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

Abstract

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.

Association between Multiscale Entropy Characteristics of Heart Rate Variability and Ischemic Stroke Risk in Patients with Permanent Atrial Fibrillation

“…In DFA, such nonstationary variation is removed by detrending [19]. In contrast, the local averaging procedure in the original MSE analysis cannot remove the effects of nonstationary variations.…”

“…In the bottom-right panel of Figure 1, the probability density function structure is affected by nonstationary and low-frequency variations included in the HRV time series. In DFA, such nonstationary variation is removed by detrending [19]. In contrast, the local averaging procedure in the original MSE analysis cannot remove the effects of nonstationary variations.…”

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

“…To our best knowledge analytical studies of DMA exist for the derivation of the scaling behaviour for fractional Gaussian noise [45,63,64] and on the ability of removing additive trends [64]. In contrast there exist relatively more analytical studies of DFA which can be classified into four categories: 1) Calculation of the scaling behaviour of the fluctuation function for specific process, namely autoregressive model of first order [65], fractional Gaussian noise [27,37,[66][67][68] and FBM [36,69,70]; 2) Derivation of the relationship between the fluctuation function and known statistical quantities, namely the autocorrelation function [66], power spectrum [12,[69][70][71][72][73][74] and variogram [75]; 3) Describing statistical properties of the fluctuation function [67,68]; 4) Illuminating the functionality of detrending [76]. Nevertheless there are still many open questions about these methods, see for example [72,77], not just about minor technical details but questions about fundamental principles and properties of detrending methods.…”

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

“…The fluctuation parameter α has been derived for fractional Gaussian noise [1,21,25] and for fractional Brownian motion [9,15,22]. The connection between the DFA and the power spectral analysis has been investigated in [9,15,16,17,24,26,27]. Also, the relationship between the fluctuation function of DFA and the autocorrelation function for stationary processes is known [10,11,12].…”