Fractionally integrated process with power-law correlations in variables and magnitudes

Composite Multiscale Partial Cross-Sample Entropy Analysis for Quantifying Intrinsic Similarity of Two Time Series Affected by Common External Factors

In this paper, we propose a new cross-sample entropy, namely the composite multiscale partial cross-sample entropy (CMPCSE), for quantifying the intrinsic similarity of two time series affected by common external factors. First, in order to test the validity of CMPCSE, we apply it to three sets of artificial data. Experimental results show that CMPCSE can accurately measure the intrinsic cross-sample entropy of two simultaneously recorded time series by removing the effects from the third time series. Then CMPCSE is employed to investigate the partial cross-sample entropy of Shanghai securities composite index (SSEC) and Shenzhen Stock Exchange Component Index (SZSE) by eliminating the effect of Hang Seng Index (HSI). Compared with the composite multiscale cross-sample entropy, the results obtained by CMPCSE show that SSEC and SZSE have stronger similarity. We believe that CMPCSE is an effective tool to study intrinsic similarity of two time series.

Quantifying interdependence using the missing joint ordinal patterns

In this paper, we develop the concept of forbidden/missing ordinal patterns into the forbidden/missing joint ordinal patterns and propose the ratio of the number of missing joint ordinal patterns (RMJPs) as a sign of interdependence. RMJP in a surrogate analysis can be used to differentiate the forbidden joint ordinal patterns from the missing joint ordinal patterns due to small sample effects. We first apply RMJP to the simulated time series: a two-component autoregressive fractionally integrated moving average process, the Hénon map, and the Rössler system using active control and discuss the effect of the length of the time series, embedding dimension, and noise contamination. RMJP has been proven to be capable of measuring the interdependence in the numerical simulation. Then, RMJP is further used on the electroencephalogram (EEG) time series for empirical analysis to explore the interdependence of brain waves. With results by RMJP obtained from a widely used open dataset of the sleep EEG time series from healthy subjects, we find that RMJP can be used to quantify the brain wave interdependence under different sleep/wake stages, reveal the overall sleep architecture, and indicate a higher level of interdependence as sleep gets deeper. The findings are consistent with existing knowledge in sleep medicine. The proposed RMJP method has shown its validity and applicability and may assist automatic sleep quantification or bring insight into the understanding of the brain activity during sleep. Furthermore, RMJP can be used on sleep EEG under various pathological conditions and in large-scale sleep studies, helping to investigate the mechanisms of the sleep process and neuron synchronization.

Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models

In this paper we show that an autoregressive fractionally integrated moving average time-series model can identify two types of motion of membrane proteins on the surface of mammalian cells. Specifically we analyze the motion of the voltage-gated sodium channel Nav1.6 and beta-2 adrenergic receptors. We find that the autoregressive (AR) part models well the confined dynamics whereas the fractionally integrated moving average (FIMA) model describes the nonconfined periods of the trajectories. Since the Ornstein-Uhlenbeck process is a continuous counterpart of the AR model, we are also able to calculate its physical parameters and show their biological relevance. The fitted FIMA and AR parameters show marked differences in the dynamics of the two studied molecules.

Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications

Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime using a nonlinear integrated model derived by applying Newton’s second law into self-regulating systems. We discover that (i) a stochastic stable fixed point exhibits self-similarity and long-term memory, while a deterministic stable fixed point usually only exhibits self-similarity, if our observation scale is large enough; (ii) stochastic/deterministic period cycles and chaos only exhibit long-term memory, but also self-similarity for even restorative delays; (iii) fractal level of a stable fixed point is controlled primarily by the wave indicators that reflect the relative strength of extrinsic to intrinsic forces: a larger absolute slope (smaller amplitude) indicator leads to higher positive dependence (self-similarity), and a relatively large amplitude indicator or an even restorative delay could make the dependence oscillate; and (iv) fractal levels of period cycles and chaos rely on the intrinsic resistance, restoration, and regulative delays. Our findings suggest that fractals of self-regulating systems can be measured by integer dimensions.

An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics

Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.

A novel way to detect correlations on multi-time scales, with temporal evolution and for multi-variables

In this paper, two new methods, Temporal evolution of Detrended Cross-Correlation Analysis (TDCCA) and Temporal evolution of Detrended Partial-Cross-Correlation Analysis (TDPCCA), are proposed by generalizing DCCA and DPCCA. Applying TDCCA/TDPCCA, it is possible to study correlations on multi-time scales and over different periods. To illustrate their properties, we used two climatological examples: i) Global Sea Level (GSL) versus North Atlantic Oscillation (NAO); and ii) Summer Rainfall over Yangtze River (SRYR) versus previous winter Pacific Decadal Oscillation (PDO). We find significant correlations between GSL and NAO on time scales of 60 to 140 years, but the correlations are non-significant between 1865–1875. As for SRYR and PDO, significant correlations are found on time scales of 30 to 35 years, but the correlations are more pronounced during the recent 30 years. By combining TDCCA/TDPCCA and DCCA/DPCCA, we proposed a new correlation-detection system, which compared to traditional methods, can objectively show how two time series are related (on which time scale, during which time period). These are important not only for diagnosis of complex system, but also for better designs of prediction models. Therefore, the new methods offer new opportunities for applications in natural sciences, such as ecology, economy, sociology and other research fields.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Generalized permutation entropy analysis based on the two-index entropic form Sq,δ

Permutation entropy (PE) is a novel measure to quantify the complexity of nonlinear time series. In this paper, we propose a generalized permutation entropy ( PEq,δ) based on the recently postulated entropic form, Sq,δ, which was proposed as an unification of the well-known Sq of nonextensive-statistical mechanics and Sδ, a possibly appropriate candidate for the black-hole entropy. We find that PEq,δ with appropriate parameters can amplify minor changes and trends of complexities in comparison to PE. Experiments with this generalized permutation entropy method are performed with both synthetic and stock data showing its power. Results show that PEq,δ is an exponential function of q and the power ( k(δ)) is a constant if δ is determined. Some discussions about k(δ) are provided. Besides, we also find some interesting results about power law.

