复杂系统的非线性时间序列分析及谱分析
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摘要
复杂性科学属于跨各个学科的新兴领域,主要研究复杂系统及其复杂性.复杂系统一般由多个简单的单元构成,且单元之间有非线性相互作用,如经济系统、交通系统、生物系统等.由于复杂系统自身的特点,传统的线性方法已不适用于研究复杂系统,因此其推动了非线性科学的发展.分形理论是非线性科学的一个重要研究分支,自相似是分形的一个重要特征.目前在经济序列、交通序列及生物序列中均已发现自相关性,且在系统内某些单元之间存在交叉相关行为.如何得到能够反映系统内在波动性的标度指数以及反映系统单元之间相互作用的交叉相关指数具有极其重要的意义.
本文首先采用多种方法研究如何去除时间序列中的附加趋势,进而得到时间序列的真实内在标度特征；其次,基于股票序列自身存在长相关性的特征,利用分形方法研究不同股票序列的自相关性及股票序列间的交叉相关性,结合K-近邻(KNN)和经验模式分解方法(EMD)进一步分析股票序列,并给出其发展趋势的预测结论；再次,生物系统较经济系统和交通系统更为复杂,特别地,人类脑部活动研究仍是极具挑战性的工作,本文对脑部电波信号的各频带间同步性问题进行了初步探索性研究,并得到某些不同于传统观点的创新性结论.
本文共分为六章内容,具体内容如下所述：
第一章为绪论,主要介绍研究背景、研究对象及主要研究方法,概述本文主要工作.
第二章为几种改进的去除趋势方法研究,主要提出时间序列波动分析方法的三种改进形式.第一、结合B-样条与MF-DFA方法,去除序列的附加指数趋势和周期趋势；第二、分别将DFA、R/S分析、DMA、CC、DCCA方法与Laplace变换相结合,去除序列的周期趋势,得到序列自身固有的相关性及序列间固有的交叉相关性；第三、提出基于V-系统的DFA方法,验证其去除序列中附加的幂指数趋势、周期趋势、组合趋势及分段函数趋势的有效性和正确性,提出一种新颖的基于完备正交函数系的时间序列趋势去除方法.
第三章为基于DCCA方法的交叉相关性研究,主要研究除趋势波动分析(DCCA)方法.第一、在理论上论证DFA及DCCA方法叠加公式的正确性,且进行试验验证；第二、针对国内外主要股票交易机构的股票序列,采用DCCA方法分析其自相关性,以及股票交易间的交叉相关性；第三、基于时间延迟的DCCA万法,分析由时间延迟而产生的股票序列间交叉相关性.
第四章为基于EMD与KNN的时间序列预测方法研究,主要研究时间序列的非参数预测方法,提出一种基于改进KNN和EMD的预测方法,称之为EMD-KNN方法.该方法无需特定的模型与参数,只需序列提供足够多的历史数据即可预测其发展趋势及未来数值,根据对比EMD-KNN方法、KNN方法和ARIMA万法的实验结果,验证其预测效果的有效性.
第五章为脑电波信号同步性研究,主要研究脑部电波信号各频带间的同步性问题,研究对象是安装于大脑皮层的电极所记录的脑电波(EEG)信号,利用傅立叶变换(FFT)和谱分析方法将EEG信号分解为五个频带：δ波、θ波、α波、σ波和β波.针对各频带的探索性研究分为两部分：第一、脑部处于不同睡眠状态时的EEG各频带间同步性研究；第二、脑部处于睡眠状态转换时的EEG各频带间同步性研究.
第六章为总结与展望,归纳本文的主要工作成果,展望后续工作的研究内容与方向.
