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Refactors and fixes sample entropy (last M values must be ignored) #1
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@DominiqueMakowski called my attention to a discrepancy between this implementation of the sample entropy and my implementation in nolds in this issue: neuropsychology/NeuroKit#75 .
I think there is a small issue in MultiscaleEntropy that causes this inconsistence: The sample entropy is the conditional probability that two pieces of the input sequence that are similar for M time steps will remain similar for M+1 time steps. If we count the number of similar template vector pairs of length M we therefore must ignore the last template vector, since it cannot be followed for another time step. If we would include it in the calculation, this would introduce a bias that underestimates the number of template vectors that remain similar for a length of M+1.
Reference: Richman and Moorman (2000), page H2042
At Dominique's hint I found similar issues with entro-py (ixjlyons/entro-py#2), pyeeg (forrestbao/pyeeg#29) and pyEntropy (nikdon/pyEntropy#15). With the suggested fix in this pull request, MultiscaleEntropy produces the same output as nolds and the R-package pracma (which I used as reference for the implementation of nolds), as well as the fixed versions of pyeeg, entro-py and pyEntropy,
Since I found the code in MultiscaleEntropy hard to understand, I took the liberty to refactor it. After I was done with that I could also identify the actual culprit in the original code. So if you would like to incorporate the fix but not my refactored version, you can alternatively pull the branch bugfix in my fork.