Quantifying cross-correlations using local and global detrending approaches

In order to quantify the long-range cross-correlations between two time series qualitatively, we introduce a new cross-correlations test QCC(m), where m is the number of degrees of freedom. If there are no cross-correlations between two time series, the cross-correlation test agrees well with the χ(m) distribution. If the cross-correlations test exceeds the critical value of the χ(m) distribution, then we say that the cross-correlations are significant. We show that if a Fourier phase-randomization procedure is carried out on a power-law cross-correlated time series, the cross-correlations test is substantially reduced compared to the case before Fourier phase randomization. We also study the effect of periodic trends on systems with power-law cross-correlations. We find that periodic trends can severely affect the quantitative analysis of long-range correlations, leading to crossovers and other spurious deviations from power laws, implying both local and global detrending approaches should be applied to properly uncover long-range power-law auto-correlations and cross-correlations in the random part of the underlying stochastic process. PACS. 05.45.Tp Time series analysis – 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion There are a number of situations where different signals exhibit cross-correlations, ranging from geophysics [1] to finance [2–14] and solid-state physics [15]. Cross-correlation functions together with auto-correlation functions are commonly used to gain insight into the dynamics of natural systems. By their definitions, these techniques should be employed only in the presence of stationarity. However, it is an important fact that many time series of physical, biological, hydrological, and social systems are non-stationary and exhibit long-range power-law correlations [16–22]. In practice, statistical properties of these systems are difficult to study due to these nonstationarities. For determining the scaling exponent of a long-range power-law auto-correlated time series in the presence of nonstationarities, the detrended fluctuation analysis (DFA) method has been developed [23] and its performance has been systematically tested for the effect of different types of trends and nonstationarities [24–27] as encountered in a wide range of different fields, such as a e-mail: [email protected] cardiac dynamics [28], economics [29], DNA analysis [30], and meteorology [31]. The square root of the detrended variance grows with time scale n as FDFA(n) ∼ nDFA , where λDFA is the DFA scaling exponent [23–26], where 1/2 < λDFA < 1, indicates the presence of power-law auto-correlations, and 0 < λDFA < 1/2 indicates the presence of long-range power-law anti-correlations. There are many realistic situations in which one desires to quantify cross-correlations between two non-stationary time series. Examples include blood pressure and heart rate [32], air temperature and air humidity, and the temporal expression data of different genes. To quantify power-law cross-correlations in non-stationary time series, a new method based on detrended covariance, called detrended cross-correlations analysis (DCCA), has been recently proposed [11]. If cross-correlations decay as a power law, the corresponding detrended covariances are either always positive or always negative, and the square root of the detrended covariance grows with time scale n as FDCCA(n) ∝ nDCCA , (1) 244 The European Physical Journal B where λDCCA is the DCCA cross-correlation exponent. If, however, the detrended covariance oscillates around zero as a function of the time scale n, there are no long-range cross-correlations. In order to investigate power-law auto-correlations and power-law cross-correlations and effects of sinusoidal periodicity on cross-correlations, first we define a periodic twocomponent fractionally autoregressive integrated movingaverage (ARFIMA) process [33–37], where each variable depends not only on its own past, but also on the past values of the other variable,