Efficient noncausal noise reduction for deterministic time series

We present a simple noncausal noise reduction algorithm for time series that consist of noisy measurements of the state vectors of a deterministic (chaotic) nonlinear system. The underlying dynamical system is assumed to be known and to operate in discrete time. The noise reduction algorithm is an iterative scheme for finding exact deterministic orbits close to the measured noisy orbits. Furthermore, we discuss cases where the solution is not the original orbit but homoclinic to it. (c) 2001 American Institute of Physics.     

非线性时间序列的动力结构突变检测的研究

基于非线性时间序列分析方法——动力学相关因子指数，提出一种新的动力结构突变的检测方法——动力学指数分割算法.通过理想时间序列试验，验证了该方法检测动力结构突变的有效性，同时发现相对少量的尖峰噪声对该方法的影响较小，但连续分布的随机白噪声对其具有一定的影响，并与传统的滑动T检验法和Yamamoto法进行比较，进而讨论它们各自的优缺点. 关键词： 动力学相关因子指数 动力学指数分割算法 噪声 滑动T检验 Yamamoto法     

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.     

Noise reduction by recycling dynamically coupled time series

We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties.     

微波管中离子张弛振荡的混沌现象

微波管中离子张弛振荡引起的噪声对其工作性能有很大影响，已成为微波管领域研究的焦点之一.用包络方程描述电子束特性，用离散的宏粒子模型描述离子特性，在此基础上编写了一维粒子模拟程序，对行波管与速调管模拟，得到张弛振荡的时间序列；将振荡看作是一个复杂非线性动力学系统的响应，分析了离子张弛振荡时间序列的功率谱、重构相图及Lyapunov指数，指出离子张弛振荡具有混沌性质，为研究离子张弛振荡的控制与带有离子振荡噪声信号的处理提供了参考. 关键词： 张弛振荡 粒子模拟 微波管 混沌     

Estimation of noise levels for models of chaotic dynamical systems

We investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series. Using Bayesian methods, we derive estimates for the two noise levels, and find that, given a good model of the dynamics, these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level, whereas a simple calculation of root mean square error badly understates the dynamical noise. We argue that this allows better estimates of the underlying dynamical time series, and so better predictions of its future and of its fundamental dynamical properties.     

Correlation dimension: A pivotal statistic for non-constrained realizations of composite hypotheses in surrogate data analysis

Currently surrogate data analysis can be used to determine if data is consistent with various linear systems, or something else (a nonlinear system). In this paper we propose an extension of these methods in an attempt to make more specific classifications within the class of nonlinear systems.

In the method of surrogate data one estimates the probability distribution of values of a test statistic for a set of experimental data under the assumption that the data is consistent with a given hypothesis. If the probability distribution of the test statistic is different for different dynamical systems consistent with the hypothesis, one must ensure that the surrogate generation technique generates surrogate data that are a good approximation to the data. This is often achieved with a careful choice of surrogate generation method and for noise driven linear surrogates such methods are commonly used.

This paper argues that, in many cases (particularly for nonlinear hypotheses), it is easier to select a test statistic for which the probability distribution of test statistic values is the same for all systems consistent with the hypothesis. For most linear hypotheses one can use a reliable estimator of a dynamic invariant of the underlying class of processes. For more complex, nonlinear hypothesis it requires suitable restatement (or cautious statement) of the hypothesis. Using such statistics one can build nonlinear models of the data and apply the methods of surrogate data to determine if the data is consistent with a simulation from a broad class of models. These ideas are illustrated with estimates of probability distribution functions for correlation dimension estimates of experimental and artificial data, and linear and nonlinear hypotheses.     

Error Covariance Matrix Estimation of Noisy and Dynamically Coupled Time Series

We estimate the covariance matrix of the errors in several dynamically coupled time series corrupted by measurement errors. We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. The estimation of the covariance matrix of the errors is made using a noise reduction algorithm that efficiently exploits the information contained jointly in the dynamically coupled noisy time series. The method is particularly powerful for short length time series with high uncertainties.     

