The Case of Ties in Bubble Entropy

Bubble entropy (bEn), like Permutation entropy (pEn), belongs in the family of entropy definitions, which exploit sorting for the estimation of the complexity in an mdimensional embedding space.In this paper, we investigate how ties (successive RR intervals with equal values, obtained from ECG recorded at sampling rates smaller than 500Hz) affect the method.Although researchers have claimed that in pEn ties have no significant effect and might even be beneficial in the context of classification, a common practical work-around is to add small perturbations.We quantify how much bEn is affected by ties, analyzing HRV series from Holter ECG data sets (Normal Sinus Rhythm and Congestive Heart Failure).The original sampling rates are between 128Hz and 250Hz.We further increase the number of ties in the HRV series by replicating values at random and compare, with statistical ANOVA F-Test, the discriminating capability of the metrics before and after the distortion.An increasing addition of ties inevitably reduces the discriminating capability of bEn, likely due to the loss of information.However, bEn still discriminates the populations efficiently and almost consistently presents the best F-value among the metrics we consider.Surprisingly, pEn slightly increases F-value for some values of m, possibly due to an initial low discriminating capability and a subsequent random behavior.