## 2 Correlation Model of Monitoring Data
### Correlation Modeling Using Partial Least Squares and Multiple Regression
#### 2.2.2 Establishment of Correlation Model Using the Partial Least Square Method  
The reconstructed variables and response variables are standardized as:  
\[\left(\mathbf{x}\right)_{i}^{*}=\frac{\left(\mathbf{x}\right)_{i}-\mu_{ \left(\mathbf{x}\right)_{i}}}{\sigma_{\left(\mathbf{x}\right)_{i}}} \tag{12}\]  
\[\mathbf{y}^{*}=\frac{\mathbf{y}-\mu_{\mathbf{y}}}{\sigma_{\mathbf{y}}} \tag{13}\]  
The response variables \(\mathbf{X}_{P}\) and the reconstructed variable \(\mathbf{Y}_{P}\) are standardized into \(\mathbf{E}_{\mathbf{0}}\) and \(\mathbf{F}_{\mathbf{0}}\), respectively:  
\[\mathbf{E}_{\mathbf{0}}=\left\{\left(\mathbf{x}\right)_{1}^{*}&\left(\mathbf{x }\right)_{2}^{*}&\cdots&\left(\mathbf{x}\right)_{p}^{*}\right\} \tag{14}\]  
\[\mathbf{F}_{\mathbf{0}}=\left\{\mathbf{y}^{*}\right\} \tag{15}\]  
The main components set of \(\mathbf{E}_{\mathbf{0}}\) and \(\mathbf{F}_{\mathbf{0}}\) can be obtained by using principal component analysis, which are denoted as \(\mathbf{T}\) and \(\mathbf{U}\), respectively:  
\[\mathbf{T}=\left\{ \begin{array}{cccccc}t_{1}&t_{2}&...&t_{h}&...&t_{A} \end{array} \right\} \tag{16}\]\[\mathbf{U}=\left\{ \begin{array}{cccccccc}u_{1}&u_{2}&...&u_{h}&...&u_{A}\end{array} \right\} \tag{17}\]  
The regression equations regarding \(\mathbf{E_{0}}\) and \(\mathbf{F_{0}}\) using partial least squares [10] and first principal components are:  
\[\mathbf{E_{0}}=t_{1}\mathbf{p_{1}}^{T}+\mathbf{E_{1}} \tag{18}\]  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]  
where the regression coefficient vector is:  
\[\mathbf{p_{1}}=\frac{\mathbf{E}_{0}^{T}t_{1}}{\left\|t_{1}\right\|^{2}} \tag{20}\]  
\[\mathbf{r_{1}}=\frac{\mathbf{E}_{0}^{T}t_{1}}{\left\|t_{1}\right\|^{2}} \tag{21}\]  
The regression equations for the \(h^{\text{th}}\) principal component are:  
\[\mathbf{E}_{h-1}=t_{h}\mathbf{p_{h}}^{T}+\mathbf{E}_{h} \tag{22}\]  
\[\mathbf{F}_{h-1}=t_{h}\mathbf{r_{h}}^{T}+\mathbf{F}_{h} \tag{23}\]  
where the regression coefficient vector is:  
\[\mathbf{p}_{h}=\frac{\mathbf{E}_{h-1}^{T}t_{h}}{\left\|t_{h}\right\|^{2}} \tag{24}\]  
\[\mathbf{r}_{h}=\frac{\mathbf{F}_{h-1}^{T}t_{h}}{\left\|t_{h}\right\|^{2}} \tag{25}\]  
The relationship between the \(h^{\text{th}}\) principal component \(t_{h}\) and the \(h^{\text{th}}\) axis of \(\mathbf{E_{0}}\) is:  
\[t_{h}=\mathbf{E}_{h-1}\mathbf{w}_{h} \tag{26}\]  
where \(\mathbf{w}_{h}\) is the unit eigenvector corresponding to the maximum eigenvalue \(\mathbf{\theta_{h}}^{2}\) of matrix \(\mathbf{E}_{h-1}^{T}\mathbf{F}_{h-1}\mathbf{F}_{h-1}^{T}\mathbf{E}_{h-1}\). According to the Equations (18), (19), (22), and (23), \(\mathbf{E_{0}}\) and \(\mathbf{F_{0}}\) can be expressed as:  
\[\mathbf{E_{0}}=t_{1}\mathbf{p_{1}}^{T}+t_{2}\mathbf{p_{2}}^{T}+\cdots+t_{h} \mathbf{p_{}}^{T}+\mathbf{E}_{h} \tag{27}\]  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]0  
Substituting Equation (26) into Equation (22) yields:  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]1  
Substituting Equations (26) and (29) into Equation (27) yields:  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]2  
Substituting Equation (30) into Equation (26) yields:  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]3\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]4  
Substituting Equation (31) into Equation (28) yields:  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]5  
Assuming the number of the principal components is \(h\) when the principal components satisfy the cross-validation principle. Meanwhile, \(\mathbf{F}_{h}\) approaches 0, which can be ignored. Equation (33) is written as:  
\[\mathbf{F_{0}}=t_{1}\mathbf{r_{1}}^{T}+\mathbf{F_{1}} \tag{19}\]6  
Substituting Equations (14) and (15) into Equation (34) yields:  
\[\mathbf{y}*=\left[(\mathbf{x})_{1}^{*}\ldots(\mathbf{x})_{i}^{*}\ldots( \mathbf{x})_{p}^{*}\right]\left[