Test your surrogate data before you test for nonlinearity

The schemes for the generation of surrogate data in order to test the null hypothesis of linear stochastic process undergoing nonlinear static transform are investigated as to their consistency in representing the null hypothesis. In particular, we pinpoint some important caveats of the prominent algorithm of amplitude adjusted Fourier transform surrogates (AAFT) and compare it to the iterated AAFT, which is more consistent in representing the null hypothesis. It turns out that in many applications with real data the inferences of nonlinearity after marginal rejection of the null hypothesis were premature and have to be reinvestigated taking into account the inaccuracies in the AAFT algorithm, mainly concerning the mismatching of the linear correlations. In order to deal with such inaccuracies, we propose the use of linear together with nonlinear polynomials as discriminating statistics. The application of this setup to some well-known real data sets cautions against the use of the AAFT algorithm.     

Re-evaluating the performance of the nonlinear prediction error for the detection of deterministic dynamics

The nonlinear prediction error in combination with appropriately generated surrogate data is widely used as a discriminating statistic for weak nonlinearity. In this paper two of the most widely used formulations of this particular measure are evaluated using data from exemplary nonlinear and linear systems. It is found that both methods, contrary to expectation and accepted utility, give rise to substantial false positive and false negative detection rates in purely linear and nonlinear systems respectively. These findings suggest that the nonlinear prediction error should be used with considerable caution. The implications of these findings for the search for nonlinearity in the human electroencephalogram are briefly discussed.     

一种时间序列的弱非线性检验方法

时间序列的非线性是判定该时间序列具有混沌特性的必要条件.提出一种基于线性和非线性AR模型归一化多步预测误差比值的非线性检验量δ_NAR,采用替代数据法来检测时间序列中的弱非线性.以Lorenz时间序列为例,分析了估计非线性检验量δ_NAR时各相关参数对弱非线性检测性能的影响.通过混沌时间序列非线性检测试验,对4种混沌时间序列中的3种,非线性检验量δ_NAR都表现出比基于AIC模型选择准则的非线性检验量相似文献   

Algorithms for generating surrogate data for sparsely quantized time series

The method of surrogate data is frequently used for a statistical examination of nonlinear properties underlying original data. If surrogate data sets are generated by a null hypothesis that the data are derived by a linear process, a rejection of the hypothesis means that the original data have more complex properties. However, we found that if an algorithm for generating surrogate data, for example, amplitude adjusted Fourier transformed, is applied to sparsely quantized data, there are large discrepancies between their power spectrum and that of the original data in lower frequency regions. We performed some simulations to confirm that these errors often lead to false rejections.In this paper, in order to prevent such drawbacks, we advance an extended hypothesis, and propose two improved algorithms for generating surrogate data that reduce the discrepancies of the power spectra. We also confirm the validity of the two improved algorithms with numerical simulations by showing that the extended null hypothesis can be rejected if the time series is produced from chaotic dynamical systems. Finally, we applied these algorithms for analyzing financial tick data as a real example; then we showed that the extended null hypothesis cannot be rejected because the nonlinear statistics or nonlinear prediction errors exhibited are the same as those of the original financial tick time series.     

基于贝叶斯序贯方法的微波效应实验设计

提出了一种基于贝叶斯验后概率的序贯检验方法,建立了检验的判别准则,给出了判别准则临界值的计算方法。在给定截尾实验次数的条件下,提出了一种截尾方案,建立了截尾判断方法。将其用于解决验证效应物在微波作用下失效概率达到给定水平的微波效应实验设计问题,给出了相应的实验方案和所需样本量的估计。最后,通过实例对上述方法的应用过程进行了说明,并和现有方法进行了分析比较。     

Linear and nonlinear time series analysis of the black hole candidate cygnus X-1

We analyze the variability in the x-ray lightcurves of the black hole candidate Cygnus X-1 by linear and nonlinear time series analysis methods. While a linear model describes the overall second order properties of the observed data well, surrogate data analysis reveals a significant deviation from linearity. We discuss the relation between shot noise models usually applied to analyze these data and linear stochastic autoregressive models. We debate statistical and interpretational issues of surrogate data testing for the present context. Finally, we suggest a combination of tools from linear and nonlinear time series analysis methods as a procedure to test the predictions of astrophysical models on observed data.     

