## 3 Methodology
### SHI Construction
#### 3.2.2 Feature Fusion  
The calculation process of the SHI was essentially a process of mapping the multi-dimensional physical characteristic information of the system collected by sensors to a one-dimensional virtual variable with a value range of \((0,1)\). The value of \(1\) indicated that the system was working normally, and the value of \(0\) indicated that the system had a total loss of function. Just like using physical characteristics to describe the health condition directly, the SHI was also based on the fusion of the available and real-state monitoring information to produce the health assessment of the system, and this method could describe the health condition of the system accurately and comprehensively. The SAE deep neural network was used to fuse the features extracted dynamically to further reduce the dimensions and extract a fusion SHI that contained the degradation characteristics of the system. SAE's method of depth-feature fusion is to form a depth network by infinite stacking, take the output of the upper layer as the input of the next layer, and then minimize the reconstruction error of the input and output signals to complete the layer-by-layer compression of the input data. The basic structure is shown in Figure 3a. Figure 2: Schematic diagram of DKPCA. The input and output of the network are \(x\) and \(\hat{x}\), hidden layer node output \(h_{j}\) and output layer node \(\hat{x}_{k}\) can be defined by Equations (5) and (6):  
\[h_{j}=f(\sum_{i=1}^{m_{0}}x_{i}w_{ij}+b_{1}), \tag{5}\]  
\[\hat{x}_{k}=f(\sum_{j=1}^{n_{0}}h_{j}w_{jk}+b_{2}), \tag{6}\]  
where \(f(\cdot)\) is the active function, \(w_{ij}\) and \(w_{jk}\) are the weight matrices, \(b_{1}\), \(b_{2}\) are the bias vectors, \(m_{0}\) is the number of feature fusion, and \(n_{0}\) is the number of input and output nodes and is equal to the \(\gamma\) calculated in Section 3.2.1 above. The coding process of the network refers to mapping n-dimension as the input sample \(a\) into m-dimensional \(h_{j}\), with the decoding function \(h_{j}\) remapping into n-dimensional \(\hat{x}_{k}\) through decoding. The \(\hat{\rho}_{j}\) represents the average activation amount of \(h_{j}\), then:  
\[\hat{\rho}_{j}=\frac{1}{T_{n}}\sum_{j=1}^{T_{n}}h_{j}, \tag{7}\]  
where \(T_{n}\) is the number of training samples. In order to avoid useless learning, the sparsity system is added to the network, and only a few nodes are allowed to be active, that is, make \(\hat{\rho}_{j}\) close to \(\rho\), and \(\rho\) is a sparse parameter approaching 0. Kullback-Leibler divergence (KL) measures the deviation degree between \(\hat{\rho}_{j}\) and \(\rho\), as shown in Equation (8):  
\[KL(\rho\|\hat{\rho}_{j})=\rho\ln\frac{\rho}{\hat{\rho}_{j}}+(1-\rho)\ln\frac{ 1-\rho}{1-\hat{\rho}_{j}}. \tag{8}\]  
The overall cost function of the \(T_{n}\) samples can be defined as Equation (9):  
\[J(w,b)=\frac{1}{T_{n}}\sum_{i=1}^{T_{n}}\|x_{i}-\hat{x}_{i}\|^{2}+\lambda\|w \|^{2}+\beta\sum_{j=1}^{m_{0}}KL(\rho\|\hat{\rho}_{j}), \tag{9}\]  
where \(\lambda\) is the weight-decay parameter and \(\beta\) is the sparsity penalty term parameter used to control the relative importance between the first reconstruction term and the second penalty term. Figure 3: Schematic diagram of feature fusion by SAE: (**a**) SAE structure; (**b**) SAE fusion. According to the feature data extracted dynamically, two SAE stacks were used to form a trestle self-encoder for second-order parameter fusion. One-level fusion: taking the initial parameters as the input and output of the network structure shown in Figure 3a, the number of hidden layer nodes was less than the network input. Network coding and decoding were conducted, and the network parameters from the input layer to the \(h_{j}\) layer were obtained. After the training, the decoding layer was removed, leaving only the input layer to the \(h_{j}\) layer, as shown in Figure 3b, and the output of the hidden layer was the first-order fusion result.