Abstract
Aims: Bubble entropy () is an entropy metric with a limited dependence on parameters. 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 for autoregressive (AR) processes shows that, for this class of processes, the relation between the first autocorrelation coefficient and 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 , which considered the cost of ordering two additional samples. We first compared it with the original 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 , 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 , which ensures that 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.
Highlights
In a nonlinear dynamical system, the average rate of divergence of the trajectories in the state space is captured by the largest Lyapunov exponent [1]
Inspired by many inconclusive results arising from practical applications of the Kolmogorov–Sinai entropy [6,7], Pincus [8] recognized that, even when only a limited amount of data is available and the system lacks stationary behavior, entropy can still be effectively employed to measure the complexity or the degree of repeatability of a time series and, indirectly, of the system that generated this series
Even though this paper examines alternatives in the definition of bubble entropy, we took the opportunity to summarize in this last section some conclusions on the comparison of bubble entropy with other popular definitions, especially in the m-dimensional space
Summary
In a nonlinear dynamical system, the average rate of divergence of the trajectories in the state space is captured by the largest Lyapunov exponent [1]. This is the rate at which the dynamical system loses information related to the initial condition or, equivalently, the rate at which information is generated [2]. Inspired by many inconclusive results arising from practical applications of the Kolmogorov–Sinai entropy [6,7], Pincus [8] recognized that, even when only a limited amount of data is available and the system lacks stationary behavior, entropy can still be effectively employed to measure the complexity or the degree of repeatability of a time series and, indirectly, of the system that generated this series. To deal with an arbitrary series of observations, Bandt and Pompe [9] suggested avoiding the problem altogether by measuring the entropy of the probability distribution of ordinal patterns, which, in the limit, provides an upper bound for the Kolmogorov–Sinai entropy [10]
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