Fast algorithm for scaling analysis with higher-order detrending moving average method

Abstract

Among scaling analysis methods based on the root-mean-square deviation from the estimated trend, it has been demonstrated that centered detrending moving average (DMA) analysis with a simple moving average has good performance when characterizing long-range correlation or fractal scaling behavior. Furthermore, higher-order DMA has also been proposed; it is shown to have better detrending capabilities, removing higher-order polynomial trends than original DMA. However, a straightforward implementation of higher-order DMA requires a very high computational cost, which would prevent practical use of this method. To solve this issue, in this study, we introduce a fast algorithm for higher-order DMA, which consists of two techniques: (1) parallel translation of moving averaging windows by a fixed interval; (2) recurrence formulas for the calculation of summations. Our algorithm can significantly reduce computational cost. Monte Carlo experiments show that the computational time of our algorithm is approximately proportional to the data length, although that of the conventional algorithm is proportional to the square of the data length. The efficiency of our algorithm is also shown by a systematic study of the performance of higher-order DMA, such as the range of detectable scaling exponents and detrending capability for removing polynomial trends. In addition, through the analysis of heart-rate variability time series, we discuss possible applications of higher-order DMA.

Detection of oriented fractal scaling components in anisotropic two-dimensional trajectories

“…We introduce methods for detailed anisotropic characterisation assuming an anisotropic 2D time-series different from such 2D fGn. In our approach, we first evaluate the angle-dependent scaling properties using higher-order DMA 21 (henceforth, called DDMA) and then decompose the observed 2D time series into two components with different orientations and scaling properties.…”

“…For instance, the practical options are the wavelet-decomposition-based method with a wavelet having vanishing moments 5 and detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) 16 , 18 and DMA 19 . Recently, we have established the theoretical foundation for DMA including higher-order DMA 6 , 15 , 20 and developed a fast implementation algorithm for DMA 21 to increase the reliability and applicability of DMA. In this study, we further extend the application of DMA to 2D trajectory analysis.…”

Multifractal Cross Wavelet Analysis

“…Other multifractal cross-correlation analysis methods include multifractal detrended cross-correlation analysis based on detrended fluctuation analysis (MFXDFA) [13], which is a multifractal version of detrended cross-correlation analysis (DCCA) [14], multifractal detrended cross-correlation analysis based on detrending moving-average analysis (MFXDMA) [15] based on multifractal detrending moving-average analysis (MF-DMA) [16] and detrending movingaverage analysis (DMA) [17][18][19][20][21][22][23][24], multifractal cross-correlation analysis (MFCCA) [25,26], and multifractal detrended partial correlation analysis (MFDPXA) [27].…”

Direct determination approach for the multifractal detrending moving average analysis

“…In real applications, one should keep it in mind that the determination of scaling ranges plays a crucial role in computing the scaling exponents [31][32][33]. These methods have been extended to many directions, such as objects in high dimensions [34][35][36][37][38], detrended cross-correlation analysis and its variants for two time analysis [39][40][41][42][43][44][45][46][47][48][49][50][51][52], detrended partial cross-correlation analysis for multivariate time series [53][54][55], and so on.…”