Beyond Benford's Law: Distinguishing Noise from Chaos

Abstract

Determinism and randomness are two inherent aspects of all physical processes. Time series from chaotic systems share several features identical with those generated from stochastic processes, which makes them almost undistinguishable. In this paper, a new method based on Benford's law is designed in order to distinguish noise from chaos by only information from the first digit of considered series. By applying this method to discrete data, we confirm that chaotic data indeed can be distinguished from noise data, quantitatively and clearly.

Quantifying non-stationarity effects on organization of atmospheric turbulent eddy motion by Benford's law

“…The different MFDE results for stationary and non-stationary vertical velocity series are consistent with the conclusion that deviations of the first digit distribution from BL vary with different scales in deterministic chaotic systems while not in the stochastic processes [39]. Herein we extend this conclusion from idealized models to practical atmospheric turbulent flows, and exploit that variation of the first digit distribution of velocity increment with scales can be taken to quantify the non-stationarity effects on organization of eddy motions in atmosphere boundary layer.…”

“…2e and f. The results indicate that the large-scale modulation is the main factor leading to the different first digit distributions of multi-scaled increments between stationary and non-stationary time series. This is consistent with the results between chaotic systems and stochastic processes in previous work [39].…”

“…Especially, the first digits conform to BL distribution (χ 2 = 13.7 < 15.51) for non-stationary time series with scale factor h = 10. It has been demonstrated that the first digit distribution from deterministic chaotic systems deviates from BL with different scales while it does not for the stochastic processes [39]. It is not difficult to understand why the first digit distribution from stationary series changes little with scales, since the atmospheric turbulence is usually fully developed during the day and then some features of the stationary records are more like those from stochastic processes.…”

“…In recent years, most applications of BL are limited to detecting whether particular datasets follow this law [33,34], detecting frauds in election and accounting [35,36], testing physical system transition [37,38]. Only recently, Li and Fu [39] used BL to develop a novel strategy to distinguish stochastic processes and chaotic systems, and found that BL can be applied to detect different dynamics hidden in measurements.…”

“…However, all the considered stochastic processes and chaotic systems in the reported work [39] are from idealized models, since the atmospheric turbulent flows are characterized by three-dimensional chaotic motions with different scales and orientations [40], whether this strategy is applicable in boundary layer turbulence? And if so, whether non-stationarity affects the organization degrees of complex eddy motions deserves further research.…”