Surrogate Data Method Research Articles

Dynamic state of low-Reynolds-number turbulent channel flow.

We numerically study the dynamic state of a low-Reynolds-number turbulent channel flow from the viewpoints of symbolic dynamics and nonlinear forecasting. A low-dimensionally (high-dimensionally) chaotic state of the streamwise velocity fluctuations emerges at a viscous sublayer (logarithmic layer). The possible presence of the chaotic states is clearly identified by orbital instability-based nonlinear forecasting and ordinal partition transition network entropy in combination with the surrogate data method.

Estimating the Significance of the Added Skill from Initializations: The Case of Decadal Predictions

Abstract A considerable part of the skill in decadal forecasts often comes from the forcings, which are present in both initialized and uninitialized model experiments. This makes the added value from initialization difficult to assess. We investigate statistical tests to quantify if initialized forecasts provide skill over the uninitialized experiments. We consider three correlation-based statistics previously used in the literature. The distributions of these statistics under the null hypothesis that initialization has no added values are calculated by a surrogate data method. We present some simple examples and study the statistical power of the tests. We find that there can be large differences in both the values and power for the different statistics. In general, the simple statistic defined as the difference between the skill of the initialized and uninitialized experiments behaves best. However, for all statistics the risk of rejecting the true null hypothesis is too high compared to the nominal value. We compare the three tests on initialized decadal predictions (hindcasts) of near-surface temperature performed with a climate model and find evidence for a significant effect of initializations for small lead times. In contrast, we find only little evidence for a significant effect of initializations for lead times longer than 3 years when the experience from the simple experiments is included in the estimation.

Investigating non-linear and stochastic hard X-ray variability of active galactic nuclei using recurrence analysis

ABSTRACT We present results of recurrence analysis of 46 active galactic nuclei (AGNs) using light curves from the 157-month catalogue of the Swift Burst Alert Telescope (BAT) in the 14–150 keV band. We generate recurrence plots and compute recurrence plot metrics for each object. We use the surrogate data method to compare all derived recurrence-based quantities to three sets of stochastic light curves with identical power spectrum, flux distribution, or both, in order to determine the presence of determinism, non-linearity, entropy, and non-stationarity. We compare these quantities with known physical characteristics of each system, such as black hole mass, Eddington ratio, and bolometric luminosity, radio loudness, obscuration, and spectroscopic type. We find that almost all AGNs in this sample exhibit substantial higher order modes of variability than is contained in the power spectrum, with approximately half exhibiting non-linear or non-stationary behaviour. We find that Type 2 AGNs are more likely to contain deterministic variability than Type 1 AGNs, while the same distinction is not found between obscured and unobscured AGNs. The complexity of variability among Type 1 AGNs is anticorrelated with Eddington ratio, while no relationship is found among Type 2 AGNs. The connections between the recurrence properties and AGN class suggest that hard X-ray emission is a probe of distinct accretion processes among classes of AGNs, which supports interpretations of changing-look AGNs and challenges the traditional unification model that classifies AGNs only on viewing angle.

Information Dynamics of Electric Field Intensity before and during the COVID-19 Pandemic.

This work investigates the temporal statistical structure of time series of electric field (EF) intensity recorded with the aim of exploring the dynamical patterns associated with periods with different human activity in urban areas. The analyzed time series were obtained from a sensor of the EMF RATEL monitoring system installed in the campus area of the University of Novi Sad, Serbia. The sensor performs wideband cumulative EF intensity monitoring of all active commercial EF sources, thus including those linked to human utilization of wireless communication systems. Monitoring was performed continuously during the years 2019 and 2020, allowing us to investigate the effects on the patterns of EF intensity of varying conditions of human mobility, including regular teaching and exam activity within the campus, as well as limitations to mobility related to the COVID-19 pandemic. Time series analysis was performed using both simple statistics (mean and variance) and combining the information-theoretic measure of information storage (IS) with the method of surrogate data to quantify the regularity of EF dynamic patterns and detect the presence of nonlinear dynamics. Moreover, to assess the possible coexistence of dynamic behaviors across multiple temporal scales, IS analysis was performed over consecutive observation windows lasting one day, week, month, and year, respectively coarse grained at time scales of 6 min, 30 min, 2 h, and 1 day. Our results document that the EF intensity patterns of variability are modulated by the movement of people at daily, weekly, and monthly scales, and are blunted during periods of restricted mobility related to the COVID-19 pandemic. Mobility restrictions also affected significantly the regularity of the EF intensity time series, resulting in lower values of IS observed simultaneously with a loss of nonlinear dynamics. Thus, our analysis can be useful to investigate changes in the global patterns of human mobility both during pandemics or other types of events, and from this perspective may serve to implement strategies for safety assessment and for optimizing the design of networks of EF sensors.

