Detrending moving average algorithm: Frequency response and scaling performances

Carbone, Anna Filomena; Kiyono, Ken

doi:10.1103/physreve.93.063309

Cited by 40 publications

(29 citation statements)

References 51 publications

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“…For DFA this equation is zero if the order of detrending is q ≥ p + 1. For DMA it is q ≥ p + 1 for odd p and q ≥ p for even p, this has also been found in [64]. For those cases the estimator of the fluctuation function is unbiased F 2 (s) = F 2 (s) because the bias is zero B(s) = 0 due to Eq.…”

Section: Vi43 External Nonstationaritysupporting

confidence: 61%

“…The current analytical understanding of DFA and DMA is at a different states. 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].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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We present a general framework of detrending methods of fluctuation analysis of which detrended fluctuation analysis (DFA) is one prominent example. Another more recently introduced method is detrending moving average (DMA). Both methods are constructed differently but are similarly able to detect long-range correlations as well as anomalous diffusion even in the presence of nonstationarities. In this article we describe their similarities in a general framework of detrending methods. We establish this framework independently of the definition of DFA or DMA but by investigating the failure of standard statistical tools applied on nonstationary time series, let these be intrinsic nonstationarities such as for Brownian pathes, or external ones due to additive trends. In particular, we investigate the sample averaged mean squared displacement of the summed time series. By modifying this estimator we introduce a general form of the so-called fluctuation function and can formulate the framework of detrending methods. A detrending method provides an estimator of the fluctuation function which obeys the following principles: The first relates the scaling behaviour of the fluctuation function to the stochastic properties of the time series. The second principles claims unbiasedness of the estimatior. This is the centerpiece of the detrending procedure and ensures that the detrending method can be applied to nonstationary time series, e.g. FBM or additive trends. Both principles are formulated and investigated in detail for DFA and DMA by using the relationship between the fluctuation function and the autocovariance function of the underlying stochastic process of the time series.

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Section: Vi43 External Nonstationaritysupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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

2019

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Multifractal Cross Wavelet Analysis

et al. 2017

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Complex systems are composed of mutually interacting components and the output values of these components usually exhibit long-range cross-correlations. Using wavelet analysis, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call multifractal cross wavelet analysis (MFXWT). We assess the performance of the MFXWT method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. For binomial multifractal measures, we find the empirical joint multifractality of MFXWT to be in approximate agreement with the theoretical formula. For bFBMs, MFXWT may provide spurious multifractality because of the wide spanning range of the multifractal spectrum. We also apply the MFXWT method to stock market indices, and in pairs of index returns and volatilities we find an intriguing joint multifractal behavior. The tests on surrogate series also reveal that the cross correlation behavior, particularly the cross correlation with zero lag, is the main origin of cross multifractality.

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“…image technology and turbulence [11][12][13][14][15]. Another popular family include the detrended fluctuation analysis (DFA) [16][17][18] and the detrending moving average analysis (DMA) [19][20][21][22]. Extensive numerical simulations display that the performance of the DMA approach is comparable to the DFA approach with slightly different priorities under different situations [23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Direct determination approach for the multifractal detrending moving average analysis

Zhou

2017

Phys. Rev. E

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In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.

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Detrending moving average algorithm: Frequency response and scaling performances

Cited by 40 publications

References 51 publications

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

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

Multifractal Cross Wavelet Analysis

Direct determination approach for the multifractal detrending moving average analysis

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