Complexity science is a new rising cross-discipline which mainly discusses the complex system and complexity including many fields. Complex system typically has a large number of simple components, and there are nonlinear interactions between the components, such as economic system, traffic system and biological system. Due to certain characteristics of complex system, it's hard to study by linear methods which promotes the development of nonlinear science. The investigation of fractal theory is very important for nonlinear science. One of the main feature of fractal is its scale-invariant. Nowadays, it has been clarified that the existence of auto-correlations in economic system, traffic system, biological system and cross-correlations between sys-tem components. It is of great importance to develop suitable techniques to capture the scaling exponents and cross-correlation exponents. In order to find the correct scaling behavior of time series, we propose diverse techniques for minimizing the effects of different superimposed trends. In this paper, auto-correlations of several stock markets and cross-correlations between different stock markets are investigated by fractal meth-ods. Furthermore, existence and determination of long-range correlations of stock time series make it possible for forecasting. A new forecasting technique based on K nearest neighbor and empirical mode decomposition method is introduced to predict stock time series. Brain activity of biological system is a dynamical system more complex than economic and traffic system. We pointed out the study of electroencephalogram(EEG) is a crucial and challenging research subject. The synchronization of different frequency bands of EEG signal is analyzed in this paper, and some radical ideas which different from traditional views are being pointed out.
Chapter1introduces the research background, object of study and main research techniques, furthermore, the important works of this paper.
In Chapter2, three modifications of fluctuation analysis are proposed. Firstly, B-spline method is combined with MF-DFA for minimizing the effect of exponential trends and periodical trends superimposed in series; secondly, Laplace transform is used in DFA, R/S analysis, DMA, CC and DCCA method for minimizing external periodi-cal trends, and resembled the correlation and cross-correlation behavior of original time series; lastly, A new detrending method based on orthogonal V-system is devoted for eliminating power-law trends, periodical trends, combined trends and piecewise func-tion trends.
Chapter3is dedicated to detrended cross-correlation analysis(DCCA). Firstly, the superposition rules are studied both in theoretical and experimental aspect. Secondly, the correlations of stock markets and cross-correlations between different stock markets are analyzed by applying DCCA technique. Finally, cross-correlations of stock markets based on time delay are discussed using time delay DCCA.
Chapter4focuses on a new non-parameter forecasting tool. The new predict-ing method base on modified K nearest neighbor and empirical mode decomposition method, called EMD-KNN method, is given in this chapter. This method does not re-quire certain models or parameters, and only sufficiently large quantities of data which represent the underlying system. For testing forecasting accuracy, the results of EMD-KNN, KNN and ARIMA method are compared.
Chapter5analyzes the synchronization of frequency bands in EEG signal. The researching object of this chapter is the EEG signal which is recorded by setting elec-trodes on human's scalp. The EEG signal can be decomposed into five frequency bands by applying fourier transform and spectral analysis:δ,θ,α,σ andβ. We mainly follow two different parts to investigate the frequency bands, first part is synchronization of fre-quency bands during different sleep stages, second part is synchronization of frequency bands in transitions of sleep stages.
Chapter6devotes to the summary and some further works.
本文首先采用多种方法研究如何去除时间序列中的附加趋势,进而得到时间序列的真实内在标度特征；其次,基于股票序列自身存在长相关性的特征,利用分形方法研究不同股票序列的自相关性及股票序列间的交叉相关性,结合K-近邻(KNN)和经验模式分解方法(EMD)进一步分析股票序列,并给出其发展趋势的预测结论；再次,生物系统较经济系统和交通系统更为复杂,特别地,人类脑部活动研究仍是极具挑战性的工作,本文对脑部电波信号的各频带间同步性问题进行了初步探索性研究,并得到某些不同于传统观点的创新性结论.
本文共分为六章内容,具体内容如下所述：
第一章为绪论,主要介绍研究背景、研究对象及主要研究方法,概述本文主要工作.
第二章为几种改进的去除趋势方法研究,主要提出时间序列波动分析方法的三种改进形式.第一、结合B-样条与MF-DFA方法,去除序列的附加指数趋势和周期趋势；第二、分别将DFA、R/S分析、DMA、CC、DCCA方法与Laplace变换相结合,去除序列的周期趋势,得到序列自身固有的相关性及序列间固有的交叉相关性；第三、提出基于V-系统的DFA方法,验证其去除序列中附加的幂指数趋势、周期趋势、组合趋势及分段函数趋势的有效性和正确性,提出一种新颖的基于完备正交函数系的时间序列趋势去除方法.
第三章为基于DCCA方法的交叉相关性研究,主要研究除趋势波动分析(DCCA)方法.第一、在理论上论证DFA及DCCA方法叠加公式的正确性,且进行试验验证；第二、针对国内外主要股票交易机构的股票序列,采用DCCA方法分析其自相关性,以及股票交易间的交叉相关性；第三、基于时间延迟的DCCA万法,分析由时间延迟而产生的股票序列间交叉相关性.
