A Two-Steps-Ahead Estimator for Bubble Entropy

Aims: Bubble entropy ([Formula: see text]) is an entropy metric with a limited dependence on parameters. [Formula: see text] does not directly quantify the conditional entropy of the series, but it assesses the change in entropy of the ordering of portions of its samples of length m, when adding an extra element. The analytical formulation of [Formula: see text] for autoregressive (AR) processes shows that, for this class of processes, the relation between the first autocorrelation coefficient and [Formula: see text] changes for odd and even values of m. While this is not an issue, per se, it triggered ideas for further investigation. Methods: Using theoretical considerations on the expected values for AR processes, we examined a two-steps-ahead estimator of [Formula: see text] , which considered the cost of ordering two additional samples. We first compared it with the original [Formula: see text] estimator on a simulated series. Then, we tested it on real heart rate variability (HRV) data. Results: The experiments showed that both examined alternatives showed comparable discriminating power. However, for values of [Formula: see text] , where the statistical significance of the method was increased and improved as m increased, the two-steps-ahead estimator presented slightly higher statistical significance and more regular behavior, even if the dependence on parameter m was still minimal. We also investigated a new normalization factor for [Formula: see text] , which ensures that [Formula: see text] [Formula: see text] when white Gaussian noise (WGN) is given as the input. Conclusions: The research improved our understanding of bubble entropy, in particular in the context of HRV analysis, and we investigated interesting details regarding the definition of the estimator.