Detrending moving average algorithm for multifractals

TL;DR

This paper introduces a generalized multifractal detrending moving average (MFDMA) algorithm for analyzing multifractal measures in one and higher dimensions, demonstrating its superior accuracy over existing methods through synthetic data and real stock market analysis.

Contribution

The paper develops a new MFDMA algorithm for multifractal analysis, extending DMA to higher dimensions and comparing its performance with existing methods.

Findings

Backward MFDMA provides the most accurate estimates of multifractal scaling exponents.

MFDMA outperforms the multifractal detrended fluctuation analysis (MFDFA).

Application to stock market data confirms multifractality in financial time series.

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.