Reverse Weighted-Permutation Entropy: A Novel Complexity Metric Incorporating Distance and Amplitude Information

Yuxing Li

doi:10.3390/ecea-5-06688

Abstract

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. WPE introduces amplitude information to weigh each arrangement pattern, it can not only better reveal the complexity of time series with a sudden change of amplitude, but it 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 incorporates the advantages of both WPE and RPE. Three simulation experiments were conducted, including mutation signal detection 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 the other three kinds of PE for time series with various lengths.

Highlights

Permutation entropy (PE) was first brought forward by Bandt and Pompe in a seminal paper [1].It introduced a simple and robust symbolic method that takes into account arrangement patterns of time series by comparing neighboring values of time series
In order to keep and enhance the advantages of WPE and reverse permutation entropy (RPE), we propose a novel complexity metric incorporating distance and amplitude information and name it reverse weighted-permutation entropy (RWPE)
RPE are relatively stable under different signal-to-noise ratios (SNRs) and close to 1 and 0, respectively; the entropies incorporating amplitude information respond to changes in SNR, the influence of noise on complexity decreases with the increase of SNR, the values of WPE and RWPE are monotonically decreasing and increasing

Summary

Introduction

Permutation entropy (PE) was first brought forward by Bandt and Pompe in a seminal paper [1]. It introduced a simple and robust symbolic method that takes into account arrangement patterns of time series by comparing neighboring values of time series. The development of PE includes two aspects: One is the expansion of its application in different fields, the other is the improvement of PE theory. WPE and RPE have their own edges in indicating the complexity of time series from different angles. In order to keep and enhance the advantages of WPE and RPE, we propose a novel complexity metric incorporating distance and amplitude information and name it reverse weighted-permutation entropy (RWPE). Three simulation experiments have been carried out to demonstrate validity of RWPE by analysis and comparison with PE, WPE and RPE

Reverse Weighted-Permutation Entropy

Simulation 1

Simulation 2

Simulation 3

Conclusions