Determinism and Non-linear Behaviour of Log-return and Conditional Volatility: Empirical Analysis for 26 Stock Markets

Determinism and non-linear behaviour in log-return and conditional volatility time series of the stock market index is examined for twenty-six countries. For this goal, the principal statistical techniques used in this study are a robust estimator of correlation dimension, a normalized non-linear prediction error, and pseudo-periodic surrogate data method. The proposed approach indicates, first, the stochastic behaviour of all log-return time series. Second, the inability of local linear, ARMA, or state- dependent noise models (such as ARCH, GARCH, and EGARCH) to describe its structure for the frontier, emerging, and developed markets. The same stochastic behaviour of conditional volatility time series, estimated by the stochastic volatility model with moving average innovations, is detected. This finding proves the efficiency of the stochastic volatility model compared with some analysed types of GARCH models for all studied markets. JEL Classification: C12, C52, D53, E44

Two-Parameter 0-1 Test for Chaos and Sample Entropy Bifurcation Diagrams for Nonlinear Oscillating Systems

This paper presents and compares two-parameter bifurcation diagrams obtained from the 0-1 test for chaos and sample entropy methods for nonlinear oscillating systems. Based on the computed diagrams it is often possible to establish a unique correspondence between the two types of diagrams. Comparison of the diagrams are presented for typical chaotic systems, including those of Lorenz, Rössler, electric arc circuits and chemical oscillating systems. The two types of bifurcation diagrams may also be related to the period-n and frequency distribution diagrams. The main goal is to address the issue of obtaining similar results from the 0-1 test for chaos and sample entropy methods and discuss difficulty in properly selecting parameters needed in the two methods. A possible link between the 0-1 test for chaos and surrogate data methods is also mentioned.

Testing nonlinearity in topological organization of functional brain networks.

Aiming to provide an argumentation on the underlying nonlinearity of the overall functional brain network via surrogate data method and graph theory. Taking the functional magnetic resonance imaging data as original data set and then shuffled the time series of each region of interest to generate surrogate data sets, corresponding original network and its 400 surrogates were obtained via computing connectivity matrixes. The results show that both the global correlation level and corresponding small-world topological characters exhibited obvious differences between the original network and its surrogates. And the following statistical testing results demonstrate their significant distinction, and this topological difference has been proved to be caused by the intrinsic nonlinear dynamics. Accordingly, the nonlinearity of the original functional network and its superior dynamical complexity have been confirmed. The results of this study could provide a novel angle into exploring the underlying mechanism of the neural brain system and offer an essential evidence in explaining complex brain activities.

A traffic data collection and analysis method based on wireless sensor network

With the rapid development of urbanization, collecting and analyzing traffic flow data are of great significance to build intelligent cities. The paper proposes a novel traffic data collection method based on wireless sensor network (WSN), which cannot only collect traffic flow data, but also record the speed and position of vehicles. On this basis, the paper proposes a data analysis method based on incremental noise addition for traffic flow data, which provides a criterion for chaotic identification. The method adds noise of different intensities to the signal incrementally by an improved surrogate data method and uses the delayed mutual information to measure the complexity of signals. Based on these steps, the trend of complexity change of mixed signal can be used to identify signal characteristics. The numerical experiments show that, based on incremental noise addition, the complexity trends of periodic data, random data, and chaotic data are different. The application of the method opens a new way for traffic flow data collection and analysis.

Coupling in complex systems as information transfer across time scales.

Complex systems such as the human brain or the Earth's climate consist of many subsystems interacting in intricate, nonlinear ways. Moreover, variability of such systems extends over broad ranges of spatial and temporal scales and dynamical phenomena on different scales also influence each other. In order to explain how to detect cross-scale causal interactions, we review information-theoretic formulation of the Granger causality in combination with computational statistics (surrogate data method) and demonstrate how this method can be used to infer driver-response relations from amplitudes and phases of coupled nonlinear dynamical systems. Considering complex systems evolving on multiple time scales, the reviewed methodology starts with a wavelet decomposition of a multi-scale signal into quasi-oscillatory modes of a limited bandwidth, described using their instantaneous phases and amplitudes. Then statistical associations, in particular, causality relations between phases or between phases and amplitudes on different time scales are tested using the conditional mutual information. As an application, we present the analysis of cross-scale interactions and information transfer in the dynamics of the El Niño Southern Oscillation. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Hypothesis Testing for Nonlinear Phenomena in the Geosciences Using Synthetic, Surrogate Data

Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged.

