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

How high frequency trading affects a market index

The relationship between a market index and its constituent stocks is complicated. While an index is a weighted average of its constituent stocks, when the investigated time scale is one day or longer the index has been found to have a stronger effect on the stocks than vice versa. We explore how this interaction changes in short time scales using high frequency data. Using a correlation-based analysis approach, we find that in short time scales stocks have a stronger influence on the index. These findings have implications for high frequency trading and suggest that the price of an index should be published on shorter time scales, as close as possible to those of the actual transaction time scale.

Stochastic nature of series of waiting times

Although fluctuations in the waiting time series have been studied for a long time, some important issues such as its long-range memory and its stochastic features in the presence of nonstationarity have so far remained unstudied. Here we find that the "waiting times" series for a given increment level have long-range correlations with Hurst exponents belonging to the interval 1/2<H<1. We also study positive-negative level asymmetry of the waiting time distribution. We find that the logarithmic difference of waiting times series has a short-range correlation, and then we study its stochastic nature using the Markovian method and determine the corresponding Kramers-Moyal coefficients. As an example, we analyze the velocity fluctuations in high Reynolds number turbulence and determine the level dependence of Markov time scales, as well as the drift and diffusion coefficients. We show that the waiting time distributions exhibit power law tails, and we were able to model the distribution with a continuous time random walk.

Removal of visual feedback lowers structural variability of inter-digit force coordination during sustained precision pinch

This study examined the effects of visual feedback on inter-digit force coordination during a precision pinch. Sixteen healthy, right-handed subjects were instructed to pinch an instrumented apparatus for 1 min with a stable force output. Visual feedback was provided for the first 30s and withdrawn for the second 30s. Detrended fluctuation analysis (DFA) and detrended cross-correlation analysis (DCCA) methods were used to quantify the time-dependent structures of each digit's force and of the force correlation between the digits. After removing visual feedback, the DFA scaling exponent, αDFA, increased from 1.10±0.12 to 1.29±0.13 for the thumb and from 0.95±0.08 to 1.33±0.13 for the index finger (F1,95=372.47, p<0.001); the DCCA scaling exponent, αDCCA, increased from 1.00±0.08 to 1.33±0.13 (t95=20.33, p<0.001). Structural changes were observed beginning with the first 5s epoch after the removal of visual feedback. The results provide evidence that removing visual feedback lowers the structural variability of inter-digit force coordination. This change is reflected in the high-level control strategy, resulting in the two digits being more tightly coupled under somatosensory feedback without visual inputs.

Microstructure identification via detrended fluctuation analysis of ultrasound signals

We describe an algorithm for simulating ultrasound propagation in random one-dimensional media, mimicking different microstructures by choosing physical properties such as domain sizes and mass densities from probability distributions. By combining a detrended fluctuation analysis (DFA) of the simulated ultrasound signals with tools from the pattern-recognition literature, we build a Gaussian classifier which is able to associate each ultrasound signal with its corresponding microstructure with a very high success rate. Furthermore, we also show that DFA data can be used to train a multilayer perceptron which estimates numerical values of physical properties associated with distinct microstructures.

Cross-correlation detection and analysis for California's electricity market based on analogous multifractal analysis

A novel method, which we called the analogous multifractal cross-correlation analysis, is proposed in this paper to study the multifractal behavior in the power-law cross-correlation between price and load in California electricity market. In addition, a statistic ρAMF-XA, which we call the analogous multifractal cross-correlation coefficient, is defined to test whether the cross-correlation between two given signals is genuine or not. Our analysis finds that both the price and load time series in California electricity market express multifractal nature. While, as indicated by the ρAMF-XA statistical test, there is a huge difference in the cross-correlation behavior between the years 1999 and 2000 in California electricity markets.

