Time Series Analysis Based On Information Entropy And Its Several Applications

The complexity and similarity of nonlinear time series widely exist in complex systems,so it is very important to measure the complexity and similarity of time series accurately.Although the research shows that the multi-scale entropy method is an effective technique to measure the complexity and similarity of time series.However,there are still three problems:(1)If the two time series are affected by the third party at the same time,the existing multi-scale entropy methods will be difficult to get their intrinsic synchronization;(2)The asynchrony between two time series is not always symmetric;(3)When the time series are short,the multi-scale entropy method may not be able to obtain accurate and effective results.For the first problem,in this thesis,we propose a new cross-sample entropy,namely the composite multiscale partial cross-sample entropy(CMPCSE),for quantifying the intrinsic similarity of two time series affected by common external factors.First,in order to test the validity of CMPCSE,we apply it to three sets of artificial data.Experimental results show that CMPCSE can accurately measure the intrinsic crosssample entropy of two simultaneously recorded time series by removing the effects from the third time series.Then CMPCSE is employed to investigate the partial cross-sample entropy of Shanghai securities composite index(SSEC)and Shenzhen Stock Exchange Component Index(SZSE)by eliminating the effect of Hang Seng Index(HSI).Compared with the composite multiscale cross-sample entropy,the results obtained by CMPCSE show that SSEC and SZSE have stronger similarity.We believe that CMPCSE is an effective tool to study intrinsic similarity of two time series.To solve the second problem,in this thesis,we propose an asymmetric multi-scale fuzzy measure cross-trend entropy(AMFMCTE)method.This method inherits the advantages of strong consistency of sample entropy,and is not parameter sensitive,that is,the calculation results will not change abruptly with the change of parameters.In addition,this method will not lead to the situation that the definition of entropy does not exist.On the other hand,this method also considers the cross-entropy of positive and negative wave series,which makes the results more abundant and reliable than the original analysis method.In order to verify the effectiveness of the method,we first use numerical examples to verify the immune effect of the method on the inherent trend of the sequence.Then we use AMFMCTE to analyze the Dow Jones index(DJI),standard &Poor's index 500(SP500),SSEC and SZSE.The results show that the stock data from the same countries have stronger synchronization.To solve the third problem,we propose two-index fuzzy measure entropy and two-index fuzzy measure cross-entropy.Compared with the original fuzzy measure entropy(Fuzzy MEn1)and fuzzy measure cross-entropy(FMCE1),we introduce the second fuzzy measure entropy(Fuzzy MEn2)and the second fuzzy measure cross-entropy(FMCE2).In order to prove that Fuzzy MEn2 and FMCE2 are meaningful,we use them to calculate the two-index fuzzy measure entropy and two-index fuzzy measure cross-entropy of three groups of simulation data respectively.The experimental results show that Fuzzy MEn2 and FMCE2 represent information that Fuzzy MEn1 and FMCE1 do not contain.In order to further test whether Fuzzy MEn2 contains valid information,we calculate the two-index fuzzy measure entropy of DJI?SP500?SSEC and SZSE,the results show that Fuzzy MEn1 can't divide the four sequences into two groups by region,but Fuzzy MEn2 can do it easily.In order to further test whether FMCE2 contains valid information,we calculate the two-index fuzzy measure cross-entropy between any two indexes of DJI?SP500?HSI?SSEC?SZSE and NAS,respectively.The results show that FMCE1 or FMCE2 alone can not accurately divide the six sequences into two groups by region.But when we consider FMCE1 and FMCE2 comprehensively,the results of two-index fuzzy measure crossentropy between any stock index and other five stock indexes can be distinguished by region.In general,FMCE2 is meaningful and can be used as supplementary information of FMCE1.To sum up,we propose three new methods to solve the three problems encountered in using the entropy based multi-scale method,and the corresponding numerical and practical examples also prove that they are effective.The main innovations of this thesis are as follows:· A composite multi-scale partial cross-sample entropy is proposed, which makes the cross-sample entropy result not affected by the third party factors;· A asymmetric multi-scale fuzzy measure cross-trend entropy is pro- posed,which provides a new method for asymmetric similarity anal- ysis;· Two-index fuzzy measure entropy and two-index fuzzy measure cross- entropy are proposed,which provide new and effective indexes for short time series analysis.