Sample entropy and surrogate data analysis for Alzheimer's disease.

Alzheimer's disease (AD) is a neurological degenerative disease, which is mainly char-acterized by the memory loss. As electroencephalogram (EEG) device is relatively cheap, portable and non-invasive, it has been widely used in AD-related studies. We proposed a method to detect the differences between healthy subjects and AD patients, which combines classical sample entropy (Sam-pEn) and surrogate data method. EEGs from 14 AD patients and 20 healthy subjects were analyzed. The results based on the original data showed that the SampEn of AD patients was significantly de-creased (p < 0.01) at electrodes c3, f3, o2 and p4, which confirmed that AD could cause complexity loss. However, using original data could be subject to human judgement, so we generated a series of surrogate data. We found that, there were significant difference of SampEn between the original time series and their surrogate data at c3 and o2 electrodes and the differences between healthy subjects and AD patients can be verified. Our method is capable of distinguishing AD patients from healthy subjects, which is consistent with the concept of physiologic complexity, and providing insights for understanding of AD.

Nonlinear Time Series Analysis of Avalanching Granular Flow Data

Abstract Dry cohesive granular flow can show pronounced nonlinear behaviour, which needs to be identified prior to the design of bulk solids handling systems. Laboratory tests yield time series data representing the flow behaviour of granular systems, but widely accepted analytical procedures for these systems have not been established yet. In this paper, the premise that avalanching flow yields nonlinear time series that can be identified by surrogate data methods is investigated. Preliminary results indicate that powder systems with a high propensity for avalanching yield nonlinear data and conversely, powder systems exhibiting good flow properties tend to yield linear time series data that can be identified by amplitude adjusted Fourier transform surrogate data methods.

The meaning of "coherent" and its quantification in coherent hemodynamics spectroscopy.

We have recently introduced a new technique, coherent hemodynamics spectroscopy (CHS), which aims at characterizing a specific kind of tissue hemodynamics that feature a high level of covariation with a given physiological quantity. In this study, we carry out a detailed analysis of the significance of coherence and phase synchronization between oscillations of arterial blood pressure (ABP) and total hemoglobin concentration ([Hbt]), measured with near-infrared spectroscopy (NIRS) during a typical protocol for CHS, based on a cyclic thigh cuff occlusion and release. Even though CHS is based on a linear time invariant model between ABP (input) and NIRS measurands (outputs), for practical reasons in a typical CHS protocol, we induce finite “groups” of ABP oscillations, in which each group is characterized by a different frequency. For this reason, ABP (input) and NIRS measurands (output) are not stationary processes, and we have used wavelet coherence and phase synchronization index (PSI), as a metric of coherence and phase synchronization, respectively. PSI was calculated by using both the wavelet cross spectrum and the Hilbert transform. We have also used linear coherence (which requires stationary process) for comparison with wavelet coherence. The method of surrogate data is used to find critical values for the significance of covariation between ABP and [Hbt]. Because we have found similar critical values for wavelet coherence and PSI by using five of the most used methods of surrogate data, we propose to use the data-independent Gaussian random numbers (GRNs), for CHS. By using wavelet coherence and wavelet cross spectrum, and GRNs as surrogate data, we have found the same results for the significance of coherence and phase synchronization between ABP and [Hbt]: on a total set of 20 periods of cuff oscillations, we have found 17 coherent oscillations and 17 phase synchronous oscillations. Phase synchronization assessed with Hilbert transform yielded similar results with 14 phase synchronous oscillations. Linear coherence and wavelet coherence overall yielded similar number of significant values. We discuss possible reasons for this result. Despite the similarity of linear and wavelet coherence, we argue that wavelet coherence is preferable, especially if one wants to use baseline spontaneous oscillations, in which phase locking and coherence between signals might be only temporary.

Long-Range Temporal Correlations of Patients in Minimally Conscious State Modulated by Spinal Cord Stimulation.

