Analysis Of Time Series Based On Diffusion Entropy

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.Fractal theory is an important research branch of nonlinear science. The main method employed by fractal theory is analyzing time series’dimension, multifractal spectra and correlation exponents. This paper mainly discusses on the calculation of scaling exponent of time series, analysis of multifractal spectra and the effect of filtering and trend on diffusion entropy analysis based on fractal theory, combined with diffusion technology.The work and innovation point of this paper is as below:This paper firstly gives a brief introduction to diffusion entropy analysis method; secondly generalizes diffusion analysis to multifractal condition and improves it combined with Fourier transform; then proposes multiscale fractal and multiscale multifractal diffusion entropy analysis and does numerical experiment on stock series; lastly plots the Legendre and large deviations spectrum and discusses the effect of filtering on the diffusion entropy analysis and multifractal diffusion entropy analysis.This paper includes seven chapters, and the detailed is as below:Chapter1introduces the research background, object of study and main research techniques, furthermore, the important works of this paper.Chapter2firstly gives a brief introduction on diffusion entropy analysis and ana- lyzes artificial fractal series (ARFIMA). We get its scaling exponent and compares to the theory value so as to verify the validity of the method.We combined Empirical Mode Decomposition(EMD) and diffusion entropy analysis to analyze Beijing congestion ex-ponent.Chapter3firstly generalizes diffusion entropy analysis to multifractal condition and presents multifractal diffusion entropy analysis by employing Renyi entropy. The method uses the techniques of diffusion process and Renyi entropy to focus on the scal-ing behaviors of regular volatility and extreme volatility respectively in developed and emerging markets. Secondly we discuss the series with periodic trends. We introduce Fourier filtering method to eliminate the trend effects and systematically investigate the multifractal long-range correlation of traffic congestion index and daily average temper-ature series. For daily average temperature series of Beijing, regular volatility and ex-treme volatility both exhibit long-range persistence. For traffic congestion index series of Beijing, regular volatility exhibits un-correlated or short correlated, while extreme volatility reveals anti-persistence.Chapter4mainly discuss the diffusion entropy analysis and multifractal diffusion entropy analysis in multiscale condition. Because traditional single and two coefficient model can’t curently describe the character of system, we proposed multiscale diffusion entropy analysis and describe the variety of series of scaling exponent by scaling expo-nent spectra.We also proposed multiscale multifractal diffusion entropy analysis. The method combines the techniques of diffusion process and Renyi entropy to focus on the scaling behaviors of stock index series using a multiscale, which allows us to extend the description of stock index variability to include the dependence on the magnitude of the variability and time scale. We find that stock index variability appears to be far more complex than reported in the studies using a fixed time scale.Chapter5devotes to the estimation method of Legendre spectrum and large devi-ations spectrum of multifractal diffusion entropy analysis from the angle of large de-viation theory and plots the multifractal spectrum of high frequency traffic flow data and stock closed price of day. We let Legendre and large deviations spectrum be given a horizontal translation and get the fair superimposition of the estimated spectra at all scales with the theoretical curve, which is the evidence of the underlying scaling prop-erties of series. This phenomenon proves nonscaling exists in the high frequency traffic data and stock data. Nonscaling is due to the extreme value of series.Traffic and stock series both reproduce non-concavely shaped spectra. In the present case, the non-concavity is due to a few very large oscillations creating locally very small exponents. Then, by spectrum continuity, these exponents create a non-concave varia-tion that cannot be observed with the Legendre spectrum.Chapter6focuses on the effect of filtering on diffusion entropy analysis and mul-tifractal diffusion entropy analysis. We discuss how polynomial filter, exponent filter and logarithmic filter affect the scale exponent and multifractal spectrum of diffusion entropy analysis. The results show:Linear filter doesn’t change the property of fractal and multifractal; Nonlinear polynomial filter affect the fractal property of series and the degree depends on the degree of polynomial. Other filtering all change the scaling ex-ponent of series except linear filter. Meanwhile, we find all kinds of filter don’t change the highest point of multifractal spectrum. The width of multifractal spectrum increases with the increase of parameter of exponent filter. The width of multifractal spectrum decreases with the decrease of parameter of logarithmic filter.In Chapter7, we summarize this thesis and design the plans for future.