Multiscale fractional-order approximate entropy analysis of financial time series based on the cumulative distribution matrix

  • Yue TengEmail author
  • Pengjian Shang
  • Jiayi He
Original Paper


In this paper, a generalized method, fractional-order approximate entropy (FOApEn), is proposed with the objective of distinguishing the complexity of time processes from statistical perspectives and characterizing differences and changes in dynamical systems. Moreover, we generalized approximate entropy (ApEn) to multiscales, which can detect complexity of time series in more scales and probe the multiscale properties containing in time series. This fractional-order approximate entropy, which provides an assessment on the multiscale complexity between measurements, is defined in terms of the FOApEn method and the multiscale method. The implementation of multiscale FOApEn is illustrated with simulated time series and financial time series. Examples taken from simulated and financial data demonstrate that tuning the fractional order allows a high sensitivity to the signal evolution and how the FOApEn for complex systems behaves on different scales and determination, which is helpful in describing the dynamics of complex systems.


Fractional calculus Multiscale Financial market Approximate entropy 



The financial supports from the funds of the Fundamental Research Funds for the Central Universities (2018JBZ104, 2019YJS193), the National Natural Science Foundation of China (61771035) and the Beijing National Science (4162047) are gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

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