Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Figure 1

(a) Frequency responses $\left|H_n\left(e^{i2\pi f}\right)\right|$ of zeroth-order centered DMA [Eq. (18)]. (b) Cutoff frequency definition. The cutoff frequency is defined as the corner frequency at which the low- and high-frequency asymptotic behaviors intersect (dashed lines).

Figure 2

Relationship between power spectral analysis and detrending-operation-based scaling analysis (DFA and DMA). Time series of long-range correlated processes displaying power-law power spectral density (PSD), i.e., $S\left(f\right)\sim f^{-\beta }$, are analyzed (from left to right, $\beta =-0.8$, 0, and $0.8)$. [(a)–(c)] PSD. [(d)–(f)] Zeroth-order centered DMA. [(g)–(i)] First-order DFA. For the PSD-based estimations of $F\left(n\right)$ (solid lines), each PSD is converted to $F\left(n\right)$ using Eq. (14) or Eq. (22). Each point is obtained by averaging over 100 samples.

Figure 3

Comparison of analytical and numerical single-frequency response functions. (a) First-order DFA (DFA1). The exact solution (solid line) is given by Eq. (29). (b) Zeroth-order centered DMA (DMA0). The exact solution (solid line) is given by Eq. (31). The circles indicate numerically estimated $F\left(n\right)$, where a single-frequency time series [Eq. (24)] with $f=0.01$ and $A=1$ is analyzed using (a) first-order DFA and (b) zeroth-order centered DMA.

Figure 4

(a) Single-frequency response functions of first-order DFA (DFA1; solid line) and zeroth-order centered DMA (DMA0; dashed line) for $A=1$ and $f=0.01$. (b) Pseudofrequency response of first-order DFA (solid line) and frequency response of zeroth-order centered DMA (dashed line) for $n=99$. (c) Pseudofrequency response (dashed line) of first-order DFA and numerically calculated Fourier amplitude spectrum of filtered white-noise time series (solid line). For the numerical results, each point is obtained by averaging over 100 samples.

Figure 5

Comparison of third-order DFA, second-order centered DMA, and fourth-order Butterworth filter detrending operations. (a) Frequency responses in frequency domain $\left(n=199\right)$. For the DFA, the pseudofrequency response [Eq. (35)] is plotted. (b) Single-frequency response functions in time domain $\left(f=0.005\right)$.

Figure 6

Frequency responses of standardized detrending operations $\left(\tilde{n}=99\right)$. To standardize the detrending operations, the standardized time scale defined as $\tilde{n}=n/r_c$ is used. (a) First-order DFA (DFA1), zeroth-order centered DMA (DMA0), and second-order Butterworth filter (BW2). (b) Third-order DFA (DFA3), second-order centered DMA (DMA2), and fourth-order Butterworth filter (BW4).

Figure 7

Time-scale distortions observed in DFA and DMA. (a) DFA. (b) Centered DMA. (c) DFA using standardized time scale $\tilde{n}$. (d) Centered DMA using $\tilde{n}$.

Figure 8

Scaling analysis of a time series given by the sum of an fGn with $H=0.2$ and a second-order polynomial trend. (a) Third-order DFA. (b) Second-order centered DMA with detrending operations identical to Savitzky-Golay filters. (c) Scaling analysis with fourth-order Butterworth filter. The dashed lines indicate lines with slope 0.2.

Figure 9

Frequency responses $\left|H_n\left(e^{i2\pi f}\right)\right|$ of $m\mathrm{th}$-order centered DMA $\left(n=99\right)$.