Testing nonlinear Markovian hypotheses in dynamical systems

We present a statistical approach for detecting the Markovian character of dynamical systems by analyzing their flow of information. Especially in the presence of noise which is mostly the case for real-world time series, the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions. We circumvent this difficulty by measuring the information flow indirectly. More precisely, we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead. As an extension of Theiler's method of surrogate data (Theiler et al., 1992) this cumulant based information flow (a function of the look-ahead r) is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear Markov processes of increasing order. This procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow. Since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets. Numerical results on artificial and real-world examples including non-chaotic, nonlinear processes, autoregressive models and noisy chaos show the effectiveness of our approach.     

Testing the assumptions of linear prediction analysis in normal vowels

In this paper we develop an improved surrogate data test to show experimental evidence, for all the simple vowels of U.S. English, for both male and female speakers, that Gaussian linear prediction analysis, a ubiquitous technique in current speech technologies, cannot be used to extract all the dynamical structure of real speech time series. The test provides robust evidence undermining the validity of these linear techniques, supporting the assumptions of either dynamical nonlinearity and/or non-Gaussianity common to more recent, complex, efforts at dynamical modeling speech time series. However, an additional finding is that the classical assumptions cannot be ruled out entirely, and plausible evidence is given to explain the success of the linear Gaussian theory as a weak approximation to the true, nonlinear/non-Gaussian dynamics. This supports the use of appropriate hybrid linear/nonlinear/non-Gaussian modeling. With a calibrated calculation of statistic and particular choice of experimental protocol, some of the known systematic problems of the method of surrogate data testing are circumvented to obtain results to support the conclusions to a high level of significance.     

Local filtering of noisy nonlinear time series

We investigate the use of different local nonlinear modelling and nonlinear filtering techniques to clean a noisy time series obtained from a deterministic chaotic systems. The methods are tested on data from the Ikeda map and the Mackey-Glass delay differential equation. We test the results of the filtered times series using the correlation dimension statistic and SNR gain. In all cases we see that local filtering has produced a new time series which is more consistent with the original clean time series.     

A suggested statistical test for measuring bivariate nonlinear dependence

We devise a new asymptotic statistical test to assess independence in bivariate continuous distributions. Our approach is based on the Cramér–von Mises test, in which the empirical process is viewed as the Kullback–Leibler divergence, that is, as the distance between the data under the independence hypothesis and the data empirically observed. We derive the theoretical characteristic function of the limit distribution of the test statistic and find the critical values through computer simulation. A Monte Carlo experiment is considered as assessing the validation and power performance of the test by assuming a bivariate nonlinear dependence structure with fat tails. Two extra examples, respectively, consider stationary and conditionally nonstationary series. Results confirm that our suggested test is consistent and powerful in the presence of bivariate nonlinear dependence even if the environment is non-Gaussian. Our case is illustrated with high-frequency data from stocks listed on the NYSE that recently experienced so-called mini-flash crashes.     

On the evidence of deterministic chaos in ECG: Surrogate and predictability analysis

The question whether the human cardiac system is chaotic or not has been an open one. Recent results in chaos theory have shown that the usual methods, such as saturation of correlation dimension D(2) or the existence of positive Lyapunov exponent, alone do not provide sufficient evidence to confirm the presence of deterministic chaos in an experimental system. The results of surrogate data analysis together with the short-term prediction analysis can be used to check whether a given time series is consistent with the hypothesis of deterministic chaos. In this work nonlinear dynamical tools such as surrogate data analysis, short-term prediction, saturation of D(2) and positive Lyapunov exponent have been applied to measured ECG data for several normal and pathological cases. The pathology presently studied are PVC (Premature Ventricular Contraction), VTA (Ventricular Tachy Arrhythmia), AV (Atrio-Ventricular) block and VF (Ventricular Fibrillation). While these results do not prove that ECG time series is definitely chaotic, they are found to be consistent with the hypothesis of chaotic dynamics. (c) 1998 American Institute of Physics.     

Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.     

A Bayesian approach to identifying structural nonlinearity using free-decay response: Application to damage detection in composites

This work discusses a Bayesian approach to approximating the distribution of parameters governing nonlinear structural systems. Specifically, we use a Markov Chain Monte Carlo method for sampling the posterior parameter distributions thus producing both point and interval estimates for parameters. The method is first used to identify both linear and nonlinear parameters in a multiple degree-of-freedom structural systems using free-decay vibrations. The approach is then applied to the problem of identifying the location, size, and depth of delamination in a model composite beam. The influence of additive Gaussian noise on the response data is explored with respect to the quality of the resulting parameter estimates.     

