Surrogate Data Method Research Articles

Detection of quadratic phase coupling from human EEG signals using higher order statistics and spectra

Interactions among neural signals in different frequency components have become a focus of strong interest in biomedical signal processing. The bispectrum is a method to detect the presence of quadratic phase coupling (QPC) between different frequency bands in a signal. The traditional way to quantify phase coupling is by means of the bicoherence index (BCI), which is essentially a normalized bispectrum. The main disadvantage of the BCI is that the determination of significant QPC becomes compromised with noise. To mitigate this problem, a statistical approach that combines the bispectrum with an improved surrogate data method to determine the statistical significance of the phase coupling is introduced. The method was first tested on two simulation examples. It was then applied to the human EEG signal that has been recorded from the scalp using international 10–20 electrodes system. The frequency domain method, based on normalized spectrum and bispectrum, describes frequency interactions associated with nonlinearities occurring in the observed EEG.

Periodic and Chaotic Motions of a Two-Bar Linkage with OPCL Controller

A two-bar linkage, which is described in differential dynamical equations, can perform nonlinear behaviors due to system parameters or external input. As a basic component of robot system, the investigation of its behavior can improve robot performance, control strategy, and system parameters. An open-plus-close-loop (OPCL) control method therefore is developed and applied to reveal and classify the complicated behaviors of a two-bar linkage. In this paper, the conception and stability of OPCL are addressed firstly. Then it is applied to the dynamical equations of two-bar linkage. Different motions including single-periodic, multiple-periodic, quasiperiodic, and chaotic motions are unfolded by numerical simulations when changing the controller parameters. Furthermore, the obtained chaotic motions are sorted out for qualitative and quantificational study using Lyapunov exponents and hypothetic possibilities of surrogate data method.

RMB Exchange Rate Forecasting in the Context of the Financial Crisis

Abstract This article offers an effective solution of forecasting the RMB exchange rate volatility during the financial crisis period. Based on the test of nonlinearity structure in the exchange rate system via the method of surrogate data, the optimal lag periods for each specific exchange rate series were computed by autocorrelation criterion (AC) approach, and then, the structure of multilayer perceptrons (MLP) and recurrent neural networks (RNN) were applied to build the homogeneous artificial neural network (ANN) model. The comparison of the forecast results of ANNs with different parameters shows that, according to the specific exchange rate series, the forecast performance of ANN models with different freedom of degrees has obvious differences in different forecast periods. The RNN model, which contains layer feedback process, has showed great ability to explain and forecast the RMB exchange rates volatility behavior. The optimal forecasting model for each RMB exchange rate volatility series has been found and explained.

Fulfillment of Retailer Demand by Using the MDL-Optimal Neural Network Prediction and Decision Policy

Prediction of demand plays a critical role in replenishment, in supply chain management. Accurate prediction of demand is a fundamental requirement and is also a great challenge to demand prediction models. This has motivated the research team to develop the minimum description length (MDL)-optimal neural network (NN) which can accurately predict retailer demands with various time lags. Moreover, a surrogate data method is proposed prior to the prediction to investigate the dynamical property (i.e., predictability) of various demand time series so as to avoid predicting random demands. In this paper, we validate the proposed ideas by a full factorial study combining its own decision rules. We describe improvements to prediction accuracy and propose a replenishment policy for a Hong Kong food wholesaler. This leads to a significant reduction in its operation costs and to an improvement in the level of retailer satisfaction.

