## 2 Modeling of Initial Health Assessment Based on MSET
### Establish Health Baseline Based on MSET
The expression is as follows:  
\[\varepsilon=\left|X_{est}-X_{obs}\right| \tag{8}\]  
By minimizing the residual vector \(\varepsilon\) and deriving the weight vector \(W\) using the least square method, the expression is as follows:  
\[\left\|\,\varepsilon\,\right\|^{2}=\left\|\,X_{obs}-X_{est}\right\|^{2}=0 \tag{9}\]  
Further expand the operation:  
\[F(w)=\sum_{i=1}^{n}\varepsilon_{i}^{2}=\varepsilon^{T}\varepsilon=(X_{obs}- DW)^{T}(X_{obs}-DW)=\sum_{i=1}^{n}\left(X_{obs}(i)-\sum_{j=1}^{m}w_{j}D_{ij} \right)^{2} \tag{10}\]  
The partial derivative of \(w_{k},k=1,2,\ldots,m\) is obtained from \(F(w)\) and set to 0, which is calculated as follows:  
\[\frac{\partial F(w)}{\partial w_{k}}=-2\sum_{i=1}^{n}\left(X_{obs}(i)-\sum_{j =1}^{m}w_{i}D_{ij}\right)D_{ik}=0 \tag{11}\]  
\[\sum_{i=1}^{n}X_{obs}(i)D_{ik}=\sum_{i=1}^{n}\sum_{i=1}^{m}w_{j}D_{ij}D_{ik}= \sum_{j=1}^{m}\big{(}\sum_{i=1}^{n}D_{ij}D_{ik}\big{)}w_{j} \tag{12}\]  
Convert the above equation to matrix form:  
\[D^{T}\cdot D\cdot W=D^{T}\cdot X_{obs} \tag{13}\]  
\[W=\left(D^{T}\cdot D\right)^{-1}\left(D^{T}\cdot X_{obs}\right) \tag{14}\]  
According to Equation (14), \(D^{T}\cdot D\) invertibility is a prerequisite for the existence of a weight vector \(W\), and a necessary but not sufficient condition for invertibility is that the number of columns \(D\) is less than the number of rows \(D\) [2 ]. In fact, in order to make the calculation accuracy of MSET high, the memory matrix needs to include as much as possible the historical observation vector that the equipment runs normally under all working conditions. Therefore, the number of columns of the memory matrix is bound to be greater than the number of rows, so the weight vector cannot be calculated by dot multiplication. Nonlinear operators \(\otimes\) can be used to solve such problems, and the matrix product can be replaced by nonlinear operators \(\otimes\) to avoid the irreversible problem of matrix multiplication. Therefore, the matrix-vector expression is:  
\[W={{{\left({{D^{T}}\otimes{D}}\right)}^{-1}}}{{\left({{D^{T}}\otimes{X_{obs}} }\right)}} \tag{15}\]  
where \(\otimes\) is the symbol of nonlinear operation, Euclidean distance is chosen as the nonlinear operator between vectors, defined as the following equation:  
\[X_{est}=DW=w_{1}X(t_{1})+w_{2}X(t_{2})+\ldots+w_{m}X(t_{m}) \tag{7}\]0  
where \(X\), \(Y\) refer to vectors. The higher the similarity between the observation vector \(X_{obs}\) and the vector \(X_{j},j=1,2,\dots,m\) in the historical memory matrix \(D\) is, the greater the corresponding weight \(w_{j},j=12,\dots,m\) will be, and the weight vector \(W\) can be calculated from this. Substitute the weight vector \(W\) into Equation (7) to calculate the estimation vector \(X_{est}\), and simplify it, as shown as follows:  
\[X_{est}=DW=w_{1}X(t_{1})+w_{2}X(t_{2})+\ldots+w_{m}X(t_{m}) \tag{7}\]1  
The estimation vector as the output of MSET represents the data condition predicted by the model when the equipment is in a normal state under the current working condition to be evaluated and can be used as the health benchmark of the current equipment operation condition. The greater the deviation between the observation vector and the estimation vector is, the further the equipment deviates from the health benchmark. When the deviation between the observation vector and the estimation vector is small, it indicates that the device is currently in a healthy state or that the degree of performance degradation is small. By condition matching, the multivariate state estimation method outputs the estimation vector after removing the influence of the condition. By quantifying the deviation degree between the observation vector and the estimation vector, combined with the health mapping equation, the purpose of evaluating the current equipment's health state is finally achieved.