时间序列的相关性及复杂性研究

发布时间：2018-03-09 10:06

本文选题：时间序列分析　切入点：分形　出处：《北京交通大学》2015年博士论文　论文类型：学位论文

【摘要】：时间序列分析作为统计学科的一个重要分支,因涉及的理论全面、方法普适、应用广泛掀起了跨学科的研究热潮。其中,时间序列的相关性和复杂性研究既是窥探真实世界复杂系统动态演化和相互作用的重要手段,也是本文的主要研究和探讨对象。非平稳性和非线性作为复杂系统时间序列的典型特征,使得构建于平稳性和线性假设的传统理论方法不再适用。在本文中,我们基于消除趋势相关分析研究非平稳时间序列相关呈现的分形或重分形结构以及多维非平稳时间序列的消除趋势相关矩阵；基于信息理论的熵值分析研究非线性时间序列的复杂性和信息流；并讨论不同自相关结构下极端事件重现区间的统计特征。本文共分七章,组织结构如下：第1章为引言部分。介绍本文的研究背景、研究意义、研究对象和主要工作概述。第2章为非平稳时间序列消除趋势相关分析。首先,通过分整自回归移动平均(ARFIMA)过程和二项式重分形级联过程检验线性相关函数、消除趋势相关分析(DCCA)与重分形消除趋势相关分析(MF-DXA)方法、高度相关分析(HXA)与重分形高度相关分析(MF-HXA)方法,并揭示DCCA与MF-DXA方法最为有效。其次,我们基于消除趋势相关分析估计局部Hurst指数构建重分形相关大偏差谱,并比较了勒让德谱与大偏差谱的异同。然后,根据勒让德谱与大偏差谱研究中国股票交易市场的重分形相关关系。上海股票交易市场与深圳股票交易市场收益率序列和波动序列均呈现重分形相关特征。第3章为多维非平稳时间序列消除趋势相关矩阵研究。首先,我们分析消除趋势相关系数与皮尔逊相关系数的联系与区别。通过消除趋势相关系数构建消除趋势相关矩阵,并从理论上推导出纯随机序列消除趋势相关矩阵的特征值分布,一定程度上弥补了随机矩阵理论分析非平稳时间序列的不足。其次,基于消除趋势协方差矩阵研究非平稳时间序列的主元素分析,从理论上证明消除趋势协方差矩阵的特征向量对应序列线性组合的系数。第4章为非线性时间序列复杂性研究。根据时间序列近邻值或状态向量内部数据大小关系确定的排列模式是时间序列的重要信息特征。基于排列模式的排列熵成为研究时间序列复杂性的重要技术手段。我们针对排列熵处理小样本序列出现的样本尺寸效应,移除固定时间延迟项,提出小样本排列熵并检验其有效性。其次,为了实现对不同概率事件侧重分析,我们提出Renyi排列熵,并研究上海股票交易市场日收盘价多种不同排列模式的复杂性。第5章为非线性时间序列信息流研究。我们提出对称性度量——排列互信息、排列交叉样本熵和排列内部构成队列(IOTA)墒,以及非对称性度量——相对信息贡献量。其中,排列互信息侧重于分析序列间的静态相互作用；排列交叉样本熵侧重于分析序列相关作用的动态持续性；排列IOTA熵侧重于分析序列的同步耦合性。由于传递熵无法描述序列间有向信息流所占的比重,我们研究相对传递熵,提出随时间推移各个子系统对整个系统的相对信息贡献量,并通过设计模型检验了以上方法的有效性。实证分析显示中国股票交易市场板块间收益率序列与波动序列存在信息交互。此外,上海和深圳股票交易市场存在有向信息流动。第6章为极端事件重现区间研究。时间序列的自相关性决定极端事件出现的频次和规律,从而影响极端事件重现区间的特征。我们重点研究重现区间的分布和相关性。首先,从理论上推导了纯随机序列极端事件服从泊松分布,重现区间服从指数分布。其次,研究ARFIMA过程产生的长期正相关序列,对于不同的阈值,重现区间均服从拉伸指数分布,并且呈现长期正相关性。然后,研究ARFIMA过程产生的长期反相关序列,发现重现区间服从指数分布,并且不存在自相关性,因此与纯随机序列的重新区间特征极其相似。最后,研究不同模型产生的短相关序列,重现区间的分布和相关性均受到模型参数影响。第7章为结论。归纳本文的主要研究成果,同时展望了下一步的探索方向。
[Abstract]:Time series analysis is an important branch of statistics, involving the theory of comprehensive, universal, wide application has been a hot research interdisciplinary. Among them, an important means of time sequence and the complexity of the real world is on complex system dynamic evolution and interaction, is the main research and discussion the object of nonstationary and nonlinear time series as the typical features of complex systems, traditional theories and methods of making construction on the stationarity and the linear hypothesis is no longer applicable. In this paper, we eliminate the trend based on correlation analysis of non-stationary time series has fractal or multifractal correlation structure and multi-dimensional non-stationary time series to eliminate the trend matrix theory; information entropy analysis of nonlinear time series based on the complexity and information flow; and discuss different autocorrelation nodes The statistical characteristics of the recurrence interval of extreme events are constructed.
This article is divided into seven chapters, and the organizational structure is as follows:
The first chapter is the introduction. It introduces the background of the research, the significance of the research, the research object and the summary of the main work.
The second chapter is the non-stationary time series to eliminate the trend related analysis. Firstly, the autoregressive integrated moving average (ARFIMA) process and the binomial multifractal cascade process test of linear correlation function, detrended correlation analysis (DCCA) and multifractal detrended correlation analysis (MF-DXA) method, height correlation analysis (HXA) and multifractal height correlation analysis (MF-HXA) method, and revealed that the DCCA and MF-DXA method is most effective. Secondly, we eliminate the trend of correlation analysis to estimate local Hurst index construction related multifractal spectrum based on large deviations, and compared the similarities and differences between Legendre spectrum and the large deviation spectrum. Then, according to the Legendre spectrum multifractal correlation study of China stock trading the market with large deviation. The Shanghai stock market and Shenzhen stock market rate of return series and volatility series showed multifractal features.
