Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average

We present a bottom-up derivation of fluctuation analysis with detrending for the detection of long-range correlations in the presence of additive trends or intrinsic nonstationarities. While the well-known detrended fluctuation analysis (DFA) and detrending moving average (DMA) were introduced ad hoc, we claim basic principles for such methods where DFA and DMA are then shown to be specific realizations. The mean-squared displacement of the summed time series contains the same information about long-range correlations as the autocorrelation function but has much better statistical properties for large time lags. However, the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities. We therefore define the fluctuation function as mean-squared displacement with weighting kernel. We require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal, which is only achieved if the weighting kernel implies detrending. We show how DFA and DMA satisfy these requirements and we extract their kernel weights.

Figure 1

Possible asymptotic scaling behavior of the path MSD for stochastic processes discussed in this article (skipping potential crossover-regimes and normalizing $R^2\left(0\right)=1$). The figure shows white noise with $\alpha =1/2$ (solid black line), fractional Gaussian noise with $1/2<\alpha <1$ (grey shaded area), Brownian motion with $\alpha =3/2$ (dashed blue line) and fractional Brownian motion with $3/2<\alpha <1$ (blue shaded area). Note that we only consider specific intervals of the Hurst parameter $H$ here; see Eq. (8).

Figure 2

Schematic representation of the replacement of the $\nu \mathrm{th}$ realization in the first segment ${\left\{{\epsilon }^{\left(\nu \right)}\left(i\right)\right\}}_{i=1}^s$ by the first realization in the $\nu \mathrm{th}$ segment ${\left\{{\epsilon }^{\left(1\right)}\left(i\right)\right\}}_{i=1+d_{\nu }}^{s+d_{\nu }}$ with $\nu \in \{1,K\}$; see Eq. (17). It is shown here for $K=3$ realizations/segments with Brownian motion as stochastic process. The first realization (solid line) represents the time series, the second (dashed line) and third realization (dotted line) are assumed to exist theoretically in the first segment; see more details in the text of this section. Here the segments have a length of $s=40$ and are disjoint with shift $d_{\nu }=(\nu -1)s$.

Figure 3

Splitting of the autocovariance function in the $\nu \mathrm{th}$ segment $\langle x(i+d_{\nu })x(j+d_{\nu })\rangle =\langle \epsilon (i+d_{\nu })\epsilon (j+d_{\nu })\rangle$ (red dotted line) of Brownian motion into the autocovariance function of the first segment $\text{Cov}(i,j)$ (black solid line) and the autocovariance difference $d_{\nu }$ (black dashed line); see Eq. (30). Here it is $j=30$. The segments have a length of $s=40$ and are disjoint with shift $d_{\nu }=(\nu -1)s$.

Figure 4

Splitting of the autocovariance function in the $\nu \mathrm{th}$ segment $\langle x(i+d_{\nu })x(j+d_{\nu })\rangle$ (red dotted line) of an AR(1) process with additive linear trend with slope $0.05$ into the autocovariance function of the first segment of the AR(1) process $\text{Cov}(i,j)$ (black solid line) and the autocovariance difference $0.{05}^2(i+d_{\nu })(j+d_{\nu })$ (black dashed line); see Eq. (39). The AR(1) process has the autocorrelation function $C\left(\right|i-j\left|\right)=0.9^{|i-j|}$. Here it is $j=30$. The segments have a length of $s=40$ and are disjoint with shift $d_{\nu }=(\nu -1)s$. Note that we use white noise instead of an AR(1) process in the text of this section. But the situation of the splitting of the autocovariance function remains.

Figure 5

The weights of the residual ${\omega }^{\left(q\right)}(i,t)$ for Eq. (66) for detrending order $q=0$ and 1 (black solid line), $q=2$ and 3 (blue dashed line) and $q=4$ and 5 (red dotted line). The segment length is $s=41$ and the time point is $t=(s+1)/2=21$. Note that for $t=(s+1)/2$ the weights are identical for $q=0$ and 1, 2, and 3 as well as 4 and 5 but not for other $t$; see Fig. 6.

Figure 6

The weights of the residual ${\omega }^{\left(q\right)}(i,t)$ for Eq. (66) for detrending order $q=0$ (black solid line), 1 (blue dashed line), 2 (red dotted line), and 3 (green dash-dotted line). The segment length is $s=41$ and the time point is $t=17$.

Figure 7

The weights of the stationary DFA fluctuation function ${\mathcal{L}}_{\text{DFA}q}$ of Eq. (73) for detrending order $q=0$ (black solid line), 1 (blue dashed line), and 2 (red dotted line). Here it is $s=41$.

Figure 8

The weights of the stationary DFA fluctuation function ${\mathcal{L}}_{\text{DFA}q}$ of Eq. (73) for detrending order $q=0$ and 1 (black solid line), 2 and 3 (blue dashed line), as well as 4 and 5 (red dotted line). Here it is $s=41$.