Two-dimensional matrix algorithm using detrended fluctuation analysis to distinguish Burkitt and diffuse large B-cell lymphoma

A detrended fluctuation analysis (DFA) method is applied to image analysis. The 2-dimensional (2D) DFA algorithms is proposed for recharacterizing images of lymph sections. Due to Burkitt lymphoma (BL) and diffuse large B-cell lymphoma (DLBCL), there is a significant different 5-year survival rates after multiagent chemotherapy. Therefore, distinguishing the difference between BL and DLBCL is very important. In this study, eighteen BL images were classified as group A, which have one to five cytogenetic changes. Ten BL images were classified as group B, which have more than five cytogenetic changes. Both groups A and B BLs are aggressive lymphomas, which grow very fast and require more intensive chemotherapy. Finally, ten DLBCL images were classified as group C. The short-term correlation exponent α1 values of DFA of groups A, B, and C were 0.370 ± 0.033, 0.382 ± 0.022, and 0.435 ± 0.053, respectively. It was found that α1 value of BL image was significantly lower (P < 0.05) than DLBCL. However, there is no difference between the groups A and B BLs. Hence, it can be concluded that α1 value based on DFA statistics concept can clearly distinguish BL and DLBCL image.

Measuring information interactions on the ordinal pattern of stock time series

The interactions among time series as individual components of complex systems can be quantified by measuring to what extent they exchange information among each other. In many applications, one focuses not on the original series but on its ordinal pattern. In such cases, trivial noises appear more likely to be filtered and the abrupt influence of extreme values can be weakened. Cross-sample entropy and inner composition alignment have been introduced as prominent methods to estimate the information interactions of complex systems. In this paper, we modify both methods to detect the interactions among the ordinal pattern of stock return and volatility series, and we try to uncover the information exchanges across sectors in Chinese stock markets.

Performance of multifractal detrended fluctuation analysis on short time series

The performance of the multifractal detrended analysis on short time series is evaluated for synthetic samples of several mono- and multifractal models. The reconstruction of the generalized Hurst exponents is used to determine the range of applicability of the method and the precision of its results as a function of the decreasing length of the series. As an application the series of the daily exchange rate between the U.S. dollar and the euro is studied.

Contrarian behavior in a complex adaptive system

Contrarian behavior is a kind of self-organization in complex adaptive systems (CASs). Here we report the existence of a transition point in a model resource-allocation CAS with contrarian behavior by using human experiments, computer simulations, and theoretical analysis. The resource ratio and system predictability serve as the tuning parameter and order parameter, respectively. The transition point helps to reveal the positive or negative role of contrarian behavior. This finding is in contrast to the common belief that contrarian behavior always has a positive role in resource allocation, say, stabilizing resource allocation by shrinking the redundancy or the lack of resources. It is further shown that resource allocation can be optimized at the transition point by adding an appropriate size of contrarians. This work is also expected to be of value to some other fields ranging from management and social science to ecology and evolution.

Evaluation of scale invariance in physiological signals by means of balanced estimation of diffusion entropy

By means of the concept of the balanced estimation of diffusion entropy, we evaluate the reliable scale invariance embedded in different sleep stages and stride records. Segments corresponding to waking, light sleep, rapid eye movement (REM) sleep, and deep sleep stages are extracted from long-term electroencephalogram signals. For each stage the scaling exponent value is distributed over a considerably wide range, which tell us that the scaling behavior is subject and sleep cycle dependent. The average of the scaling exponent values for waking segments is almost the same as that for REM segments (∼0.8). The waking and REM stages have a significantly higher value of the average scaling exponent than that for light sleep stages (∼0.7). For the stride series, the original diffusion entropy (DE) and the balanced estimation of diffusion entropy (BEDE) give almost the same results for detrended series. The evolutions of local scaling invariance show that the physiological states change abruptly, although in the experiments great efforts have been made to keep conditions unchanged. The global behavior of a single physiological signal may lose rich information on physiological states. Methodologically, the BEDE can evaluate with considerable precision the scale invariance in very short time series (∼10^{2}), while the original DE method sometimes may underestimate scale-invariance exponents or even fail in detecting scale-invariant behavior. The BEDE method is sensitive to trends in time series. The existence of trends may lead to an unreasonably high value of the scaling exponent and consequent mistaken conclusions.

