Denoising Nonlinear Time Series Using Singular Spectrum Analysis and Fuzzy Entropy

We present a hybrid singular spectrum analysis(SSA) and fuzzy entropy method to filter noisy nonlinear time series.With this approach,SSA decomposes the noisy time series into its constituent components including both the deterministic behavior and noise,while fuzzy entropy automatically differentiates the optimal dominant components from the noise based on the complexity of each component.We demonstrate the effectiveness of the hybrid approach in reconstructing the Lorenz and Mackey-GIass attractors,as well as improving the multi-step prediction quality of these two series in noisy environments.     

观测窗中多元非线性时间序列的分析

从观察的角度去了解原有系统的特性，不同程度上会受到观测手段的限制．针对一个观测窗口获得的多元时间序列，利用一种能够更好地体现多元时间序列复杂的时空结构的多元奇异系统分析方法，研究多元序列的相关维数和最大李雅普诺夫指数的变化．利用多层感知器神经网络模型分析其可预测性．研究发现，随着观测窗中多元时间序列的变量数目的增加，反映出的复杂性有所提高，稳定性会产生波动，且相应的可预测性也会产生一些差异． 关键词：     

Application of the nonlinear time series prediction method of genetic algorithm for forecasting surface wind of point station in the South China Sea with scatterometer observations

The present work reports the development of nonlinear time series prediction method of genetic algorithm(GA) with singular spectrum analysis(SSA) for forecasting the surface wind of a point station in the South China Sea(SCS) with scatterometer observations.Before the nonlinear technique GA is used for forecasting the time series of surface wind,the SSA is applied to reduce the noise.The surface wind speed and surface wind components from scatterometer observations at three locations in the SCS have been used to develop and test the technique.The predictions have been compared with persistence forecasts in terms of root mean square error.The predicted surface wind with GA and SSA made up to four days(longer for some point station) in advance have been found to be significantly superior to those made by persistence model.This method can serve as a cost-effective alternate prediction technique for forecasting surface wind of a point station in the SCS basin.     

Cross-correlation between time series of vehicles and passengers

We study in this paper a cross-correlation between time series of vehicles and passengers collected in the ferry-boat system (sea route that connects the city of Salvador and Itaparica island, Bahia, Brazil), this study is based on the detrended cross-correlation analysis (DCCA) method. The DCCA method is designed to investigate power-law cross correlations between different simultaneously recorded time series in the presence of nonstationarity. Here in this paper we show that is possible to discriminate cross-correlation between vehicles and passengers and also identify seasonal components.     

一种新的时间序列确定性辨识方法

提出一种新的基于辛几何谱的时间序列确定性检测方法，通过计算原始时间序列及其替代数据的辛几何谱，利用非参数Mann-Whitney 秩和检验方法，可有效地辨别确定性混沌过程和随机过程.通过对常见的随机过程，Lorenz，Rssler，Mackey-Glass，高维耦合方程的仿真检验，说明了方法的有效性. 并通过用Santa Fe测试集中两个序列检验了其对实际时间序列的适用性，最后研究不同数据长度和不同强度噪声对该方法性能的影响说明了其鲁棒性.     

Multifractal properties of Chinese stock market in Shanghai

In this article, we apply three methods of multifractal analysis, partition function method, singular spectrum method and multifractal detrended fluctuation analysis method, to analyze the closing index fluctuations of Shanghai stock market during the past seven years. We have found that Shanghai stock market has weak multifractal features and there are long-range power-law correlations between index series. The shapes of singular spectrums do not change with time scales and their strengths weaken when the scales shorten. But when the orders of partition function increase, the strengths of multifractal increase, the singular spectrums become rougher and the general Hurst exponents decrease. These results provide solid and important values for further study on the dynamic mechanism of stock market price fluctuation.     

