## 4 Comparison Studies with LASSO and PCR Methods  
### LASSO Regression Method  
According to LASSO regression, as shown in Equation (4), the \(\lambda\) parameter controls the sparsity. The smaller the \(\lambda\), the more features (strain gauge locations) are selected, which means the sparsity is low. \[J_{\beta}=\frac{1}{2N}\sum_{i=1}^{N}\left(f_{i}-s_{i}^{\mathrm{T}}\beta\right)^ {2}+\lambda P_{a}(\beta) \tag{4}\]  
in which  
\[P_{a}(\beta)=\sum_{j=1}^{p}|\beta_{j}|, \tag{5}\]  
where \(N\) is the number of observed objects (strain gauge locations in this study), and the coefficient \(\beta\) is obtained by optimizing the minimum value of the loss function. \(s\) and \(f\) imply the strain data and the load data, respectively. When \(\lambda=0.00001\), the relative errors between the predicted and actual values of the No. 1 loading actuator are presented in Figure 12. Moreover, the normalized regression formula is given as follows:  
\[ \begin{array}{l}f_{1}=0.0114+0.0717s_{1}+0.4115s_{2}+0.1874s_{3}+0.0417s_{4} +7.87\times 10^{-5}s_{5}\\ +0.2687s_{6}-0.1108s_{7}+8.54\times 10^{-5}s_{8}-0.0424s_{9}-0.1720s_{10}-0.1168 s_{11}\\ +0.0739s_{12}-4.58\times 10^{-5}s_{13}\end{array} \]  
Among them, \(f_{1}\) represents the operation load of the No. 1 loading actuator, and \(s_{1}\sim s_{13}\) are the strain value vectors corresponding to 1\(\sim\)13 monitoring points. The predicted errors of the No. 2 loading actuator are given in Figure 13, and its normalized regression formula is as follows:  
\[ \begin{array}{l}f_{2}=0.018+0.0979s_{1}-0.0108s_{2}+0.142s_{3}+0.0684s_{4}+0. 2037s_{5}\\ +0.0044s_{6}+0.2872s_{7}-0.1527s_{8}+0.0422s_{9}+0.2309s_{10}-\\ 1.65\times 10^{-4}s_{11}+0.2147s_{12}+1.1\times 10^{-4}s_{13}\end{array} \]  
Figure 11: Expected and predicted value errors of No. 2 load actuator. Among them, \(f_{2}\) indicates the predicted operation load of the No. 2 loading actuator, and \(s_{1}\sim s_{13}\) are the strain value vectors corresponding to 1-13 monitoring points. As it manifests from Figures 12 and 13, the LASSO regression selection has many features; all 13 monitoring points are selected, and the load prediction accuracy meets the accuracy of engineering applications. When \(\lambda=0.023\), the load predicted errors of the No. 1 loading actuator are depicted in Figure 14, and its normalized regression formula is  
\[f_{1}=-0.003+0.9677s_{6}+0.0109s_{8},\]  
in which \(f_{1}\) denotes the predicted loads of the No. 1 loading actuator, and \(s_{6},s_{8}\) imply the strain value vectors corresponding to strain monitoring points 6 and 8, respectively. Figure 12: Expected and predicted load errors of No. 1 load actuator. Figure 13: Expected and predicted load errors of No. 2 load actuator. The load predicted errors of the No. 2 loading actuator are illustrated in Figure 15, and its normalized regression formula is  
\[f_{2}=0.0181+0.7583_{\mathcal{S}7}+0.2449_{\mathcal{S}12},\]  
in which \(f_{2}\) is the predicted load of the No. 2 loading actuator, and \(s_{7},s_{12}\) are the strain value vectors corresponding to strain monitoring points 7 and 12, respectively. The results of Figures 14 and 15 show that the LASSO regression sparsity is larger when the value of \(\lambda\) is larger, which means that more strain gauge locations will be utilized. In comparison with the results of \(\lambda=0.00001\), the load prediction accuracy of the monitoring point is obviously degraded when the monitoring point is less. The load regression equation of the No. 1 loading actuator selects strain monitoring points 6 and 8, while the No. 2 loading actuator chooses strain monitoring points 7 and 12. It can be concluded that the LASSO method requires both the strain data and the loading data for strain monitoring point selection, which decreases its efficiency. Moreover, the LASSO method individually optimizes the strain monitoring locations for each load actuator. Thus, the determined  
Figure 14: Expected and predicted load errors of No. 1 load actuator. Figure 15: Expected and predicted load errors of No.