Study On The Correlation And Information Entropy Of Complex Time Series

In the real world,complex systems are composed of various simple components,while these simple components operate across multiple temporal and spatial scales and interact with each other.Complex systems show multilayered structures and self organi-zation,etc.,which makes the analysis and understanding of complex systems more diffi-cult.To analyze the output time series of complex systems is one of important methods to investigate and understand the inner mechanism and dynamics of complex systems.This thesis studies the non-stationary time series of complex systems and explores the prop-erties of complex systems from the aspects of correlation,complexity,synchronization,time irreversibility of time series and their variability on the multiple time scales.In this thesis,we mainly study the auto-correlations,cross-correlations and the large deviation spectrum of cross-correlation based on multifractal theory and detrended analysis meth-ods;introduce various entropies and develop these entropies to multivariate and mul-tiscale analysis to investigate the complexity and synchronization between time series;discuss the properties of segments based on the entropic segmentation method;detect the time asymmetry and its change on the multiple time scales based on the probability distribution theory.The main contents of this thesis include five parts,and the detailed work can be listed as follows:1.We propose the multidimensional scaling based on different dissimilarity mea-sures.We creatively propose these MDS methods based ?DCCA dissimilarity and DTW dissimilarity and employ them to compare the dissimilarities between stock markets.As a comparison,the more traditional Euclidean dissimilarity is employed to MDS in or-der to provide a reference.We decide to confront MDS with an alternative visualization method,"Unweighed Average" clustering method,for comparison.The MDS analysis and "Unweighed Average" clustering method are employed based on the same dissimilar-ity.Through the results we find that MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior,while the MDS anal-ysis based on aDCCA dissimilarity can provide more detailed and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dis-similarity.The MDS analysis based on DTW dissimilarity indicates more knowledge about the correlation between stock markets particularly and interestingly.Meanwhile,it reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity.2.We investigate the inner properties of segments and multiscale characteristics based on entropic segmentation method.We introduce an entropic segmentation algo-rithm and apply it to decompose the financial sequences into compositionally homoge-neous domains.From the view of segmentation position and segment length,we investi-gate the statistical properties of the segments and reveal some important and interesting conclusions and information hidden in these time series of stock markets.Then,we focus on the study of the intrinsic properties for each segment in the time series from two as-pects:time irreversibility and correlation.The fluctuations on the time irreversibility and the scaling exponent all support that the segments present compositional heterogeneity and verify the segmentation.Besides,we combine the entropic segmentation algorithm with recently developed multiscale techniques.By using multiscale entropy,multiscale time irreversibility and multiscale detrended fluctuation analysis,we reveal some inter-esting knowledge of original series and the shuffled series based on the segmentation method on the aspects of complexity,time irreversibility and correlation properties and get a better understanding about the dynamics of stock markets.3.We focus on the multiscale synchronisms between non-stationary time series.Multiscale joint permutation entropy(MJPE)is proposed to study the synchronism be-tween two complex time series from the view of ordinal pattern and multiple scales.First,we use the Rossler system using active control,two-component ARFIMA processes to test the effectiveness of MJPE and also study the effect of noise on MJPE.Through the empirical analysis on financial time series and traffic time series,we find these MJPE results for financial time series are consistent with the actual situation of the synchro-nism and correlation between stock indices.Meanwhile,the results for traffic time series suggest the need for studying the synchronism from the perspective of multiple scales and point out the different synchronisms for traffic time series of weekdays and week-ends.Moreover,MJPE is applied to the sleep EEG data of healthy subjects and shows its validity on the sleep EEG data.Then we further apply MJPE on sleep EEGs from subjects under pathological conditions.The synchronization index also properly reveals their sleep architectures,with consistent trends of the synchronization through the nights.4.We concentrate on the multifractal cross-correlation analysis of non-stationary time series based on large deviation estimates.We propose the large deviation analysis for cross-correlation between two time series.From the angle of large deviations theory,we apply an adaptive nonparametric algorithm combined with the classical roughness exponent based on DCCA to estimate the large deviations spectrum between signals at each scale.First,we generate two ARFIMA series and test the effectiveness of the es-timation procedure of large deviations spectrum on the simulated series in multifractal time.Meanwhile,it also casts doubt on the scaling invariant hypothesis.Then,we turn to the analysis of traffic signals and show how the proposed estimation algorithm reveals unobserved properties of these signals.The non-concavity announces its presence in the analysis between traffic time series and has been verified to associate with traffic acci-dents.Besides,removing traffic accidents data for the time series does not destroy the nature of multifractal properties hidden in the cross-correlation between these sequences.Next,we study the scaling invariant hypothesis on the cross-correlations between traffic signals and provide a way to measure the scale invariance of cross-correlations.It can be found that there is no scale invariance property for the cross-correlations.Finally,the generating mechanism of multifractality of cross-correlation is studied.5.We discuss the multivariate time series analysis.First we introduce the mul-tivariate multiscale sample entropy(MMSE)to evaluate the complexity in multiple data channels over different time scales.We illustrate the necessity of MMSE method by com-paring MMSE results with the multiscale sample entropy(MSE)results on original and shuffled traffic time series respectively.MMSE is capable of revealing the long-range cor-relations and the weekday and weekend patterns containing in traffic signals.Then,we propose multivariate multiscale permutation entropy(MMPE)and multivariate weighted multiscale permutation entropy(MWMPE)and apply these methods to the simulated trivariate time series which are compose of white noise and 1/f noise to test the validity of multivariate methods.It can be found from the simulated series that MWMPE is able to measure the complexity of the multichannel data accurately and reflect more informa-tion about the multivariate time series as well as hold a better robustness.Then MMPE and MWMPE methods are employed to the financial time series:closing prices and trade volume,from different areas.MWMPE method is capable of differentiating these stock markets,detecting their multiscale structure and reflecting more information containing in the financial time series.Finally,we propose the multivariate predicting method and discuss the prediction performance of multivariate time series by comparison with uni-variate time series and K-nearest neighbor(KNN)nonparametric regression model.The predicting results for traffic time series by multivariate predicting method are better and more accurate than those based on univariate time series and KNN model.