## 2 Multiscale Entropy Theory  
Multiscale entropy (ME) is a method utilized for analyzing the complexity of time-series data [20]. The complexity of signals is measured at different time scales, providing more comprehensive information compared to traditional single-scale methods. The ME theory was initially proposed by [PERSON] et al. in 2002 [21], aimed at addressing the limitations associated with measuring complexity using a single scale. The computational process involves the following steps: (1) signal data are constructed at different time scales. Initially, a series of new signals is generated by averaging the samples of the original signal inside windows. These generated signal series represent different time scales. (2) The sample entropy is calculated at each time scale for each downsampled signal to obtain entropy values at various time scales. A smaller sample entropy value indicates stronger regularity in the signal and better diagnostic capability, while a larger value suggests greater randomness and poorer diagnostic capability in the signal. The specific computation steps are outlined as follows:  
1. A downsampled signal of the vibration data is constructed. Given a signal of \(X=\{x_{1},x_{2},\ldots,x_{N}\}\), the downsampled sequence (\(X^{(\tau)}\)) at a time scale of \(\tau\) is constructed as follows: \[X^{(\tau)}=\{y_{1},y_{2},\ldots,y_{N/\tau}\}\] (1) where \(y_{j}=\frac{1}{\tau}\sum_{i=(j-1)\tau+1}^{j\tau}x_{i}\) and \(\tau\) is the size of the time window.
2. The constructed signal is used to calculate the sample entropy. For each downsampled signal (\(X^{(\tau)}\)), the sample entropy is calculated using two parameters: \(m\) (the embedding dimension) and \(r\) (the similarity tolerance). A vector (\(X^{(\tau)}\)) of length \(m\) of the following sequence is constructed: \(\mathbf{u}_{i}=\{y_{i},y_{i+1},\ldots,y_{i+m-1}\},\quad i=1,2,\ldots,N/\tau-m\). Then, the distance between the elements of the vector is determined (\(d(\mathbf{u}_{i},\mathbf{u}_{j})=\max_{k=0,\ldots,m-1}|y_{i+k}-y_{j+k}|\)). The similarity ratio (\(C^{m}(r)\)) is defined, which is calculated as follows: \[C^{m}(r)=\frac{1}{N/\tau-m}\sum_{j=1}^{N/\tau-m}\theta(r-d(\mathbf{u}_{i}, \mathbf{u}_{j}))\] (2) where \(\theta(x)\) is the indicator function. The value is 1 when \(x\) exceeds 0 and 0 otherwise.  
The approximation ratio (\(B^{m}(r)\)) is calculated, and the likelihood ratio is the ratio of the approximate number to the total number. It is calculated as follows:  
\[B^{m}(r)=\frac{1}{N/\tau-m}\sum_{i=1}^{N/\tau-m}C_{i}^{m}(r) \tag{3}\]  
Then, the dimension is increased by \(1\) to become \(m+1\), and the above steps are repeated to obtain \(B^{m+1}(r)\). It is calculated as follows:  
\[B^{m+1}(r)=\frac{1}{N/\tau-m-1}\sum_{i=1}^{N/\tau-m-1}C_{i}^{m+1}(r) \tag{4}\]  
Finally, the sample entropy is defined as follows:  
\[\text{SampEn}(m,r,N)=-\ln\biggl{(}\frac{B^{m+1}(r)}{B^{m}(r)}\biggr{)} \tag{5}\]
3. Finally, the ME is calculated. For each value of \(\tau\), the corresponding SampEn is calculated to obtain the ME. It is calculated as follows:  
\[\text{MSE}=\{\text{SampEn}(X^{(1)}),\text{SampEn}(X^{(2)}),\ldots,\text{SampEn }(X^{(\tau_{N})})\} \tag{6}\]