第四章为基于EMD与KNN的时间序列预测方法研究,主要研究时间序列的非参数预测方法,提出一种基于改进KNN和EMD的预测方法,称之为EMD-KNN方法.该方法无需特定的模型与参数,只需序列提供足够多的历史数据即可预测其发展趋势及未来数值,根据对比EMD-KNN方法、KNN方法和ARIMA万法的实验结果,验证其预测效果的有效性.
第五章为脑电波信号同步性研究,主要研究脑部电波信号各频带间的同步性问题,研究对象是安装于大脑皮层的电极所记录的脑电波(EEG)信号,利用傅立叶变换(FFT)和谱分析方法将EEG信号分解为五个频带：δ波、θ波、α波、σ波和β波.针对各频带的探索性研究分为两部分：第一、脑部处于不同睡眠状态时的EEG各频带间同步性研究；第二、脑部处于睡眠状态转换时的EEG各频带间同步性研究.
第六章为总结与展望,归纳本文的主要工作成果,展望后续工作的研究内容与方向.
Complexity science is a new rising cross-discipline which mainly discusses the complex system and complexity including many fields. Complex system typically has a large number of simple components, and there are nonlinear interactions between the components, such as economic system, traffic system and biological system. Due to certain characteristics of complex system, it's hard to study by linear methods which promotes the development of nonlinear science. The investigation of fractal theory is very important for nonlinear science. One of the main feature of fractal is its scale-invariant. Nowadays, it has been clarified that the existence of auto-correlations in economic system, traffic system, biological system and cross-correlations between sys-tem components. It is of great importance to develop suitable techniques to capture the scaling exponents and cross-correlation exponents. In order to find the correct scaling behavior of time series, we propose diverse techniques for minimizing the effects of different superimposed trends. In this paper, auto-correlations of several stock markets and cross-correlations between different stock markets are investigated by fractal meth-ods. Furthermore, existence and determination of long-range correlations of stock time series make it possible for forecasting. A new forecasting technique based on K nearest neighbor and empirical mode decomposition method is introduced to predict stock time series. Brain activity of biological system is a dynamical system more complex than economic and traffic system. We pointed out the study of electroencephalogram(EEG) is a crucial and challenging research subject. The synchronization of different frequency bands of EEG signal is analyzed in this paper, and some radical ideas which different from traditional views are being pointed out.
Chapter1introduces the research background, object of study and main research techniques, furthermore, the important works of this paper.
In Chapter2, three modifications of fluctuation analysis are proposed. Firstly, B-spline method is combined with MF-DFA for minimizing the effect of exponential trends and periodical trends superimposed in series; secondly, Laplace transform is used in DFA, R/S analysis, DMA, CC and DCCA method for minimizing external periodi-cal trends, and resembled the correlation and cross-correlation behavior of original time series; lastly, A new detrending method based on orthogonal V-system is devoted for eliminating power-law trends, periodical trends, combined trends and piecewise func-tion trends.
Chapter3is dedicated to detrended cross-correlation analysis(DCCA). Firstly, the superposition rules are studied both in theoretical and experimental aspect. Secondly, the correlations of stock markets and cross-correlations between different stock markets are analyzed by applying DCCA technique. Finally, cross-correlations of stock markets based on time delay are discussed using time delay DCCA.
Chapter4focuses on a new non-parameter forecasting tool. The new predict-ing method base on modified K nearest neighbor and empirical mode decomposition method, called EMD-KNN method, is given in this chapter. This method does not re-quire certain models or parameters, and only sufficiently large quantities of data which represent the underlying system. For testing forecasting accuracy, the results of EMD-KNN, KNN and ARIMA method are compared.
Chapter5analyzes the synchronization of frequency bands in EEG signal. The researching object of this chapter is the EEG signal which is recorded by setting elec-trodes on human's scalp. The EEG signal can be decomposed into five frequency bands by applying fourier transform and spectral analysis:δ,θ,α,σ andβ. We mainly follow two different parts to investigate the frequency bands, first part is synchronization of fre-quency bands during different sleep stages, second part is synchronization of frequency bands in transitions of sleep stages.
Chapter6devotes to the summary and some further works.
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