Relationships of exponents in two-dimensional multifractal detrended fluctuation analysis

摘要

Multifractal detrended fluctuation analysis (MF-DFA) is a generalization of the conventional multifractalnanalysis. It is extended from the detrended fluctuation analysis (DFA) which is developed for the purposenof detecting long-range correlation and fractal property in stationary and nonstationary time series. The MFDFAnand some corresponding relationships of the exponents have been subsequently extended to the twodimensionalnspace.We reexamine two extended relationships in this study and demonstrate that: (i) The invaliditynof the relationship h(q) ≡ H for two-dimensional fractional Brownian motion, and h(q = 2) ≡ H between thenHurst exponent H and the generalized Hurst exponent h(q) in the two-dimensional case. Two more logicalnrelationships are proposed instead as h(q = 2) = H for the stationary surface and h(q = 2) = H + 2 for thennonstationary signal. (ii) The invalidity of the expression τ (q) = qh(q) − Df stipulating the relationship betweennthe standard partition-function-based multifractal exponent τ (q) and the generalized Hurst exponent h(q) in thentwo-dimensional case. Reasons for its invalidity are given from two perspectives.