On the Efficacy of the Combined Use of the Cross-Bicoherence with Surrogate Data Technique to Statistically Quantify the Presence of Nonlinear Interactions

The cross-bispectrum is an approach to detect the presence of quadratic phase coupling (QPC) between different components in bivariate signals. Quantification of QPC is by means of the cross-bicoherence index (CBI). The major limitations of the CBI are that it favors only the strongly coupled signals and its accuracy becomes compromised with noise and low coupling strength. To overcome this limitation, a statistical approach which combines CBI with a surrogate data method to determine the statistical significance of the QPC derived from bivariate signals is introduced. We demonstrate the accuracy of the proposed approach using simulation examples which are designed to test its robustness against noise contamination as well as varying levels of phase coupling and data lengths. Comparisons were made to the traditional CBI and the method based on the use of cross-bispectrum followed by a surrogate data technique. Our results show that the cross-bicoherence with surrogate data technique outperforms the two other methods compared in both sensitivity and specificity, and provides an unbiased and statistical approach to determining the presence of QPC in bivariate signals. These results are in contrast to our recent study where the auto-bispectrum combined with surrogate data approach had the best performance. Application of this approach to renal hemodynamic data was applied to renal stop flow pressure data obtained in the nephrons of the normotensive (N = 18) and hypertensive (N = 15) rats. We found significant nonlinear interactions between nephrons only when they are derived from the same cortical renal artery. The accuracy was 100% and verified by comparing the results to the known vascular connectivity between nephrons.

Chaotic Patterns in Aeroelastic Signals

This work presents the analysis of nonlinear aeroelastic time series from wing vibrations due to airflow separation during wind tunnel experiments. Surrogate data method is used to justify the application of nonlinear time series analysis to the aeroelastic system, after rejecting the chance for nonstationarity. The singular value decomposition (SVD) approach is used to reconstruct the state space, reducing noise from the aeroelastic time series. Direct analysis of reconstructed trajectories in the state space and the determination of Poincaré sections have been employed to investigate complex dynamics and chaotic patterns. With the reconstructed state spaces, qualitative analyses may be done, and the attractors evolutions with parametric variation are presented. Overall results reveal complex system dynamics associated with highly separated flow effects together with nonlinear coupling between aeroelastic modes. Bifurcations to the nonlinear aeroelastic system are observed for two investigations, that is, considering oscillations‐induced aeroelastic evolutions with varying freestream speed, and aeroelastic evolutions at constant freestream speed and varying oscillations. Finally, Lyapunov exponent calculation is proceeded in order to infer on chaotic behavior. Poincaré mappings also suggest bifurcations and chaos, reinforced by the attainment of maximum positive Lyapunov exponents.

Stationarization via surrogates

The surrogate data method, classically used for non-linearity tests, amounts to the use ofsome constrained noise providing a reference for statistical testing. It is revisited here asa method for stationarization and this feature is put forward in the context ofnon-stationarity testing. The stationarization property of surrogates is first explored in atime–frequency perspective and used for devising a test of stationarity relative to anobservation time. Then, more general forms of surrogates are developed, directly intime–frequency or mixed domains of representation (ambiguity and time-lag domainsincluded), and it is shown how they allow for other tests of non-stationary features:detection of the existence of a transient in some noise; assessment of non-stationarycross-correlations.

EFFECT OF JUMP DISCONTINUITY FOR PHASE-RANDOMIZED SURROGATE DATA TESTING

In this paper we discuss two modifications of the surrogate data method based on phase randomization, see [Theiler et al., 1992]. By construction, phase randomized surrogates are circular stationary. In this respect they differ from the original time series. This can cause level inaccuracies of surrogate data tests. We will illustrate this. These inaccuracies are caused by end to end mismatches of the original time series. In this paper we will discuss two approaches to remedy this problem: resampling from subsequences without end to end mismatches and data tapering. Both methods can be understood as attempts to make non-circular data approximately circular. We will show that the first method works quite well for a large range of applications whereas data tapering leads only to improvements in some examples but can be very unstable otherwise.

Assessment of the dynamical properties in EDM process—detecting deterministic nonlinearity of EDM process

Time series of gap state were often used as feedback signal in electrical discharge machining (EDM) adaptive control systems. However, models precisely describing the EDM process have never been built because of the once believed stochastic nature of the EDM process. In this case, the power of adaptive controls in EDM had not been fully brought into play. Before building a feasible model, it is prerequisite to determine whether an efficient stable EDM process is nonlinear or linear, deterministic or stochastic. The main purpose of this paper is to investigate the deterministic nonlinearity of the process. A discriminating method was first provided to judge states in the gap at sampling intervals from voltage and current. Gap state was then statistically quantified from a train of discriminated states at sampling intervals within a specified period of time. Based on a time series of gap state data, we took use of surrogate data method to detect the nonlinearity of the process. From the results of two kinds of tests, it can be concluded that the deterministic nonlinearity of the process reflected by gap states is intrinsic.

