Relationships of exponents in two-dimensional multifractal detrended fluctuation analysis

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