Nonlinear Dynamics

, Volume 98, Issue 3, pp 2147–2170 | Cite as

Complexity and uncertainty analysis of financial stock markets based on entropy of scale exponential spectrum

  • Boyi ZhangEmail author
  • Pengjian Shang
Original paper


Research on complexity and uncertainty of nonlinear signals has great significance in dynamic system analysis. In order to further analyze the detailed information from the financial data to acquire deeper insight into the complex system and improve the ability of prediction, we propose the entropy of scale exponential spectrum (EOSES) as a new measure. Combined with multiscale theory, we get multiscale EOSES. The scale exponential spectrum (SES) is derived from the scale exponent of detrended fluctuation analysis. As for the entropy, we choose Rényi entropy and fractional cumulative residual entropy to compare and analyze the results. Simulated data and financial time series are used to obtain further in-depth information on the EOSES. Compared with traditional methods, we find that Rényi EOSES over moving window can provide more details of complexity which include fractal structure and scale properties. Also, it reduces the influence of degree of fitting polynomial and has higher noise immunity. In addition, through the SES and EOSES, we can better research the properties and stages of stock markets and distinguish stock markets with different characteristics.


Complexity Rényi entropy Scale exponent Financial time series 



The financial support from the Fundamental Research Funds for the Central Universities (Grant No. 2018JBZ104) and the National Natural Science Foundation of China (Grant No. 61771035) are gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingChina

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