Study On Complexity And Irreversibility Of Nonlinear Time Series

Complex system consists of many complex components,which is influenced and regulated by the interaction of multiple time and space scales.Theoretically and practically,most complex systems show characteristics such as multiple lay-ers,nonlinear and self-organization,making it difficult to analyze the operation mechanism of dynamic systems directly.For this reason,people conduct auxiliary research by observing the time series generated by complex systems.However,in practice,time series are often nonlinear and nonstationary in nature,which makes theoretical methods based on stationary and linear assumptions no longer appli-cable.Therefore,we study the real signals of complex systems from two aspects of theory and application,using relevant nonlinear analysis methods and statisti-cal models to characterize the similarity,complexity and irreversibility of nonlinear and nonstationary time series.In this paper,we explore the complex structure of nonlinear and nonstationary time series based on the probability distribution in-formation,cumulative residual information,information entropy theory,and time irreversibility method,in an attempt to explain its potential characteristics and mechanism of action.This paper includes the following four aspects:1.Similarity and cumulative residual information of time series.We improve and generalize it based on distribution entropy（Dist En）.On the one hand,we propose Dist En based on similarity matrix by combining kronecker delta similarity with Chebyshev distance.The model takes the fluctuation of series into account,which is more in line with the actual situation than the traditional Dist En method.The research shows that this method can accurately describe the dynamics of ran-dom series and chaotic series in complex systems,and performs better than the traditional method in distinguishing different types of series.On the other hand,due to the defects of Dist En algorithm,we propose cumulative residual distribution entropy（CRDE）,based on the cumulative residual information and information entropy,which is combined with Rényi entropy to obtain cumulative residual Rényi distribution entropy（CRRDE）.This new measurement approach is fundamentally different from all existing entropy measurement methods in it using the cumulative residual distribution of random variables rather than its density function.We give the definition of CRRDE under continuous and discrete conditions,and prove the accuracy of this method by the classical distribution function.In theory,we study some properties of CRRDE and give the proof.Finally,we apply CRRDE to the analysis of financial time series.The results show that CRRDE is highly sensitive to the fluctuation of the series.2.Research on the complexity of time series based on dispersion entropy.First,we extend the original box-counting dimension（BCD）and dispersion entropy（DE）methods to the dynamic plane for the first time,and propose box-counting di-mension and discrete entropy dynamic plane（BCDDE）.We analyze the statistical properties of time series through their respective advantages.On the one hand,the fractal dimension is used to quantify the irregularity of the series.On the other hand,the series related information is captured by analyzing the different mapping patterns after state space reconstruction and quantifying their distribution.The results show that the information structure of different time series can be identified by the position of points on the dynamic plane.Secondly,an inverse dispersion entropy（IDE）algorithm is proposed to analyze the complexity information relative to the classical DE.On this basis,multiscale inverse dispersion entropy（MIDE）is given to study the richer internal properties in dynamic systems.In addition,inspired by the nature of fractional calculus,this section proposes fractional order IDE（FIDE）and s^αfractional order IDE（SFIDE）.We collected heart rate fluctua-tion data from both healthy and unhealthy subjects.The experimental results show that DE and IDE can provide a more complete understanding of the complexity of signals.Moreover,the MIDE method can distinguish health status,pathological status and aging mode.SFIDE is more sensitive to the change of series than FIDE,and can provide important clues for nonlinear heart rate signals,which is of great significance in cardiac medical diagnosis.3.Study on time irreversibility.Firstly,we study the multiscale irreversible index based on segmentation.Specifically,we use time irreversibility indexes,such as Porta index（P%）,Guzik index（G%）and Ehler index（E）etc.to study the time irreversibility of the simulation series on the segmentation scale.Then,in the empirical analysis of financial time series,the time irreversible presentation of stock markets in different regions is given,so as to further facilitate series identification and classification.In addition,two irreversible detection methods,Kullback Leibler divergence（KLD）and weighted KLD（WKLD）based on horizontal visibility graph（HVG）,are studied and compared.The validity and applicability of the WKLD model are verified by experiments on uniform distribution U（0,1）,1/f noise,frac-tal Brownian motion（f Bm）,ARFIMA,logistic map and other series.In addition,we apply the WKLD method to analyze the disturbance signal（DST index）of the earth’s magnetic field activity.We deeply study the irreversibility of the DST index in different periods.The results show that the time irreversibility of DST index is significant,and the degree of irreversibility changes according to the mutation of geomagnetic signal.This study provides clues for characterizing the nonlinear char-acteristics of magnetic storm signals,so as to detect emergencies more accurately.4.Statistical complexity of nonstationary time series.We propose the distance components of multivariable time series in the multidimensional space（MMDISCO）to estimate the complexity of multivariable time series.According to the classical analysis of variance（ANOVA）,we consider the discrete measure of univariate or multivariable response based on all paired distances between sample elements,cal-culate the distance components（DISCO）by using the distance indexα,and then decompose the total dispersion of the sample.Compared with the traditional com-plexity measurement method,MMDISCO not only gives the total dispersion,but also calculates the complexity within the sample components and the similarity be-tween components.The effectiveness of this method in detecting different series features is verified by simulated data.Then the complexity of sleep EEG signals at different sleep stages is studied.It is found that the complexity of the SWS stage is higher,and the similarity between the N1 stage and the REM stage is stronger.As we all know,multiscale sample entropy（MSE）is an effective method to characterize the information feedback of time series on different time scales.Because the traditional coarse-graining process inevitably leads to the loss of information,we study the coarse-graining in which skewness and kurtosis replace the mean,and then calculate the sample entropy of the new series.This method is called MSE based on skewness and kurtosis(MSE_skew&MSE_kurt).Simulation experiments show that mseskew and msekurt can more accurately describe the complexity of the signal,especially the 1/f noise and white noise series have no information loss on a large scale.For the ECG signal complexity analysis of different pathological states and healthy fetuses,it is found that MSE_skewand MSE_kurtcan distinguish them by using the fluctuation information in the series,and it is verified that the ECG signal complexity of healthy fetuses is high and stable on most scales.Finally,we propose multiscale high-order entropy based on roughness grain exponents（MHER）,which aims to identify dynamics,scale,and oscillatory infor-mation.Like graph entropy,MHER can be regarded as a powerful tool to evalu-ate the complex characteristics of time series structure.Through simulation series analysis,we reveal that MHER can capture the dynamic properties of the system.Although the oscillation amplitude decreases naturally,the range of roughness grain exponents remains stable in the whole scale,which is the characteristic expression of scale invariance.This method provides a broad prospect for studying the com-plexity of dynamic systems.