Permutation entropy and its variants for measuring temporal dependence

Xin Huang*, Han Lin Shang, David Pitt

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
35 Downloads (Pure)


Permutation entropy (PE) is an ordinal-based non-parametric complexity measure for studying the temporal dependence structure in a linear or non-linear time series. Based on the PE, we propose a new measure, namely permutation dependence (PD), to quantify the strength of the temporal dependence in a univariate time series and remedy the major drawbacks of PE. We demonstrate that the PE and PD are viable and useful alternatives to conventional temporal dependence measures, such as the autocorrelation function (ACF) and mutual information (MI). Compared to the ACF, the PE and PD are not restricted in detecting the linear or quasi-linear serial correlation in an autoregression model. Instead, they can be viewed as non-parametric and non-linear alternatives since they do not require any prior knowledge or assumptions about the underlying structure. Compared to MI estimated by k-nearest neighbour, PE and PD show added sensitivity to structures of relatively weak strength. We compare the finite-sample performance of the PE and PD with the ACF and the MI estimated by k-nearest neighbour in a number of simulation studies to showcase their respective strengths and weaknesses. Moreover, their performance under non-stationarity is also investigated. Using high-frequency EUR/USD exchange rate returns data, we apply the PE and PD to study the temporal dependence structure in intraday foreign exchange.
Original languageEnglish
Pages (from-to)442-477
Number of pages36
JournalAustralian and New Zealand Journal of Statistics
Issue number4
Publication statusPublished - Dec 2022

Bibliographical note

© 2022 The Authors. Australian & New Zealand Journal of Statistics published by John Wiley & Sons Australia, Ltd on behalf of Statistical Society of Australia. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


  • Discretization
  • EUR/USD exchange rate
  • Financial time series analysis
  • High-frequency return
  • Non-stationarity
  • Ordinal pattern


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