Multi-Scale Permutation Entropy: A Complexity Measure for Discriminating Two-Phase Flow Dynamics

Abstract

We propose an improved permutation entropy method, i.e., multi-scale permutation entropy (MSPE), for discriminating two-phase flow dynamics. We first take the signals from different typical dynamical systems as examples to demonstrate the effectiveness of the methods. In particular, we compute the MSPE values of sinusoidal signal, logistic, Lorenz and Chen chaotic signals and their signals with white Gaussian noise added. We find that the MSPE method can be an effective tool for analyzing the time series with distinct dynamics. We finally calculate the multi-scale permutation entropy and rate of MSPE from 66 groups of conductance fluctuating signals and find that these two measures can be used to identify different flow patterns and further explore dynamical characteristics of gas-liquid flow patterns. These results suggest that the MSPE can potentially be a useful tool for revealing the dynamical complexity of two-phase flow on different scales.

Multi-Scale Permutation Entropy Based on Improved LMD and HMM for Rolling Bearing Diagnosis

“…Then, the phase space reconstruction of the PF1 is carried out by using the MI and the FNN to calculate the delay time and embedding dimension, and then we can set the scale to obtain the MPE of PF1. In Figure 3, MI [17][18][19][20] determines the optimal delay time.…”

“…In the formula: ( ) h x and ( ) h y respectively correspond to ( ) x i and ( ) y j of the entropy in the specified system and measure the average amount of information; ( , ) h x y is a joint information entropy. In Figure 3, MI [17][18][19][20] determines the optimal delay time. x = {x i , i = 1, 2, · · · , N} represents a group of signals, the probability density function of the point is p x [x(i)], the signal is mapped to the probability y = y j , j = 1, 2, · · · , N .…”

Design of Grid Multiscroll Chaotic Attractors via Transformations

“…Among them, PE algorithm is a proper choice to estimate the complexity of a time series accurately and rapidly. Recently, multiscale permutation entropy (MPE) was proposed to analyze the complexity of time series for its faster computation speed [Fan et al, 2013]. In this paper, we will employ MPE algorithm to measure the complexity of the chaotic sequence generated by different grid multiscroll chaotic attractors.…”

Permutation Entropy for Random Binary Sequences

“…PE is easily implemented and is computationally much faster than other comparable methods, such as Lyapunov exponents, while also being robust to noise [18], which makes it more and more popular [19][20][21][22][23][24][25][26]. In theory, the authors of [19] proposed a generalized PE based on a recently postulated entropic form and compared it to the original PE.…”

“…In theory, the authors of [19] proposed a generalized PE based on a recently postulated entropic form and compared it to the original PE. Fan et al [20] proposed a multiscale PE as a new complexity measure of nonlinear time series and the authors of [21] generalized it by introducing weights. Unakafova et al [22] discussed the relationship between PE and Kolmogorov-Sinai entropy in the one-dimensional case, et al In application, the authors of [23] used PE to quantify the nonstationarity effect in the vertical velocity records.…”