Some Studies On Complex Time Series

Complex time series are prevalent in our daily lives,where the financial series is closely related to us.Because of human or natural factors,these complex systems often cause extreme events,and the time series generated by complex systems are often non-stationary.In this paper,we mainly study the complexity,correlation and the time reversibility of the non-stationary time series.Based on historical research,several new statistical models are proposed and applied to the real financial market time series.Firstly,we study the correlation of time series based on the detrended fluctuation analysis,and propose an improved generalized multivariatedetrended fluctuation analysis(MNDFA)method,which depends on the scaling of the time series,that is,the change of the length of selected time series.In traditionalDFA method,we obtained the influence of the sequence segmentation interval variables,and it inspiresus to propose a new model MNDFA to discuss the scaling of time series size N towards fluctuation function.We analyze the exponential relationship between A(N)and N.Theeffectiveness of the procedure is verified by numerical experiments with both artificial and stockreturns series.Results show that the proposed MNDFA method contains more significant informationof series compared to traditional DFA method.Our analysis and finite-size effect test demonstrate that an appropriate choice of the time series size can avoidunnecessary influences,and also make the testing results more accurate.We then study the properties of complex nonlinear dynamical systems based on the entropy of time series.Based on the traditional permutation entropy(PE)method,we propose a maximum permutation entropy(PMAE)method.Compared with the traditional entropy of the evaluation sequence,the results show that the maximum permutation entropy method can help to reveal the complexity and the potential correlation of time series.The process of calculating of classical PE method which uses the sample entropy is changed.The entropy model is used to calculate the entropy value,which can reflect the complexity of the series more clearly.Furthermore,we propose the multi-scale maximum permutation entropy(MPMAE)method and the improved composite multi-scale maximum permutation entropy(RMPMAE)method.We also compare and analyze two kinds of multi-scale processes.In order to study the relationship oftwo variables that reflect the complexity of the dynamic system from different perspectives,we compare the DFA values and PMAE values of the non-stationary sequences.Wefind that the trends of the two indicators are similar in some parts.Although the research point of view is different,they can well reflect the internal properties of non-stationary sequences.Finally,we study the complexity of nonlinear systems based on local irreversibility measurement of time series,and propose several new measurementsabout time reversibility.Based on the traditional holistic measurement of time series,local time irreversibility measurements are proposed,which can express the irreversibility of time series more clearly and directly.We also compare the relationship between different measurements and analyze the similaritiesand differences.In addition,the multi-scale process of the seriesis studied,and the relationship between the irreversible measurement and the scale factor of the sequence is discussed.Through this experiment,we can get a greatstudy within the complex properties of the sequence and make a good foundation for further research.