## 2 Correlation Model of Monitoring Data
### Influence of Response Variable Number on Reconstruction Error  
In order to discuss the influences of the different number of response variables, three additional cases are discussed, which are:  
\[\mathbf{X}_{P5}=\left\{\left(\mathbf{x}\right)_{1}\quad\left(\mathbf{x}\right)_{2 }\quad\left(\mathbf{x}\right)_{3}\quad\left(\mathbf{x}\right)_{4}\quad\left( \mathbf{x}\right)_{5}\quad\right\} \tag{51}\]  
\[\mathbf{X}_{P6}=\left\{\left(\mathbf{x}\right)_{1}\quad\left(\mathbf{x}\right) _{2}\quad\left(\mathbf{x}\right)_{3}\quad\left(\mathbf{x}\right)_{4}\quad \left(\mathbf{x}\right)_{5}\quad\left(\mathbf{x}\right)_{6}\quad\right\} \tag{52}\]  
\[\mathbf{X}_{P7}=\left\{\left(\mathbf{x}\right)_{1}\quad\left(\mathbf{x}\right) _{2}\quad\left(\mathbf{x}\right)_{3}\quad\left(\mathbf{x}\right)_{4}\quad \left(\mathbf{x}\right)_{5}\quad\left(\mathbf{x}\right)_{6}\quad\left(\mathbf{x }\right)_{7}\quad\right\} \tag{53}\]  
The correlation model using \(\mathbf{X}_{P5}\) is:  
\[\mathbf{y}_{6}(i)=1.33\left(\mathbf{x}(i)\right)_{1}+0.10\left(\mathbf{x}(i) \right)_{2}-0.03\left(\mathbf{x}(i)\right)_{3}-0.08\left(\mathbf{x}(i)\right) _{4}-0.20\left(\mathbf{x}(i)\right)_{5}-32.82 \tag{54}\]  
The contrast curves between reconstructed values and actual values are shown in Figure 9. The average absolute error is 0.80 MPa and the relative error is 2.45%. The correlation model using \(\mathbf{X}_{P6}\) is:  
\[\mathbf{y}_{7}(i)=1.33\left(\mathbf{x}(i)\right)_{1}+0.11\left(\mathbf{x}(i) \right)_{2}-0.04\left(\mathbf{x}(i)\right)_{3}-0.11\left(\mathbf{x}(i)\right) _{4}-0.16\left(\mathbf{x}(i)\right)_{5}+0.08\left(\mathbf{x}(i)\right)_{6}-3 0.95 \tag{55}\]  
Figure 8: The reconstructed value and actual value using data from 7 May 2014 to 14 May 2014: (**a**) the curve of the reconstructed value and the actual value; and (**b**) the curve of the reconstruction error. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Data Acquisition Time** & \(\boldsymbol{e_{a}}\) (MPa) & \(\boldsymbol{e_{b}}\) \\ \hline
2014.05.07–2014.05.10 & 0.89 & 2.71\% \\
2014.05.07–2014.05.12 & 0.65 & 1.99\% \\
2014.05.07–2014.05.14 & 0.22 & 0.68\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The summarized errors using stress monitoring data from different time periods. Figure 9: The reconstructed value and actual value using response variables \(\mathbf{X}_{P5}\): (**a**) the curve of the reconstructed value and the actual value; and (**b**) the curve of reconstruction error. The contrast curves between reconstructed values and actual values are shown in Figure 10. The average absolute error is 0.80 MPa and the relative error is 2.44%. The correlation model using \(\mathbf{X}_{P7}\) is  
\[\mathbf{y}_{8}=1.30(\mathbf{x})_{1}+0.09(\mathbf{x})_{2}-0.07(\mathbf{x})_{3}-0.08(\mathbf{x})_{4}-0.19(\mathbf{x})_{5}+0.12(\mathbf{x})_{6}+0.13(\mathbf{x}) _{7}-26.93 \tag{56}\]  
The contrast curves between the reconstructed values and the actual values are shown in Figure 11. The average absolute error is 0.77 MPa and the relative error is 2.36%. The errors of the reconstructed information using different number of the response variables are summarized in Table 4.  
It can be seen from Table 4 that the errors become smaller when the number of response variables become larger, in which the average absolute error and relative error are minimum when \(\mathbf{X}_{P7}\) is chosen as the response variables. However, there is only a slight difference between these discussed cases, and the number of the response variables with the same level of correlation coefficients influencing the errors of the reconstructed variables. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Response Variable** & \(\boldsymbol{e_{a}}\) (MPa) & \(\boldsymbol{e_{b}}\) \\ \hline \(\mathbf{X}_{P1}\) & 0.89 & 2.71\% \\ \(\mathbf{X}_{P5}\) & 0.80 & 2.45\% \\ \(\mathbf{X}_{P6}\) & 0.80 & 2.44\% \\ \(\mathbf{X}_{P7}\) & 0.77 & 2.36\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The summarized errors using different number of the response variables.