Detrended fluctuation analysis for fractals and multifractals in higher dimensions

Abstract

One-dimensional detrended fluctuation analysis (1D DFA) and multifractal detrended fluctuation analysis (1D MF-DFA) are widely used in the scaling analysis of fractal and multifractal time series because of being accurate and easy to implement. In this paper we generalize the one-dimensional DFA and MF-DFA to higher-dimensional versions. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. The twodimensional MF-DFA is also adopted to analyze two images from nature and experiment and nice scaling laws are unraveled.

Multifractal detrended cross-correlation analysis for two nonstationary signals

“…(7). This observation is analogous to the case of higher-dimensional detrended fluctuation analysis [7].…”

“…For a single nonstationary time series, the detrended fluctuation analysis (DFA) can be adopted to explore its long-range autocorrelations [4,5] and multifractal features [6]. The DFA method can also be extended to investigate higher-dimensional fractal and multifractal measures [7].…”

“…The trend function is pre-chosen in different function forms [7]. The simplest function could be a plane u(i, j) = ai + bj + c, which is adopted to test the validation of the method.…”

Comparison of detrending methods for fluctuation analysis

“…Its non-detrending predecessors are Hurst's rescaled range analysis [2] and fluctuation analysis (FA) [3]. DFA was later generalized for higher order detrending [4], multifractal analysis [5], separate analysis of sign and magnitude series [6], and data with more than one dimension [7]. Its features have been studied in many articles [8,9,10,11,12,13].…”

Multiresolution analysis of fluctuations in non-stationary time series through discrete wavelets

“…Among these are, de-trended fluctuation analysis and its variants [1,2] and the wavelet transform [3,4] based multiresolution analysis [5,6]. These methods and earlier methods [7,8] have found wide application in analysis of correlations and characterization of scaling behavior of time-series data in, physiology, finance, and natural sciences [9,10,11,12,13,14,15,16,17,18,19]. Recently, the relative merits of MF-DFA and a variety of other approaches to characterize fluctuations have been carried out [20].…”