Financial time series analysis using Total-CApEn and Avg-CApEn with cumulative histogram matrix

Approximate entropy (ApEn) is an index that reflects the overall characteristics of a signal from the point of the complexity of time series. And increasing improved methods have been proposed in recent years. The traditional computing method of cross-approximate entropy (Cross-ApEn) is limited by tolerance r; in order to reduce the influence of r on accuracy, we proposed an adaptive method called cumulative histogram method (CHM) to gain a range of Cross-ApEn values. We calculate total cross-approximate entropy (Total-CApEn), average cross-approximate entropy (Avg-CApEn) and the standard deviation of cross-approximate entropy (SD-CApEn) to distinguish simulated data and financial stock data. Because CHM is a function related with the length of the time series N and the dimension m, the choice of N is a very important problem. We find that Cross-ApEn almost doesn't change much after N is 400, therefore 400 is a fairly suitable length. And we verify the advantages of CHM through many aspects, such as effectiveness test, length test, entropy plane construction, moving window construction and so on.