水体多环芳烃组分识别小样本分析方法研究

多环芳烃(PAHs)是一类在自然环境中常见且广泛存在的有毒有害有机物。其主要来源有自然界的各种微生物以及植物的生物合成，富含植被区域的天然火灾，火山的喷发物，化石燃料以及人为工业碳氢化合物的不完全燃烧和运输过程中的石油泄漏等。多环芳烃的毒性较为强烈，具有生物致癌性，遗产毒性和致突变性。它对于人体呼吸系统，循环系统，神经系统有着多方面的危害，是一种重要的有机污染物，因此有必要对多环芳烃的现场监测和分析方法进行研究。目前对于多环芳烃的分析方法主要有化学分析法和光谱分析法。化学分析法包含有前处理的化学滴定法，液相色谱法(LC)，高效液相色谱法(HPLC)，气相色谱质谱法（GC-MS）；光谱学分析法涉及紫外吸收光谱，荧光光谱和三维荧光光谱等。三维荧光光谱同时获得激发波长和发射波长的信息，因而包含的光学信息十分丰富，灵敏度高，光谱特征显著，在实际水体的现场检测和水体样本混合组分的快速研究有明显的优势。常见的三维荧光光谱解析方法有平行因子分析法(PARAFAC)，多维偏最小二乘法(N-PLS)等。平行因子分析是分析多环芳烃重叠三维荧光光谱的一种有效方法。但有时由于多种组分的荧光较弱，它对三维荧光光谱的欠定分析并不能得到令人满意的结果。为了从两个样品中提取更多的成分，提出一种基于奇异值分解(SVD)和PARAFAC的方法。首先对每个观测样本进行奇异值分解，根据累积贡献率选取合适的奇异值，构造新的伪样本来突出微弱的荧光信号。然后，将两个观测样品及其对应的伪样品输入PARAFAC，恢复组分光谱。为验证所提方法的有效性，对三组不同荧光强度的多环芳烃重叠三维荧光光谱进行了分析。结果表明，从两个混合样品中提取并识别出6个多环芳烃的纯组分光谱，其分辨发射和激发光谱与标准光谱的相似性均在0.80以上。     

Noise reduction in multiple-echo data sets using singular value decomposition

A method is described for denoising multiple-echo data sets using singular value decomposition (SVD). Images are acquired using a multiple gradient- or spin-echo sequence, and the variation of the signal with echo time (TE) in all pixels is subjected to SVD analysis to determine the components of the signal variation. The least significant components are associated with small singular values and tend to characterize the noise variation. Applying a "minimum variance" filter to the singular values suppresses the noise components in a way that optimally approximates the underlying noise-free images. The result is a reduction in noise in the individual TE images with minimal degradation of the spatial resolution and contrast. Phantom and in vivo results are presented.     

解析动态光谱的奇异值分解最小二乘法的改进

本文改进了处理动态光谱的奇异值分解最小二乘法（SVDLS），通过对模拟的蛋白质二级结构的动态圆二色谱和小牛胸腺DNA在电化学过程中动态紫外光谱的分析，表明该方法不仅给出动态过程中所包含的组分数，而且可同时获得各组分的光谱和各组分分数的分布曲线，是多种动态光谱数据处理的有效的无模型方法。     

On the intrinsic time scales involved in synchronization: a data-driven approach

We address the problem of detecting, from scalar observations, the time scales involved in synchronization of complex oscillators with several spectral components. Using a recent data-driven procedure for analyzing nonlinear and nonstationary signals [Huang, Proc. R. Soc. London A 454, 903 (1998)], we decompose a time series in distinct oscillation modes which may display a time varying spectrum. When applied to coupled oscillators with multiple time scales, we found that motions are captured in a finite number of phase-locked oscillations. Further, in the synchronized state distinct phenomena as phase slips, anti-phase or perfect phase locking can be simultaneously observed at specific time scales. This fully data-driven approach (without a priori choice of filters or basis functions) is tested on numerical examples and illustrated on electric intracranial signals recorded from an epileptic patient. Implications for the study of the build-up of synchronized states in nonstationary and noisy systems are pointed out.     

