Some Studies On Nonstationary Time Series

Based on the study of the complex systems, some unique properties of them could be revealed in a better way through the theory of fractal and multifractal. The study on the multifractal theory of nonstationary time series has been one of the mature and popular research subjects, and there are also diverse methods aiming at the study of the multifractal of time series. What is more, the correlation analysis of the subsystems in the whole system provides rich information for accurate description of the complex systems. Therefore, the research on the correlations of different signals is under continuously exploring.In this paper, aiming at the analysis of multifractal, we first present a new method named multifractal signal fluctuation analysis (MSFA), which makes some references to the theory of0-1test on the analysis of chaos systems. With the knowledge of0-1test, we describe the specific procedure of the new method, and particularly illustrate some parts of the equations. Secondly, we use binomial multifractal series (BMS) with typically multifractal character to generate varying multifractal degrees of sequences, and put them into the use of scientific inspection on the new method, verifying the effectiveness and validity of the model. In the meantime, we also use the multifractal detrended fluctuation analysis (MF-DFA) into the analyzing artificial sequences above and obtain the corresponding analysis results. MF-DFA method has been largely applied in practice, and the results have been proven to be accurate and robust. According to the results, we know that in the MSFA, functions have diverse effects with different sign of exponent. Then, with the MSFA and the classical MF-DFA, we analyze the multifractal property of traffic flow series and find that MSFA method is also appropriate for analysis of traffic flow. It has some advantages that0-1test dose not have, and the dependency of its sensitiveness to the degrees of multifractal is much more obvious.DCCA method studies the character of cross-correlation between different signals. Cross-correlation coefficient has been accordingly proposed for indicating the presence/absence and type of cross-correlation, which can quantify the cross-correlation strength. We investigate what effect kinds of linear and nonlinear filters (transformation) have on the cross-correlation coefficient before using it to the actual data, and focus on analyzing the effect of polynomial class, difference class and logarithmic class. In order to assure the accuracy and reliability of the results, we choose two-component ARFIMA process to generate series with different strength of coupling which can get different values of coefficient. Then, we transform the series from different groups, and analyze impact on the coefficient. The results prove that, for the ARFIMA series with perfect cross-correlation (anti-cross-correlation or absence of cross-correlation), the effect of three kinds of transformations to the results of the coefficient is almost faint. However, for the ARFIMA series with inadequate positive (negative) correlation, cubic transformation presents better results than quadric transformation, and difference transform has a great influence on the process of detrending, and for the logarithmic transformation, different values of positive constant have different degrees of influence on the coefficient.At last, we analyze the cross-correlation of traffic flow time series and financial time series respectively. We find that the correlation of traffic flow series changes with the time scales. It is much more stable and presents higher degree of positive correlation when we take financial time series into consideration. Further analysis shows that, at the same time scale, the level of cross-correlation between Nasdaq and S&P500Composite Index is higher when compared with the correlation of Dow Jones and Nasdaq.