Detection of a long-range correlation with an adaptive detrending method

We propose a methodology of estimating the scaling exponent for a long-range correlation in a nonstationary time series from the perspective of the regression analysis. By an adaptive degree determination of a regression polynomial, the proposed methodology is designed to properly remove various types of trends embedded in the nonstationary signal so that the scaling exponent can be estimated without artificial crossovers. To show the validity of the proposed methodology, we applied it to the detrended fluctuation analysis and tested it out against correlated data superimposed by various types of trends. It turned out that, unlike the conventional technique, our approach was capable of eliminating artificial crossovers. We also discuss the statistical characteristics of the proposed method with regard to the estimation of the scaling exponent.

Scaling of seismic memory with earthquake size

It has been observed that discrete earthquake events possess memory, i.e., that events occurring in a particular location are dependent on the history of that location. We conduct an analysis to see whether continuous real-time data also display a similar memory and, if so, whether such autocorrelations depend on the size of earthquakes within close spatiotemporal proximity. We analyze the seismic wave form database recorded by 64 stations in Japan, including the 2011 "Great East Japan Earthquake," one of the five most powerful earthquakes ever recorded, which resulted in a tsunami and devastating nuclear accidents. We explore the question of seismic memory through use of mean conditional intervals and detrended fluctuation analysis (DFA). We find that the wave form sign series show power-law anticorrelations while the interval series show power-law correlations. We find size dependence in earthquake autocorrelations: as the earthquake size increases, both of these correlation behaviors strengthen. We also find that the DFA scaling exponent α has no dependence on the earthquake hypocenter depth or epicentral distance.

Statistical tests for power-law cross-correlated processes

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.

Modeling asset price processes based on mean-field framework

We propose a model of the dynamics of financial assets based on the mean-field framework. This framework allows us to construct a model which includes the interaction among the financial assets reflecting the market structure. Our study is on the cutting edge in the sense of a microscopic approach to modeling the financial market. To demonstrate the effectiveness of our model concretely, we provide a case study, which is the pricing problem of the European call option with short-time memory noise.

Asymmetric Lévy flight in financial ratios

Because financial crises are characterized by dangerous rare events that occur more frequently than those predicted by models with finite variances, we investigate the underlying stochastic process generating these events. In the 1960s Mandelbrot [Mandelbrot B (1963) J Bus 36:394-419] and Fama [Fama EF (1965) J Bus 38:34-105] proposed a symmetric Lévy probability distribution function (PDF) to describe the stochastic properties of commodity changes and price changes. We find that an asymmetric Lévy PDF, L, characterized by infinite variance, models several multiple credit ratios used in financial accounting to quantify a firm's financial health, such as the Altman [Altman EI (1968) J Financ 23:589-609] Z score and the Zmijewski [Zmijewski ME (1984) J Accounting Res 22:59-82] score, and models changes of individual financial ratios, ΔX(i). We thus find that Lévy PDFs describe both the static and dynamics of credit ratings. We find that for the majority of ratios, ΔX(i) scales with the Lévy parameter α ≈ 1, even though only a few of the individual ratios are characterized by a PDF with power-law tails X(i)(-1-α) with infinite variance. We also find that α exhibits a striking stability over time. A key element in estimating credit losses is the distribution of credit rating changes, the functional form of which is unknown for alphabetical ratings. For continuous credit ratings, the Altman Z score, we find that P(ΔZ) follows a Lévy PDF with power-law exponent α ≈ 1, consistent with changes of individual financial ratios. Estimating the conditional P(ΔZ|Z) versus Z, we demonstrate how this continuous credit rating approach and its dynamics can be used to evaluate credit risk.

