A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum

• We study the differences, the advantages and the applicabilities of 2D-MFDFA and 2D-MFDMA.
• Compare 2D-MFDFA and 2D-MFDMA to detect the inheritance and development of DFA and DMA.
• Do the comparative analysis of 2D-MC by the 2D-MFDFA and 2D-MFDMA from six aspects for the first time.
• Provide a guidance on the choice and parameter settings of two methods in the real applications.

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 signal. They have been generalized to deal with two-dimensional and higher-dimensional fractal signals. This paper gives a brief introduction of the two-dimensional multifractal detrended fluctuation analysis (2D-MFDFA) and two-dimensional multifractal detrended moving average (2D-MFDMA) algorithm, and a detailed description of the application of the two-dimensional fractal signal processing by using the two methods. By applying the 2D-MFDFA and 2D-MFDMA to the series generated from the two-dimensional multiplicative cascading process, we systematically do the comparative analysis to get the advantages, disadvantages and the applicabilities of the two algorithms for the first time from six aspects such as 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 qq-orders and the calculation amount. The results provide a valuable reference on how to choose the algorithm from 2D-MFDFA and 2D-MFDMA, and how to make the schemes of the parameter settings of the two algorithms when dealing with specific signals in practical applications.