| Summary: | <i>Aims</i>: Bubble entropy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula>) is an entropy metric with a limited dependence on parameters. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula> 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 <i>m</i>, when adding an extra element. The analytical formulation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula> for autoregressive (AR) processes shows that, for this class of processes, the relation between the first autocorrelation coefficient and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula> changes for odd and even values of <i>m</i>. While this is not an issue, per se, it triggered ideas for further investigation. <i>Methods</i>: Using theoretical considerations on the expected values for AR processes, we examined a two-steps-ahead estimator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula>, which considered the cost of ordering two additional samples. We first compared it with the original <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula> estimator on a simulated series. Then, we tested it on real heart rate variability (HRV) data. <i>Results</i>: The experiments showed that both examined alternatives showed comparable discriminating power. However, for values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>10</mn><mo><</mo><mi>m</mi><mo><</mo><mn>20</mn></mrow></semantics></math></inline-formula>, where the statistical significance of the method was increased and improved as <i>m</i> increased, the two-steps-ahead estimator presented slightly higher statistical significance and more regular behavior, even if the dependence on parameter <i>m</i> was still minimal. We also investigated a new normalization factor for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula>, which ensures that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mi>E</mi><mi>n</mi><mo> </mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> when white Gaussian noise (WGN) is given as the input. <i>Conclusions</i>: 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.
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