异常心电节律VT和VF信号的复杂性分析

在替代随机数据假设检验的基础上,提出了针对确定性混沌信号的改进非线性算法,并应用于心脏中的异常节律心电信号分析.指出：室性心动过速(VT)和心室纤颤(VF)是不同于随机信号的,是具有复杂非线性特性的混沌信号.进而在信号定性分析的基础上,从非线性动力学的角度,提出新的复杂度和复杂率的定义和相关的检测方法,对VT和VF进行了定量分析.结果表明,异常心电节律VT和VF信号的定性和定量分析是客观的和准确可靠的. 关键词： 复杂离散度 Lempel-Ziv复杂度 复杂率 平均复杂度 复杂饱和度     

一种超混沌系统的加密特性分析

把欠采样的思想用于混沌保密通信系统的设计中，对Lorenz系统及一种典型的超混沌系统的时间序列进行了分析. 研究发现，加密系统的安全性不仅取决于系统维数，而且还与采样间隔的选取有关. 用VWK非线性检验方法和替代数据检验方法对上述混沌加密系统在不同采样间隔时的输出信号进行了检验. 关键词： 混沌加密 时间序列分析 VWK非线性检验 替代数据检验     

Anderson localization or nonlinear waves: a matter of probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.     

Benchmarking Transfer Entropy Methods for the Study of Linear and Nonlinear Cardio-Respiratory Interactions

Transfer entropy (TE) has been used to identify and quantify interactions between physiological systems. Different methods exist to estimate TE, but there is no consensus about which one performs best in specific applications. In this study, five methods (linear, k-nearest neighbors, fixed-binning with ranking, kernel density estimation and adaptive partitioning) were compared. The comparison was made on three simulation models (linear, nonlinear and linear + nonlinear dynamics). From the simulations, it was found that the best method to quantify the different interactions was adaptive partitioning. This method was then applied on data from a polysomnography study, specifically on the ECG and the respiratory signals (nasal airflow and respiratory effort around the thorax). The hypothesis that the linear and nonlinear components of cardio-respiratory interactions during light and deep sleep change with the sleep stage, was tested. Significant differences, after performing surrogate analysis, indicate an increased TE during deep sleep. However, these differences were found to be dependent on the type of respiratory signal and sampling frequency. These results highlight the importance of selecting the appropriate signals, estimation method and surrogate analysis for the study of linear and nonlinear cardio-respiratory interactions.     

Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems

This paper discusses a class of unexpected irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation that explains the underlying principles. Recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation, provided that the system has a particular natural frequency distribution. The present work introduces a general theory that provides a mathematical foundation and a physical explanation for the near irreversibility phenomena observed and reported in previous publications. Inspired by the properties of probability distribution functions, the general formulation developed here is based on particular properties of harmonic series, which form the common basis of linear dynamic system models. The results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response. In addition to uncovering a new class of dynamic system properties, the results have far-reaching implications in engineering applications where classical vibration damping or absorption techniques may not be effective. Furthermore, the results also support the notion of nearly irreversible energy transfer in conservative linear systems, which until now has been a concept associated exclusively with nonlinear systems.     

Conditional independence graph for nonlinear time series and its application to international financial markets

Conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series, which extend the graphical modeling approach based on partial correlation. The vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series. The conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure. Furthermore, a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence. The efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations. Finally, we show how the method can be applied to international financial markets to investigate the nonlinear independence structure.     

Deterministic nonlinearity in ventricular fibrillation

We provide numerical evidence that the electrocardiogram data collected from pigs during induced ventricular fibrillation cannot be described by a monotonic nonlinear transformation of linearly filtered noise. To establish this we use surrogate techniques and apply two test statistics: (1) the Takens' maximum likelihood estimator of the Grassberger-Procaccia correlation dimension and (2) an improved correlation dimension estimation routine. The improved dimension estimates provide evidence that the correlation dimension of the underlying dynamics during the episode of VF in the first 30 s is slightly less than 6. This result is consistent and reproducible among subjects. (c) 2000 American Institute of Physics.