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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.

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.

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