Aims: Bubble Entropy (bEn) is an entropy metric, with limited dependence on parameters. This is a clear advantage with respect to other estimators, for which the selection of the values of the parameters employed to obtain an estimate is critical. bEn does not quantify directly the entropy rate of the series (like Approximate Entropy does), but of the ordering of its samples. An analytical formulation of bEn for autoregressive (AR) processes was recently made available and showed, that, at least for this class of processes, the relation between the first autocorrelation coefficient and bEn changes for odd and even values of m. While this is not an issue, per se, it triggered the idea that further refinements of the definition might be possible.
Methods: Using theoretical considerations on the expected values for AR processes, we examined a two steps ahead estimator of bEn, which depends on twice the conditional entropy of the ordering of the samples. We first compared it with the original bEn estimator on simulated series, obtained from autoregressive processes. The check was mainly aimed at validating our theoretical expectations. Then, we tested it on real heart rate variability (HRV) signals obtained from the Physionet Normal Synus Rhythm (NSR) and Congestive Heart Failure (CHF) databases, two populations which are effectively discriminated using entropy.
Results: Experiments showed that both examined alternatives showed comparable discrimination power between NSR and CHF. However, for values of m>10, the two steps ahead estimator presented slightly higher statistical significance and a more regular behavior, even if the dependence on the parameter m is still minimal.
Conclusions: The research moved further steps in the understating of bubble entropy, in particular in the context of HRV analysis. The two steps ahead estimator, while a minor refinement, should not be ignored in our future research directions.