Multiscale multifractal analysis of traffic signals to uncover richer structures

Multifractal detrended fluctuation analysis (MF-DFA) is the most popular method to detect multifractal characteristics of considerable signals such as traffic signals. When fractal properties vary from point to point along the series, it leads to multifractality. In this study, we concentrate not only on the fact that traffic signals have multifractal properties, but also that such properties depend on the time scale in which the multifractality is computed. Via the multiscale multifractal analysis (MMA), traffic signals appear to be far more complex and contain more information which MF-DFA cannot explore by using a fixed time scale. More importantly, we do not have to avoid data sets with crossovers or narrow the investigated time scales, which may lead to biased results. Instead, the Hurst surface provides a spectrum of local scaling exponents at different scale ranges, which helps us to easily position these crossovers. Through comparing Hurst surfaces for signals before and after removing periodical trends, we find periodicities of traffic signals are the main source of the crossovers. Besides, the Hurst surface of the weekday series behaves differently from that of the weekend series. Results also show that multifractality of traffic signals is mainly due to both broad probability density function and correlations. The effects of data loss are also discussed, which suggests that we should carefully handle MMA results when the percentage of data loss is larger than 40%.

Fractional process as a unified model for subdiffusive dynamics in experimental data

We show how to use a fractional autoregressive integrated moving average (FARIMA) model to a statistical analysis of the subdiffusive dynamics. The discrete time FARIMA(1,d,1) model is applied in this paper to the random motion of an individual fluorescently labeled mRNA molecule inside live E. coli cells in the experiment described in detail by Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)] as well as to the motion of fluorescently labeled telomeres in the nucleus of live human cells (U2OS cancer) in the experiment performed by Bronstein et al. [Phys. Rev. Lett. 103, 018102 (2009)]. It is found that only the memory parameter d of the FARIMA model completely detects an anomalous dynamics of the experimental data in both cases independently of the observed distribution of random noises.

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.

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.

Minding impacting events in a model of stochastic variance

We introduce a generalization of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time is below a certain threshold, , and another one when the local standard deviation outnumbers . In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterized by large values of the Hurst exponent (H>0.8), which are ubiquitous features in complex systems.

Comparison between response dynamics in transition economies and developed economies

In developed economies, the sign of the price increment influences the volatility in an asymmetric fashion--negative increments tend to result in larger volatility (increments with larger magnitudes), while positive increments result in smaller volatility. We explore whether this asymmetry extends from developed economies to European transition economies and, if so, how such asymmetry changes over time as these transition economies develop and mature. We analyze eleven European transition economies and compare the results with those obtained by analyzing U.S. market indices. Specifically, we calculate parameters that quantify both the volatility asymmetry and the strength of its dependence on prior increments. We find that, like their developed economy counterparts, almost all transition economy indices exhibit a significant volatility asymmetry, and the parameter γ characterizing asymmetry fluctuates more over time for transition economies. We also investigate how the association between volatility and volatility asymmetry varies by type of market. We test the hypothesis of a negative correlation between volatility and volatility asymmetry. We find that, for developed economies, γ experiences local minima during (i) "Black Monday" on October 19, 1987, (ii) the dot-com bubble crash in 2002, and (iii) the 2007-2009 global crisis while for transition economies, γ experiences local maxima during times of economic crisis.

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.

Normal heartbeat series are nonchaotic, nonlinear, and multifractal: new evidence from semiparametric and parametric tests

We present new evidence that normal heartbeat series are nonchaotic, nonlinear, and multifractal. In addition to considering the largest Lyapunov exponent and the correlation dimension, the results of the parametric and semiparametric estimation of the long memory parameter (long-range dependence) unambiguously reveal that the underlying process is nonstationary, multifractal, and has strong nonlinearity.

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.

Dynamic scaling approach to study time series fluctuations

We propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface-growth problem. In this framework, the fluctuation sampling time interval plays the role of time variable, whereas the physical time is treated as the analog of spatial variable. In this way we found that the fluctuations of many real-world time series satisfy the analog of the Family-Viscek dynamic scaling ansatz. This finding permits us to use the powerful tools of kinetic roughening theory to classify, model, and forecast the fluctuations of real-world time series.

Understanding volatility correlation behavior with a magnitude cross-correlation function

We propose an approach for analyzing the basic relation between correlation properties of the original signal and its magnitude fluctuations by decomposing the original signal into its positive and negative fluctuation components. We use this relation to understand the following phenomenon found in many naturally occurring time series: the magnitude of the signal exhibits long-range correlation, whereas the original signal is short-range correlated. The applications of our approach to heart rate variability signals and high-frequency foreign exchange rates reveal that the difference between the correlation properties of the original signal and its magnitude fluctuations is induced by the time organization structure of the correlation function between the magnitude fluctuations of positive and negative components. We show that this correlation function can be described well by a stretched-exponential function and is related to the nonlinearity and the multifractal structure of the signals.