Boris Podobnik

Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series.

Here we propose a new method, detrended cross-correlation analysis, which is a generalization of detrended fluctuation analysis and is based on detrended covariance. This method is designed to investigate power-law cross correlations between different simultaneously recorded time series in the presence of nonstationarity. We illustrate the method by selected examples from physics, physiology, and finance.

Cross-correlations between volume change and price change

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 rate R̃, 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 |R̃|, and their relationship to price changes |R|. We analyze 14,981 daily recordings of the Standard and Poors (S & P) 500 Index over the 59-year period 1950–2009, and find power-law cross-correlations between |R| and |R̃| by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between |R| and |R̃|, we estimate the tail exponent α̃ of the probability density function P(|R̃|) ∼ |R̃|−1−α̃ 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 estimate α̃, we calculate the time intervals τq between events where R̃ > q. We demonstrate that τ̃q, the average of τq, obeys τ̃q ∼ qα̃. We find α̃ ≈ 3. Furthermore, by aggregating all τq values of 28 global financial indices, we also observe an approximate inverse cubic law.

Detrended cross-correlation analysis for non-stationary time series with periodic trends

Noisy signals in many real-world systems display long-range autocorrelations and long-range cross-correlations. Due to periodic trends, these correlations are difficult to quantify. We demonstrate that one can accurately quantify power-law cross-correlations between different simultaneously recorded time series in the presence of highly non-stationary sinusoidal and polynomial overlying trends by using the new technique of detrended cross-correlation analysis with varying order l of the polynomial. To demonstrate the utility of this new method —which we call DCCA-l(n), where n denotes the scale— we apply it to meteorological data.

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 long-range 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.

Calling patterns in human communication dynamics

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 cr-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.

Common scaling patterns in intertrade times of U. S. stocks

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.

Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multiscale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the U.S. dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the multifractal DCCA method fails.

Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes

We investigate how simultaneously recorded long-range power-law correlated multivariate 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.

Linking agent-based models and stochastic models of financial markets

It is well-known that financial asset returns exhibit fat-tailed distributions and long-term memory. These empirical features are the main objectives of modeling efforts using (i) stochastic processes to quantitatively reproduce these features and (ii) agent-based simulations to understand the underlying microscopic interactions. After reviewing selected empirical and theoretical evidence documenting the behavior of traders, we construct an agent-based model to quantitatively demonstrate that “fat” tails in return distributions arise when traders share similar technical trading strategies and decisions. Extending our behavioral model to a stochastic model, we derive and explain a set of quantitative scaling relations of long-term memory from the empirical behavior of individual market participants. Our analysis provides a behavioral interpretation of the long-term memory of absolute and squared price returns: They are directly linked to the way investors evaluate their investments by applying technical strategies at different investment horizons, and this quantitative relationship is in agreement with empirical findings. Our approach provides a possible behavioral explanation for stochastic models for financial systems in general and provides a method to parameterize such models from market data rather than from statistical fitting.

Influence of corruption on economic growth rate and foreign investment

We analyze the dependence of the Gross Domestic Product (GDP) per capita growth rates on changes in the Corruption Perceptions Index (CPI). For the period 1999–2004 for all countries in the world, we find on average that an increase of CPI by one unit leads to an increase of the annual GDP per capita growth rate by 1.7%. By regressing only the European countries with transition economies, we find that an increase of CPI by one unit generates an increase of the annual GDP per capita growth rate by 2.4%. We also analyze the relation between foreign direct investments received by different countries and CPI, and we find a statistically significant power-law functional dependence between foreign direct investment per capita and the country corruption level measured by the CPI. We introduce a new measure to quantify the relative corruption between countries based on their respective wealth as measured by GDP per capita.