| 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. |
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