Effects of Polynomial Trends on Detrending Moving Average Analysis

Abstract

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by § Corresponding author. 1550034-1Fractals Downloaded from www.worldscientific.com by EMORY UNIVERSITY on 07/28/15. For personal use only.Y. -H. Shao et al. the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.

Detrending moving average algorithm: Frequency response and scaling performances

“…Let us consider the sum of two uncorrelated time series x A (i) and x B (i), the superposition law of the mean square deviation holds [40]:…”

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

“…Another more recently developed method is called detrending moving average (DMA), see [44,45] and for a good overview see [46]. This method is also well-performing [31,32,[47][48][49][50] with strong applications in analyzing financial data [51][52][53][54][55] and fractal structures [56][57][58]. In [59] a fast algorithm has been proposed which drastically decreases the computation time of DMA.…”

Long-range temporal correlation in Auditory Brainstem Responses to Spoken Syllable/da/

“…Where s is the window size, and θ is the position parameter varying in the range [0,1]. Hence, the moving average θ = 0 is called the backward moving average, θ = 0.5 corresponds to the centered moving average, and θ = 1 refers to the forward moving average [71,72].…”

“…On the other hand, MFDFA contains discontinuity in its internal algorithm for capturing local trends leading to discrepancy in computed scaling exponents. Therefore, MFDMA has been proposed to quantify the statistical properties of mono (multi) fractal time series [70][71][72]. In addition, several robust methods have been also proposed to remove trends such as Fourier Detrended Fluctuations Analysis (FDFA) [73], Adaptive Detrending method (AD) [74], Singular Value Decomposition (SVD) [67,75], and Empirical Mode Decomposition (EMD) [76].…”