PID: a PDF-induced distance based on permutation cross-distribution entropy

  • Jiayi HeEmail author
  • Pengjian Shang
  • Yali Zhang
Original Paper


In this paper, a PDF-induced distance (PID) based on permutation cross-distribution entropy is proposed to measure the dissimilarity between complex time series. It overcomes the effects of spatial distance by focusing on similar local fluctuation patterns. It also corrects the disadvantage of being insensitive to symmetric skewness distributions. We have applied PID to synthetic data and financial time series. The Euclidean distance is employed as a reference. In simulated experiments, eight signals generated from four models are detected, and results are presented by hierarchical clustering analysis via PID, which is correctly clustered and superior to other methods. Then, PID is applied to the real-world financial time series. Eight stocks in the global financial markets are employed. It reveals that they are clearly divided based on their financial backgrounds. The cophenetic correlation coefficient is used to measure the quality of solutions. It reveals that the PID is convincing and superior to the Euclidean distance. In addition, as a classic method for non-stationary time series, detrended cross-correlation analysis (DCCA) is adopted to compare the superiority of PID. Although experiments show that DCCA does perform well, it is inferior to PID. Also, we conduct additional experiments on the effects of non-Gaussian noises. Excitingly, PID can still cluster signals accurately after adding uniformly distributed noises, and the cophenetic correlation coefficient reaches 0.9788.


Financial time series Distribution entropy Dissimilarity measure Probability distribution function 



The financial supports from the funds of the Fundamental Research Funds for the Central Universities (2019YJS193).

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.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

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