The Fractal And Chaotic Analysis Of Time Series

Time series has become a hot topic of researchers all over the world, and it's mainly concerned with the understanding and deepening of the category and concept of determinacy or randomness, regularity or irregularity, simplicity or complexity, and so on. So the study is of great importance in nature science. In this paper, we mainly discuss the multi-fractal nature of time series, the effects of different trends to time series and the cross-correlation behaviors of two groups of time series. Especially, we design a class of new algorithm to investigate the cross-correlation behaviors of two groups of time series, which provide theoretical fundamental for distinguishing the degree of cross-correlation behaviors.In Chapter 1, we introduce the study background, motivation, significance and the main results of this paper, in addition, chaos and fractal theories which are needed.In Chapter 2, a smoothing algorithm based on the Discrete Laplace Transform (DFT) is proposed to minimize the effect of exponential trends and distortion in the log-log plots obtained by MF-DFA techniques. The effectiveness of the technique is demonstrated on monofractal and multifractal data corrupted with exponential trends.In Chapter 3, in order to investigate the causing mechanism of the cross-correlation behaviors, together with the effects of auto-correlation and trends on cross-correlation behaviors, we construct two-component fractionally auto-regressive integrated moving a,vera,ge(ARFIMA) model. Furthermore, we suggest the con-cept of multi-fractal cross-scaling exponents and cross-singularity spectrum, then we give the quantitative relations between them by applying Legendre trans-form. Finally, we investigate the influence of power-law trends, logarithm function trends and exponential trends on the cross-correlation behaviors of two groups of ARFIMA model.In Chapter 4. we demonstrate the technology of separating multi-fractal time series into positive fluctuation signals and the negative fluctuation signals. There exist long-range cross-correlations between the positive fluctuation signals and the negative fluctuation signals, which derived from two adjacent lanes respectively. It is the theory and algorithm foundation for judging the cross-correlation properties.In Chapter 5, the chaos forecasting method based on the weight of distance is suggested. We improve the LinSay neighboring points selection technology based on the largest Lyapunov exponent forecasting theory. The predicted values are, in general, in good agreement with the observed ones within 15 days.In Chapter 6, a smoothing algorithm based on the chaotic singular value decomposition (CSVD) is proposed to minimize the effect of exponential trends and distortion in the log-log plots obtained by MF-DFA techniques. The effectiveness of the technique is demonstrated on monofractal and multifractal data corrupted with exponential trends.In Chapters 7, the main contributions of thesis are concluded and some future research issues are presented.