## 3 Methodology
### Feature Fusion  
Through the SDAE compression features, to a certain extent, can improve the efficiency of features, and the characteristics of advantage can be found, but because of the existence of redundancy, the optimal feature attributes together is not necessarily the optimal and sensitive feature subset. The sensitive features that can represent the running state of the bearing should be monotonically related to the degradation process of the bearing and have low redundancy with other features, since irrelevant and redundant features will affect the prediction efficiency and accuracy, eliminating those features that cannot provide sufficient fault information has become the main task in the feature selection stage. For the measurement of correlation, most studies use the method of Person linear correlation coefficient [31; 32] to measure the correlation, so as to select the features with higher contribution rate. However, the nonlinear relationship between variables is often ignored, resulting in the unreliability of measurement results, because the complex relationship between general variables cannot be modeled by a single function. The MIC [33] solves the above problems very well. The MIC is suitable for measuring the linear and nonlinear relationship between variables in the measurement data, and mining the non functional dependency relationship between variables. Based on these goals, MIC is used to measure the relationship between features and degradation trend and the correlation between features. The MIC is calculated mainly by mutual information (MI) and meshing method. MI is used to measure the degree of correlation between variables. Given variable \(A=\{a_{i},i=1,2,\ldots n\}\) and variable \(B=\{b_{i},i=1,2,\ldots n\}\), where \(n\) is the number of samples, the mutual information \((MI)\) is defined as follows:  
\[MI(A,B)=\sum_{a\in A}\sum_{b\in B}p(a,b)\log\frac{p(a,b)}{p(a)p(b)} \tag{11}\]  
where \(P(a,b)\) is the joint probability density of \(A\) and \(B\), and \(P(a)\) and \(P(b)\) are the boundary probability densities of \(A\) and \(B\), respectively. Suppose set \(D=\{(a_{i},b_{i}),i=1,2,\ldots n\}\) is a set of finite ordered pairs. It defines a division \(G\), which is used to divide the value range of variable \(A\) into \(x\) segments and divide the value range of variable \(B\) into \(y\) segments. \(G\) is a grid with a size of \(x\times y\). Calculate \(MI(A,B)\) within each grid partition obtained, since the same grid can be divided several ways. The maximum value of \(MI(A,B)\) under different division methods is chosen as the \(MI\) value of a division \(G\). The maximum mutual information formula of \(D\) under a division is defined as \(MI^{*}(D,x,y)=\max MI(D|G)\), where \(D|G\) denotes data \(D\) are divided by \(G\). The maximum information coefficient \((MIC)\) uses \(MI\) to indicate the quality of the grid; a feature matrix is formed by maximum normalized \(MI\) values under different divisions. The feature matrix is defined as \(M(D)_{x,y}\) and the formula is :  
\[M(D)_{x,y}=\frac{MI^{*}(D,x,y)}{\log\min\{x,y\}} \tag{12}\]  
\(MIC\) is defined as:  
\[MIC(D)=\max_{xy<B(n)}\{M(D)_{x,y}\} \tag{13}\]where \(n\) is the sample size of the sample and \(B(n)\) is a function of sample size and represents the upper limit of the grid \(x\times y\). Generally, \(\omega\)\((1)\leq B(n)\leq o\left(n^{1-\varepsilon}\right)\), \(0<\varepsilon<1\). In this paper, the MIC is used to measure the correlation between features and degradation trend as well as features and features. In essence, it is a normalized maximum MI with the value interval of \([0,1]\). **Definition 1** (The correlation-\(Mic_{\text{FR}}\)). : _Suppose feature set \(F=\left\{f_{1t},f_{2t},\ldots,f_{mt};R_{1t},R_{2t},\ldots,R_{mt}\right\}\), \(m\) is the feature number, \(t\) is the time series, \(R_{mt}\) is the RUL corresponding to the feature \(f_{m}\) at time \(t\).