Spinal cord stimulation (SCS) has been shown to improve the consciousness levels of patients with disorder of consciousness (DOC). However, the underlying mechanisms of SCS remain poorly understood. This study recorded resting-state electroencephalograms (EEG) from 16 patients with minimally conscious state (MCS), before and after SCS, and investigated the mechanisms of SCS on the neuronal dynamics in MCS patients. Detrended fluctuation analysis (DFA), combined with surrogate data method, was employed to measure the long-range temporal correlations (LRTCs) of the EEG signals. A surrogate data method was utilized to acquire the genuine DFA exponents (GDFAE) reflecting the genuine LRTCs of brain activity. We analyzed the GDFAE in four brain regions (frontal, central, posterior, and occipital) at five EEG frequency bands [delta (1–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz), and gamma (30–45 Hz)]. The GDFAE values ranged from 0.5 to 1, and showed temporal and spatial variation between the pre-SCS and the post-SCS states. We found that the channels with GDFAE spread wider after SCS. This phenomenon may indicate that more cortical areas were engaged in the information integration after SCS. In addition, the GDFAE values increased significantly in the frontal area at delta, theta, and alpha bands after SCS. At the theta band, a significant increase in GDFAE was observed in the occipital area. No significant change was found at beta or gamma bands in any brain region. These findings show that the enhanced LRTCs after SCS occurred primarily at low-frequency bands in the frontal and occipital regions. As the LRTCs reflect the long-range temporal integration of EEG signals, our results indicate that information integration became more “complex” after SCS. We concluded that the brain activities at low-frequency oscillations, particularly in the frontal and occipital regions, were improved by SCS.

Surrogate Data Method Requires End-Matched Segmentation of Electroencephalographic Signals to Estimate Non-linearity.

The aim of the study is to clarify the impact of the strong cyclic signal component on the results of surrogate data method in the case of resting electroencephalographic (EEG) signals. In addition, the impact of segment length is analyzed. Different non-linear measures (fractality, complexity, etc.) of neural signals have been demonstrated to be useful to infer the non-linearity of brain functioning from EEG. The surrogate data method is often applied to test whether or not the non-linear structure can be captured from the data. In addition, a growing number of studies are using surrogate data method to determine the statistical threshold of connectivity values in network analysis. Current study focuses on the conventional segmentation of EEG signals, which could lead to false results of surrogate data method. More specifically, the necessity to use end-matched segments that contain an integer number of dominant frequency periods is studied. EEG recordings from 80 healthy volunteers during eyes-closed resting state were analyzed using multivariate surrogate data method. The artificial surrogate data were generated by shuffling the phase spectra of original signals. The null hypothesis that time series were generated by a linear process was rejected by statistically comparing the non-linear statistics calculated for original and surrogate data sets. Five discriminating statistics were used as non-linear estimators: Higuchi fractal dimension (HFD), Katz fractal dimension (KFD), Lempel-Ziv complexity (LZC), sample entropy (SampEn) and synchronization likelihood (SL). The results indicate that the number of segments evaluated as non-linear differs in the case of various non-linear measures and changes with the segment length. The main conclusion is that the dependence on the deviation of the segment length from full periods of dominant EEG frequency has non-monotonic character and causes misleading results in the evaluation of non-linearity. Therefore, in the case of the signals with non-monotonic spectrum and strong dominant frequency, the correct use of surrogate data method requires the signal length comprising of full periods of the spectrum dominant frequency. The study is important to understand the influence of incorrect selection of EEG signal segment length for surrogate data method to estimate non-linearity.

Comparison of statistical tests in effective connectivity analysis of ECoG data

BackgroundThe effects of statistical testing on the results of multivariate autoregressive (MVAR)-based effective connectivity analysis have not been adequately investigated, and it is still unclear which statistical test can provide the most accurate results. New methodsUsing simulated and real electrocorticographic (ECoG) data, we investigated the performance of three nonparametric statistical tests – Monte Carlo permutation, bootstrap resampling, and surrogate data method in MVAR-based effective connectivity analysis. Receiver operating characteristic (ROC) analysis and area under the ROC curve (AUC) were used to assess the performance of each statistical test method. In addition, we found optimal p-values for each method based on ROC analysis. Finally, we investigated the application of statistical tests on partial directed coherence analysis of ECoG data collected in a patient with epilepsy. ResultsThe bootstrap statistical test performed more accurately than other methods. The surrogate method slightly outperformed the Monte Carlo permutation method. Optimal p-values of statistical tests depended on signal-to-noise ratio (SNR) of data, and its value increased by reducing SNR of data. By considering the typical SNR range of electrophysiological data, we recommended an optimal p-value range for the application of each statistical test method. Comparison with existing methodsLimited studies have investigated the performance of statistical tests for MVAR-based effective connectivity analysis. For the first time, we have investigated the effects of baseline connections on the various performances of statistical tests. ConclusionsWe recommend utilizing the bootstrap statistical test with p-value between 0.05 and 0.1 for effective connectivity analysis of ECoG data.

