Multi-scale Transition Network Analysis Of Time Series

The output of a complex system(time series)contains a lot of information.Through the time series analysis,one can obtain its statistical properties such as the mean,variance and autocorrelation coefficient;and through complex network analysis,one can find its structural characteristics and evolutionary behaviors.These properties provide us a multi-dimensional picture for the time series.Up to now one of the basic task for investigating dynamical process is to construct a general model that can reproduce simultaneously the multi-dimensional picture.The preliminary step is to describe the multi-dimensional picture in a unified theoretical framework.As one of the solution candidates,we proposed a concept called multi-scale transition matrix,which turns out to be able to bridge the statistical and the structural properties of time series.What is more,in the calculations of the multi-scale transition matrix and the Hurst exponent the limited number of data induce generally systematic errors.We proposed also a method called unbiased detrended fluctuation analysis,providing a powerful method to estimate Hurst exponents for short time series,and a theoretical basis for multi-scale transition network analysis of time series.The contribution in the present thesis is three-fold.First,multi-scale transition matrix analysis of stochastic processes.First,we separate the range of time series from the minimal to maximal value into several states,and distribute elements of time series according to the corresponding state space.Then get transition probability matrix under the different time delay,and finally obtain the sequence of transition matrix,called multi-scale transition matrix.Theoretical analysis shows that the eigenvector corresponding to the largest eigenvalue of the multi-scale transition matrix is consistent with the probability distribution function of time series.The upper bound of the autocorrelation function is determined by the second largest eigenvalue of the multi-scale transition matrix.The eigenvector corresponding to the second eigenvalue gives the relaxation behavior of the system.The autocorrelation structure and evolution of time series are obtained by the multi-scale transition matrix.In order to demonstrate,we use the simulated data of the autoregressive conditional heteroscedasticity model and fractional Brownian motion,the calculated results of these time series are consistent with the theoretical analysis.Then,we choose the high frequency data of Shenzhen component index and the word length sequence of the novel Remembrance of Things Past as the empirical examples.Second,multi-scale transition matrix analysis of chaotic time series.In this paper,the results of the multi-scale transition matrix in the Logistics model,Tent model and Lorentz system show that the entropy outdegree and entropy indegree of the multi-scale transition network have the same trend with the Lyapunov exponent.The fitting slope of the eigenvalues of the multi-scale transition matrix in a certain scale interval is opposite to the trend of the Lyapunov exponent.The relaxation behavior of the system is given by the eigenvector corresponding to the second largest eigenvalue of the multi-scale transition matrix.The multi-scale transition matrix gives the autocorrelation structure and evolution behavior of the system in chaotic and non-chaotic states.Third,long-range correlation of short time series.Long-range correlation reflects the scale invariance contained in time series,and the scale invariance is generally measured by the Hurst exponent.There are many methods of estimation,among which the well-known method is de-trended fluctuation analysis(DFA),but the estimation effect of this method is limited by the length of time series.In consideration of a time series in reality is generally not too long,we propose an improvement of DFA,and obtain a new estimation method called UDFA.Extensive calculations show its high performance in evaluating long-range correlation behavior in a very short time series(length of 300～500).As a typical example,the proposed method is used to monitor evolution of fractal gait rhythm of a volunteer.Rich patterns are found in the evolutionary process.To sum up,we combine the statistical and structural properties of time series in a unified theoretical method.In this unified theory,not only the probability distribution,autocorrelation coefficient and other statistical properties of time series are given,but also the internal network structure and evolution of various systems are obtained,which provides a unified theoretical method for us to explore the internal characteristics of time series in different fields.At the same time,we also propose an improvement of De-trended Analysis Fluctuation,solve the limitation that the method needs a long length of time series.The improved method performs well in estimating the long-range correlation of short time series,which lays a theoretical foundation for the multi-scale transition network of short time series,at the same time,the results let us have a new understanding of gait rhythm and rich behaviors are found in the evolutionary process.