Interspike interval embedding of chaotic signals

According to a theorem of Takens [Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1981), Vol. 898], dynamical state information can be reproduced from a time series of amplitude measurements. In this paper we investigate whether the same information can be reproduced from interspike interval (ISI) measurements. Assuming an integrate-and-fire model coupling the dynamical system to the spike train, there is a one-to-one correspondence between the system states and interspike interval vectors of sufficiently large dimension. The correspondence implies in particular that a data series of interspike intervals, formed in this manner, can be forecast from past history. This capability is demonstrated using a nonlinear prediction algorithm, and is found to be robust to noise. A set of interspike intervals measured from a simple neuronal circuit is studied for deterministic structure using a prediction error statistic. (c) 1995 American Institute of Physics.     

Using the modified sample entropy to detect determinism

A modified sample entropy (mSampEn), based on the nonlinear continuous and convex function, has been proposed and proven to be superior to the standard sample entropy (SampEn) in several aspects. In this Letter, we empirically investigate the ability of the mSampEn statistic combined with surrogate data method to detect determinism. The effects of the datasets length and noise on the proposed method to differentiate between deterministic and stochastic dynamics are tested on several benchmark time series. The noise performance of the mSampEn statistic is also compared with the singular value decomposition (SVD) and symplectic geometry spectrum (SGS) based methods. The results indicate that the mSampEn statistic is a robust index for detecting determinism in short and noisy time series.     

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.     

Estimation of initial conditions and parameters of a chaotic evolution process from a short time series

Tracing back to the initial state of a time-evolutionary process using a segment of historical time series may lead to many meaningful applications. In this paper, we present an estimation method that can detect the initial conditions, unobserved time-varying states and parameters of a dynamical (chaotic) system using a short scalar time series that may be contaminated by noise. The technique based on the Newton-Raphson method and the least-squares algorithm is tolerant to large mismatch between the initial guess and actual values. The feasibility and robustness of this method are illustrated via the numerical examples based on the Lorenz system and Rossler system corrupted with Gaussian noise.     

Surrogate analysis of volatility series from long-range correlated noise

Detrended fluctuation analysis (DFA) [C.-K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons, H.E. Stanley, Nature 356 (1992) 168] of volatility series has been proposed to identify possible nonlinear/multifractal signatures in the given empirical sample [Y. Ashkenazy, P.Ch. Ivanov, S. Havlin, C.-K. Peng, A.L. Goldberger, H.E. Stanley, Phys. Rev. Lett. 86 (2001) 1900; Y. Ashkenazy, S. Havlin, P. Ch. Ivanov, C.-K. Peng, V. Schulte-Frohlinde, H.E. Stanley, Physica A. 323 (2003) 19; T. Kalisky, Y. Ashkenazy, S. Havlin, Phys. Rev. E 72 (2005) 011913]. Long-range volatility correlation can be an outcome of static as well as dynamical nonlinearity. In order to argue in favor of dynamical nonlinearity, surrogate testing is used in conjunction with volatility analysis [Y. Ashkenazy, P.Ch. Ivanov, S. Havlin, C.-K. Peng, A.L. Goldberger, H.E. Stanley, Phys. Rev. Lett. 86 (2001) 1900; Y. Ashkenazy, S. Havlin, P. Ch. Ivanov, C.-K. Peng, V. Schulte-Frohlinde, H.E. Stanley, Physica A. 323 (2003) 19; T. Kalisky, Y. Ashkenazy, S. Havlin, Phys. Rev. E 72 (2005) 011913]. In this brief communication, surrogate testing of volatility series from long-range correlated monofractal noise and their static, invertible nonlinear transforms is investigated. Long-range correlated noise is generated from fractional auto regressive integrated moving average (FARIMA) (0, d, 0), with Gaussian and non-Gaussian innovations. We show significant deviation in the scaling behavior between the empirical sample and the surrogate counterpart at large time-scales in the case of FARIMA (0, d, 0) with non-Gaussian innovations whereas no such discrepancy was observed in the case of Gaussian innovations. The results encourage cautious interpretation of surrogate analysis of volatility series in the presence of non-Gaussian innovations.     

