Study And Application Of Weighted Trend Sample Entropy And Multifractal Cross-correlation Of Time Series

The time series from complex systems have the characteristics of nonstationary and nonlinear.More and more researchers are interested in studying the complexity and long-range correlation of nonstationary time series.Entropy is a common tool that can be used to measure the complexity and irregularity of series.Fractal and multifractal methods are effective tools to explore the long-range correlation and fractal characteristics of non-stationary time series.In recent years,a large number of entropy analysis methods and fractal methods have emerged to study the complexity and long-range correlation of non-stationary time series.However,the existing entropy analysis method and the multifractal method are not efficient and unsuitable for all time series.Therefore,the development of higher efficiency,smaller error and stronger applicability of entropy analysis method and multifractal method are an important research direction.In this thesis,based on the moving fitting window and time-weighted function,we propose a new complexity measurement method,multiscale weighted trend sample entropy(MWTSE),which incorporates symbolic representation and similarity measure.The entropy calculated by this method can describe the complexity at different time scales.We verify the effectiveness of the MWTSE method through two artificial experiments.On the one hand,we applied the MWTSE to calculate the entropy value of the multifractal binomial measure(BMFs)sequence,Gaussian sequence and the synthetic series generated by them.The results show that the synthetic sequence has different entropy values at different scales,which further proves that the MWTSE method can capture the complexity of the sequence at different scales.On the other hand,by adding linear trend,quadratic trend,cubic trend and sinusoidal trend to the original Gaussian series,respectively,and using the MWTSE method to calculate the entropy values of the original Gaussian sequence and the sequence after adding the trend,respectively.In order to facilitate comparison,we also use the existing algorithms,i.e.,multiscale sample entropy based on symbol representation and similarity measures(MSEBSS)and multiscale trend sample entropy(MTSE),to obtain the corresponding entropy values.The comparison results show that the MWTSE method can truly reflect the complexity of the sequence without being affected by the above four trends.Compared with the MTSE,the results show that the MWTSE method has obvious advantages in dealing with the three trends and sinusoidal trends,and is less affected by the trends.Later,we applied MWTSE method to successfully distinguish seven stock time series from three financial markets.The significance test based on entropy value reveals significant differences between stock series from different financial markets.Fractal and multifractal methods are widely used to study univariate time series.However,there are few works that are concerned with the study of multifractal cross-correlation analysis for multivariate time series.In this thesis,the multivariate multifractal detrended cross-correlation analysis(MMXDFA)is firstly proposed to investigate the multifractal features in multivariate time series.MMXDFA may produce oscillations in the fluctuation function and spurious cross correlations.In order to overcome these problems,we then propose the multivariate multifractal temporally weighted detrended cross-correlation analysis(MMTWXDFA).An innovation of MMTWXDFA is the application of the signed Manhattan distance to calculate the local detrended covariance function.To evaluate the performance of the MMXDFA and MMTWXDFA methods,we apply them on some artificially generated multivariate series.Several numerical tests demonstrate that both methods can identify their fractality,but MMTWXDFA can detect long-range cross correlation and more accurately quantify the levels of cross correlation between two multivariate series.To verify the applicability of the MMTWXDFA in real multivariate time series,we attempt to apply the univariate case of MMTWXDFA,called as the multifractal time-weighted detrend correlation analysis(MFTWXDFA)method,to study the cross correlation between air pollutants(PM10,NOX)and meteorological factors(temperature,pressure,wind speed,relative humidity)in urban and rural areas of Hong Kong.The results on the dataset from 1 January 2005 to 31 December 2014 in urban and rural areas of Hong Kong show the existence of multifractal crosscorrelation between all pairs of pollutants and meteorological factors in both urban and rural areas.Different from the results obtained by the previous multifractal detrended cross-correlation analysis(MFDCCA),we found that the multifractality of cross-correlation between PM10and(temperature,pressure)is more obvious in urban area.The multifractal strength of cross-correlation between NOXand wind speed is very weak in either urban or rural area.Furthermore,the MF-TWXDFA crosscorrelation coefficient ?MF-TWXDFAcan capture negative correlation between pollutants and meteorological factors.For PM10,?MF-TWXDFAin urban area is less than or close to that in rural area with respect to these four meteorological factors.The ?MF-TWXDFAof NOXin urban and rural areas shows more complex patterns for varied meteorological factors.Compared with MFDCCA,MF-TWXDFA can provide much richer information about the relationships between pollutants and meteorological factors.