Bubble Entropy of Fractional Gaussian Noise and Fractional Brownian Motion

Abstract

Aims: Bubble Entropy (bEn) is a metric which links the complexity of the series to the cost of sorting its samples, with limited dependence on parameters. Fractional Brownian motion (fBm) is a long-memory process, which has largely been used in modeling heart rate variability (HRV). fBm displays ephemeral regularities and periodicity at multiple time scales, which then vanish to reform differently. In here we tested if the continuously growing or decaying trends in fBm, which hint a broad range of swaps necessary for sorting, lead to maximal values of bEn. Methods: We synthetically generated realizations of fBm (10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> samples), along with its increments, the fractional Gaussian noise (fGn), a time-discrete process. The Hurst exponent H, on which fBm and fGn are parameterized, was varied in the entire range (0,1). bEn was computed with m ranging up to 200 (typically beyond the scope of other entropy metrics). Results: For fGn, a stationary process, bEn showed a very small, if minimal, dependence on m. Empirically, it scaled as H/2 + 3/4. At low values of m, the dependence was more significant for fBm, a non-stationary process. When m grew, bEn approached a constant value. Conclusions: bEn behaves like a scaling estimator for stationary Gaussian long-memory processes, but less so when non-stationarity becomes relevant (as it is for HRV).