Quantifying signals with power-law correlations: A comparative study of detrended fluctuation analysis and detrended moving average techniques

Limei Xu, Plamen Ch. Ivanov, Kun Hu, Zhi Chen, Anna Carbone, and H. Eugene Stanley

Phys. Rev. E 71, 051101 – Published 6 May 2005

Abstract

Detrended fluctuation analysis (DFA) and detrended moving average (DMA) are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals. We systematically study the performance of different variants of the DMA method when applied to artificially generated long-range power-law correlated signals with an a priori known scaling exponent $α_{0}$ and compare them with the DFA method. We find that the scaling results obtained from different variants of the DMA method strongly depend on the type of the moving average filter. Further, we investigate the optimal scaling regime where the DFA and DMA methods accurately quantify the scaling exponent $α_{0}$ , and how this regime depends on the correlations in the signal. Finally, we develop a three-dimensional representation to determine how the stability of the scaling curves obtained from the DFA and DMA methods depends on the scale of analysis, the order of detrending, and the order of the moving average we use, as well as on the type of correlations in the signal.

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Received 7 August 2004

DOI:https://doi.org/10.1103/PhysRevE.71.051101

Authors & Affiliations

Limei Xu¹, Plamen Ch. Ivanov¹, Kun Hu¹, Zhi Chen¹, Anna Carbone², and H. Eugene Stanley¹

¹Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
²INFM and Physics Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

