## 4 Case Study
### Debutanizer Column
#### 4.2.2 Performance Test  
Similarly, in the EPLS, VSEPLS, and LV-VSPLS methods, the number of base models is equal to the number of sample spaces generated under resampling, which is 15, 30, and 45, respectively. In the MLV-VSEPLS method, the number of resampling times is 5, 10, and 15, and the number of base models is 15, 30, and 45, respectively. Table 5 shows the performance of the full ensemble-based soft sensor in the debutanizer column test set under different diversity perturbation mechanisms. From the experimental results in Table 5, for the debutanizer column, the prediction accuracy of the EPLS method using only the input data perturbation is very low because of the strong nonlinearity of the seven process variables, and the prediction accuracy is improved to some extent by adding input variable perturbation, but it is far from meeting the requirements. Based on the kernel trick, under the framework of the full ensemble model, the accuracy of the soft sensor ensemble model has been improved to a certain extent. However, a single kernel function is used to construct the latent variable subspace, so even with the increase in resampling times in the LV-VSEPLS method, the precision improvement space of the soft sensor model is limited. By comparison, in the MLV-VSEPLS method, multiple diversity enhancement mechanisms are used in the process of subsample set generation, initial variable space, and latent variable subspace construction for diversified base models. Under the fully ensemble framework, the MLV-VSEPLS method has achieved better results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Kernel Function**} & \multirow{2}{*}{**Resampling Times**} & \multirow{2}{*}{**Number of Subspaces**} & \multirow{2}{*}{**RMSE**} & \multirow{2}{*}{**\(\mathbf{R^{2}}\)**} \\ \cline{5-6}  & & & 12 & 12 & 0.1555 & 0.1345 \\ \multirow{2}{*}{EPLS} & \multirow{2}{*}{-} & 30 & 30 & 0.1553 & 0.1300 \\  & & **45** & **45** & **0.1554** & **0.1321** \\ \hline \multirow{2}{*}{VSEPLS} & \multirow{2}{*}{-} & 15 & 15 & 0.1713 & 0.1250 \\  & & 30 & 30 & 0.1621 & 0.1293 \\ \multirow{2}{*}{VSEPLS} & \multirow{2}{*}{-} & **45** & **45** & **0.1613** & **0.1295** \\ \hline \multirow{2}{*}{LV-VSEPLS} & Gaussian kernel function & 15 & 15 & 0.0646 & 0.8576 \\ \multirow{2}{*}{LV-VSEPLS} & Laplace kernel function & 15 & 15 & 0.0527 & 0.9053 \\  & & 30 & 30 & 0.0519 & 0.9081 \\ \multirow{2}{*}{VSEPLS} & Cauchy kernel function & 15 & 15 & 0.0552 & 0.8960 \\  & & 30 & 30 & 0.0511 & 0.9110 \\ \multirow{2}{*}{VSEPLS} & Cauchy kernel function & **45** & **45** & **0.0497** & **0.9159** \\ \hline \multirow{2}{*}{MLV-VSEPLS} & Multiple kernel functions & 5 & 15 & 0.0516 & 0.9142 \\ \multirow{2}{*}{MLV-VSEPLS} & Multiple kernels functions & 10 & 30 & 0.0513 & 0.9138 \\ \multirow{2}{*}{MLV-VSEPLS} & Multiple kernels functions & **15** & **45** & **0.0503** & **0.9195** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance of fully ensemble soft sensor models in test sets under different diversity perturbation mechanisms. However, due to the existence of poor performance and redundant base models, the full ensemble method used above cannot achieve the optimal performance. Therefore, in order to improve the performance of the ensemble soft measurement models, the ensemble pruning work is essential. Similarly, a comparative study of ensemble pruning methods is designed to display the superiority of the multi-objective optimization selection method based on GWO. A multi-objective optimization method is used to decode one of the Pareto optimal solutions under 45 base models, and the target variables are predicted on the test set; the experimental results are shown in Table 6.  
In order to more clearly show the performance, the box plots in Figure 7 are used to show comparatively the RMSE obtained with the selective ensemble and the full ensemble under the statistical results of 25 experiments.