We investigate how simultaneously recorded long-range power-law correlated multi-variate signals cross-correlate. To this end we introduce a two-component ARFIMA stochastic process and a two-component FIARCH process to generate coupled fractal signals with long-range power-law correlations which are at the same time long-range cross-correlated. We study how the degree of cross-correlations between these signals depends on the scaling exponents characterizing the fractal correlations in each signal and on the coupling between the signals. Our findings have relevance when studying parallel outputs of multiple-component of physical, physiological and social systems.
Many empirical data are characterized by long-range power-law auto-correlations as well as by long-range cross-correlations. Such scale-invariant organization in both auto-correlations and cross-correlations can be observed either for the data variables or their absolute values [1,2,3,4,5,6,7,8,9]. Scale-invariant power-law auto-correlations in stochastic variables can be modeled by the fractionally autoregressive integrated moving-average process (ARFIMA) [10,11]:
where d ∈ (-0.5, 0.5) is a scaling parameter, ǫ t denotes independent and identically distributed (i.i.d.) Gaussian variables with ǫ t = 0 and ǫ 2 t = 1, a n (d) are the weights defined by a n (d) = d Γ(n -d)/(Γ(1 -d)Γ(n + 1)), where Γ denotes the Gamma function and n is the time scale. We denote the autocorrelation function for x t as A(x t , x t-n ) ≡ A(n). For d = 0 the generated variable x t becomes random.
To account for power-law cross-correlations between two variables x t and y t , where each variable is itself power-law auto-correlated, we propose a twocomponent ARFIMA stochastic process defined by two stochastic variables x t and y t . Each of these variables at any time depends not only on its own past values but also on past values of the other variable:
where ǫ t and ǫt denote i.i.d. Gaussian variables with ǫ t = ǫt = 0 and ǫ 2 t = ǫ2 t = 1, a n (d 1 ) and a n (d 2 ) are the weights defined in Eq. ( 1) through the scaling parameters d 1 and d 2 (0 ≤ d 1,2 < 0.5), and W is a free parameter controlling the coupling strength between x t and y t (0.5 ≤ W ≤ 1). We denote the cross-correlation function between x t and y t as C(x t , y t-n ) ≡ C(n). For different values of W a different degree of cross-correlation between the variables x t and y t is observed. For example, for the case when W = 1, the process defined in Eqs. (2a)-(2d) reduces to two decoupled ARFIMA processes defined in Eq. ( 1). Thus, when W = 1 the long-range cross-correlations between x t and y t vanish, while both x t and y t remain long-range power-law auto-correlated.
In Fig. 1(a) we show segments of the time series x t and y t generated by the process defined in Eq. (2a)-(2d) with parameters W = 0.8 and d 1 = d 2 = 0.4. Both variables exhibit a very similar comovement. In Fig. 1(b) we show the auto-correlation functions A(n) for x t and y t , as well as the cross-correlation function C(x t , y t-n ) ≡ C(n). These three curves practically overlap [Fig. 1 parameters d 1 and d 2 .
Motivated by the fact that for linear processes the auto-correlation function does not change under randomization of the Fourier phase [13,14] how this phase-randomization procedure affects the degree of cross-correlation between x t and y t . First, we perform a Fourier transform of the original time series, e.g. x t , preserving the Fourier amplitudes but randomizing the Fourier phases. Then, we perform an inverse Fourier transform and obtain a surrogate (linearized) time series xt . Applying this phase-randomization procedure to both time series x t and y t generated by the two-component ARFIMA process in Eq. ( 2), we calculate the two auto-correlation functions for xt and ỹt , as well as their cross-correlation function C(x t , ỹt-n ). As expected, the autocorrelation functions remain unchanged after Fourier phase randomization, but the cross-correlation function C(x t , ỹt-n ) completely vanishes [Fig. 2].
Next, we investigate the case when the scaling parameters d 1 and d 2 are fixed, while the coupling parameter W varies. In Fig. 3
α=0.6 α=0.9
Fig. 4. DFA scaling curves for the time series x t and y t generated by the twocomponent ARFIMA process in Eqs. (2a)-(2d), where d 1 = 0.4 and d 2 = 0.1. For W = 1, x t and y t are decoupled and thus not cross-correlated, and x t behaves as the ARFIMA process in Eq. ( 1) defined only by the scaling parameter d 1 , while y t becomes a separate ARFIMA process defined only by the scaling parameter d 2 .
For W = 1, the scaling properties of x t depend on both parameters d 1 and d 2 .
When W = 0.5, the DFA correlation exponent α for x t becomes equal to the DFA correlation exponent for y t . The DFA exponent for |y t | does not depend on W .
the processes x t and y t are decoupled and thus not cross-correlated. In this case, x t behaves as a power-law auto-correlated ARFIMA process controlled by only the scaling parameter d 1 , with the DFA correlation exponent equals α = 0.5 + d 1 = 0.9. Similarly, y t becomes a separate ARFIMA process (decoupled from x t ) which is controlled only by the scaling parameter d 2 , where α = 0.5 + d 2 = 0.6. We find that with decreasing value of W (from 1 to 0.5), x t becomes a mixture of two ARFIMA processes and the DFA correlation exponent α gradually decreases towards α = 0.6 corresponding to the y t process, controlled by parameter d 2 = 0.1. In contrast to x t , for the process y t the DFA correlation exponent α virtually does not change with varying the coupling parameter W .
We next consider a separat
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