The Correlation And Complexity Analysis Of Time Series

As a crucial branch of statistics, time series analysis has attracted much attention in many disciplines due to the comprehensive theories, universal methods and its wide range of applications. Especially, the correlation and complexity analysis on time series make it possible for us to gain insight into the organization structure and the interactions among the components of real-world complex systems, thus become the focus of this thesis. The time series of complex systems are usually characterized by non-stationarity and nonlinearity, hence, many techniques constructed on the assumptions of stationarity or linearity are not applicable. In this thesis, we study the fractal or multifractal struc-ture of cross correlations for non-stationary time series by detrended cross-correlation analysis, and analyze the detrended cross-correlation matrix for multidimensional time series with non-stationarity. We further study the complexity and information flow of nonlinear time series in the frame of information theory. In the end, we analyze the return intervals of extreme events in presence of diverse autocorrelation structures.This thesis consists of7chapters, which is organized as follows:In Chapter1, we briefly introduce the research background, the significance, the objects and the main works of this thesis.In Chapter2, we focus on the detrended cross-correlation analysis for non-stationary time series. We test the linear cross-correlation function, detrended cross-correlation analysis (DCCA), multifractal detrended cross-correlation analysis (MF-DXA), height cross-correlation analysis (HXA) and multifractal height cross-correlation analysis (MF-HXA) methods by the autoregressive fractional integrated moving average (ARFIMA) process and the binomial multifractal cascade process, where DCCA and MF-DXA seem to be most effective. Therefore, we estimate the local Hurst exponents by de-trended cross-correlation analysis then construct the large deviations spectrum of mul-tifractal cross correlations. We apply the Legendre spectrum and the large deviations spectrum to analyze the multifractal cross correlations in Chinese stock markets. The return series and volatility series of Shanghai and Shenzhen markets both present mul-tifractal cross correlations.In Chapter3, we study the detrended cross-correlation matrix for multidimensional time series with non-stationarities. We firstly analyze the relationship between the de-trended cross-correlation coefficient and Pearson’s cross-correlation coefficient. Then, we construct the detrended cross-correlation matrix by the detrended cross-correlation coefficients between each pair of series. We theoretically derive the distribution of eigenvalues of detrended cross-correlation matrix for purely random series. We also explore the principal components analysis on non-stationary time series, and theoreti-cally prove that the eigenvectors of detrended covariance matrix correspond to the co-efficients of linear combinations.In Chapter4, we study the complexity of nonlinear time series. The information of time series can be delivered by the permutations of neighboring values or the data in state vectors. Permutation entropy is one of the most popular techniques to measure the complexity of time series. We propose modified permutation entropy for small-sized series to cope with the sample size effects, and the models we set verify the effectiveness of this proposal. Furthermore, we propose Renyi permutation entropy, and emphasize on the complexity of different permutations that have different probabilities.In Chapter5, we focus on the information flow between nonlinear time series. We propose the symmetric methods including the permutation mutual information, permu-tation cross-sample entropy and permutation inner composition alignment (IOTA) en-tropy, and also the asymmetric measures of relative transfer entropy and relative contri-bution. The permutation mutual information gives the static interactions between time series. The permutation cross-sample entropy emphasizes on the dynamic persistence of time series. The permutation IOTA entropy gives the degree of coupling between time series. We also introduce the relative transfer entropy to study the contribution of one sub-system on the information flow to another sub-system. We further analyze the contributions of sub-systems to the whole system. We design models and verify the effectiveness of these methods, then study the return series and volatility series of Chinese stock plates, and also the bidirectional information flow between Shanghai and Shenzhen markets.In Chapter6, We study the return intervals of extreme events. The autocorrelation of time series affects the occurrence of extreme events, and therefore, the characteri-zations of return intervals. We focus on the distribution and the correlation of return intervals. Firstly, we theoretically derive that the extreme events of purely random se-ries follow Poisson distribution and the return intervals follow exponential distribution. Next, we study the long-range persistent series by ARFIMA process. For different thresholds, the return intervals yield stretched exponential distribution, and also present long-range persistence. Then, we analyze the long-range anti-persistent series also by ARFIMA process. We find that the return intervals yield exponential distribution and present no correlation, hence, the return intervals behave very similar to the return in- tervals of purely random series. At last, we discuss the short-range correlated series, and find both the distribution and the correlation of return intervals are affected by the parameters of underlying models.In Chapter7, we summarize this thesis and design the plans for future.