## 3 Methodology
### Feature Fusion
The correlation between any feature \(f_{i}\) and the RUL \(R_{i}\) is defined as \(MIC_{F_{i}R_{i}}\) and denoted as \(MIC_{F_{i}R_{i}}\), the larger the value of \(MIC_{F_{i}R_{i}}\) is, then \(f_{i}\) is the strong correlation feature. On the contrary, \(f_{i}\) is a weak correlation feature; If the value of \(MIC_{F}\) is 0, \(f_{i}\) is an independent feature._  
**Definition 2** (The redundancy-\(Mic_{\text{FF}}\)). : _The redundancy ( a kind of correlation) between any feature \(f_{i}\) and feature \(f_{j}\) is defined as \(MIC_{F_{i}F_{j}}\), denotation as \(MIC_{F_{j}}\), the higher the value of \(MIC_{\text{FF}}\), the stronger the substitutability between \(f_{i}\) and \(f_{j}\), namely the stronger the redundancy; the value of \(MIC_{\text{FF}}\) is 0, indicating that \(f_{i}\) and \(f_{j}\) are independent of each other._  
Calculating the redundancy \(MIC_{F_{i}F_{j}}\) between features in feature subset \(F_{SDAE}\), the \(MIC-FF\) matrix will be obtained that can measure the correlation between features. Finding the minimum values for each column in the \(MIC-FF\) matrix and combining these minimum values into a set \(m_{FF}=\left\{m_{FF0},m_{FF1},\ldots,m_{FFn}\right\}\), where each column corresponds to one feature, and there are \(n\) columns in this matrix. The maximum value will as the \(FF-threshold\). Counting the number of elements in each column that are less than the threshold value, combining the numbers into a set \(N_{FF}=\left\{Num_{FF0},N_{FF1},\ldots,N_{FFn}\right\}\), sorting the numbers to find the median. If the number of values is better than the median, the features corresponding to this column will be the elements of the De-redundant feature subset \({}_{DE}MIC_{FF}\). Similarly, the correlation \(MIC_{\text{FR}}\) is obtained for the elements in subset \({}_{DE}MIC_{FF}\), so as to obtain the degree of influence of those features on the degradation state after the redundancy is removed. **Definition 3** (The monotonicity-\(Mon_{\text{F-DT}}\)). : _In order to measure the performance of the feature more comprehensively, monotonicity is used as one of the measures. We will measure the monotonicity of features and degradation tendency, denoted as \(Mon_{F-DT}\)._  
\(Mon_{\text{F-DT}}=\left| \begin{array}{cc}\frac{F_{\text{F-D}}}{L-1}-\frac{F_ {\text{F-D}}}{L-1}\end{array} \right|\)_, \(F_{\text{S}}\) is the different feature sequence, \(L\) is the sample length of the full life cycle, and when \(Mon_{F-DT}=1\), the feature and degradation tendency are completely monotonous._  
**Definition 4** (The Standard-\(Std_{\text{DI}}\)). : _Both feature measures \(MIC_{\text{FR}}\) and \(Mon_{\text{F-DT}}\) are limited within the range of \([0,1]\). They are positively correlated with the performance of candidate features and are suitable as feature selection measures. The linear combination of the above two indexes is taken as the selection standard of bearing degradation index, denoted as \(Std_{DI}\)._  
\(Std_{DI}=\frac{MIC_{\text{FR}}+Mon_{\text{F-DT}}}{2}\)_based on the \(MIC_{\text{FR}}\) value and \(Mon_{\text{F-DT}}\) value of all features in feature subset \({}_{DE}MIC_{\text{FF}}\) are obtained. If \(Std_{DI}>0.5\), the feature will be selected into the optimal DI set \(Opt_{DI}\)._  
Optimal DI Set Construction  
A subset of \(F_{SDAE}\) is composed of low-dimensional feature expression based on SDAE feature compression method. We will further de-redundant the features in the subset \(F_{SDAE}\). Based on the MIC feature selection method [34], a de-redundancy feature subset \({}_{DE}MIC_{\text{FF}}\) can be obtained by de-redundancy operation. After the de-redundant feature subset \({}_{DE}MIC_{\text{FF}}\) is obtained, we will calculate the correlation and monotonicity of the features in the \({}_{DE}MIC_{\text{FF}}\) subset, and measure the sensitivity of the DI based on the proposed \(Std_{DI}\) standard, which ultimately constitutes the optimal DI set \(Opt_{DI}\). The steps for obtaining the set \(Opt_{DI}\) are shown in Algorithm 1.