Research On Structural Complexity And Correlation Of Nonlinear Time Series

Complex systems can be seen everywhere in the real world,and analyzing the time series from these systems is the main means to spy on the operating mechanism and intrinsic characteristics of the system.This paper mainly studies the complexity of nonlinear time series and the correlation algorithm of variables.On the basis of entropy theory,a new nonlinear statistical physical model is extended and applied to the real financial system and transportation system.First,we study the time series complexity problem based on entropy measure.We propose generalized permutation entropy combined with diversity index,and combined with multi-scale algorithm and weight information of redistribution order pattern,we propose multi-scale generalized permutation entropy and weighted diversity index permutation entropy.When the permutation criterion is used to map the signal to sequential patterns,it will lead to loss of amplitude information between sequences.To this end,we introduce a weighting algorithm and a sliding window to recapture the amplitude information of the sequence and improve the ability to detect signal mutation.Then The original information entropy is extended to the multivariate system,and the multivariate and multiscale distribution entropy is proposed.Using the phase space reconstruction theory,the distance information between vectors is extracted to measure the complexity of multivariate time series.Secondly,we construct the complexity-entropy causality plane by using "information" and "imbalance"."Information" and "imbalance" are two kinds of complexity measures,which describe the complexity of the system from different perspectives:"Information" is the amount of information needed to describe the behavior of the system;"Imbalance" is the distance between the probability distribution of the system and the uniform distribution.We incorporate the coarse-grained algorithm into the original entropy plane,and the joint power spectrum information from the sequence distribution is observed at different scales.After that,the multi-scale entropy plane is extended to the multi-variable system,and the complexity-entropy causality plane of the multi-variable multi-scale distribution entropy is proposed.The numerical simulation results show that the method uses the distance information of the state space,Does not depend on parameter settings.It can analyze traffic sequences with two data channels and give classification.Then,based on the non-extended Tsallis entropy,we propose the cumulative residual power spectral entropy.The power spectral entropy does not need to rely on preset parameters,it uses the Fourier transform to transform the signal from the time domain to the frequency domain,and then combines the spectral information to measure the complexity of the sequence.Compared with the traditional sample entropy,permutation entropy and distribution entropy,the cumulative residual power spectral entropy can reflect the difference between sequences no matter how the signal length changes,and the results are consistent with the change of parameters.Then we optimize the process of estimating the probability density distribution,considering the sequential structure of the sequence and propose a multi-scale cumulative residual permutation distribution entropy.This method extracts the distance information between vectors and reconstructs,replacing the original sequence with sequential patterns to reflect its Internal features.Since the permutation entropy is based on the sequential pattern of the sequence,sorting according to the size of the value realizes the complexity detection of the sequence,but it only considers the size of the value and loses detailed amplitude information,and the measurement for the equivalent case is not accurate enough.Based on these problems,we combine the concept of symbolic dynamics and propose the cumulative residual discrete entropy,using the normal cumulative distribution function to transform the original data into a new sequence with only a few distinct elements,this method does not require the calculation of each embedding vector.Amplitudes are sorted and are very sensitive to amplitude transformations.In empirical analysis,combined with hierarchical clustering analysis,it can reveal richer market classifications.Finally,on the basis of correlation theory,we propose the K-Medoids clusteringbased maximum information coefficient(KM-MIC)to measure the degree of correlation between variables.The original maximum information coefficient method needs to traverse two variables in two each grid division in the dimensional space scatterplot can be implemented by enumeration method for small-scale data sets,but it greatly increases the computational difficulty for larger-scale data sets.We use K-Medoids clustering to improve The grid division in the original algorithm finds the center value in the cluster as the center point of the cluster,and reduces the influence of outliers and convex data on the calculation results.The numerical simulation results show that KM-MIC maintains the superior property of the maximum information coefficient:Generality and Equitability,that is,it can capture a wide range of correlations not limited to linear,periodic,etc.,and the results are consistent with increasing noise.At the same time,KM-MIC takes less computation time to determine random relationships,the dependent data scale is smaller,and the random variables in the large data set can be quickly screened out.