## 2 Proposed Fault Diagnosis Method
### Feature Extraction Based on MPE  
[PERSON] and [PERSON] [11] proposed PE, which is based on a comparison of adjacent data without considering the specific value of the data. PE can effectively reduce noise interference and computational complexity. The specific calculation principles are as follows:  
For the time-domain signal sequence \(x(i)\) with length \(N\), by reconstructing its phase space, the following time series could be obtained:  
\[\left\{ \begin{array}{l}X(1)=\{x(1),x(1+t),\cdots,x(1+(m-1)t]\\ \vdots\\ X(i)=\{x(i),x(i+t),\cdots,x(i+(m-1)t]\\ \vdots\\ X(N-(m-1)t)=\{x(N-(m-1)t),x(N-(m-2)t),\cdots,x(N)\}\end{array} \right. \tag{3}\]  
where \(m\) is the embedding dimension, and \(t\) is the time delay. The arbitrary sequence \(X(i)=\{x(i),x(i+t),\cdots,x(i+(m-1)t]\) is rearranged in ascending order as shown in Equation (4). \[X(i)=\{x(i+(j_{1}-1)t)\leq x(i+(j_{2}-1)t)\leq\cdots\leq x(i+(j_{m}-1)t] \tag{4}\]  
If \(x(i+(j_{a}-1)t)=x(i+(j_{b}-1)t)\) exists, it is arranged according to the values of \(j_{a}\) and \(j_{b}\). For example, when \(j_{a}<j_{b}\) is satisfied, \(x(i+(j_{a}-1)t)\leq x(i+(j_{b}-1)t)\). Therefore, for any sequence \(x(i)\), all reconstructed vectors can obtain a set of symbol sequences. \[S(l)=\{j_{1},j_{2},\cdots j_{m}\} \tag{5}\]  
where \(l=1,2,\cdots,m!\). For \(m\) different symbols \(\{j_{1},j_{2},\ldots j_{m}\}\), there are \(m!\) different permutations, which is \(m!\) different symbol sequences. For each symbol sequence, the probability of its occurrence is \(p_{f}\). [PERSON]'s entropy defines the PE of the symbol sequence corresponding to the \(k\) reconstruction vector of time series \(x(i)\), as shown in Equation (6). \[H_{p}(m)=-\sum_{l=1}^{m!}p_{f}\ln(p_{f}) \tag{6}\]  
when \(p_{f}=\frac{1}{m! }\), \(H_{p}(m)\) achieves the maximum value \(\ln(m! )\), and \(H_{p}(m)\) is standardized in turn:  
\[H_{p}=H_{p}(m)/\ln(m!)\]The larger the \(H_{P}\), the more random the signal sequence, and the more dispersed the signal energy. The smaller the \(H_{P}\), the more regular the signal sequence, the more concentrated the signal energy, and the higher the fault probability. On the basis of PE, [PERSON] et al. [14] proposed multiscale permutation entropy (MPE) to measure the complexity and randomness of time series at different scales. MPE is the PE of the time series on different scales, which adds a process of coarsening time series, as shown in Equation (8):  
\[y^{\lambda}(j)=\frac{1}{\lambda}\sum_{i=(j-1)\lambda+1}^{j\lambda}x(i) \tag{8}\]  
where \(\lambda\) represents the scale factor and \(y^{\varepsilon}(j)\) represents coarse-grained data arrangement. The parameter selection considerably influences the calculation result, so before feature extraction, it is necessary to determine three important parameters: embedding dimension \(m\), time delay \(t\), and scale factor \(\lambda\). When \(m\) is too large, the calculation of the phase space is complex, and the calculation time increases. When \(m\) is too small, the reconstructed information is insufficient for the extraction and detection of mutation signals. According to [PERSON] and [PERSON] [11], \(m\) ranges from 3 to 7. Figure 2 shows the PE values of Gaussian white noise in different embedding dimension. Figure 2 shows that the smaller the embedding dimension, the larger the PE value. Therefore, the embedding dimension should not be too small. In our experimental study, \(m=5\) was selected. Figure 2 also shows that the MPE of Gaussian white noise decreases monotonously with the increase in the scale factor. The relationship between PE and embedding dimension under different time delays is shown in Figure 3. It can be seen that time delay has little effect on time series, \(t=1\) was chosen in this paper. Scale factor \(\lambda\) determines the PE characteristics of the signal at the corresponding scale. Generally, as proposed in [PERSON] et al. [37], the maximum scale factor is usually more than 10. Figure 3: PE of Gauss white noise with different time delays.