Improved DOSY NMR data processing by data enhancement and combination of multivariate curve resolution with non-linear least square fitting

The quality of DOSY NMR data can be improved by careful pre-processing techniques. Baseline drift, peak shift, and phase shift commonly exist in real-world DOSY NMR data. These phenomena seriously hinder the data analysis and should be removed as much as possible. In this paper, a series of preprocessing operations are proposed so that the subsequent multivariate curve resolution can yield optimal results. First, the baseline is corrected according to a method by Golotvin and Williams. Next, frequency and phase shift are removed by a new combination of reference deconvolution (FIDDLE), and a method presented by Witjes et al. that can correct several spectra simultaneously. The corrected data are analysed by the combination of multivariate curve resolution with non-linear least square regression (MCR-NLR). The MCR-NLR method turns out to be more robust and leads to better resolution of the pure components than classic MCR.     

Analyzing multiple nonlinear time series with extended Granger causality

Identifying causal relations among simultaneously acquired signals is an important problem in multivariate time series analysis. For linear stochastic systems Granger proposed a simple procedure called the Granger causality to detect such relations. In this work we consider nonlinear extensions of Granger's idea and refer to the result as extended Granger causality. A simple approach implementing the extended Granger causality is presented and applied to multiple chaotic time series and other types of nonlinear signals. In addition, for situations with three or more time series we propose a conditional extended Granger causality measure that enables us to determine whether the causal relation between two signals is direct or mediated by another process.     

基于奇异值分解和中位数绝对偏差的拉曼成像数据去噪方法

拉曼成像是一种无损伤、无需标记的光谱成像技术,它可以提供样品的不同组分的分子指纹信息以及空间分布特征,相比其他成像技术有着更重要的应用。但是拉曼散射的截面积小,灵敏度低,加上在很多实验中为了观察某些组分的动态分布而缩短扫描时间,导致最终得到的成像数据被噪声干扰,因此往往需要对信号进行去噪处理。常规的算法一般都是基于一个给定的数学模型对光谱进行处理,容易造成过滤波,使得信号失真;另外,在处理拉曼成像数据时,常规算法往往是对数据进行逐条光谱去噪,从而忽略了多条光谱之间的相互关系,导致最终的拉曼图像仍然受许多噪点干扰。因此,提出了一种基于奇异值分解和中位数绝对偏差的拉曼成像的信号处理方法,用于拉曼成像数据的去噪处理。该方法首先对拉曼成像数据进行奇异值分解,获得一个奇异值矩阵与两个正交矩阵;然后通过中位数绝对偏差法对奇异值矩阵中的各奇异值进行离群值检测,选取前k个被连续标记的离群值作为要保留的奇异值,并将其余的奇异值赋值为零,得到新的奇异值矩阵;最后用新的奇异值矩阵与两个正交矩阵重新求解得到去噪后的拉曼成像数据。实验中,首先验证了中位数绝对偏差法确定前k个奇异值的正确性,其次分别从处理后的图像质量和信号波形两方面对比了该算法与常规算法的去噪效果。结果证明,中位数绝对偏差法可以快速地确定出合理的k值大小,而且,依据该k值处理后的成像数据不仅在成像质量上消除了大量的噪点,使得组分的空间分布特征清晰可见,也在信号波形上较完美地保留了微小谱峰,并恢复光谱信号。该算法不同于常规算法,能同时对整个拉曼成像数据进行处理,并保留光谱之间的统计特征,是一种更加有效的拉曼成像数据的去噪方法。     

Multifractal Detrended Fluctuation Analysis of Interevent Time Series in a Modified OFC Model

We use multifractal detrended fluctuation analysis(MF-DFA) method to investigate the multifractal behavior of the interevent time series in a modified Olami-Feder-Christensen(OFC) earthquake model on assortative scale-free networks.We determine generalized Hurst exponent and singularity spectrum and find that these fluctuations have multifractal nature.Comparing the MF-DFA results for the original interevent time series with those for shuffled and surrogate series,we conclude that the origin of multifractality is due to both the broadness of probability density function and long-range correlation.     