基于分数阶最大相关熵算法的混沌时间序列预测

为提高最大相关熵算法对混沌时间序列的预测速度和精度,提出了一种新的分数阶最大相关熵算法.在采用最大相关熵准则的基础上,利用分数阶微分设计了一种新的权重更新方法.在alpha噪声环境下,采用新的分数阶最大相关熵算法对Mackey-Glass和Lorenz两类具有代表性的混沌时间序列进行预测,并分析了分数阶的阶数对混沌时间序列预测性能的影响.仿真结果表明:与最小均方算法、最大相关熵算法以及分数阶最小均方算法三类自适应滤波算法相比,所提分数阶最大相关熵算法在混沌时间序列预测中能够有效地抑制非高斯脉冲噪声干扰的影响,具有较快收的敛速度和较低的稳态误差.     

Model-based detector and extraction of weak signal frequencies from chaotic data

Detecting a weak signal from chaotic time series is of general interest in science and engineering. In this work we introduce and investigate a signal detection algorithm for which chaos theory, nonlinear dynamical reconstruction techniques, neural networks, and time-frequency analysis are put together in a synergistic manner. By applying the scheme to numerical simulation and different experimental measurement data sets (Henon map, chaotic circuit, and NH(3) laser data sets), we demonstrate that weak signals hidden beneath the noise floor can be detected by using a model-based detector. Particularly, the signal frequencies can be extracted accurately in the time-frequency space. By comparing the model-based method with the standard denoising wavelet technique as well as supervised principal components analysis detector, we further show that the nonlinear dynamics and neural network-based approach performs better in extracting frequencies of weak signals hidden in chaotic time series.     

Duffing系统随机相位抑制混沌与随机共振并存现象的机理研究

针对随机相位作用的Duffing混沌系统, 研究了随机相位强度变化时系统混沌动力学的演化行为及伴随的随机共振现象. 结合Lyapunov指数、庞加莱截面、相图、时间历程图、功率谱等工具, 发现当噪声强度增大时, 系统存在从混沌状态转化为有序状态的过程, 即存在噪声抑制混沌的现象, 且在这一过程中, 系统亦存在随机共振现象, 而且随机共振曲线上最优的噪声强度恰为噪声抑制混沌的参数临界点. 通过含随机相位周期力的平均效应分析并结合系统的分岔图, 探讨了噪声对混沌运动演化的作用机理, 解释了在此过程中随机共振的形成机理, 论证了噪声抑制混沌与随机共振的相互关系.     

Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents

A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is certtral to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Roessler system.     

