Generalized phase permutation entropy algorithm based on two-index entropy

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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.