Inferring characteristic timescales from the effect of autoregressive dynamics on detrended fluctuation analysis

Abstract

Exploiting recent progress in the theoretical understanding of detrended fluctuation analysis (DFA), we use the non-asymptotic properties of the fluctuation function in order to extract more information from time series data than just its Hurst exponent. In particular, we can identify exponential relaxation and oscillation periods and estimate their specific values. We illustrate the strength of this method through applications to climate data. Thereby, we determine the relaxation time of the atmospheric response to perturbations. We also find by DFA a period length of the dominant frequency mode of the El Niño Southern Oscillation to be 3.3 years.

A simple decomposition of European temperature variability capturing the variance from days to a decade

“…In [26] the authors have shown that this method has the ability to uncover characteristic timescales in time series. The idea is to fit the theoretical fluctuation function of autoregressive models of different orders: AR(1) models for relaxation times, and AR(2) for noisy oscillations.…”

“…where the noise term t X is the output of some process on a much smaller timescale. The relaxation time r is determined by the AR-parameter c. The shape of the distribution of t X is not important for our model [26], however, it seems to be well approximated by a Gaussian distribution. The variance σ 2 is a free parameter.…”

“…(C1) as proposed in [13] and performed in [12] and [26]. The kernel L 1 in DFA-1 is (C2) The correlation function of AR(1) is known to be…”

“…We present the foundations of our method in section 3, summarizing previous work in [26]. In section 4 we postulate our model for weather and macroweather variability in Europe and extend our method to processes with more than one characteristic timescale.…”

An ARFIMA-based model for daily precipitation amounts with direct access to fluctuations

“…These data sets are a random selection of the databases (Deutscher Wetterdienst (DWD) 2018; European Climate Assessment and Dataset 2018) by Klein Tank et al (2002) and (Met Office Hadley Centre 2018) chosen such that the observed period spans more than 70 years, the data sets are nearly complete and the assumption of weak seasonality is satisfied. Prominent annual cyclicity can be evaluated by deviations of the scaling of, e. g., the DFA fluctuation function from linearity (Meyer and Kantz 2019). For an automatized estimation of the stationarity we employed the regression values of the linear fit of the fluctuations quantifying functions.…”