The Correlation And Information Entropy Analysis Of Time Series

Complex systems in the real world is a structure composed of simple units. These units interact with each other, operate on multiple temporal and spatial scales, show up a complex graphical, multi-level structure and self organization, etc., these makes the understanding and characterization of complex system very difficult. One effective method is to analyze the output time series of complex systems, which will help us understand the evolution mechanism and its inherent dynamic interactions. Due to the non-stationarity and nonlinearity is the typical characteristic of a complex system time series, the traditional methods which based on stationary and linear hypothesis is no longer applicable. The main objective of this thesis is to study the correlation, coupling and complexity characteristics of non-stationary and nonlinear time series. Specifical-ly, we study on the cross-correlation between non-stationary time series, explore the structure features of large deviations spectrum, discuss the coupling and complexity of nonlinear time series based on information entropy theory.This thesis is divided into 6 chapters, which is organized as follows:In Chapter 1, we briefly introduce the research background, the objects, the signif-icance and the main works of this thesis.In Chapter 2, we focus on the cross-correlation analysis of non-stationary time series and the dynamic performance of multifractal cross-correlation at multiple time scales. Traditional Pearson correlation coefficient a priori assumed time series is sta-tionary, here we propose multiscale DCCA cross-correlation coefficient, which is used to measure the cross-correlation features between non-stationary time series. When ap-plied it to analyze the financial time series, a strong cross correlation was found between the stock indices from a same country. On the other hand, we propose a method of mul-tiscale multifractal detrended cross-correlation analysis (MM-DCCA) for analyzing the dependence of multifractal cross-correlation properties on multiple time scales. In the empirical analysis, the cross correlation features described by Hurst surface is more abundant, and MM-DCCA method provides a more faithful and more interpretable description of the dynamic mechanism between financial time series than traditional MF-DCCA. We also propose to reduce the scale ranges to analyze short length time series, and some inherent properties which remain hidden when a wide range is used may exhibit perfectly in this way.In Chapter 3, the large deviations spectrum of non-stationary time series was inves-tigated. The scaling exponent a obtained by the MF-DFA method is an important index to measure the long range auto-correlation of time series. If a is nonconstant, then the original time series is multifractal. It is of great practical significance to study the distri-bution structure of the multifractal scale index (multifractal spectrum). In this chapter, the large deviations spectrum was compared with traditional Legendre spectrum. The results show that Legendre spectrum is structurally blind to subtle features like non-concavity or, to a certain extent non scaling of the distributions. We illustrate the large deviations spectrum on both artificial (ARFIMA model) and traffic speed time series, and verify that this kind of approach is able to reveal significant information (represents some traffic characteristics here) that remains hidden with Legendre spectrum. In the mean time, the multiscale analysis of the large deviations spectrum was conducted to quantify the presence or absence of scale invariant phenomenon in the study of traffic speed signals.Chapter 4 focuses on the coupling between nonlinear time series. We propose the asymmetric methods cross permutation entropy (CPE) and its extension weighted cross permutation entropy (WCPE). Different from the traditional transfer entropy, CPE is able of inferring the coupling strength between two time series together with their di-rectionality. The results show that CPE is efficient in the analysis of cross-correlation between nonlinear signals. Besides, in the analysis of financial time series, we found the coupling between stock markets in a same country are generally stronger than what would be expected from different countries. To address some of CPE’s limitations, mainly its inability to differentiate between distinct patterns of a certain motif and the sensitivity of patterns close to the noise floor, we give the method of WCPE. It shows more stable and reliable results than CPE does when applied it to spiky data and AR(1) processes. Besides, we adapt the CPE method to infer the coupling between short-length time series by freely changing the time delay, and test it with Gaussian random series and random walks. The modified method shows the advantages in reducing de-viations of entropy estimation compared with the conventional one.Chapter 5 analyzes the complexity of nonlinear time series. Multiscale entropy (MSE) is effective in quantifying the degree of unpredictability of time series in different time scales. We apply the method in quantifying the complexities of sleep EEG in different sleep stages. Both the simulated signals (logistic map) and the EEGs with different sleep stages are calculated and compared using three types of coarse-graining procedure:including MSEμ(mean), MSEσ2(variance) and MSEskew(skewness). The simulated results show that the generalized MSE (including MSEσ2 and MSEskew) can identify the differences in chaotic more easily with less fluctuation of entropy values in different time scales. As for the analysis of human sleep EEG, we found:(1) at small scales (<0.04sec), the entropy is higher during wakefulness and increasing time scales. (2) At large scales (0.25sec-2sec) in contrast, entropy is higher during deep sleep and lower with increasing time scales.Chapter 6 will be devoted to the summary and some further works.