Multiscale analysis of financial time series by Rényi distribution entropy

It is of great interests to analyze the complexity or dependence of time series of the daily stock closing price in different regions. Entropy is typically a fundamental technique to explore the complexity of time series. The well−studied irregularity measure, sample entropy (SampEn), and a more recently proposed complexity measure, distribution entropy (DistEn) are treated as classification features. In our study, based on Rényi entropy, which has been recently suggested as a measure of complexity in nonlinear systems, we propose multiscale Rényi distribution entropy (MRDE) to research the complexity of nonlinear time series over multiple time scales. Considering the validity and accuracy, Rényi entropy obtains more information than the Shannon entropy for time series mingled with much noise like financial time series, which is mainly reflected in the choice of parameters for frequent events and rare events. Both the simulated signals (logistic map and Gaussian noise) and the financial time series with three different stock markets are calculated and compared using an innovative method in different situations. For different stock markets, the entropy decreases as the scale factor increases, except when the parameter is the maximum in the special case. Meanwhile, We illustrate the necessity and advantage of Rényi distribution entropy (RDE) method by comparing RDE results with the SampEn results about stability and consistency on synthetic data and financial time series. It is applied to measure the dependence by detecting the length of the time series and special algorithm parameters.