Research On The Complexity And Causality Of Nonlinear Time Series And Its Applications

In the era of informationization with rapid development,data collection,pro-cessing and analysis play a vital role.As an important research direction of statis-tics,statistical data analysis has attracted wide attention of scientists from different fields because of its rich theoretical backgrounds and extensive applications.Among them,the nonlinearity and correlationship measurements of time series can help us better understand the internal mechanism and dynamic evolution of complex sys-tems objectively existing in the real world,which is also the main research of this dis-sertation.In the dissertation,we study the complexity of nonlinear non-stationary time series,the performance and differences in different time scales based on cumu-lative residual information,probability theory and information entropy;we detect the causal relationship between time series based on empirical mode decomposition and phase consistency;we analyze the fractal structures of time series based on fractal theory and generalized entropy.The main works of this dissertation include:1.Cumulative residual information of time series.Firstly,we generalize the cu-mulative residual Kullback-Leibler information(CRKL)into fractional order based on Tsallis entropy,namely fractional CRKL(FCRKL).In theoretical aspect,we discuss the relationship between FCRKL and CRKL as well as Fisher information.We also give some properties and theorems of FCRKL,and then prove them.In addition,we define the discrete version of FCRKL for calculation.The effectiveness of FCRKL has been proved by numerous simulations.Then we apply FCRKL to financial markets to analyze dissimilarity between stock indices.It turns out that FCRKL can distinguish the differences between stock indices and specific financial crisis in different areas and periods.On the other hand,we combine cumulative residual distribution and distribution entropy(DistEn)to solve the drawbacks of DistEn.The newly proposed measurement is defined as cumulative residual distri-bution entropy(CRDE).CRDE can analyze the distance distribution and amplitude difference simultaneously between vectors obtained from phase space reconstruction,and thus can capture more comprehensive dynamic information of the time series.At the same time,multiscale CRDE(MCRDE)is able to accurately describe the variability of Gaussian white noise and 1/f noise in different scales compared with multiscale DistEn.Moreover,MCRDE reveals the difference of heart rate interval time series under different pathological conditions,and verifies that the physiolog-ical mechanism of healthy young individuals is the most complex,because their body can be adjusted according to the changes of external environment,while the physiological mechanism of elderly and disease individuals is relatively fragile and less complex.2.The causality analysis of time series in different time scales.Causal decom-position method is initially utilized to study the causality between GDP series of ten major countries in recent 48 years.Firstly,we apply ensemble empirical mode decomposition(EEMD)to decompose the GDP change rate series into four IMFs and one residual trend,and then calculate their variance contributions and average periods respectively.Research shows that the economic growth fluctuation of most countries is dominated by the short-term period of 3-5 years.Afterwards,the direc-tion and strength of causal relationship between each pair of IMFs are calculated by the covariance principle of cause and result and phase consistency theory.The di-rection and strength of causality vary to different degrees among countries in terms of different dominant periods,especially in the short and medium long periods.This phenomenon also implies that the economic and trade exchanges and related policies from country to country change with time.However,under the condition of long period,the causality between countries obviously become weaker,which shows that the economic fluctuation of a country in the long run mainly depends on itself development policy and environment.3.The tolerance calculation method of multivariate sample entropy(MVSE)and multivariate multiscale sample entropy(MMSE).The selection of tolerance parameter r has a significant influence on the proportion of similar patterns.The traditional method to calculate the tolerance parameter r is relatively simple,which may be not accurate enough to measure the probabilities of newly generated pat-terns,and therefore we may obtain inaccurate results.Based on such issue,we estimate all possible r values by cumulative histogram method(CHM)and estab-lish MVSE/MMSE profile to measure the complexity of the series more comprehen-sively.By comparing the adaptive MVSE based on CHM results with the traditional MVSE results of different series,it is verified that MVSE based on CHM can not only be used to calculate short time series,but the deviation of results is small with the increasing length of simulated series.At the same time,we define two auxil-iary measures AvgMMSE and SDMMSE to measure the complexity at different scales.AvgMVSE and SDMVSE are utilized for short time series.In addition,MMSE based on CHM can clearly reveal information change of different stock in-dices over time,and retain the volatility characteristics of each channel of stock indices in terms of closing price and trading volume.4.Complexity-entropy causality plane(CECP)and its fractional structures analysis.Firstly,we extend the original CECP to multivariate case and propose multivariate multiscale complexity entropy causality plane(MMCECP).The sta-tistical properties of multivariate time series are analyzed by calculating the mul-tivariate permutation entropy and multivariate statistical complexity measure.We demonstrate the validity and immunity to noise of by different types of simulated data.In the application of financial time series,MMCECP can distinguish the stock indices from developed or emerging countries.On the other hand,we use power spectral entropy(PSD)in place of permutation entropy(PE)to analyze the dynamic characteristics of time series in frequency domain.The primary merit of PSE algorithm is that it does not need to select appropriate parameters.Then the original CECP is generalized to entropy curve analysis by Tsallis and Renyi entropy.The limit values of generalized entropy and complexity measure are given when the parameters tend to 0 and infinity respectively.In the qth complexity-entropy curve,the shapes of entropy curve with respect to chaotic and random series are distin-guishable.The curve of chaotic time series is open whereas the curve of random one is closed.In the rth complexity-entropy curve,we can differentiate different types of series by the curvature of the curve.The curvature of the curve with long-range autocorrelation exists positive values whereas the curvature of uncorrelated series is negative.Meanwhile,the relationship between the two kinds of generalized en-tropy can also be observed by Tsallis-Renyi entropy curve.In the analysis of sleep ECG and EEG signals,it is verified that SWS possesses the highest complexity,and EEG signals can distinguish different sleep states more accurately than ECG signals.Last but not least,we propose a new complexity measure,dispersion Lempel-Ziv complexity(DLZC).Compared with the original LZC,DLZC can project the orig-inal series to the multivariate symbolic sequence,which can attenuate the loss of information.Numerical simulations of different groups validate that DLZC and normalized dispersion entropy(NDE)are not sensitive to parameters and varying degrees of noise.Moreover,we establish a new complexity-entropy plane with NDE,in which NDE is X-axis and DLZC is Y-axis.Chaotic series with three different properties are separable in the aggregation region of complexity-entropy plane,and their multiscale features are obviously different,which proves the effectiveness of the proposed method.In the application of heart rate interval time series,we analyze the complexity of heart rate data under different pathological conditions and exam-ine the correlations between ages,diseases and heart rate series.Results show that healthy young individuals have the most significant complexity whereas aging and disease can moderate the complexity of cardiovascular system.This phenomenon indicates that the individual's ability for adapting external environment is weak-ened with the increase of age or the interference of some diseases,which can lead to the decrease of complexity.