Multiscale rescaled range analysis of EEG recordings in sevoflurane anesthesia

OBJECTIVE

The Hurst exponent (HE) is a nonlinear method measuring the smoothness of a fractal time series. In this study we applied the HE index, extracted from electroencephalographic (EEG) recordings, as a measure of anesthetic drug effects on brain activity.

METHODS

In 19 adult patients undergoing sevoflurane general anesthesia, we calculated the HE of the raw EEG; comparing the maximal overlap discrete wavelet transform (MODWT) with the traditional rescaled range (R/S) analysis techniques, and with a commercial index of depth of anesthesia - the response entropy (RE). We analyzed each wavelet-decomposed sub-band as well as the combined low frequency bands (HEOLFBs). The methods were compared in regard to pharmacokinetic/pharmacodynamic (PK/PD) modeling, and prediction probability.

RESULTS

All the low frequency band HE indices decreased when anesthesia deepened. However the HEOLFB was the best index because: it was less sensitive to artifacts, most closely tracked the exact point of loss of consciousness, showed a better prediction probability in separating the awake and unconscious states, and tracked sevoflurane concentration better - as estimated by the PK/PD models.

CONCLUSIONS

The HE is a useful measure for estimating the depth of anesthesia. It was noted that HEOLFB showed the best performance for tracking drug effect.

SIGNIFICANCE

The HEOLFB could be used as an index for accurately estimating the effect of anesthesia on brain activity.

Financial factor influence on scaling and memory of trading volume in stock market

We study the daily trading volume volatility of 17,197 stocks in the US stock markets during the period 1989-2008 and analyze the time return intervals τ between volume volatilities above a given threshold q. For different thresholds q, the probability density function P(q)(τ) scales with mean interval 〈τ〉 as P(q)(τ)=〈τ〉(-1)f(τ/〈τ〉), and the tails of the scaling function can be well approximated by a power law f(x)∼x(-γ). We also study the relation between the form of the distribution function P(q)(τ) and several financial factors: stock lifetime, market capitalization, volume, and trading value. We find a systematic tendency of P(q)(τ) associated with these factors, suggesting a multiscaling feature in the volume return intervals. We analyze the conditional probability P(q)(τ|τ(0)) for τ following a certain interval τ(0), and find that P(q)(τ|τ(0)) depends on τ(0) such that immediately following a short (long) return interval a second short (long) return interval tends to occur. We also find indications that there is a long-term correlation in the daily volume volatility. We compare our results to those found earlier for price volatility.

Self-similarity of higher-order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).

Coupling detrended fluctuation analysis for analyzing coupled nonstationary signals

When many variables are coupled to each other, a single case study could not give us thorough and precise information. When these time series are stationary, different methods of random matrix analysis and complex networks can be used. But, in nonstationary cases, the multifractal-detrended-cross-correlation-analysis (MF-DXA) method was introduced for just two coupled time series. In this article, we have extended the MF-DXA to the method of coupling detrended fluctuation analysis (CDFA) for the case when more than two series are correlated to each other. Here, we have calculated the multifractal properties of the coupled time series, and by comparing CDFA results of the original series with those of the shuffled and surrogate series, we can estimate the source of multifractality and the extent to which our series are coupled to each other. We illustrate the method by selected examples from air pollution and foreign exchange rates.