Spectral and informational analysis of seismicity: An application to the 1996–2012 seismicity of the Northern Caucasus–Azerbaijan part of the greater Caucasus–Kopet Dag region

The time series analysis of seismic sequences needs proper methodologies that allow us to capture the main features of the time dynamics of earthquakes. Among these features, the identification of periodicities along with the quantification of their intensity represents an important task, concerning the detection of regular dynamical behaviours, with clear implications for earthquake prediction. In the present study, we applied three different methods to investigate the time dynamics of the seismic activity of the Northern Caucasus–Azerbaijan part of the Greater Caucasus–Kopet Dag region. We analysed the monthly number of earthquakes which occurred between 1996 and 2012 by means of: (i) the robust estimation of the periodogram, (ii) the singular spectrum analysis (SSA), and (iii) the Fisher–Shannon method. Two main significant periodicities are detected: 102 months and 20 months. The first corresponds actually to the long-term variation of the monthly seismic activity of the area, while the second represents the more intense cyclic component. Periodicities of 7 and 30 months are also identified, but with a lower intensity than the 20-month periodicity. The Fisher–Shannon method has revealed that the long-term variation of the series is also characterized by higher organization and lower degree of disorder. The present study shows how the application of methods from statistical mechanics could contribute to unveil dynamical features in seismicity.     

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.     

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.     

Two-dimensional Fourier transform of arbitrarily sampled NMR data sets

A new procedure for Fourier transform with respect to more than one time variable simultaneously is proposed for NMR data processing. In the case of two-dimensional transform the spectrum is calculated for pairs of frequencies, instead of conventional sequence of one-dimensional transforms. Therefore, it enables one to Fourier transform arbitrarily sampled time domain and thus allows for analysis of high dimensionality spectra acquired in a short time. The proposed method is not limited to radial sampling, it requires only to fulfill the Nyquist theorem considering two or more time domains at the same time. We show the application of new approach to the 3D HNCO spectrum acquired for protein sample with radial and spiral time domain sampling.     

Using chaotic forcing to detect damage in a structure

In this work we develop a numerical test for Holder continuity and apply it and another test for continuity to the difficult problem of detecting damage in structures. We subject a thin metal plate with incremental damage to the plate changes, its filtering properties, and therefore the phase space trajectories of the response chaotic excitation of various bandwidths. Damage to the plate changes its filtering properties and therefore the phase space of the response. Because the data are multivariate (the plate is instrumented with multiple sensors) we use a singular value decomposition of the set of the output time series to reduce the embedding dimension of the response time series. We use two geometric tests to compare an attractor reconstructed from data from an undamaged structure to that reconstructed from data from a damaged structure. These two tests translate to testing for both generalized and differentiable synchronization between responses. We show loss of synchronization of responses with damage to the structure.     

DWT-iPLS在烃源岩漫反射光谱数据处理中的应用

红外漫反射光谱技术被广泛地应用于粉末样品的定性、定量测量中。但是由于粉末样品自身的特点,颗粒度、密度、表面粗糙程度等几何参数的影响,使得漫反射光谱数据的信噪比很低、背景干扰很大。因此需要一种有效的方法对漫反射光谱数据进行预处理来提高信噪比,消除背景干扰。文章采用了离散小波变换对红外漫反射光谱进行了预处理,有效地消除了光谱中的高频噪声和低频背景的干扰,并结合iPLS(间隔偏最小二乘回归)方法进行线性回归分析,建立了用于复杂样品体系组分分析的建模方法(DWT-iPLS)。并以烃源岩的红外漫反射光谱为例,将DWT-iPLS应用于数据的预处理及建立数学模型,其结果与未用DWT预处理的方法相比较,准确度有明显的提高,证明了此方法是一种快速有效的定量分析的建模方法。