Nonlinear Time Series Analysis By Means Of Multi-scale Transition Network Approaches

In the era of big data,the exploring of data complexity has entered a new stage of rapid development.Complex networks have become an important method for describ-ing the interdependent structures of complex systems,which help to obtain useful infor-mation from seemingly chaotic data.Time series analysis based on complex networks is one of the important applications in network science,which has made great progress in recent years.Dynamical properties of time series are characterized by investigating network structures.At present,several methods such as recurrence networks,visibility graphs,and transition networks have been proposed,characterizing the complexity of time series from different perspectives.In the traditional symbolic analysis based on chaos theory,the transition frequencies between symbols have not received much at-tention.In addition,most of the transition network methods only focus on structural features at a particular spatial scale.In this thesis,we will propose a novel multi-scale transition network approach,and verify the practicality through analyzing experimental data such as f MRI and EEG of the brain.Specifically,the findings of this thesis are the following.1.Unlike traditional complex networks for static time series analysis,this thesis proposes a multi-scale transition network method from the perspective of symbolic anal-ysis,taking into account the temporal transitions between phase space regions.In this network,nodes are symbol vectors,and edges are the transition frequencies between nodes.Network entropyε_mis proposed to characterize the non-uniform transitions be-tween states of the network.The order of networks increases when the symbol vector length m is increased,such that the deterministic structures of the underlying system continue to be retained in the higher order network,while observational noise depart gradually.Therefore,the calculation ofε_mshows high robustness in characterizing dy-namics which overcomes the noise effects.Using three typical systems as examples,including Logistic map,Hénon map and R?ssler,we demonstrate thatε_msuccessfully characterizes the bifurcation diagrams when changing the corresponding control pa-rameters.The advantage of this method is that we choose a very small number of phase space partition K and simultaneously increasing the symbol vector length m,which overcomes the shortcomings of histogram based static methods.ε_mshows a high cor-relation with the traditional Lyapunov exponents.The study of the temporal complexity of brain data such as f MRI,EEG,and ECG validate the practicality of network methods.2.Due to the effects of both the deterministic components of chaotic systems and finite length of time series,there are many forbidden or missing nodes in the result-ing multi-scale time series networks.This thesis further proposes complementary net-works to characterize the importance of missing edges.Statistical complexity measures from three different perspectives are proposed to characterize the differences between time series networks and that of uniformly distributed random ones.In particular,(1)the node-wise transition complexity n WTC?(2)node averaged transition complexity n AT C?(3)dispersion complexity which measures the difference between the connec-tions of each node and its average connection(TDC).We demonstrate that the com-plexity of complementary networks can not only describe the bifurcation diagrams,but also clearly characterize the hysteresis phenomenon between the chaotic state and syn-chronous phase-locked state in the coupled R?ssler oscillators.The method character-izes difference between the chaotic states and periodic ones of the system.These results based on EEG data show that there is a significant statistical complexity difference be-tween the groups of epileptic patients and healthy controls.3.All edges of each node in a multi-scale transition network contain both forward and backward temporal information.This thesis reveals the nonlinear characteristics of the system by characterizing the asymmetry of forward and backward edge distribu-tions,which serves as an important criterion for time series reversibility.Quantitative measures of Kullback-Leibler and Jensen-Shanon divergence are proposed in the fol-lowing to aspects:Firstly,the differences in the distributions of the original network’s forward and backward edges?Secondly,the differences in the distributions of network edges between the original and the reversed time series.The results suggest that the distributions of temporal forward and backward edges in the network show symmetry for reversible systems,where both KL and JS divergence values are converged to zeros.In contrast,significant asymmetry exists in irreversible systems,where both KL and JS divergence values are non-zero,making it an important criterion for nonlinear charac-teristics.From the viewpoint of the differences between the distributions of forward and backward connected edges of networks,we point out that EEG data of healthy indi-viduals have the characteristics of linear Gaussian white noise,while epileptic seizure patients have significant nonlinear characteristics.A consistent conclusion is found for ECG data that there is significant asymmetry in the arrhythmia time series networks,while symmetry in the edge distributions of normal sinus rhythm networks of healthy controls.In summary,this thesis proposes a novel multi-scale transition network approach for time series analysis and structures are characterized by network entropy and statis-tical complexity measures.These methods successfully describe the bifurcation pro-cesses of typical dynamics,and their practicality is verified by brain data.The results of this thesis provide novel insights and ideas for time series prediction and spatial com-plexity in high dimensional systems.