Multifractal detrending moving-average cross-correlation analysis

Zhi-Qiang Jiang (蒋志强) and Wei-Xing Zhou (周炜星)

Phys. Rev. E 84, 016106 – Published 21 July 2011

Abstract

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 $h_{x y}$ 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 $h_{x y} (q)$ since its $h_{x y} (2)$ 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.

Received 7 May 2011

DOI:https://doi.org/10.1103/PhysRevE.84.016106

Authors & Affiliations

Zhi-Qiang Jiang (蒋志强)^1,2 and Wei-Xing Zhou (周炜星)^1,2,3,*

¹School of Business, East China University of Science and Technology, Shanghai 200237, China
²Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China
³School of Science, East China University of Science and Technology, Shanghai 200237, China

^*wxzhou@ecust.edu.cn

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Vol. 84, Iss. 1 — July 2011

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Figure 1
Multifractal detrended cross-correlation analysis of bivariate fractional Brownian motions. Comparisons are performed among three MFXDMA algorithms with $θ = 0$ , 0.5, and 1 and the MFXDFA algorithm. The results in (c)–(f) are averaged over 100 repeated simulations. (a) A typical example of BFBM with $H_{x x} = 0.1$ , $H_{y y} = 0.5$ , and $ρ = 0.3$ . (b) Power-law dependence of the fluctuation functions $F_{x y} (q, s)$ of the BFBM shown in (a) with respect to the scale $s$ for $q = - 4$ , $q = 0$ , and $q = 4$ . The straight lines are the best power-law fits to the data. The results have been translated vertically for better visibility. (c) Scaling exponents $h_{x y} (q)$ with the theoretical values as a dashed line for $H_{x x} = H_{y y} = 0.8$ and $ρ = 0.5$ (top) and $H_{x x} = 0.1$ , $H_{y y} = 0.5$ , and $ρ = 0.3$ (bottom). (d) Independence of the scaling exponents $h_{x y}$ with respect to the cross-correlation coefficient $ρ$ for $H_{x x} = H_{y y} = 0.8$ (top) and $H_{x x} = 0.1$ and $H_{y y} = 0.5$ (bottom). (e) Differences $Δ h_{x y} (q)$ between the estimated scaling exponents $h_{x y}$ and the theoretical exponents $H_{x y}$ for BFBMs, where $H_{x x} = 0.1$ is fixed, $H_{y y}$ varies from 0.1 to 0.9, and $ρ$ takes different values. (f) Differences $Δ h_{x y} (q)$ between $h_{x y}$ and $H_{x y}$ for BFBMs with $H_{x x} = H_{y y}$ varying from 0.1 to 0.9 and different $ρ$ values.Reuse & Permissions
Figure 2
Multifractal detrended cross-correlation analysis of two-component ARFIMA processes. Comparisons are performed among three MFXDMA algorithms with $θ = 0$ , 0.5, and 1 and the MFXDFA method. (a) Power-law dependence of the fluctuation functions $F_{x y} (q, s)$ with respect to the scale $s$ for $q = - 4$ , $q = 0$ , and $q = 4$ for the process in Eq. (20) with $d_{1} = d_{2} = 0.4$ . The straight lines are the best power-law fits to the data. The results have been translated vertically for better visibility. (b) Scaling exponents $h_{x y} (q)$ for the process in Eq. (20) with $d_{1} = d_{2} = 0.4$ . (c) Differences $Δ h_{x y}$ between the estimated scaling exponents $h_{x y}$ and the theoretical exponents $H_{x y}$ for the process in Eq. (20) with different $d$ values where $d_{1} = d_{2} = d$ . (d) Power-law dependence of the fluctuation functions $F_{x y} (q, s)$ with respect to the scale $s$ for $q = - 4$ , $q = 0$ , and $q = 4$ for the process in Eq. (22) with $d_{1} = 0.1$ and $d_{2} = 0.4$ . (e) Scaling exponents $h_{x y} (q)$ for the process in Eq. (22) with $d_{1} = 0.1$ and $d_{2} = 0.4$ . (f) Differences $Δ h_{x y}$ between $h_{x y}$ and $H_{x y}$ for the process in Eq. (22) with different $d$ values where $d_{1} = d_{2} = d$ .Reuse & Permissions
Figure 3
Multifractal detrended cross-correlation analysis of two cross-correlated binomial measures generated from the $p$ model with $p_{x} = 0.3$ and $p_{y} = 0.4$ . Comparisons are performed among three MFXDMA algorithms with $θ = 0$ , 0.5, and 1 and the MFXDFA method. (a) Power-law dependence of the fluctuation functions $F_{x y} (q, s)$ with respect to the scale $s$ for $q = - 4$ , $q = 0$ , and $q = 4$ . The straight lines are the best power-law fits to the data. The results have been translated vertically for better visibility. (b) Scaling exponents $h_{x y} (q)$ with the theoretical values as a dashed line. The insets show the $h_{x y} (q)$ curves and the corresponding $[h_{x x} (q) + h_{y y} (q)] / 2$ curves, verifying the relation $h_{x y} (q) = [h_{x x} (q) + h_{y y} (q)] / 2$ . (c) Multifractal mass exponents $τ (q)$ obtained from the MFXDMA and MFXDFA methods, with the theoretical curve shown as a dashed line. (d) Multifractal spectra $f (α)$ with respect to the singularity strength $α$ for the four methods. The dashed curve is the theoretical multifractal spectrum. (e) Differences $Δ h_{x y} (q)$ between the estimated mass exponents and their theoretical values for the three algorithms: MFXDFA, MFXDMA with $θ = 0$ , and MFXDMA with $θ = 1$ . (f) Differences $Δ τ (q)$ between the estimated mass exponents and their theoretical values for the three algorithms: MFXDFA, MFXDMA with $θ = 0$ , and MFXDMA with $θ = 1$ .Reuse & Permissions
Figure 4
Multifractal detrended cross-correlation analysis of (a–c) the return time series and (d–f) the volatility time series for the DJIA index and the NASDAQ index. Comparisons are performed among three MFXDMA algorithms with $θ = 0$ , 0.5, and 1 and the MFXDFA method. (a, d) Dependence of the fluctuation functions $F_{x y} (q, s)$ with respect to the scale $s$ for $q = - 4$ , $q = 0$ , and $q = 4$ . The straight lines are the best power-law fits to the data. The results have been translated vertically for better visibility. (b, e) Scaling exponents $h_{x y} (q)$ with respect to $q$ . (c, f) Multifractal spectra $f_{x y} (α)$ with respect to the singularity strength $α$ .Reuse & Permissions

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