White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy

Nonlinear noise excitation in nuclear magnetic resonance is a form of nonlinear spectroscopy which exploits the nonlinear susceptibilities in a very direct way. The nonlinear susceptibilities are defined by perturbation theory in the frequency domain. In nonlinear system analysis, on the other hand, the system response is described by a Volterra series in the time domain. The kernels of the Volterra functionals carry the information about the system and are to be determined by experiment.The series expansion of a molecular, atomic or nuclear system response is derived in quantum mechanics by time dependent perturbation theory, leading to a Volterra series with time ordered, triangular kernels. The kernels are multi-dimensional products of decaying exponentials, which describe coherence decays of particular density matrix elements. The Fourier transforms of the triangular Volterra kernels are the susceptibilies, which are formally identical in NMR spectroscopy and nonlinear optical spectroscopy. The nonlinear susceptibilities are multi-dimensional spectra, which in NMR spectroscopy reveal the spin communication pathways. These are established by various forms of single quantum coherence connectivities, such as indirect coupling, chemical exchange, cross-relaxation, dipolar and quadrupolar coupling.If the functionals of the Volterra series are orthogonalized with respect to Gaussian white noise excitation, the Wiener series results. The Wiener kernels can be derived by multi-dimensional cross-correlation of the system response with different powers of the Gaussian white noise excitation.Cross-correlation of the transverse magnetization response to noise excitation in NMR leads to multi-dimensional time functions, the Fourier transforms of which closely resemble the nonlinear susceptibilities. The cross-correlation spectra differ from the susceptibilities in the governing Liouvillean and the dynamic density matrix, which are affected by saturation for continuous excitation. Cross-correlation spectra and susceptibilities converge for vanishing excitation power. Therefore the cross-correlation spectra are referred to as stochastic susceptibilities.In stochastic NMR spectroscopy only odd order susceptibilities exist for transverse magnetization. The first nonlinear order is the third, and the nonlinear spectral information is derived from the third order susceptibility. Higher order susceptibilities are not feasible to derive from experimental data. An important share of the nonlinear information is found on the six subdiagonal 2D cross-sections through the third order susceptibility. These cross-sections arise in three pairs, which carry distinct information, separated according to longitudinal magnetization and population effects, zero quantum coherences, and double quantum coherences.In practice a nonlinear 3D spectrum is computed from experimental data by an algorithm in the frequency domain, which yields access to selected regions in the 3D spectrum. This spectrum is the symmetrized stochastic third order susceptibility. All its sub-diagonal 2D cross-sections are equivalent. They are the average of the six different sub-diagonal 2D cross-sections through the asymmetric third order susceptibility.The stochastic excitation technique in NMR is characterized by several unique attributes. (1) There is no minimum time for a data acquisition cycle, so that, at the expense of signal-to-noise ratio, strong samples can be investigated faster with stochastic NMR than with pulsed FT NMR. (2) Stochastic excitation tests the sample extensively, and measures a maximum amount of information in a single experiment. This feature is of particular interest for investigation of short-lived samples and of samples with little a priori information. (3) An experiment with stochastic excitation is simple to perform, but the data processing is more complex than in FT spectroscopy. (4) The nonlinear information about spin communication pathways is derived for individual frequency regions only, which are identified in the stochastic ID spectrum. This information is located primarily on the sub-diagonal 2D cross-sections through the third order susceptibility. (5) Stochastic NMR spectra derived from random noise excitation are contaminated by systematic noise. In the sub-diagonal 2D cross-sections the noise is reduced by filtering and symmetrization during data processing. (6) Sub-diagonal 2D cross-sections are sensitive to experimental phase distortions in one direction only. They are readily adjusted in phase with the same parameters as the ID spectrum. (7) Stochastic multi-dimensional spectra can be computed at variable resolution from one and the same set of raw data.So far stochastic NMR spectroscopy is not applied routinely in analytical spectroscopy. More practical experience is needed to evaluate its merits in comparison with Fourier transform NMR.Stochastic excitation is distinguished from continuous wave and sparsely pulsed excitation by low input power in connection with large bandwidth. This important property cannot be exploited in high resolution NMR in liquids, because excitation power is not a restricting factor in this case. The situation is different in NMR imaging, where large field gradients require large bandwidths and the excitation power becomes a point of concern. For this reason stochastic RF excitation is being investigated in NMR imaging.The multi-dimensional cross-correlation functions obtained from random noise excitation generally are contaminated by systematic noise. The occurrence of systematic noise can be avoided if pseudo-random excitation is used in combination with a transformation of the system response to obtain the kernels. This technique is used successfully in Hadamard spectroscopy, where the linear Volterra kernel is the Hadamard transform of the linear response functional. Nonlinear transformations^(220,221) for retrieval of nonlinear kernels have not yet been realized in NMR spectroscopy.The cross-correlation technique underlying the data evaluation in stochastic nonlinear system analysis is equivalent to interferometry in optical spectroscopy. The Michelson interferometer is the most prominent optical correlator. The time resolution of the kernels derived by cross-correlation is determined by the inverse bandwidth of the excitation. With the Michelson interferometer a time resolution of 10⁻¹⁴ s is achieved in IR spectroscopy. Since the IR correlogramm is Fourier transformed for spectral analysis, the time resolution cannot be exploited otherwise. For analysis of fast time dependent processes a two-dimensional interferometer should be constructed, which performs a 2D cross-correlation of the system response to two in general different noise inputs. One input pumps the time dependent process, the other is used to investigate the time dependence spectroscopically. This technique is introduced by the name of ‘two-dimensional interferometry’. It uses low excitation power, but provides high time resolution at large response energy. Related work is pursued in nonlinear optical spectroscopy with incoherent excitation. In this area the use of broad band lasers is investigated for generation of echoes and for correlation based measurements of relaxation times.     

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.     

From nonlinearity to causality: statistical testing and inference of physical mechanisms underlying complex dynamics

Principles and applications of statistical testing as a tool for inference of underlying mechanisms from experimental time series are discussed. The computational realizations of the test null hypothesis known as the surrogate data are introduced within the context of discerning nonlinear dynamics from noise, and discussed in examples of testing for nonlinearity in atmospheric dynamics, solar cycle and brain signals. The concept is further generalized for detection of directional interactions, or causality in bivariate time series.