Eur. Phys. J. B 71, 243-250 (2009)
https://doi.org/10.1140/epjb/e2009-00310-5

Quantifying cross-correlations using local and global detrending approaches

¹ Faculty of Civil Engineering, University of Rijeka, 51001 Rijeka, Croatia
² Zagreb School of Economics and Management, 10000 Zagreb, Croatia
³ Center for Polymer Studies and Department of Physics, Boston University, Boston, MA, 02215, USA
⁴ Institute of Computer Science, Martin Luther University, Halle, Germany
⁵ Physics Department, Faculty of Science, University of Zagreb, Bijenička c. 32, 10000 Zagreb, Croatia
⁶ Lancaster Environment Centre, Lancaster University, Farrer Avenue, Lancaster, LA1 4YQ, UK
⁷ Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
⁸ Harvard Medical School and Division of Sleep Medicine, Brigham and Women's Hospital, Boston, MA, 02115, USA

Corresponding author: ^a bp@phy.hr

Received: 18 May 2009
Revised: 27 July 2009
Published online: 17 September 2009

Abstract

In order to quantify the long-range cross-correlations between two time series qualitatively, we introduce a new cross-correlations test Q_CC(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 χ²(m) distribution. If the cross-correlations test exceeds the critical value of the χ²(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.