Detrended fluctuation analysis and the difference between external drifts and intrinsic diffusionlike nonstationarity

Detrended fluctuation analysis (DFA) has been shown to be an effective method to study long-range correlation of nonstationary series. In principle, DFA considers F_{DFA}^{2}(s), the mean of variance around the local polynomial fit in segments with length s, and then estimates the scaling exponent α_{DFA} in F_{DFA}(s)∼s^{α_{DFA}} with varying s. Usually, the methodological studies of DFA focus on its effect on removing the drift due to the external trends. Only few paid attention to nonstationary series without drift, such as fractional Brownian motion (FBM) with nonstationarity due to its intrinsic dynamics. Both of these types of nonstationarity can shift the local mean by drift or diffusion and can be treated as the additive nonstationarity eliminable by the additive decomposition. In this study, we limit our discussion to such additive nonstationarity and furthermore specifically distinguish these two types of nonstationarity, namely the drift and the intrinsic diffusionlike nonstationarity. To understand how DFA works for the intrinsic diffusionlike nonstationarity, we take FBM as the example and seek for the answers to two fundamental questions: (1) what DFA removes from FBM; and (2) why DFA can handle such intrinsic diffusionlike nonstationarity, in contrast to methods only applicable to stationary series such as the fluctuation analysis. A crucial condition, i.e., statistical equivalence among all segments, is proposed and checked in the fluctuation analysis and DFA. As shown, the crucial condition is a natural requirement for the connection between DFA and autocorrelation function. With the help of the crucial condition, our study analytically and numerically demonstrates for the intrinsic diffusionlike nonstationary series that (1) rather than the nonstationarity as thought, DFA actually removes the difference among all segments; (2) the detrended segments fulfill the crucial condition so that the average over segments becomes equivalent to the ensemble average over realizations. These answers are also true for series with a drift. Thus, we provide a unified perspective to refresh the understanding of how DFA works on nonstationarity and underpin the mathematical ground of DFA.