Jiang Zhi-Qiang, Zhou Wei-Xing
School of Business, East China University of Science and Technology, Shanghai, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jul;84(1 Pt 2):016106. doi: 10.1103/PhysRevE.84.016106. Epub 2011 Jul 21.
There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross correlations. The multifractal detrended cross-correlation analysis (MFDCCA) approaches can be used to quantify such cross correlations, such as the MFDCCA based on the detrended fluctuation analysis (MFXDFA) method. We develop in this work a class of MFDCCA algorithms based on the detrending moving-average analysis, called MFXDMA. The performances of the proposed MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents h(xy) extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent of the cross correlation is independent of the cross-correlation coefficient between two time series, and the MFXDFA and centered MFXDMA algorithms have comparative performances, which outperform the forward and backward MFXDMA algorithms. For two-component autoregressive fractionally integrated moving-average processes, we also find that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q<0 and underperforms when q>0. We apply these algorithms to the return time series of two stock market indexes and to their volatilities. For the returns, the centered MFXDMA algorithm gives the best estimates of h(xy)(q) since its h(xy)(2) is closest to 0.5, as expected, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract rational multifractal nature.
在一些复杂系统中，存在多种信号同时被记录的情况，这些系统呈现出长期的幂律交叉相关性。多重分形去趋势交叉相关性分析（MFDCCA）方法可用于量化此类交叉相关性，例如基于去趋势波动分析（MFXDFA）方法的MFDCCA。在这项工作中，我们开发了一类基于去趋势移动平均分析的MFDCCA算法，称为MFXDMA。通过对由双变量分数布朗运动、两分量自回归分数整合移动平均过程和二项测度生成的时间序列对进行广泛的数值实验，将所提出的MFXDMA算法的性能与MFXDFA方法进行了比较，这些时间序列对具有多重分形性质的理论表达式。在所有情况下，从MFXDMA和MFXDFA算法中提取的标度指数h(xy)都非常接近理论值。对于双变量分数布朗运动，交叉相关性的标度指数与两个时间序列之间的交叉相关系数无关，并且MFXDFA和居中的MFXDMA算法具有可比的性能，优于向前和向后的MFXDMA算法。对于两分量自回归分数整合移动平均过程，我们还发现MFXDFA和居中的MFXDMA算法具有可比的性能，而向前和向后的MFXDMA算法性能稍差。对于二项测度，向前的MFXDMA算法表现出最佳性能，居中的MFXDMA算法表现最差，并且当矩阶q<0时，向后的MFXDMA算法优于MFXDFA算法，而当q>0时表现不如MFXDFA算法。我们将这些算法应用于两个股票市场指数的收益率时间序列及其波动率。对于收益率，居中的MFXDMA算法给出了h(xy)(q)的最佳估计，因为其h(xy)(2)最接近0.5，正如预期的那样，并且MFXDFA算法具有第二好的性能。对于波动率，向前和向后的MFXDMA算法给出了相似的结果，而居中的MFXDMA和MFXDFA算法未能提取出合理的多重分形性质。
