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2D Fourier Transform Based Analysis Comparing the DFA with the DMA

Even if they can be outperformed by other methods, the detrended fluctuation analysis (DFA) and the detrended moving average (DMA) are widely used to estimate the Hurst exponent because they are based on basic notions of signal processing. For the last years, a great deal of interest has been paid to compare them and to better understand their behaviors from a mathematical point of view. In this paper, our contribution is the following: we first propose to express the square of the so-called fluctuation function as a 2D Fourier transform (2D-FT) of the product of two matrices. The first one is defined from the instantaneous correlations of the signal while the second, called the weighting matrix, is representative of each method. Therefore, the 2D-FT of the weighting matrix is analyzed in each case. In this study, differences between the DFA and the DMA are pointed out when the approaches are applied on non-stationary processes.