A comparative study of multifractal detrended fluctuation analysis and multifractal detrended moving average algorithm to estimate the multifractal spectrum
Authors: Xi Cai-Ping  , Zhang Shu-Ning  , Xiong Gang  , Zhao Hui-Chang  ,   School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China;,   School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China;,   School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Published: December 2020

Multifractal detrended fluctuation analysis (MFDFA) and multifractal detrended moving average (MFDMA) algorithm have been established as two important methods to estimate the multifractal spectrum of the one-dimensional random fractal signals. They have been generalized to deal with two-dimensional and higher-dimensional fractal signals. This paper gives a brief introduction of the two algorithms, and a detail description of the numerical experiments on the one-dimensional time series by using the two methods. By applying the two methods to the series generated from the binomial multiplicative cascades (BMC), we systematically carry out comparative analysis to get the advantages, disadvantages and the applicability of the two algorithms, for the first time so far as we know, from six aspects: the similarities and differences of the algorithm models, the statistical accuracy, the sensitivities of the sample size, the selection of scaling range, the choice of the q-orders, and the calculation amount. For one class of signals, the larger the sample size, the more accurate the estimated multifractal spectrum. Selection of appropriate scaling range affects the statistical accuracy in comparison of the two methods for almost all examples. The presence of scale invariance should be checked by first running the two methods over a large scaling range (e.g., from 10 to (N+1)/11 in this paper) and then plot log10 (Fq (scale)) against log10 (scale). In the MFDFA-m (m is the polynomial order, and in this paper m=1) method, the scaling range can be selected from {m + 2, 10} to N/10, N is the sample size of the time series. In the MFDMA algorithm, the scaling range should be from 10 to (N+1)/11. It is favorable to have an equal spacing between scales and the number of the scales should be larger than 10 and usually be selected from 20 to 40. The q-orders should consist of both positive and negative q's. When |q| = 5, the calculated results will not be sensitive with the increase of Δq from 0.05 to 1. If Δq = 0.1, the calculation error will be relatively small when 0 q|≤ 10. With the increase of |q|, the width of the multifractal spectrum will obviously become wider when 0 q|≤10 and the change will be smaller when |q|≥20. If |q| continues to increase, the local fluctuations will approach zero when the scale is small. The critical steps exist in the calculation of local trends for the MFDFA-m and the running moving average for the MFDMA. If the sample size N is fixed and the scale is relatively small, the runtime of the critical steps of MFDFA-1 will be longer than that of MFDMA. When the scale increases from 4 to N/4, it will be shorter than that of MFDMA. Results provide a valuable reference on how to choose the algorithm between MFDFA and MFDMA, and how to make the schemes of the parameter setting of the two algorithms when dealing with specific signals in practical applications.