Detection of crossover time scales in multifractal detrended fluctuation analysis

Abstract

Fractal is employed in this paper as a scale-based method for the identification of the scaling behavior of time series. Many spatial and temporal processes exhibiting complex multi(mono)-scaling behaviors are fractals. One of the important concepts in fractals is crossover time scale(s) that separates distinct regimes having different fractal scaling behaviors. A common method is multifractal detrended fluctuation analysis (MF-DFA). The detection of crossover time scale(s) is, however, relatively subjective since it has been made without rigorous statistical procedures and has generally been determined by eye balling or subjective observation. Crossover time scales such determined may be spurious and problematic. It may not reflect the genuine underlying scaling behavior of a time series. The purpose of this paper is to propose a statistical procedure to model complex fractal scaling behaviors and reliably identify the crossover time scales under MF-DFA. The scaling-identification regression model, grounded on a solid statistical foundation, is first proposed to describe multi-scaling behaviors of fractals. Through the regression analysis and statistical inference, we can (1) identify the crossover time scales that cannot be detected by eye-balling observation, (2) determine the number and locations of the genuine crossover time scales, (3) give confidence intervals for the crossover time scales, and (4) establish the statistically significant regression model depicting the underlying scaling behavior of a time series. To substantive our argument, the regression model is applied to analyze the multi-scaling behaviors of avian-influenza outbreaks, water consumption, daily mean temperature, and rainfall of Hong Kong. Through the proposed model, we can have a deeper understanding of fractals in general and a statistical approach to identify multi-scaling behavior under MF-DFA in particular.

Decomposing Multifractal Crossovers

“…We also developed qSRA-FMF, a moment- and FMF-based variant of the segmented line regression method (Ge and Leung, 2013) in order to obtain an MSE-based reference for discerning (i) additive from (ii) non-additive forms of signal genesis (see Figures 2A,B, respectively). Goodness-of-fit statistics—for given bimodal characteristics—were used as a guide in choosing between estimates obtained by the SFD-FMF or qSRA-FMF methods, as respectively valid.…”

“…However, disregarding their moment-wise nature (Ge and Leung, 2013) distorts the acquired H ( q )s with breakpoints falling outside the proper SR. This generates uneven error propagation across the moments in the case of standard multifractal methods (Ludescher et al, 2011), or introduces a slight but global error when FMF methods are applied.…”

“…In particular, recent works have either focused mostly on monofractal multimodality (Kuznetsov et al, 2013) or applied a monofractal approach to multimodal handling of multifractal signals (Ge and Leung, 2013; Rego et al, 2013; Figure 2A). …”

Statistical properties of the seismic noise field: influence of soil heterogeneities

“…For a proper determination of the transition time intervals, we made use of the scaling-identification regression model by a statistical interference procedure proposed by Ge & Leung (2013). The linear fit to the average squared soil displacement is segmented in a statistical sense based on the join point regression model (Muggeo 2003), with each linear segment representing a distinct regime of the multiscaling behaviour of the time-series under study.…”

Study on nature of crossover phenomena with application to gearbox fault diagnosis

“…To estimate the locations of crossover points, the piecewise-linear regression models are typically used in the PTP and ICM. It contains primarily two types (grid-search method [34,35] and Hudson's method [36,37]). The Hudson's method can estimate the crossover points on the continuous observation scale.…”