The third chapter is the research trend of the correlation matrix to eliminate the multidimensional nonstationary time series analysis. First, we eliminate the connection and difference between the correlation coefficient and Pearson correlation coefficient. By eliminating the correlation coefficient to construct the detrended correlation matrix, and theoretically deduced pure random sequence to eliminate the trend of the correlation matrix of value distribution, to a certain extent to make up for the lack of random matrix theory analysis of non-stationary time series analysis. Secondly, the elimination of the main elements of the covariance matrix of the trend based on non-stationary time series, from the theory that eliminate coefficient linear combination corresponding feature vector sequence of the trend of the covariance matrix.
The fourth chapter is the study of nonlinear time series complexity. Determined according to time sequence neighbor value or state vector data within the size of the relationship between the arrangement mode is an important feature of time series. The permutation entropy pattern has become an important technology of time series based on complexity. We focused on sample size effect of the small sample sequence permutation entropy, remove the fixed time delay, the small sample permutation entropy and to test its effectiveness. Secondly, in order to realize the different probability events focus on analysis, we propose Renyi permutation entropy, and study the complexity of the Shanghai stock market trading day closing price of a variety of different patterns.
The fifth chapter is the study of nonlinear time series information flow. We propose the symmetry measure - permutation mutual information, cross entropy and sample arrangement arrangement of internal queue (IOTA) content, and non symmetry measure relative information contribution. Among them, the static interaction arrangement focuses on the analysis of mutual information between sequences; dynamic cross sample arrangement focus on the analysis of entropy sequence correlation function continuity; synchronous coupling arrangement focuses on the analysis of IOTA entropy sequence. Because the transfer entropy can describe sequence between the proportion to the flow of information on the relative proportion, we proposed transfer entropy, over time the various subsystems of the whole system with relative information quantity and effectiveness the above method is verified with the design model. The empirical analysis shows Chinese stock market plate returnseries and volatility series has information interaction. In addition, Shanghai and Shenzhen There is a flow of information to the stock market in Shenzhen Stock Exchange.
The sixth chapter is the study of extreme events. Since the return interval correlation decision rule of the extreme event frequency and time series, thus affecting the characteristics of extreme event interval. We focus on return distribution and correlation of interval. Firstly, the pure random sequence of extreme events of Poisson distribution is deduced in theory, return interval of exponential distribution second, a long-term positive correlation of ARFIMA sequence generation process, for different thresholds, return interval are subject to tensile index distribution, and presents a long-term positive correlation. Then, research on ARFIMA process to produce long-term anti related sequences, that return interval obey the exponential distribution, and there is no correlation, therefore re interval characteristics and pure random sequence is very similar. Finally, a short sequence of different models, to reproduce the distribution and correlation of interval are affected by the model parameters. Ring.
The seventh chapter is the conclusion. The main research results of this paper are summed up, and the direction of the next step is prospected.

【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O211.61;O151.21

【共引文献】