The Volatility Based On Higher-order Moment And Multiscale Complexity Of Financial Time Series

The output of many complex systems in real society reflects the dynamic characteristics of the complex system itself to a certain extent,such as fluctuation of stock index,complex physiological structure of human body,and the changing of traffic conditions.The research object of this paper is financial time series,and many methods of time series analysis are used to study the dynamic characteristics of complex systems.We take the financial markets of many countries and regions as the research object,study and discuss the series of stock index,and measure some statistical characteristics of them.This paper extends and improves three research methods of financial stock time series:one is the analysis of time series complexity based on entropy.Entropy is one of the common methods for estimating the complexity of time series.Entropy has the advantage of profound physical background and variety,which can better capture the non-linear relationship between systems.The second method is the detrend fluctuation analysis based on the high-order moments.The detrend fluctuation analysis method is one of the effective methods for estimating the correlation of non-stationary time series.It can effectively reduce the non-stationarity of the sequence generated by local trends.The third is the study of local irreversibility of time series.Time irreversibility is one of the important characteristics of non-stationary time series.It can detect the non-linear dynamic system and characterize the unstable state of the system.This article is divided into five chapters,which is organized as follow:In Chapter 1,we briefly introduce the research background,research methods,research significance and main work.In Chapter 2,we propose the concept of transfer entropy coefficient based on transfer entropy,which extends the transfer entropy to the concept of multiscale.This transfer entropy coefficient provides a method for evaluating the measured multiscale information flow.It is defined on the basis of the transfer entropy method and the multiscale method.We analyze and discuss the application of transfer entropy coefficient method to financial time series and simulated generated series respectively.The experimental results of financial and simulation data both show that the dynamic mechanism of complex systems can not be detected by transfer entropy of single scale.Transfer entropy coefficient method is used to analyze the effect of time scale on the transfer entropy of two time series at the same time.In Chapter 3,we study a method for analyzing the volatility of time series,detrending fluctuation analysis(DFA).In this chapter,we extend the detrend fluctuation analysis method to the third and fourth moments,analyze the DFA of the higher moments,and study the characteristics and the richer internal structure of the sequence on the higher moments.Third-order moments reflect the asymmetric degree of data distribution,and fourth-order moments describe whether the peak value is abrupt or flat.They reflect the fluctuation characteristics of higher-order moments in different aspects.In Chapter 4,we extend the study of irreversibility of time series to multiscale,and study the local dynamic invariance of time series under time reversal.Global reversible sequences are not always locally reversible.Local irreversibility algorithm can describe the relationship between global and local irreversibility of time series.The local irreversibility algorithm can effectively discuss the characteristics of local irreversible fluctuations of time series with the change of scale.This method is applied to simulated data generated by ARFIMA process and logical mapping,showing how irreversible functions respond to multiple scales.This method has also be applied to a series of financial markets in different countries such as the United States,China and Europe.The local irreversibility of different markets has obvious characteristics.The simulation and real data support the rationality and adaptability of local irreversibility.In Chapter 5,a generalized approximate entropy model is proposed,and the cumulative histogram method is used to generate r,which is driven by the data itself and more accurate.The short sequence is also applicable.And based on fractional calculus,in order to make it more sensitive to the dynamic changes of series,we introduce the fractional calculus into the approximate entropy,and extend the fractional order approximate entropy analysis to multi-scale to study the richer dynamic characteristics of time series at multiscale.Multiscale fractional order approximate entropy is based on fractional order approximate entropy method and multiscale method,which provides the evaluation and measurement of multiscale complexity.We use simulated data and real stock data to prove the realization of multiscale fractional order approximate entropy.Simulations and examples in stock show that the model is highly sensitive to signal evolution,and how the fractional order approximate entropy of complex systems exhibits different characteristics in different scales,which is helpful to describe the dynamic characteristics of complex systems.In Chapter 6,we summarized the full paper.