Multifractal Height Cross-Correlation Analysis: A New Method for Analyzing Long-Range Cross-Correlations

We introduce a new method for detection of long-range cross-correlations and multifractality – multifractal height cross-correlation analysis (MF-HXA) – based on scaling of qth order covariances. MF-HXA is a bivariate generalization of the height-height correlation analysis of Barabasi & Vicsek [Barabasi, A.L., Vicsek, T.: Multifractality of self-affine fractals, Physical Review A 44(4), 1991]. The method can be used to analyze long-range cross-correlations and multifractality between two simultaneously recorded series. We illustrate a power of the method on both simulated and real-world time series. The research of long-range dependence and multifractality has been growing significantly in recent years with application to a wide range of disciplines [1–10]. Recently, the examination of long-range cross-correlations has become of interest as it provides additional information about the examined processes. Carbone [11] generalized the detrending moving average (DMA) method for higher dimensions. Podobnik & Stanley [12] adjusted the detrended fluctuation analysis for two time series and introduced the detrended cross-correlation analysis (DCCA). Zhou [13] further generalized the method and introduced the multifractal detrended cross-correlation analysis (MFDXA). Jiang & Zhou [14] then implemented moving average filtering to MF-DXA algorithm creating MF-X-DMA. In this paper, we introduce two new methods for an analysis of long-range cross-correlations – the multifractal height cross-correlation analysis (MF-HXA) and its special case of the height cross-correlation analysis (HXA). To analyze long-range cross-correlations, we generalize the q-th order height-height correlation function for two simultaneously recorded series. Let us consider two series Xt and Yt with time resolution ν and t = ν, 2ν, ..., νbTν c, where bc is a lower integer sign. For better legibility, we denote T ∗ = νbTν c, which varies with ν, and we write the τ -order difference as ∆τXt ≡ Xt+τ − Xt and ∆τXtYt ≡ ∆τXt∆τYt. Height-height covariance function is then defined as Kxy,q(τ) = ν T ∗ T∗/ν ∑ t=1 |∆τXtYt| q 2 ≡ 〈|∆τXtYt| q 2 〉 (1) where time interval τ generally ranges between ν = τmin, . . . , τmax. Scaling relationship between Kxy,q(τ) and the generalized bivariate Hurst exponent Hxy(q) is obtained as Kxy,q(τ) ∝ τ xy. (2) For q = 2, the method can be used for the detection of long-range cross-correlations solely and we call it the height cross-correlation analysis (HXA). Obviously, MFHXA reduces to the height-height correlation analysis of Barabasi et al. [15] for Xt = Yt. Note that it makes sense to analyze the scaling according to Eq. 2 only for detrended series Xt and Yt and only for q > 0 [5]. A type of detrending can generally take various forms – polynomial, moving averages and other filtering methods – and is applied for each time resolution ν separately. The bivariate Hurst exponent 0 < Hxy(2) < 1 has similar properties and interpretation as a univariate Hurst exponent. ForHxy(2) > 0.5, the series are cross-persistent so that a positive (a negative) value of ∆Xt∆Yt is more statistically probable to be followed by another positive (negative) value of ∆Xt+1∆Yt+1. Conversely for Hxy(2) < 0.5, p-1 ar X iv :1 20 1. 34 73 v2 [ qfi n. ST ] 1 9 Ja n 20 12 Ladislav Kristoufek1,2 the series are cross-antipersistent so that a positive (a negative) value of ∆Xt∆Yt is more statistically probable to be followed by a negative (a positive) value of ∆Xt+1∆Yt+1. Note that even two pairwise uncorrelated processes can be cross-persistent. The expected values of the bivariate Hurst exponents have been partly discussed in [12–14]. It has been shown that Hxy(q) = Hx(q) +Hy(q) 2 (3) for all q > 0 for pairwise uncorrelated and correlated processes. We present some new insights into this relation. To better understand the behavior of the bivariate Hurst exponent, we use a standard multifractal formalism [16]. Consider processes Xt and Yt are multifractal with generalized Hurst exponents Hx(q) and Hy(q) so that 〈|∆τXt|〉 ∝ τ qHx(q) (4)