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Reverse Weighted-Permutation Entropy: A Novel Complexity Metric Incorporating Distance and Amplitude Information
1  School of Automation and Information Engineering, Xi’an University of Technology, Xi’an 710048, Shaanxi, China


Permutation entropy (PE), as one of the effective complexity metrics to represent the complexity of time series, has the merits of simple calculation and high calculation efficiency. In view of the limitations of PE, weighted-permutation entropy (WPE) and reverse permutation entropy (RPE) were proposed to improve the performance of PE, respectively. WPE introduces amplitude information to weigh each arrangement pattern, it not only can better reveal the complexity of time series with a sudden change of amplitude, but also has better robustness to noise; by introducing distance information, RPE is defined as the distance to white noise, it has the reverse trend to traditional PE and has better stability for time series of different lengths. In this paper, we propose a novel complexity metric incorporating distance and amplitude information and name it reverse weighted-permutation entropy (RWPE), which incorporated the advantages of both WPE and RPE. Four simulation experiments were conducted including entropy curve testing with two probabilities, mutation signal complexity testing, robustness testing to noise based on complexity, and complexity testing of time series with various lengths. The simulation results show that RWPE can be used as a complexity metric, which has the ability to accurately detect the abrupt amplitudes of time series and has better robustness to noise. Moreover, it also shows greater stability than other three kinds of PE for time series with various lengths.

Keywords: permutation entropy (PE); weighted-permutation entropy (WPE); reverse permutation entropy (RPE); reverse weighted-permutation entropy (RWPE); time series; complexity
Comments on this paper
Feiyan Liu
Thank you very much for your interesting paper.

The RWPE seems much better than PE according to the results of the simulations in this paper.
However, I am a little confused, from the formula of RWPE, RWPE is not an entropy.

Best Wishes,
Yuxing Li
Thank you very much for your comments.

In general, PE is defined based on Shannon entropy. In this paper, RWPE is defined as the distance to white noise, which is consistent with the following reference.

Bandt, C. A New Kind of Permutation Entropy Used to Classify Sleep Stages from Invisible EEG Microstructure. Entropy 2017, 19, 197.

Kind regards,