Detrending moving average algorithm for multifractals

Abstract

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.

13 pages, 3 figures, 2 tables. We provide the MATLAB codes for the one-dimensional and two-dimensional MFDMA Algorithms

References in corpus (10)

• Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Non-stationary Time Series
• Multifractal detrended cross-correlation analysis for two nonstationary signals
• Comparison of detrending methods for fluctuation analysis
• Detrended fluctuation analysis for fractals and multifractals in higher dimensions
• Detrending Moving Average variance: a derivation of the scaling law
• Algorithm to estimate the Hurst exponent of high-dimensional fractals
• Detrended fluctuation analysis of intertrade durations
• Quantifying bid-ask spreads in the Chinese stock market using limit-order book data: Intraday pattern, probability distribution, long memory, and multifractal nature
• Empirical shape function of limit-order books in the Chinese stock market
• Statistical properties of daily ensemble variables in the Chinese stock markets

Cited by in corpus (2)

• Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series
• Long-term correlations and multifractal nature in the intertrade durations of a liquid Chinese stock and its warrant