【作者】 王晶；

【导师】 商朋见；

【作者基本信息】 北京交通大学 ， 统计学， 2015， 博士

【摘要】 真实世界中的复杂系统是由相互影响的内在机制控制着,这些内在机制在多重时间和空间尺度上运行,表现出复杂的多组成成分、多层次结构和突现性等特点,这使得理解和刻画复杂系统变得困难。一个重要的方法是分析复杂系统的输出时间序列,来研究其内在的动态演化机制和交互作用。本文的主要研究对象是时间序列的相关性和复杂性,并且着重于其在多重尺度上的变异性分析。由于复杂系统的输出序列具有高度的非平稳性和非线性性,因此传统构建于平稳性和线性假设的理论方法不再适用。在本文中,我们基于去趋势波动分析,探讨非平稳时间序列的重分形相关性以及非线性滤波对它的影响,同时研究此相关性在多重尺度上的变化情况；基于信息理论中的熵,研究时间序列的复杂性及其对多重尺度的依赖性,分析各种趋势的影响及解决方案,同时还讨论时间序列间的多尺度耦合性。本文分为七章,组织结构如下：第一章为引言部分。介绍本文的研究背景、研究对象、研究意义以及主要工作。第二章为非平稳时间序列的重分形交叉相关分析以及非线性滤波对重分形性质的影响。我们提出基于统计矩的重分形交叉相关分析(MFSMXA)来研究时间序列间的交叉相关性和交叉重分形性。我们使用重分形时序进行验证,并引进传统的重分形去趋势波动交叉分析(MFXDFA)和重分形去趋势滑动平均交叉分析(MFXDMA)作为对比模型,发现MFSMXA方法得到了与两传统模型相当的结果,并与理论曲线几乎一致,可靠性较高。另外,我们分析了线性滤波、非线性多项式滤波和对数滤波对时序重分形性质的影响,基于上述分析方法在单个时序上的分析,使用人工信号和交通信号。通过对比滤波前后时序的重分形谱,我们发现：线性滤波几乎不改变时序的重分形性；非线性多项式滤波在一定程度上有所影响,影响程度取决于多项式滤波的最高次幂；对于对数滤波,随着补偿因子取值的减小,重分形谱显著缩水,谱宽明显变小。第三章为非平稳时间序列的重分形相关性在不同尺度上的变异性分析。我们不仅集中研究时间序列的重分形性质,更加重视的是此性质对于不同时间尺度的依赖性。我们提出多尺度重分形分析修正模型,并应用于分析交通信号,发现其具有更加复杂的结构和更加丰富的信息,而这是固定时间尺度的传统MFDFA方法无法获得的。更加重要的是,通过此修正模型,我们无需回避那些具有交叉点的数据或者是在处理过程中人为的缩小研究尺度。Hurst曲面提供了在不同时间尺度上的局部标度指数谱,这便于我们对交叉点进行定位。通过对比去除周期趋势前后交通信号的Hurst曲面,我们发现交叉点主要来源于信号的周期趋势。另外,工作日和周末交通信号的Hurst曲面表现不同,反映其重分形相关性质的不同。我们还分析了交通信号的重分形产生机制,它主要来源于宽概率分布和相关性的改变。最后,通过讨论数据缺失对Hurst曲面的影响,我们发现当缺失百分比高于40%时,分析结果需谨慎处理。第四章研究了不同时间序列的多尺度复杂性变化。MSE系列方法目前已是度量时序复杂度的主要方法。我们应用MSE系列方法先后分析了白噪声和1/f噪声、不同标度指数的长程相关时间序列以及金融序列,并发现其各自复杂度曲线的变化特点。基于排列熵,我们提出多尺度排列熵分析(MSPE),从时序的内在排列角度来讨论其复杂度。通过在交通拥堵指数(TCI)上的实证分析,我们发现MSPE方法能有效度量其复杂度,并发现工作日TCI指数的复杂度曲线不同于周末的,这有助于我们后期更深入的研究。第五章探讨了非线性趋势对时间序列复杂度度量的影响及解决方法。信号中携带的趋势会显著影响其复杂度度量,我们基于经验模式分解(EMD)和傅里叶技术(FT),提出EMD-RCMSE和FT-RCMSE模型。通过模拟信号验证了两模型的有效性。另外,我们将其应用到具有复杂趋势的交通信号,研究表明：(1)交通信号的实际复杂度高于传统MSE方法求得的；(2)工作日模式和周末模式(具有不同的趋势组合)显著影响分析结果；(3)信号复杂度在每一天内随时间不断变化,这是由于人类活动程度的变化造成的。第六章研究了时间序列间的多尺度耦合性。基于度量极短时序间耦合性的内在构成排列(IOTA)法,我们提出分段IOTA模型(即SIOTA)来度量长时间序列间的局部耦合性强度和全局耦合性强度,从不同角度度量两时间序列之间的交互作用。通过分析中国股市的上证指数(SZ)和深成指数,并引入对比模型交叉样本熵,我们发现两股指间的耦合强度很高。另外,我们分析了美国股指DJIA和另外六个国家地区股指之间的耦合性。发现DJIA和SZ之间的耦合强度最低,而与德国DAX和英国FTSE100之间的耦合强度很高。同时研究发现当市场大波动时,局部耦合指数会下降。而随着近年来全球合作化的不断发展,各股市之间的耦合强度逐渐增强。第七章为总结和展望。简要总结了本文的主要结果,同时展望了下一步的工作计划。更多还原

