Scaling range of power laws that originate from fluctuation analysis

Abstract

We extend our previous study of scaling range properties performed for detrended fluctuation analysis (DFA) [Physica A 392, 2384 (2013)] to other techniques of fluctuation analysis (FA). The new technique, called modified detrended moving average analysis (MDMA), is introduced, and its scaling range properties are examined and compared with those of detrended moving average analysis (DMA) and DFA. It is shown that contrary to DFA, DMA and MDMA techniques exhibit power law dependence of the scaling range with respect to the length of the searched signal and with respect to the accuracy R^{2} of the fit to the considered scaling law imposed by DMA or MDMA methods. This power law dependence is satisfied for both uncorrelated and autocorrelated data. We find also a simple generalization of this power law relation for series with a different level of autocorrelations measured in terms of the Hurst exponent. Basic relations between scaling ranges for different techniques are also discussed. Our findings should be particularly useful for local FA in, e.g., econophysics, finances, or physiology, where the huge number of short time series has to be examined at once and wherever the preliminary check of the scaling range regime for each of the series separately is neither effective nor possible.

Power-law cross-correlations estimation under heavy tails

“…The detrended covariance f 2 DCCA (s, j) is obtained for each box j of length s and it is further averaged over all boxes of length s to get F 2 DCCA (s) as an estimated covariance for scale s. For the power-law cross-correlated processes, we have F 2 DCCA (s) ∝ s 2Hxy . There are various ways of treating overlapping and non-overlapping boxes for scales s as the method can become computationally highly demanding [35,36,[60][61][62]. Due to this fact, we use non-overlapping boxes with a minimum scale of 10, a maximum scale of T /5 and a step between s equal to 10 in the simulations.…”

Asymptotic scaling properties and estimation of the generalized Hurst exponents in financial data

“…[1][2][3][4][5]). Many works have been dedicated to its empirical characterization [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], reporting strong evidence of its presence in financial markets. Several models have been proposed [24][25][26][27][28][29][30][31][32][33] to reproduce these empirical facts.…”

Direct determination approach for the multifractal detrending moving average analysis

“…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]. 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.…”