## 3 The Proposed Method
### Mscae
The decoding part in the fully connected block performs a dimensional expansion operation on the extracted HI, and the results obtained can be expressed as:  
\[X_{d}{=}f_{r}\Big{(}W_{fd}\hat{y}+b_{fd}\Big{)} \tag{13}\]  
where \(X_{d}\) denotes the output of the decoding part in the fully connected block, \(W_{fd}\) and \(b_{fd}\) denote the weight and bias of the decoding part in the fully connected block, respectively, and \(f_{r}\) is the ReLU activation function. After obtaining \(X_{d}\), it is dimensionally changed using the Reshape layer, and the transformed shape is the same as the shape of the input data in the Flatten layer. \(L\) decoding blocks reconstruct the results of the Reshape layer, i.e., three parallel upsampling and convolution pathways. Suppose \(D_{n,m}^{i}\) denotes the \(n\)-th channel data of the upsampling layer in the \(m\)-th encoding block of the \(i\)-th\((i=1,2,3)\) pathway. Then, the operation of the upsampling layer can be expressed as:  
\[Z_{n,m}^{i,d}=\text{U}\Big{(}D_{n,m}^{i},u_{m},1_{m}\Big{)} \tag{14}\]  
where \(Z_{n,m}^{i,d}\) is the result of the upsampling layer, \(U\) denotes the upsampling operation, and \(u_{m}\) and \(l_{m}\) represent the upsampling window size and stride of the \(m\)-th decoding block, respectively. In a decoding block, the upsampling layer is connected after the convolution layer, and the convolution result of the \(m\)-th decoding block can be expressed as:  
\[D_{k,m+1}^{i}=f_{r}\left(\sum_{t=1}^{N_{m}^{d}}Z_{n,m}^{i,d}*w_{k,n,m}^{i,d}+b _{m}^{i,d}\right) \tag{15}\]  
where \(D_{k,m+1}^{i}\) is the \(k\)-th channel data of the output of the \(m\)-th decoding block, \(N_{m}^{d}\) denotes the total number of channels of the input data of the \(m\)-th decoding block, \(w_{k,n,m}^{i,d}\) denotes the weight of the \(k\)-th convolutional kernel in the \(m\)-th decoding block, \(b_{m}^{i,d}\) is the bias, and \(f_{r}\) is the ReLU activation function. The reconstruction of the input data is obtained after processing by \(L\) decoding blocks. Assume that the \(i\)-th input data of MSCAE is \(x_{i}=[x_{i,1},x_{i,2},\dots,x_{i,l}]\), and the corresponding real RUL label is \(y_{i}\), \(i=1,2,\dots B\). \(l\) is the length of the input data, and \(B\) is the total life of the bearing. Then, the HI extracted by MSCAE is \(\hat{y}_{i}\), and the reconstruction of the input data is \(\hat{x}_{i}=[\hat{x}_{i,1},\hat{x}_{i,2},\dots\hat{x}_{i,l}]\). In this paper, we use the composite loss function to evaluate the HI extraction capability of MSCAE. The composite loss consists of two parts: one is the error between the input data and the reconstructed data, and the other is the error between the HI and the true RUL. The composite loss function is calculated as follows. \[J_{\theta}=\frac{1}{2}\sum_{i=1}^{B}\|x_{i}-\hat{x}_{i}\|^{2}+\frac{\
u}{2}\sum_{ i=1}^{B}\|y_{i}-\hat{y}_{i}\|^{2} \tag{16}\]  
where \(\theta\) denotes the internal parameters of MSCAE and \(\
u\) is a scaling factor to adjust the weight between the two errors. Compared with the traditional AE, the composite loss function constructed by MSCAE combines the supervised learning approach with the unsupervised learning approach for model training, which fully utilizes the degradation information of the bearings and enhances the extraction capability of the model HI. In the training phase of the model, the vibration data and labels of the training bearings are used to optimize the internal parameter \(\theta\) using the back-propagation algorithm to minimize the composite loss. In the testing phase of the model, the vibration data of the testing bearing serve as the input, and the HI is obtained using the encoder part of the trained MSCAE.