【Abstract】 Real-world complex systems are regulated by interacting mechanisms that operate across multiple temporal and spatial scales, showing up various components, multilay-ered structures and emergence, etc. These make the characterization and understanding of complex systems more difficult. One important approach is to analyze the output time series of complex systems, which helps us understand the underlying dynamic mechanisms and interactions. The object of this study is characterizing the correlations and complexity of time series, and focusing on their dependence on multiple scales. As nonstationarity and nonlinearity exist in time series, traditional theories and meth-ods based on stationarity and linearity are no longer proper to apply. In this thesis, we mainly investigate the multifractal correlations, effects from nonlinear filters and the de-pendence of correlations on time scales, based on detrended fluctuation analysis; study the complexity and its dependence on multiple scales based on entropy from informa-tion theory, in addition, we also discuss the effect of trends on complexity measurement, and the coupling between time series.This thesis consists of7chapters, which is structured as follows:In Chapter1, we briefly introduce the research background, object, significance and the main works of this thesis.In Chapter2, we focus on the multifractal cross-correlation analysis of nonstation-ary time series and the effect on multifractality from various nonlinear filters. We pro-pose multifractal cross-correlation analysis based on statistical moments (MFSMXA), to investigate the long-term cross-correlations and cross-multifractality between time series generated from complex system. Efficiency of this method is shown on multi-fractal series, comparing with the well-known multifractal detrended cross-correlation analysis (MFXDFA) and multifractal detrending moving average cross-correlation anal-ysis (MFXDMA). All three models obtain comparable results, and similar to theoretical curves. In addition, we investigate how various linear and nonlinear filters affect the multifractal properties of both artificial and traffic signals, using the multifractal anal-ysis based on statistical moments, with comparison with multifractal detrended fluctu-ation analysis. Specifically, we study the effect of three common types of transforms: linear, nonlinear polynomial, and logarithmic filters. We compare the multifractal scal-ing properties of signals before and after these filters. It is shown that linear filters do not change the multifractal properties, while the effects of nonlinear polynomial depend on the power of the polynomial filters. In addition, for logarithmic filters, the widths of multifractal spectra change significantly even for much smaller values of the offset parameter.In Chapter3, we concentrate on the variations of multifractal correlations of non-stationary time series on different time scales. We focus not only on that traffic signals have multifractal properties, but also that how such properties depend on the time s-cale. Via the proposed modified multiscale multifractal analysis, traffic signals appear to be far more complex and contain more information which MFDFA cannot explore by using a fixed time scale. More importantly, we do not have to avoid data sets with crossovers or narrow the investigated time scales, which may lead to biased results. Instead, the Hurst surface provides a spectrum of local scaling exponents at different scale ranges, which helps us easily position the crossovers. Through comparing Hurst surfaces for signals before and after removing periodical trends, we find periodicities of traffic signals are the main source of the crossovers. Besides, the Hurst surface of the weekday series behaves differently from that of the weekend series, indicating that the difference of multifractality for the two series. Results also show that multifractality of traffic signals is due to both broad probability density function and correlations. At last, the effects of data loss are also discussed, which suggests that we should carefully handle the results when the percentage of data loss is larger than40%.In Chapter4, we investigate the dependence of complexity for various time series on multiple scales. Multiscale entropy analysis (MSE) and its refined models (CMSE&RCMSE) are main methods to quantify the complexity of time series. We applied MSEs to analyze artificial signals (white noise and l/f noise), different long-term correlated series and financial time series, and obtain their complexity curves. Based on permu-tation entropy, we introduce multiscale permutation entropy analysis (MSPE), to study the complexity from internal arrangement of time series. We employ MSPE method to investigate complexities of traffic congestion series, and obtain that the complexities of weekend traffic time series differ from that of the workday time series, which may be good for future research.In Chapter5, we discuss the effect of nonlinear trends on complexity of time se-ries and its solution. Various trends carried for time series may significantly affect the measurement of complexity. We introduce the EMD-RCMSE and FT-RCMSE models based on empirical mode decomposition (EMD) and Fourier techniques (FT), to elimi-nate the effects of trends. Both models are validated by artificial noises. Furthermore, we applied the two models to analyze traffic signals with multiple trends, and obtain some interesting results:(1) Traffic signals are more complex than the results showing from MSE models;(2) Weekday and weekend patterns (different combination of trend-s) greatly affect the results;(3) Complexity indices change with time at each day, due to the degree of human activities.In Chapter6, we study the coupling/interactions between time series on multiple scales. Based on Inner composition alignment (IOTA), measuring the coupling for very short series, we propose a segmented IOTA (SIOTA) model to quantify the coupling degree for long series. SIOTA provides two important exponents:local coupling degree and global coupling degree, which measure the interactions between times series from different aspects. We apply SlOTA to analyze Shangzheng (SZ) and Shencheng stock indices, and obtain that there are strong coupling between them, comparing with cross-sample entropy. In addition, SIOTA is then applied to investigate interactions between stock market indices of America and six different countries or regions, and obtain many interesting results. We find that the coupling of DJIA and SZ are the weakest, while the coupling between DJIA and DAX/FTSE100are the strongest. We also find that local coupling degrees may decrease when big fluctuations exist. However, the coupling degrees gradually increase as the development of global cooperation in recent years.In Chapter7, we summarize this thesis and the future work.更多还原