Some Studies On Fractal Theory And Time Series Analysis

Fractal theory is a new branch of nonlinear science, whose subjects include un-smoothed, nondifferentiable geometry and long-range correlated structure in nonlinear system. Applying the studies of fractal dimension, multifractal spectrum and correla-tion exponent on substantive characteristics are the main methods of fractal theory. This thesis investigates the fractal characteristic parameters. Topics cover the properties of self-similar sets with positive Lebesgue measure, the effect of signal transformation on multifractal spectrum, the basic feature of detrended fluctuation analysis method and data denoising.This thesis is organized as follows.Chapter 1 introduces the background and research status of the multifractal spec-trum theroy and detrended fluctuation analysis method, the relationship between them, and the main results of this thesis.Chapter 2 is concerned with the question whether each self-similar set on R1 with positive Lebesgue measure contains an interval. It is shown that the question is true for two instances:(1) Let E(f1, f2) be the self-similar set with respect to the similitudes fi(x)=Cix+ei defined on R1, with ci∈(0,1), ei∈R1, i=1,2, then E(f(f1,f2) contains an interval if and only if E(f1,f2) has positive Lebesgue measure; (2) Let E=E(c,Ψ) be the self-similar set with respect to the similitudes fi(x)=cx+cei, c∈(0,1), ei∈Ψ,i= 1,2,..., m defined on R1. Suppose that the setΨc,∞is uniformly discrete, but different expansions inΨc,∞are permitted to be equal. Then E contains an interval if and only if E has positive Lebesgue measure.Chapter 3 investigates the properties of multifractal spectrum. With the data gen-erated by a multiplicative process, the maximum and minimum of Holder exponent are given for the data and its polynomial transformation. Moreover, the perturbation of logarithmic transformation on multifractal spectrum is analyzed and the result that the linear transform makes no difference to multifractal spectrum for random fractal data is shown. Furthermore, the effects of polynomial trend, seasonal trend and exponential trend on multifractal spectrum are analyzed.Chapter 4 researches the algorithm of detrended fluctuation analysis method. By defining the local fluctuation function, that the data normalization makes no difference to result of DFA exponent is shown and the effect of data measuring accuracy on DFA exponent is analyzed. Moreover, by introducing time delay DFA, it is shown that the time delay DFA fluctuation function and the conventional DFA fluctuation function are equal in probability for stationary process.Chapter 5 studies the detrended cross-correlation analysis method(DCCA). As a generation of DFA method, the DCCA method investigates the cross-correlations between different time series. let{xi},{yi} be Gaussian niose process and let{Xi}ni=1, {yi}ni=1 be the time series from the process, respectively. Then, axy≤a=max(ax,ay), where ax, ay be the respective correlated exponent of{Xi}ni=1 and{yi}ni=1,axy be the cross-correlated exponent of{Xi}ni=1 and{yi}ni=1.Chapter 6 proposes a smoothing algorithm based on chaos theory and singular value decomposition. By inosculating the singular value decomposition technique with phase space reconstruction theory, the chaos singular value decomposition method (CSVD) is presented to minimize the effect of noise in fractal data. The effectiveness is demon-strated on DFA method and DCCA method.In chapter 7, the main contributions of this thesis are concluded and some future research issues are presented.