Multifractal detrending moving-average cross-correlation analysis

There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross correlations. The multifractal detrended cross-correlation analysis (MFDCCA) approaches can be used to quantify such cross correlations, such as the MFDCCA based on the detrended fluctuation analysis (MFXDFA) method. We develop in this work a class of MFDCCA algorithms based on the detrending moving-average analysis, called MFXDMA. The performances of the proposed MFXDMA algorithms are compared with the MFXDFA method by extensive numerical experiments on pairs of time series generated from bivariate fractional Brownian motions, two-component autoregressive fractionally integrated moving-average processes, and binomial measures, which have theoretical expressions of the multifractal nature. In all cases, the scaling exponents h(xy) extracted from the MFXDMA and MFXDFA algorithms are very close to the theoretical values. For bivariate fractional Brownian motions, the scaling exponent of the cross correlation is independent of the cross-correlation coefficient between two time series, and the MFXDFA and centered MFXDMA algorithms have comparative performances, which outperform the forward and backward MFXDMA algorithms. For two-component autoregressive fractionally integrated moving-average processes, we also find that the MFXDFA and centered MFXDMA algorithms have comparative performances, while the forward and backward MFXDMA algorithms perform slightly worse. For binomial measures, the forward MFXDMA algorithm exhibits the best performance, the centered MFXDMA algorithms performs worst, and the backward MFXDMA algorithm outperforms the MFXDFA algorithm when the moment order q<0 and underperforms when q>0. We apply these algorithms to the return time series of two stock market indexes and to their volatilities. For the returns, the centered MFXDMA algorithm gives the best estimates of h(xy)(q) since its h(xy)(2) is closest to 0.5, as expected, and the MFXDFA algorithm has the second best performance. For the volatilities, the forward and backward MFXDMA algorithms give similar results, while the centered MFXDMA and the MFXDFA algorithms fail to extract rational multifractal nature.

Revealing competitive behaviours in music by means of the multifractal detrended fluctuation analysis: application to Bach's Sinfonias

The one-, two- and three-dimensional multifractal detrended fluctuation analysis (MF-DFA) was applied to Bach's Sinfonias, which are characterized by the superposition of three different voices. Each voice, represented as a time series, can be considered as a component of a one-, two- or three-dimensional vector. The one-dimensional MF-DFA was applied to any single voice, while the two- and three-dimensional MF-DFA was applied to the couples of voices and to the triple, respectively. Each voice is characterized by a multifractal degree (MD), indicated by the range of the generalized Hurst exponents; the higher the MD, the larger the amount of heterogeneity and irregularity. Competitive scaling multifractal behaviours in Bach's Sinfonias were revealed; although one (or two) voices showed a relatively high MD, the other two voices, or voice, are characterized by a low MD. Nevertheless, the overall effect of the Sinfonia, measured by the MD of the triple, tends towards homogeneity, or at least to an average between the different competitive scaling behaviour shown by the different voices.

Index cohesive force analysis reveals that the US market became prone to systemic collapses since 2002

Background

The 2007–2009 financial crisis, and its fallout, has strongly emphasized the need to define new ways and measures to study and assess the stock market dynamics.

Methodology/Principal Findings

The S&P500 dynamics during 4/1999–4/2010 is investigated in terms of the index cohesive force (ICF - the balance between the stock correlations and the partial correlations after subtraction of the index contribution), and the Eigenvalue entropy of the stock correlation matrices. We found a rapid market transition at the end of 2001 from a flexible state of low ICF into a stiff (nonflexible) state of high ICF that is prone to market systemic collapses. The stiff state is also marked by strong effect of the market index on the stock-stock correlations as well as bursts of high stock correlations reminiscence of epileptic brain activity.

Conclusions/Significance

The market dynamical states, stability and transition between economic states was studies using new quantitative measures. Doing so shed new light on the origin and nature of the current crisis. The new approach is likely to be applicable to other classes of complex systems from gene networks to the human brain.

Quantifying and modeling long-range cross correlations in multiple time series with applications to world stock indices

We propose a modified time lag random matrix theory in order to study time-lag cross correlations in multiple time series. We apply the method to 48 world indices, one for each of 48 different countries. We find long-range power-law cross correlations in the absolute values of returns that quantify risk, and find that they decay much more slowly than cross correlations between the returns. The magnitude of the cross correlations constitutes "bad news" for international investment managers who may believe that risk is reduced by diversifying across countries. We find that when a market shock is transmitted around the world, the risk decays very slowly. We explain these time-lag cross correlations by introducing a global factor model (GFM) in which all index returns fluctuate in response to a single global factor. For each pair of individual time series of returns, the cross correlations between returns (or magnitudes) can be modeled with the autocorrelations of the global factor returns (or magnitudes). We estimate the global factor using principal component analysis, which minimizes the variance of the residuals after removing the global trend. Using random matrix theory, a significant fraction of the world index cross correlations can be explained by the global factor, which supports the utility of the GFM. We demonstrate applications of the GFM in forecasting risks at the world level, and in finding uncorrelated individual indices. We find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices, which is relevant information for world managers in reducing their portfolio risk. Finally, we argue that this general method can be applied to a wide range of phenomena in which time series are measured, ranging from seismology and physiology to atmospheric geophysics.