Investigation of Nonlinearity in Hyperspectral Imagery Using Surrogate Data Methods

Although hyperspectral remotely sensed data are believed to be nonlinear, they are often modeled and processed by algorithms assuming that the data are realizations of some linear stochastic processes. This is likely due to the reason that either the nonlinearity of the data may not be strong enough, and the algorithms based on linear data assumption may still do the job, or the effective algorithms that are capable of dealing with nonlinear data are not widely available. The simplification on data characteristics, however, may compromise the effectiveness and accuracy of information extraction from hyperspectral imagery. In this paper, we are investigating the existence of non- linearity in hyperspectral data represented by a 4-m Airborne Visible/Infrared Imaging Spectrometer image acquired over an area of coastal forests on Vancouver Island. The method employed for the investigation is based on the statistical test using surrogate data, an approach often used in nonlinear time series analysis. In addition to the high-order autocorrelation, spectral angle is utilized as the discriminating statistic to evaluate the differences between the hyperspectral data and their surrogates. To facilitate the statistical test, simulated data sets are created under linear stochastic constraints. Both simulated and real hyperspectral data are rearranged into a set of spectral series where the spectral and spatial adjacency of the original data is maintained as much as possible. This paper reveals that the differences are statistically significant between the values of discriminating statistics derived from the hyperspectral data and their surrogates. This indicates that the selected hyperspectral data are nonlinear in the spectral domain. Algorithms that are capable of explicitly addressing the nonlinearity are needed for processing hyperspectral remotely sensed data.

Correntropy as a novel measure for nonlinearity tests

Nonlinearity tests have become an essential step in system analysis and modeling due to the computational demands and complexity of analysis involved in nonlinear modeling. Standard nonlinear measures are either too complicated to estimate accurately (such as Lyapunov exponents and correlation dimension), or not able to capture sufficient but not necessary conditions of nonlinearity (such as time asymmetry). Correntropy is a kernel-based similarity measure which contains the information of both statistical and temporal structure of the underlying dataset. The capability of preserving nonlinear characteristics makes correntropy a suitable candidate as a measure for determining nonlinear dynamics. Moreover, since correntropy makes use of kernel methods, its estimation is computationally efficient. Using correntropy as the test statistic, nonlinearity tests based on the null hypothesis that signals of interest are realizations of linear Gaussian stochastic processes are carried out via surrogate data methods. Experiments performed on linear Gaussian, linear non-Gaussian, and nonlinear systems with varying in-band noise levels, data lengths, and kernel sizes confirm that correntropy can be employed as a discriminative measure for detecting nonlinear characteristics in time series. Results of tests performed on data collected from natural systems are in agreement with findings in time series analysis literature.

Some theoretical properties of phase-randomized multivariate surrogates

We study the multivariate surrogate data method of Prichard and Theiler [D. Prichard and J. Theiler, Generating surrogate data for time series with several simultaneously measured variables, Phys. Rev. Lett. 73 (1994), pp. 951–954.]. This approach is an extension of the univariate surrogate data method introduced in Theiler et al. [J. Theiler, S. Eubank, A. Longtin, B. Galdrikan, and J.D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D 58 (1992), pp. 77–94.]. Surrogate data is a resampling method for linear time series that is based on phase-randomized Fourier transforms. Theoretical properties of the multivariate method are derived. We show that the conditional distribution of surrogates is asymptotically normal. Furthermore, it is shown that surrogate data is the only method to construct tests that have constant level on the hypothesis of multivariate stationary Gaussian circular processes. We argue that this result approximately holds for non-circular processes under some regularity conditions. This entails that other bootstrap methods are not available for this general null hypothesis. Bootstrap will work only for more restricted hypotheses. Our results generalize results of Chan [K.S. Chan, On the validity of the method of surrogate data, Fields Inst. Commun. 11 (1997), pp. 77–97.] for the univariate setting.

