Research And Application Of Entropy Theory In Financial Time Series

Complex systems have characteristics such as non-stationarity and randomness.In recent years,time series generated by complex systems have attracted much attention.Entropy has been proposed as a major means of transmitting complex system information,such as permutation entropy,approximate entropy,Jensen-Shannon divergence,Kullback-Leibler divergence and so on,and these kinds of entropy are evolved from information entropy.This paper mainly improves different types of entropy,and explores the application of entropy theory from many aspects,so we can analyze the financial time series.We propose four methods to study the complexity of time series.The first method is to extend the permutation entropy and Jensen-Shannon divergence(DJS)to the fractional order.We combine them to construct the entropy plane and find that the entropy plane can distinguish time series.Then we explore time series under multiscale,and observe that logistic map and Henon map are straight lines,while burgers map and stock index are curves.The second method is to use the adaptive cumulative histogram method(CHM)to solve the problem of determining the tolerance parameter r in approximate entropy,improve the accuracy of cross-approximate entropy(Cross-ApEn),and study the complexity of time series by generating a series of Cross-ApEn secondary measures.We test the validity of the time series and find that Cross-ApEn is not affected by white noise.We also perform a length test on it and find that Cross-ApEn is stable after the series length reaches 400.We use traditional methods and CHM to construct the entropy plane to improve the accuracy of series classification,and then construct a moving window to explore the macro changes in the financial stock market.The third method is to combine Kullback-Leibler divergence with higher moments,and use horizontal visibility graph(HVG)to calculate probability to explore the irreversibility of time series.Through experiments,we find that the image shapes of a=3.85 logistic map and a=0.25,b=0.52 Henon map are consistent with the stock data.Then,the time series is subjected to an overlay coarse-grained process,and it is found that the Kullback-Leibler divergence is relatively stable when the scale is 8.The fourth method is to calculate DJS and the complete mean divergence(Dm)using permutation pattern(PP)and HVG.The length tests of ARFIMA data and stock data show that both DJS and Dm decrease with the increase of time series length.In the dynamic analysis of time series,we find that the dynamic changes of other stocks are relatively stable,except that AAPL shows a downward trend.Then,the multiscale is introduced to study time series,and it is observed that PP is not easy to be disturbed even if the complexity of the series is increased.We propose Score to analyze the utility of stock and its practical significance,and find that AAPL has the highest score.