Detrending moving average (DMA)  is a widely used method to quantify the correlation of non-stationary signals. We generalize DMA to multifractal detrending moving average (MFDMA) , and then extend one-dimensional MFDMA to two-dimensional version. In the paper, we elaborate one-dimensional and two-dimensional MFDMA theoretically and apply the methods to synthetic multifractal measures  . We find that the numerical estimations of the multifractal scaling exponent \tau(q) and the multifractal spectrum f(\alpha) are in good agreement with the theoretical values. We also compare the performance of MFDMA with MFDFA, and report that MFDMA is superior to MFDFA when apply them to analysis the properties of one-dimensional and two-dimensional multifractal measures .
The detrending moving average (DMA) algorithm is a widely used technique to quantify the long-term correlations of non-stationary time series and the long-range correlations of fractal surfaces, which contains a parameter \theta determining the position of the detrending window. We develop multifractal detrending moving average (MFDMA) algorithms for the analysis of one-dimensional multifractal measures and higher-dimensional multifractals, which is a generalization of the DMA method. The performance of the one-dimensional and two-dimensional MFDMA methods is investigated using synthetic multifractal measures with analytical solutions for backward (\theta=0), centered (\theta=0.5), and forward (\theta=1) detrending windows . We find that the estimated multifractal scaling exponent \tau(q) and the singularity spectrum f(\alpha) are in good agreement with the theoretical values. In addition, the backward MFDMA method has the best performance, which provides the most accurate estimates of the scaling exponents with lowest error bars, while the centered MFDMA method has the worse performance. It is found that the backward MFDMA algorithm also outperforms the multifractal detrended fluctuation analysis (MFDFA). The one-dimensional backward MFDMA method is applied to analyzing the time series of Shanghai Stock Exchange Composite Index and its multifractal nature is confirmed .
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Detrending
The detrending
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algorithm
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method
technique
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correlation
long-term correlations
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signals. We generalize DMA to
time series and the long-range correlations of fractal surfaces, which contains a parameter \theta determining the position of the detrending window. We develop
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, and then extend
algorithms for the analysis of
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MFDMA to two-dimensional version. In the paper, we elaborate
multifractal measures and higher-dimensional multifractals, which is a generalization of the DMA method. The performance of the
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theoretically and apply the methods to
methods is investigated using
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with analytical solutions for backward (\theta=0), centered (\theta=0.5), and forward (\theta=1) detrending windows
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numerical estimations of the
estimated
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multifractal
singularity
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We also compare the performance of MFDMA with MFDFA, and report that MFDMA is superior to MFDFA when apply them to analysis the properties of one-dimensional and two-dimensional multifractal measures
In addition, the backward MFDMA method has the best performance, which provides the most accurate estimates of the scaling exponents with lowest error bars, while the centered MFDMA method has the worse performance. It is found that the backward MFDMA algorithm also outperforms the multifractal detrended fluctuation analysis (MFDFA). The one-dimensional backward MFDMA method is applied to analyzing the time series of Shanghai Stock Exchange Composite Index and its multifractal nature is confirmed