## 2 Methods
### Amplitude-Aware Permutation Entropy  
[PERSON] proposed the concept of Permutation Entropy (PE) in 2002 [31]. Currently PE is widely used in the analysis of complex time series signals to measure the complexity of a nonlinear and non-stationary signal.  
Suppose a given time series \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{N}\}\), and for each time point \(t\), embed signal \(\mathbf{x}\) into \(d\)-dimensional space to obtain reconstructed vector \(\mathbf{X}_{t}^{d,l}=\{x_{t},x_{t+l},\ldots,x_{t+(d-2)l},x_{t+(d-1)l}\}\), \(t=1,2,\ldots,N-(d-1)l\) where \(d\) and \(l\) represent embedded dimension and time delay, respectively. The magnitude of elements in each vector \(\mathbf{X}_{t}^{d,l}\) is arranged in ascending order, namely \(\left\{x_{t+(j_{1}-1)l},x_{t+(j_{2}-1)l},\ldots,x_{t+(j_{d-1}-2)l},x_{t+(j_{d-1 })l}\right\}\), where \(j_{i}\) represents the order of elements in the reconstructed vector \(\mathbf{X}_{t}^{d,l}\). Therefore, when the embedding dimension is \(d\), there are a total of \(d!\) permutations, and the \(i\)th permutation order is demoted as \(\pi_{i}\).  
The occurrence probability of \(\pi_{i}\) is expressed as:  
\[p(\pi_{i})=\frac{f(\pi_{i})}{N-d+1} \tag{3}\]  
where \(f(\pi_{i})\) is a function of counting the number of occurrences of the permutation order \(\pi_{i}\). Whenever the permutation order of the elements inside \(\mathbf{X}_{t}^{d,l}\) is \(\pi_{i}\), \(f(\pi_{i})\) increases by 1. The definition of permutation entropy is as follows,  
\[PE(\mathbf{x},d,l)=-\sum_{\pi_{i}=1}^{\pi_{i}=d!}p(\pi_{i})\ln p(\pi_{i}) \tag{4}\]  
However, there are two main problems in describing complex time series by permutation entropy. First, traditional PE only considers the order of time series amplitude, but ignores the amplitude information of the corresponding elements in the time series. Second, the effect of elements with equal amplitude on PE value in time series is not clearly explained.  
In view of this, literature [27] proposed AAPE to increase PE's sensitivity to amplitude and frequency of time series. Different from the PE algorithm, AAPE algorithm considers the mean value of signal amplitudes and the deviation between amplitudes, and introduces the counting rule of replacing \(f(\pi_{i})\) in PE with relatively normalized probability.  
Figure 1: Coarse-graining procedure diagram of multiscale entropy (MSE). (**a**) Scale factor \(\tau{=}\) 2; (**b**) Scale factor \(\tau{=}\) 3.  
Assuming that the initial value of \(p(\pi_{i}^{dJ})\) is 0, for the time series \(X_{t}^{dJ}\) in the process of \(t\) increasing from 1 to \(N-d+1\), whenever the permutation order is \(\pi_{i}^{dJ},p(\pi_{i}^{dJ})\) has to be updated.  
\[p(\pi_{i}^{dJ})=p(\pi_{i}^{dJ})+\left(\frac{A}{d}\sum\limits_{k=1}^{d}\left|x_{t +(k-1)l}\right|+\frac{1-A}{d-1}\sum\limits_{k=2}^{d}\left|x_{t+(k-1)l}-x_{t+(k- 2)l}\right|\right) \tag{5}\]  
where, \(A\in[0,1]\) is the adjustment coefficient, and the weight of signal amplitude mean and amplitude deviation is adjusted, generally 0.5. So, \(p(\pi_{i}^{dJ})\) is the probability of \(\pi_{i}^{dJ}\) appearing in the whole time series.  
\[p(\pi_{i}^{dJ})=\frac{p(\pi_{i}^{dJ})}{\sum\limits_{t=1}^{N-d+1}\left(\frac{A} {d}\sum\limits_{k=1}^{d}\left[x_{t+(k-1)l}\right|+\frac{1-A}{d-1}\sum\limits_{ k=2}^{d}\left|x_{t+(k-1)l}-x_{t+(k-2)l}\right|\right)} \tag{6}\]  
The AAPE value of time series is expressed as  
\[AAPE(d,l,n)=-\sum\limits_{\pi_{k}=1}^{\pi_{k}=d!}p(\pi_{k})\ln p(\pi_{k}) \tag{7}\]