Time-lag cross-correlations in collective phenomena

We study long-range magnitude cross-correlations in collective modes of real-world data from finance, physiology, and genomics using time-lag random matrix theory. We find longrange magnitude cross-correlations i) in time series of price fluctuations, ii) in physiological time series, both healthy and pathological, indicating scale-invariant interactions between different physiological time series, and iii) in ChIP-seq data of the mouse genome, where we uncover a complex interplay of different DNA-binding proteins, resulting in power-law cross-correlations in xij , the probability that protein i binds to gene j, ranging up to 10 million base pairs. In finance, we find that the changes in singular vectors and singular values are largest in times of crisis. We find that the largest 500 singular values of the NYSE Composite members follow a Zipf distribution with exponent ≈ 2. In physiology, we find statistically significant differences between alcoholic and control subjects. Copyright c © EPLA, 2010 Introduction. – Many complex systems are part of even larger systems where the constituent complex systems mutually interact [1–3], giving rise to the appearance of “collective modes” [4–7]. Stochastic interactions among related systems are reflected by the presence of cross-correlations, and here we address the question of whether these cross-correlations in the collective modes exhibit power-law scale-invariant properties. Zero-lag cross-correlations in the collective modes of empirical time series were analyzed by using random matrix theory (RMT) [6–9]. Recently, RMT became very successful in the analysis of cross-correlations between stock price changes, since cross-correlation matrices and associated covariance matrices play important roles in portfolio management [10,11]. A variety of studies reported the properties of the cross-correlation matrix C of price changes [6–14]. RMT enables a comparison between the cross-correlation matrix obtained from N empirical time series each of length T and a perfectly random matrix W , called a Wishart matrix, obtained from N mutually uncorrelated time series each of length T [15]. By analyzing cross-correlations between price changes of the members of the S&P 500 index, it was found that 98% of the eigenvalue spectrum of the correlation 1Heartbeat interval time series, e.g., is one among many time series comprising the functioning human. matrix C follows the Gaussian orthogonal ensemble of a perfectly random matrix [7,8]. Recently, time-lag generalizations of RMT were proposed [16–21]. However, only short-range crosscorrelations were found. To quantify long-range collective movements in correlated data sets, we apply time-lag RMT (TLRMT) to the magnitude of three selected examples of real-world data: i) finance, ii) physiology, and iii) genomics. Consider the N -variable time series X = {Xi,t : i= 1, . . . , N ; t= 1, . . . , T} of length T , where i indexes the series number, and t denotes the time. The crosscorrelation matrices for this time series and for the magnitude time series are Cij(∆t)≡ 〈Xi,tXj,t+∆t〉− 〈Xi,t〉〈Xj,t+∆t〉 σiσj , (1) C̃ij(∆t)≡ 〈|Xi,t||Xj,t+∆t|〉− 〈|Xi,t|〉〈|Xj,t+∆t|〉 σ̃iσ̃j . (2) Here σi, σj , σ̃i, and σ̃j denote the standard deviations of Xi,t, Xj,t+∆t, |Xi,t|, and |Xj,t+∆t|, respectively, and 〈. . .〉 denotes the time average. In order to quantify cross-correlations for varying lags ∆t, we compute the largest singular values λL(∆t) and λ̃L(∆t) of the cross-correlation matrices C(∆t) and C̃(∆t)