Here we propose a method, based on detrended covariance which we call detrended cross-correlation analysis (DXA), to investigate power-law cross-correlations between different simultaneously-recorded time series in the presence of non-stationarity. We illustrate the method by selected examples from physics, physiology, and finance.

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In finance, one usually deals not with prices but with growth rates R, defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rateR, the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes |R|, and their relationship to price changes |R|. We analyze 14, 981 daily recordings of the Standard and Poor's (S & P) 500 Index over the 59-year period 1950-2009, and find power-law cross-correlations between |R| and |R| by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these crosscorrelations. Motivated by the relationship between |R| and |R|, we estimate the tail exponentα of the probability density function P(|R|) ∼ |R| −1−α for both the S & P 500 Index as well as the collection of 1819 constituents of the New York Stock Exchange Composite Index on 17 July 2009. As a new method to estimateα, we calculate the time intervals τ q between events whereR > q. We demonstrate thatτ q , the average of τ q , obeysτ q ∼ qα. We findα ≈ 3. Furthermore, by aggregating all τ q values of 28 global financial indices, we also observe an approximate inverse cubic law. econophysics | finance | volatility There is a saying on Wall Street that "it takes volume to move stock prices." A number of studies have analyzed the relationship between price changes and the trading volume in financial markets (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). Some of these studies (1, 3-6) have found a positive relationship between price change and the trading volume. In order to explain this relationship, Clarke assumed that the daily price change is the sum of a random number of uncorrelated intraday price changes (3), so predicted that the variance of the daily price change is proportional to the average number of daily transactions. If the number of transactions is proportional to the trading volume, then the trading volume is proportional to the variance of the daily price change.The cumulative distribution function (cdf) of the absolute logarithmic price change |R| obeys a power lawIt is believed (15-18) that α ≈ 3 ("inverse cubic law"), outside the range α < 2 characterizing a Lévy distribution (18,19). A parallel analysis of Q, the volume traded, yields a power law (20-28)To our knowledge, the logarithmic volume change-R and its relation to the logarithmic price change R-has not been analyzed, and this analysis is our focus here. For each of the 1,819 companies and 28 indices, we calculate over the time interval of one day the logarithmic change in price S(t), Data Analyzedand also the logarithmic change in trading volume Q(t) (29),For each of the 3,694 time series, we also calculate the absolute values |R t | and |R t | and define the "price volatility" (30) and "volume volatility," respectively,andwhere σ R ≡ ( |R t | 2 − |R t | 2 ) 1/2 and σR ≡ ( |R t | 2 − |R t | 2 ) 1/2 are the respective standard deviat...

We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U.S. economy over a period of 4 yrs. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution, (ii) when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form, (iii) the intertrade times exhibit power-law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales, in agreement with the correlation behavior of the absolute price returns for the corresponding company, and (iv) the magnitude series of intertrade time increments is characterized by long-range power-law correlations suggesting the presence of nonlinear features in the trading dynamics, while the sign series is anticorrelated at small scales. Our results suggest that independent of industry sector, market capitalization and average level of trading activity, the series of intertrade times exhibit possibly universal scaling patterns, which may relate to a common mechanism underlying the trading dynamics of diverse companies. Further, our observation of long-range power-law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock, support the hypothesis that the dynamics of transaction times may play a role in the process of price formation.

For stationary time series, the cross-covariance and the cross-correlation as functions of time lag n serve to quantify the similarity of two time series. The latter measure is also used to assess whether the cross-correlations are statistically significant. For nonstationary time series, the analogous measures are detrended cross-correlations analysis (DCCA) and the recently proposed detrended cross-correlation coefficient, ρ(DCCA)(T,n), where T is the total length of the time series and n the window size. For ρ(DCCA)(T,n), we numerically calculated the Cauchy inequality -1 ≤ ρ(DCCA)(T,n) ≤ 1. Here we derive -1 ≤ ρ DCCA)(T,n) ≤ 1 for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of ρ(DCCA) within which the cross-correlations become statistically significant. For overlapping windows we numerically determine-and for nonoverlapping windows we derive--that the standard deviation of ρ(DCCA)(T,n) tends with increasing T to 1/T. Using ρ(DCCA)(T,n) we show that the Chinese financial market's tendency to follow the U.S. market is extremely weak. We also propose an additional statistical test that can be used to quantify the existence of cross-correlations between two power-law correlated time series.

Modern technologies not only provide a variety of communication modes (e.g., texting, cell phone conversation, and online instant messaging), but also detailed electronic traces of these communications between individuals. These electronic traces indicate that the interactions occur in temporal bursts. Here, we study intercall duration of communications of the 100,000 most active cell phone users of a Chinese mobile phone operator. We confirm that the intercall durations follow a power-law distribution with an exponential cutoff at the population level but find differences when focusing on individual users. We apply statistical tests at the individual level and find that the intercall durations follow a power-law distribution for only 3,460 individuals (3.46%). The intercall durations for the majority (73.34%) follow a Weibull distribution. We quantify individual users using three measures: out-degree, percentage of outgoing calls, and communication diversity. We find that the cell phone users with a power-law duration distribution fall into three anomalous clusters: robot-based callers, telecom fraud, and telephone sales. This information is of interest to both academics and practitioners, mobile telecom operators in particular. In contrast, the individual users with a Weibull duration distribution form the fourth cluster of ordinary cell phone users. We also discover more information about the calling patterns of these four clusters (e.g., the probability that a user will call the c r -th most contact and the probability distribution of burst sizes). Our findings may enable a more detailed analysis of the huge body of data contained in the logs of massive users.human dynamics | phone user categorization | social science | nonlinear dynamics | social networks U nderstanding the temporal patterns of individual human interactions is essential in managing information spreading and in tracking social contagion. Human interactions (e.g., cell phone conversations and e-mails) leave electronic traces that allow the tracking of human interactions from the perspective of either static complex networks (1-6) or human dynamics (7). Because static networks only describe sequences of instantaneous interacting links, temporal networks in which the temporal patterns of interacting activities for each node are recorded have recently received a considerable amount of research interest (8, 9). Investigations of interevent intervals between two consecutive interacting actions, such as e-mail communications (7, 10), shortmessage correspondences (11-13), cell phone conservations (14, 15), and letter correspondences (16-18), indicate that human interactions have non-Poissonian characteristics. Previous studies were conducted either on aggregate samples (14,15,19) or on a small group of selected individuals (7,(10)(11)(12)(16)(17)(18), but the communication behavior of individuals is not well understood.We study the complete voice information for cell phone users supplied by a Chinese cell phone operator and study the interevent time...

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PACS 89.75.Da -Systems obeying scaling laws PACS 89.90.+n -Other topics in areas of applied and interdisciplinary physics Abstract -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.