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Vol. 71, Iss. 5 — May 2005

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Images

Figure 1
A comparison of the scaling behavior obtained from the DMA, WDMA-1, DFA-0, and DFA-1 methods for artificially generated power-law correlated signals with a scaling exponent $α_{0}$ . The length of the signals is $N = 2^{20}$ . Scaling curves $F (n)$ versus scale $n$ for (a) an anticorrelated signal with $α_{0} = 0.2$ , (b) an uncorrelated signal with $α_{0} = 0.5$ , and (c) a positively correlated signal with $α_{0} = 0.8$ . At small scales, all methods exhibit a weak crossover, which is more pronounced for anticorrelated signals. At large scales, the $F (n)$ curves obtained from DMA, WDMA-1, and DFA-0 exhibit a clear crossover to a flat region for all signals, independent of the type of correlations. No such crossover is observed for the scaling curves obtained from the DFA-1 method, suggesting a more accurate estimate of the scaling exponent $α_{0}$ at large scales.Reuse & Permissions
Figure 2
A comparison of the performance of the different scaling methods (DMA, WDMA-1, DFA-0, DFA-1, and DFA-2) when applied to artificially generated signals with long-range power-law correlations. Here $α_{0}$ is the correlation exponent of the generated signals and $α$ is the exponent value estimated using different methods. For all methods we obtain $α$ by fitting the corresponding scaling curves $F (n)$ in the range $n ∊ [10^{2}, 10^{4}]$ . Flat regions indicate the limitations of the methods in accurately estimating the degree of correlations in the generated signals, as the “output” exponent $α$ remains unchanged when the “input” exponent $α_{0}$ is varied.Reuse & Permissions
Figure 3
A comparison of the local scaling exponent $α_{loc}$ as a function of the scale $n$ for the DMA, WDMA-1, DFA-0, and DFA-1 methods. We consider signals of length $N = 2^{20}$ and varying values of the correlation exponent $α_{0}$ . The local scaling exponent $α_{loc}$ quantifies the stability of the scaling curves $F (n)$ (see Fig. 1) and is expected to exhibit small fluctuations around a constant value $α_{0}$ if $F (n)$ is well fitted by a power-law function. $α_{0}$ is denoted by horizontal dotted lines. Symbols denote the estimated values of $α_{loc}$ and represent average results from 50 realizations of artificial signals for each value of the “input” scaling exponent $α_{0}$ . Deviations from the horizontal lines at small or at large scales indicate limitations of the methods to accurately quantify the built-in correlations in different scaling ranges.Reuse & Permissions
Figure 4
Values of the local scaling exponent $α_{loc}$ as a function of the scale $n$ obtained from 20 different realizations of artificial anticorrelated signals with an identical scaling exponent $α_{0} = 0.2$ .Reuse & Permissions
Figure 5
Values of the local scaling exponent $α_{loc}$ as a function of the scale $n$ obtained from 20 different realizations of artificial uncorrelated signals with an identical scaling exponent $α_{0} = 0.5$ .Reuse & Permissions
Figure 6
Values of the local scaling exponent $α_{loc}$ as a function of the scale $n$ obtained from 20 different realizations of artificial positively correlated signals with an identical scaling exponent $α_{0} = 0.8$ .Reuse & Permissions
Figure 7
Probability density of the estimated values of $α_{0} - δ < α_{loc} < α_{0} + δ$ , where $δ = 0.02$ for a varying scale range $n$ and for different values of the “input” correlation exponent $α_{0}$ . Separate panels show the performance of the DMA, WDMA-1, DFA-0 and DFA-1 methods, respectively, based on 50 realizations of correlated signals for each value of $α_{0}$ . The probability density values $p$ are presented in color, with the darker color corresponding to higher values as indicated in the vertical columns next to each panel. A perfect scaling behavior would correspond to dark-colored columns spanning all scales $n$ for each value of $α_{0}$ .Reuse & Permissions
Figure 8
A comparison of the local scaling exponent $α_{loc}$ as a function of the scale $n$ for the DMA, CDMA, DFA-0, and DFA-1 methods. We consider signals of length $N = 2^{20}$ and varying values of the correlation exponent $α_{0}$ . The local scaling exponent $α_{loc}$ quantifies the stability of the scaling curves $F (n)$ and is expected to exhibit small fluctuations around a constant value $α_{0}$ if $F (n)$ is well fitted by a power-law function. $α_{0}$ is denoted by horizontal dotted lines. Symbols denote the estimated values of $α_{loc}$ and represent average results from 50 realizations of artificial signals for each value of the “input” scaling exponent $α_{0}$ . Deviations from the horizontal lines at small or at large scales indicate limitations of the methods to accurately quantifying the built-in correlations in different scaling ranges. Error bars represent the standard deviation for each average value of $α_{loc}$ at different scales $n$ , and determine the accuracy of each method.Reuse & Permissions
Figure 9
Probability density of the estimated values of $α_{0} - δ < α_{loc} < α_{0} + δ$ , where $δ = 0.02$ for a varying scale range $n$ and for different values of the “input” correlation exponent $α_{0}$ . The two panels show the performance of the DMA and CDMA methods, respectively, based on 50 realizations of correlated signals for each value of $α_{0}$ . The probability density values $p$ are presented in color, with the darker color corresponding to higher values, as indicated in the vertical columns next to each panel. A perfect scaling behavior would correspond to dark-colored columns spanning all scales $n$ for each value of $α_{0}$ .Reuse & Permissions
Figure 10
Local scaling exponent $α_{loc}$ as a function of the scale $n$ for the WCDMA method. We consider signals of length $N = 2^{20}$ and varying values of the correlation exponent $α_{0}$ . The expected value of the exponent $α_{0}$ is denoted by horizontal dotted lines. Symbols denote the estimated values of $α_{loc}$ and represent average results from 50 realizations of artificial signals for each value of the “input” scaling exponent $α_{0}$ . Deviations from the horizontal lines at small or at large scales indicate limitations of the methods to accurately quantify the built-in correlations in different scaling ranges. Error bars represent the standard deviation for each average value of $α_{loc}$ at different scales $n$ , and determine the accuracy of the method.Reuse & Permissions
Figure 11
A comparison of the local scaling exponent $α_{loc}$ as a function of the scale $n$ obtained from (a) the WCDMA method and (b) the DFA-1 method. Symbols denote the estimated values of $α_{loc}$ calculated as in Fig. 10 for different “input” scaling exponents $α_{0} > 1$ . Error bars representing the standard deviation around the average $α_{loc}$ are smaller for the DFA-1 method at all scales $n$ , indicating that the DFA-1 method provides more reliable results.Reuse & Permissions
Figure 12
Probability density of estimated values of $α_{0} - δ < α_{loc} < α_{0} + δ$ , where $δ = 0.01$ for varying scale range $n$ and for different values of the “input” correlation exponent $α_{0}$ . The two panels show the performance of the DMA and DFA-1 methods, respectively, based on 50 realizations of correlated signals for each value of $α_{0}$ . The probability density values $p$ are presented in color, with darker color corresponding to higher values as indicated in the vertical columns next to each panel. A perfect scaling behavior would correspond to dark-colored columns spanning all scales $n$ for each value of $α_{0}$ .Reuse & Permissions
Figure 13
A comparison of the local scaling exponent $α_{loc}$ as a function of the scale $n$ for the WDMA- $ℓ$ method with different order $ℓ = 2, \dots, 5$ of the weighted moving average. We consider signals of length $N = 2^{20}$ and varying values of the correlation exponent $α_{0}$ . The local scaling exponent $α_{loc}$ quantifies the stability of the scaling curves $F (n)$ (see Fig. 1), and is expected to exhibit small fluctuations around a constant value $α_{0}$ if $F (n)$ is well fitted by a power-law function. $α_{0}$ is denoted by horizontal dotted lines. Symbols denote the estimated values of $α_{loc}$ and represent average results from 50 realizations of artificial signals for each value of the “input” scaling exponent $α_{0}$ . For small values of $ℓ$ at small and intermediate scales $n$ , WDMA- $ℓ$ accurately reproduces the scaling behavior of signals with $0.4 < α_{0} < 0.8$ , while for large $ℓ$ , the scaling behavior of anticorrelated signals with $α_{0} < 0.4$ are better reproduced at small scales.Reuse & Permissions
Figure 14
Probability density of estimated values of $α_{0} - δ < α_{loc} < α_{0} + δ$ , where $δ = 0.02$ for the varying scale range $n$ and for different values of the “input” correlation exponent $α_{0}$ . Separate panels show the performance of the WDMA-2, WDMA-3, WDMA-4, and WDMA-5 methods, respectively, based on 50 realizations of correlated signals for each value of $α_{0}$ . The probability density values $p$ are presented in color, with the darker color corresponding to higher values, as indicated in the vertical columns next to each panel. A perfect scaling behavior would correspond to dark-colored columns spanning all scales $n$ for each value of $α_{0}$ .Reuse & Permissions
Figure 15
Plot of the moving average filter kernel in the time (a) and in the frequency domain (b), respectively.Reuse & Permissions
Figure 16
Plot of the function ${∣ H (f) ∣}^{2}$ for the simple moving average with $n = 4$ , ${∣ H_{4} (f) ∣}^{2}$ ; for the weighted moving average, with $n = 4$ and $ℓ = 2$ , ${∣ H_{4, 2} (f) ∣}^{2}$ , respectively; for the weighted moving average with $n = 4$ and $ℓ = 4$ ${∣ H_{4, 4} (f) ∣}^{2}$ , respectively.Reuse & Permissions

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