Cocaine-Induced Changes in Low-Dimensional Attractors of Local Field Potentials in Optogenetic Mice.

Optogenetically evoked local field potential (LFP) recorded from the medial prefrontal cortex (mPFC) of mice during basal conditions and following a systemic cocaine administration were analyzed. Blue light stimuli were delivered to mPFC through a fiber optic every 2 s and each trial was repeated 100 times. As in the previous study, we used a surrogate data method to check that nonlinearity was present in the experimental LFPs and only used the last 1.5 s of steady activity to measure the LFPs phase resetting induced by the brief 10 ms light stimulus. We found that the steady dynamics of the mPFC in response to light stimuli could be reconstructed in a three-dimensional phase space with topologically similar “8”-shaped attractors across different animals. Therefore, cocaine did not change the complexity of the recorded nonlinear data compared to the control case. The phase space of the reconstructed attractor is determined by the LFP time series and its temporally shifted versions by a multiple of some lag time. We also compared the change in the attractor shape between cocaine-injected and control using (1) dendrogram clustering and (2) Frechet distance. We found about 20% overlap between control and cocaine trials when classified using dendrogram method, which suggest that it may be possible to describe mathematically both data sets with the same model and slightly different model parameters. We also found that the lag times are about three times shorter for cocaine trials compared to control. As a result, although the phase space trajectories for control and cocaine may look similar, their dynamics is significantly different.

Fetus and mother are a deterministic chaos：Analysis of surrogate data method

Fetus and mother are a deterministic chaos：Analysis of surrogate data method

Nonlinear dynamic analysis of Dα signals for type I edge localized modes characterization on JET with a carbon wall

In this paper, the dynamic characteristics of type-I ELM time-series from the JET tokamak, the world’s largest magnetic confinement plasma physics experiment, have been investigated. The dynamic analysis has been focused on the detection of nonlinear structure in Dα radiation time series. Firstly, the method of surrogate data has been applied to evaluate the statistical significance of the null hypothesis of static nonlinear distortion of an underlying Gaussian linear process. Several nonlinear statistics have been evaluated, such us the time delayed mutual information, the correlation dimension and the maximal Lyapunov exponent. The obtained results allow us to reject the null hypothesis, giving evidence of underlying nonlinear dynamics. Moreover, no evidence of low-dimensional chaos has been found; indeed, the analysed time series are better characterized by the power law sensitivity to initial conditions which can suggest a motion at the ‘edge of chaos’, at the border between chaotic and regular non-chaotic dynamics. This uncertainty makes it necessary to further investigate about the nature of the nonlinear dynamics. For this purpose, a second surrogate test to distinguish chaotic orbits from pseudo-periodic orbits has been applied. In this case, we cannot reject the null hypothesis which means that the ELM time series is possibly pseudo-periodic. In order to reproduce pseudo-periodic dynamical properties, a periodic state-of-the-art model, proposed to reproduce the ELM cycle, has been corrupted by a dynamical noise, obtaining time series qualitatively in agreement with experimental time series.

Nonlinear dynamics of laser systems with elements of a chaos: Advanced computational code

A general, uniform chaos-geometric computational approach to analysis, modelling and prediction of the non-linear dynamics of quantum and laser systems (laser and quantum generators system etc) with elements of the deterministic chaos is briefly presented. The approach is based on using the advanced generalized techniques such as the wavelet analysis, multi-fractal formalism, mutual information approach, correlation integral analysis, false nearest neighbour algorithm, the Lyapunov’s exponents analysis, and surrogate data method, prediction models etc There are firstly presented the numerical data on the topological and dynamical invariants (in particular, the correlation, embedding, Kaplan-York dimensions, the Lyapunov’s exponents, Kolmogorov’s entropy and other parameters) for laser system (the semiconductor GaAs/GaAlAs laser with a retarded feedback) dynamics in a chaotic and hyperchaotic regimes.