| Complex system is a nonlinear system formed by the interaction of independent units.The time series of its output is one of the important methods to reveal its internal mechanism and operating mechanism.Fractal theory is an important branch of nonlinear science,and fractal analysis provides a mathematical formal system for describing the space-time and dynamic structure of complex systems,which usually uses fractal scaling exponent to depict the system state characteristics.In this dissertation,based on the fractal theory,we mainly study the fractal(multifractal)structure of multivariate time series.We establish multivariate multifractal detrend fluctuation analysis method based on empirical mode decomposition(EMD-MV-MFDFA)and the multiscale multivariate multifractal detrend fluctuation analysis(MMV-MFDFA)method,which are applied to the stock market research,then the influence of filtering on multiscale multifractal auto-correlation is also analyzed.The main work of this paper is as follows:First of all,in order to detect real scale characteristics of multivariate time series with external trend,the multivariate multifractal detrend fluctuation analysis method based on empirical mode decomposition(EMD-MV-MFDFA)is established.The mothod is aimed at dealing with the phenomenon that external trend easily leads to the pseudo-intersection of the fluctuation function graph,by introducing the empirical model decomposition(EMD)algorithm to perform local feature decomposition on the time series,and the remainder representing the monotonic trend is eliminated;then the multivariate multifractal detrend fluctuation analysis(MV-MFDFA)method is used to extract the fractal scaling exponent of the time series fluctuation component,which can judge the auto-correlation of multivariate time series at different time scales.The simulation data confirms that the method can effectively remove the time series of drab trend influence and avoid the occurrence of pseudo-intersection.The empirical analysis reveals the different long-range auto-correlation of the Asian,American and European stock markets in positive and negative orders or large and small fluctuations.Secondly,the multiscale multifractal detrend fluctuation analysis of multivariate time series(MMV-MFDFA)method is established.The limitation of the MV-MFDFA method is verified by the binomial multifractal(BMF)model,that is,the scale range set in advance cannot detect the intersection of time series in the small-scale range.In response to this limitation,a sliding window is used instead of the original fixed window.By setting different window widths and moving lengths,a generalized three-dimensional Hurst surface with quasi-continuous variation is formed to characterize the time series in different scale and different order single fractal(multifractal)characteristics,and accurately locate intersections.Secondly,autoregressive fractal integrated moving average(ARFIMA)and BMF models were used to generate test time series to verify the effectiveness of the proposed method.Finally,it is applied to the study of the stock market again and compared with the results of the MV-MFDFA method.The results show that the MMV-MFDFA method can not only reproduce the results of the MV-MFDFA method,but also describe the dynamic changes of the multifractal auto-correlation with time scale in detail.Finally,the influence of filtering on multiscale multifractal auto-correlation is studied.Based on the MMV-MFDFA method,the generalized average distance between Hurst surfaces is defined to measure the difference of Hurst surfaces.Then the ARFIMA and BMF models are used to construct the simulation time series.The results show that the linear filtering does not affect the single fractal and multifractal scale characteristics of time series,and the degree of influence of polynomial filtering increases with order increasing,moreover,the exponential filtering and logarithmic filtering have different effects depending on parameter values. |
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