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SCIENTIA SINICA Informationis, Volume 49, Issue 9: 1205-1216(2019) https://doi.org/10.1360/N112018-00197

Generalized phase permutation entropy algorithm based on two-index entropy

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  • ReceivedAug 6, 2018
  • AcceptedDec 13, 2018
  • PublishedAug 30, 2019

Abstract

Generalized permutation entropy $({\rm~PE}_{q,\delta})$ with appropriate parameters can amplify minor changes in a system; however, the phase of the signal contains more critical information than the amplitude. This paper introduces the phase information into the generalized permutation entropy and proposes the generalized phase permutation entropy $({\rm~PPE}_{q,\delta})$ algorithm. Moreover, we verify the advantages of ${\rm~PPE}_{q,\delta}$ in detecting the dynamic changes in the system, analyze the influence of $q$, $\delta$ selection on the dynamic change in the system, and explore the effect of data length and noise for ${\rm~PPE}_{q,\delta}$. Finally, the ${\rm~PPE}_{q,\delta}$ is applied to analyze abnormal ECG signals. When the values of $q$, $\delta$ are the same, the ${\rm~PPE}_{q,\delta}$ has a more significant effect on the detection of the same dynamic change. Whether for use in a logistic map or in detecting abnormal ECG signal dynamic changes, when $q>0$ and $\delta>0$, the effect of ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ can be improved by decreasing $q$ value, increasing $\delta$ value, or simultaneously changing both values. Furthermore, the change in data length has no effect for ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$, and both are robust to noise.


Funded by

国家自然科学基金(11574191,11674208)


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  • Figure 1

    (Color online) Comparison of two kinds of entropy for the logistic map dynamic changes detection when $m=3$, $\tau=1$. (a) Generalized permutation entropy $({\rm~PE}_{q,\delta})$; (b) generalized phase permutation entropy $({\rm~PPE}_{q,\delta})$

  • Figure 2

    (Color online) Detecting dynamic changes in logistic map using ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ when $m=5$, $\tau=1$, $0<q\leq2$, $\delta=1$. (a) ${\rm~PE}_{q,1}$ when $0<q\leq2$; (b) ${\rm~PPE}_{q,1}$ when $0<q\leq2$

  • Figure 3

    (Color online) Detecting dynamic changes in logistic map using ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ when $m=5$, $\tau=1$, $-2<q\leq0$, $\delta=1$. (a) The ${\rm~PE}_{q,1}$ when $-2<q\leq0$; (b) the ${\rm~PPE}_{q,1}$ when $-2<q\leq0$

  • Figure 4

    (Color online) Detecting dynamic changes in logistic map using ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ when $m=5$, $\tau=1$, $q=1$, $0<\delta\leq2$. (a) The ${\rm~PE}_{1,\delta}$ when $0<\delta\leq2$; (b) the ${\rm~PPE}_{1,\delta}$ when $0<\delta\leq2$

  • Figure 5

    (Color online) Detecting dynamic changes in logistic map using ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ when $m=5$, $\tau=1$, $q=1$, $-2<\delta\leq0$. (a) The ${\rm~PE}_{1,\delta}$ when $-2<\delta\leq0$; (b) the ${\rm~PPE}_{1,\delta}$ when $-2<\delta\leq0$

  • Figure 6

    (Color online) Detecting dynamic changes in logistic map with different value of $q$, $\delta$ using ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ when $m=5$, $\tau=1$. (a) The generalized permutation entropy $({\rm~PE}_{q,\delta})$; (b) the generalized phase permutation entropy $({\rm~PPE}_{q,\delta})$

  • Figure 7

    (Color online) The ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ in different length of data and signal-to-noise ration (SNR) when $m=5$, $\tau=1$. (a) The sequence of Gaussian wave packets; (a1) enlarged view of each wave packet in a Gaussian wave packet sequence; (b) the ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ in different length of data when $m=5$, $\tau=1$; (c) the ${\rm~PE}_{q,\delta}$ and ${\rm~PPE}_{q,\delta}$ in different SNR when $m=5$, $\tau=1$

  • Figure 8

    (Color online) ${\rm~PE}_{q,\delta}$ vs. ${\rm~PPE}_{q,\delta}$ in the case of the VT signal when $m=5$, $\tau=1$. (a) The VT signal;protect łinebreak (b) comparison between ${\rm~PE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VT signal; (c) comparison between ${\rm~PPE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VT signal

  • Figure 9

    (Color online) ${\rm~PE}_{q,\delta}$ vs. ${\rm~PPE}_{q,\delta}$ in the case of the VFL signal when $m=5$, $\tau=1$. (a) The VFL signal;protect łinebreak (b) comparison between ${\rm~PE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VFL signal; (c) comparison between ${\rm~PPE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VFL signal

  • Figure 10

    (Color online) ${\rm~PE}_{q,\delta}$ vs. ${\rm~PPE}_{q,\delta}$ in the case of the VF signal when $m=5$, $\tau=1$. (a) The VF signal;protect łinebreak (b) comparison between ${\rm~PE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VF signal; (c) comparison between ${\rm~PPE}_{q,\delta}$ with different parameters $q$, $\delta$ for the VF signal

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