise decreases with the signal subsampling scale while it stays at the same level for most of scales of a 1/f process. Since a 1/f process is known to have higher complexity (defined by the higher self similarity) than Gaussian noise, the diverging MSE of a 1/f noise and the Gaussian noise appears to support that MSE may provide an approximate approach to measure system complexity.
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