Using recurrence plot for determinism analysis of EEG recordings in genetic absence epilepsy rats

ObjectiveUnderstanding the transition of brain activity towards an absence seizure is a challenging task. In this paper, we use recurrence quantification analysis to indicate the deterministic dynamics of EEG series at the seizure-free, pre-seizure and seizure states in genetic absence epilepsy rats. MethodsThe determinism measure, DET, based on recurrence plot, was applied to analyse these three EEG datasets, each dataset containing 300 single-channel EEG epochs of 5-s duration. Then, statistical analysis of the DET values in each dataset was carried out to determine whether their distributions over the three groups were significantly different. Furthermore, a surrogate technique was applied to calculate the significance level of determinism measures in EEG recordings. ResultsThe mean (±SD) DET of EEG was 0.177±0.045 in pre-seizure intervals. The DET values of pre-seizure EEG data are significantly higher than those of seizure-free intervals, 0.123±0.023, (P<0.01), but lower than those of seizure intervals, 0.392±0.110, (P<0.01). Using surrogate data methods, the significance of determinism in EEG epochs was present in 25 of 300 (8.3%), 181 of 300 (60.3%) and 289 of 300 (96.3%) in seizure-free, pre-seizure and seizure intervals, respectively. ConclusionsResults provide some first indications that EEG epochs during pre-seizure intervals exhibit a higher degree of determinism than seizure-free EEG epochs, but lower than those in seizure EEG epochs in absence epilepsy. SignificanceThe proposed methods have the potential of detecting the transition between normal brain activity and the absence seizure state, thus opening up the possibility of intervention, whether electrical or pharmacological, to prevent the oncoming seizure.

Statistical Approach to Quantify the Presence of Phase Coupling Using the Bispectrum

The bispectrum is a method to detect the presence of phase coupling between different components in a signal. The traditional way to quantify phase coupling is by means of the bicoherence index, which is essentially a normalized bispectrum. The major drawback of the bicoherence index (BCI) is that determination of significant phase coupling becomes compromised with noise and low coupling strength. To overcome this limitation, a statistical approach that combines the bispectrum with a surrogate data method to determine the statistical significance of the phase coupling is introduced. Our method does not rely on the use of the BCI, where the normalization procedure of the BCI is the major culprit in its poor specificity. We demonstrate the accuracy of the proposed approach using simulation examples that are designed to test its robustness against noise contamination as well as varying levels of phase coupling. Our results show that the proposed approach outperforms the bicoherence index in both sensitivity and specificity and provides an unbiased and statistical approach to determining the presence of quadratic phase coupling. Application of this new method to renal hemodynamic data was applied to renal stop flow pressure data obtained from normotensive (N = 7) and hypertensive (N = 7) rats. We found significant nonlinear interactions in both strains of rats with a greater magnitude of coupling and smaller number of interaction peaks in normotensive rats than hypertensive rats.

Testing for Nonlinearity in Functional Near-Infrared Spectroscopy of Brain Activities by Surrogate Data Methods

Methods of contemporary physics are increasingly important for biomedical research. For a multitude of diverse reasons there exists a gap between the practitioners of biomedicine and modern physics methodologies. In this work, the technique of surrogate data has been used as a method to test for the linearity or nonlinearity of biomedical functional near-infrared spectroscopy (fNIRS) signals observing brain activities. Throughout three different surrogate tests, the third-order autocovariance, the asymmetry resulting from time reversal, and the delay vector variance, the dynamic response of brain activities through fNIRS biomedical signals is very likely to be a nonlinear system.

