Research On The Correlation And Complexity Of Nonlinear Time Series

Complex systems from the real world present typical characteristics of complex structures,nonlinearity,multiple layers and multiple scales,etc,making it difficult to interpret their operating modes or underlying mechanisms by studying the complex systems themselves.Therefore,analyzing the output time series of complex systems becomes an important way to explore their underlying mechanisms.Generally,the time series from complex systems are nonlinear and nonstationary,which makes traditional methods based on the assumptions of stationarity and linearity no longer applicable.This thesis mainly focuses on the nonstationary and nonlinear time series generated by real complex systems,using statistical physics and nonlinear models to study their complex characteristics,including correlation,complexity and time irreversibility,etc,in an attempt to providing important clues for revealing the inherent characteristics and further exploration of complex systems.The main content of this thesis includes four parts,as follows:1.Correlation and multifractal correlation.We propose the multivariate detrended fluctuation analysis(MVDFA),and study the interrelationship between the multivariate system and initial variates when the initial variates are mutually independent,mutually correlated and from different systems.Besides,we apply MVDFA to analyze the long-range autocorrelation properties of multivariate financial time series,which confirms the validity and feasibility of MVDFA.In addition,we propose a variance-weighted multi-fractal analysis(WMA),and extract the theoretical formula of the multifractal scaling exponent of the classic binomial multifractal model under weighting.Numerical simulations and empirical analyses illustrate that WMA performs better than the classic unweighted method in distinguishing different signals.2.Information complexity.We generate the cumulative residual entropy(CRE)to the case of fractional order and propose a novel measure of information entropy,named fractional CRE(FCRE).Theoretically,we study the connections of fractional CRE to the CRE and classic differential entropy,and prove some properties of the FCRE.Besides,we show that the FCRE can be estimated by the empirical FCRE of sample data,and provide the proof that the empirical FCRE converges to the true FCRE.A central limit theorem for the empirical FCRE of random samples from the exponential distribution is derived.Numerical simulations support the validity of the empirical estimation of FCRE.Last,we apply FCRE to analyze the information complexity of financial data,finding that FCRE is more applicable and can detect financial crises more accurately.3.Time irreversibility.We study the time irreversibility of sleep EEG signals for the first time by analyzing the time irreversibility of sleep EEG signals at different sleep stages,thoroughly studying the correlation between the irreversibility of sleep signals and age,gender and body mass index of healthy subjects,and the contributions of high-frequency and low-frequency components of signal to its time irreversibility.It turns out that slow-wave sleep has the highest degree of time irreversibility.Only age shows a significant impact on the irreversibility of sleep EEG signals,and the loss of time irreversibility with aging might be due to the decrease of irreversibility in slow oscillations.4.Recurrence.A recurrence means that the recurrent state is somehow similar to a former state in phase space,which reflects the self-similarity between trajectories.Based on the recurrence plot(RP)and recurrence quantification analysis(RQA),we combine them with empirical mode decomposition,to thoroughly study the frequency-and time-evolving recurrent complexity of the intrinsic spatial structures of traffic flow from the time-frequency domain.It turns out that the recurrent structure of traffic flow is quasiperiodic and dominated by its components of medium-and low-frequencies.The signal after removing high-frequency noise presents higher complexity of recurrence,and the time-evolving RQA measures can characterize and detect dynamic transitions between different states of the signal more accurately.