Detrending moving average algorithm for multifractals

The detrending moving average (DMA) algorithm is a widely used technique to quantify the long-term correlations of nonstationary time series and the long-range correlations of fractal surfaces, which contains a parameter θ determining the position of the detrending window. We develop multifractal detrending moving average (MFDMA) algorithms for the analysis of one-dimensional multifractal measures and higher-dimensional multifractals, which is a generalization of the DMA method. The performance of the one-dimensional and two-dimensional MFDMA methods is investigated using synthetic multifractal measures with analytical solutions for backward (θ=0), centered (θ=0.5), and forward (θ=1) detrending windows. We find that the estimated multifractal scaling exponent τ(q) and the singularity spectrum f(α) are in good agreement with the theoretical values. In addition, the backward MFDMA method has the best performance, which provides the most accurate estimates of the scaling exponents with lowest error bars, while the centered MFDMA method has the worse performance. It is found that the backward MFDMA algorithm also outperforms the multifractal detrended fluctuation analysis. The one-dimensional backward MFDMA method is applied to analyzing the time series of Shanghai Stock Exchange Composite Index and its multifractal nature is confirmed.

Volatility, irregularity, and predictable degree of accumulative return series

Recently it was shown that financial time series are not completely random process but exhibit long-term or short-term dependences, which offer promises for predictability. However, we do not clearly understand the potential relationship between serial structure and predictability. This paper proposed a framework to magnify the correlations and regularities contained in financial time series through constructing accumulative return series. This method can help us distinguish the real world financial time series from random-walk process effectively by examining the change patterns of volatility, Hurst exponent, and approximate entropy. Furthermore, we have found that the predictable degree increases continually with the increasing length of accumulative return. Our results suggest that financial time series are predictable to some extent and approximate entropy is a good indicator to characterize the predictable degree of financial time series if we take the influence of their volatility into account.

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 R, 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 cross-correlations. Motivated by the relationship between |R| and |R|, we estimate the tail exponent alpha of the probability density function P(|R|) approximately |R|(-1-alpha) 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 alpha, we calculate the time intervals tau(q) between events where R > q. We demonstrate that tau(q), the average of tau(q), obeys tau(q) approximately q(alpha). We find alpha approximately 3. Furthermore, by aggregating all tau(q) values of 28 global financial indices, we also observe an approximate inverse cubic law.

Asymmetry in power-law magnitude correlations

Time series of increments can be created in a number of different ways from a variety of physical phenomena. For example, in the phenomenon of volatility clustering-well-known in finance-magnitudes of adjacent increments are correlated. Moreover, in some time series, magnitude correlations display asymmetry with respect to an increment's sign: the magnitude of |x_{i}| depends on the sign of the previous increment x_{i-1} . Here we define a model-independent test to measure the statistical significance of any observed asymmetry. We propose a simple stochastic process characterized by a an asymmetry parameter lambda and a method for estimating lambda . We illustrate both the test and process by analyzing physiological data.

Multifractal detrended cross-correlation analysis for two nonstationary signals

We propose a method called multifractal detrended cross-correlation analysis to investigate the multifractal behaviors in the power-law cross-correlations between two time series or higher-dimensional quantities recorded simultaneously, which can be applied to diverse complex systems such as turbulence, finance, ecology, physiology, geophysics, and so on. The method is validated with cross-correlated one- and two-dimensional binomial measures and multifractal random walks. As an example, we illustrate the method by analyzing two financial time series.