A test method for weak nonlinearity in time series

Nonlinearity is necessary for time series to be treated as chaotic time series. A new test statistic for nonlinearity, which is based on the ratio of the multistep normalized prediction error with respect to linear AR models and nonlinear AR models, is used to detect the weak nonlinear components contained in time series by the surrogate data method. Taking example for Lorenz time series, the effect of related parameters for test statistic estimation is analyzed. By the nonlinearity tests for chaotic time series, the proposed test statistic δNAR has better discrimination power for weak nonlinearity than the test statistic δAIC based on AIC rules and the zeroth order nonlinear prediction error δZP, which shows that the proposed test statistic has strong adaptive abilities for time series. And, for different time series, the parameters with best nonlinearity discrimination performance are kept constant. The stabilization of parameters facilitates the nonlinearity test for other time series.

EVIDENCE CONSISTENT WITH DETERMINISTIC CHAOS IN HUMAN CARDIAC DATA: SURROGATE AND NONLINEAR DYNAMICAL MODELING

Whether or not the human cardiac system is chaotic has long been a subject of interest in the application of nonlinear time series analysis. The surrogate data method, which identifies an observed time series against three common kinds of hypotheses, does not provide sufficient evidence to confirm the existence of deterministic chaotic dynamics in cardiac time series, such as electrocardiogram data and pulse pressure propagation data. Moreover, these methods fail to exclude all but the most trivial hypothesis of linear noise. We present a recently suggested fourth algorithm for testing the hypotheses of a noise driven periodic orbit to decide whether these signals are consistent with deterministic chaos. Of course, we cannot exclude all other alternatives but our test is certainly stronger than the those applied previously. The algorithmic complexity is used as the discriminating statistic of the surrogate data method. We then perform nonlinear modeling for the short-term prediction between ECG and pulse data to provide further evidence that they conform to deterministic processes. We demonstrate the application of these methods to human electrocardiogram recordings and blood pressure propagation in the fingertip of seven healthy subjects. Our results indicate that bounded aperiodic determinism exists in both ECG and pulse time series. The addition of (the inevitable) dynamic noise means that it is not possible to conclude the underlying system is chaotic.

Multivariate Nonlinear Analysis and Prediction of Shanghai Stock Market

This study attempts to characterize and predict stock returns series in Shanghai stock exchange using the concepts of nonlinear dynamical theory. Surrogate data method of multivariate time series shows that all the stock returns time series exhibit nonlinearity. Multivariate nonlinear prediction methods and univariate nonlinear prediction method, all of which use the concept of phase space reconstruction, are considered. The results indicate that multivariate nonlinear prediction model outperforms univariate nonlinear prediction model, local linear prediction method of multivariate time series outperforms local polynomial prediction method, and BP neural network method. Multivariate nonlinear prediction model is a useful tool for stock price prediction in emerging markets.

Testing for Statistical Significance in Bispectra: A Surrogate Data Approach and Application to Neuroscience

Interactions among neural signals in different frequency bands have become a focus of strong interest in neuroscience. Bispectral analysis, a type of higher order spectral analysis, provides us with the ability to investigate such nonlinear interactions. Based on the fact that the bispectrum of a linear Gaussian process is zero, a surrogate data method was proposed to test the null hypothesis that the original data were generated by a linear Gaussian process. The method was first tested on two simulation examples. It was then applied to local field potential recordings from a monkey performing a visuomotor task. The analysis reveals nonzero bispectra for beta and gamma band activities in the premotor cortex. The rigorous statistical framework proves essential in establishing these results.

Multifractal detrended fluctuation analysis of derivative and spot markets

We investigate the multifractal properties of price increments in the cases of derivative and spot markets. Through the multifractal detrended fluctuation analysis, we estimate the generalized Hurst and the Renyi exponents for price fluctuations. By deriving the singularity spectrum from the above exponents, we quantify the multifractality of a financial time series and compare the multifractal properties of two different markets. The different behavior of each agent-group in transactions is also discussed. In order to identify the nature of the underlying multifractality, we apply the method of surrogate data to both sets of financial data. It is shown that multifractality due to a fat-tailed distribution is significant.