Multiscale fractional order generalized information of financial time series based on similarity distribution entropy

This paper addresses a novel multiscale fractional order distribution entropy based on a similarity matrix (MFS-DistEn) approach to quantify the information of time series on multiple time scales. It improves the metric method of distance matrix in the original DistEn algorithm and further defines the similarity degree between each vector so that we could measure the probability density distribution more accurately. Besides, the multiscale distribution entropy based on similarity matrix combines the advantages of both the multiscale analysis and DistEn and is able to identify dynamical and scale-dependent information. Inspired by the properties of Fractional Calculus, we select the MFS-DistEn notation as the main indicator to present the relevant properties. The characteristics of the generalized MFS-DistEn are tested in both simulated nonlinear signals generated by the autoregressive fractionally integrated moving-average process, logistic map, and real world data series. The results demonstrate the superior performance of the new algorithm and reveal that tuning the fractional order allows a high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The improved similarity DistEn still has relatively lower sensitivity to the predetermined parameters and decreases with an increase of scale.