## 4 Case Study
### Fermentation Process for Penicillin Production
#### 4.1.2 Performance Test
Next, comparative experiments are conducted on the full ensemble method and the selective ensemble method based on different selection criteria and ensemble pruning methods. In the SE-MLV-VSPLS algorithm, the quantitative parameters of accuracy or diversity are used as the criteria for ensemble pruning, and the GWO algorithm is applied to optimize the selection of the base model, which can maximize the parameters. In the MOSE-MLV-VSPLS method proposed in this study, after multi-objective optimization, one of the Pareto optimal solutions is selected and decoded to the ensemble for the soft sensor. Table 3 shows the performance of each model under the experiment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \begin{tabular}{c} **Kernel** \\ **Function** \\ \end{tabular} & \
\begin{tabular}{c} **Resampling** \\ **Times** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Subspaces** \\ \end{tabular} & \multirow{2}{*}{**RMSE**} & \multirow{2}{*}{\(\mathbf{R^{2}}\)} \\ \hline \multirow{4}{*}{VSEPLS} & & 12 & 12 & 0.02794 & 0.9982 \\  & - & 30 & 30 & 0.02793 & 0.9982 \\  & & 45 & 45 & 0.02791 & 0.9982 \\  & & **60** & **60** & **0.02790** & **0.9982** \\ \hline \multirow{4}{*}{LV-VSEPLS} & Gaussian & 12 & 12 & 0.01416 & 0.9979 \\  & kernel & 30 & 30 & 0.01406 & 0.9986 \\  & function & 45 & 45 & 0.01378 & 0.9986 \\  & \(\mathbf{60}\) & **60** & **0.01375** & **0.99980** \\ \cline{2-6}  & \multirow{4}{*}{Poly kernel} & 12 & 12 & 0.01410 & 0.9987 \\  & & 30 & 30 & 0.01466 & 0.9990 \\  & function & **45** & **45** & **0.01395** & **0.9994** \\  & & 60 & 60 & 0.01451 & 0.9989 \\ \cline{2-6}  & \multirow{4}{*}{Cauchy kernel} & 12 & 12 & 0.01466 & 0.9977 \\  & & 30 & 30 & 0.01464 & 0.9987 \\  & & 45 & 45 & 0.01470 & 0.9987 \\  & & **60** & **60** & **0.01441** & **0.9990** \\ \hline \multirow{4}{*}{MLV-VSEPLS} & Multiple & 4 & 12 & 0.01267 & 0.9998 \\  & kernel & 10 & 30 & 0.01223 & 0.9998 \\ \cline{1-1}  & function & 15 & 45 & 0.01220 & 0.9999 \\ \cline{1-1} \cline{2-6}  & \multirow{2}{*}{20} & **60** & **0.011934** & **0.9998** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._The data show that the same effect of the full ensemble is obtained with a smaller number of base models after ensemble pruning is completed based on a single indicator of accuracy or diversity. In contrast, after pruning with dual objectives of accuracy and diversity, the redundant base models are effectively eliminated, the complexity of the ensemble modeling is reduced, and the optimal prediction performance is obtained with a smaller number of base models. Figure 5 shows the performance comparison of each soft sensor built based on four different methods; the horizontal axis in the figure represents the sample points, while the vertical axis represents the values of penicillin concentration. Method 1 is MOSE-MLV-VSPLS, method 2 is MLV-VSEPLS, method 3 is LV-VSEPLS, and method 4 is VSEPLS. From the prediction trend curves, there is a significant difference between the real values and the predicted output values obtained using the VSEPLS and LV-VSEPLS methods. In contrast, almost no deviation can be found between the predicted output values and the real output values under the MOSE-MLV-VSPLS method, and these two curves fit well with each other. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Ensemble Type** & **Method** & **Criteria** & \
\begin{tabular}{c} **Number of** \\ **Subspaces** \\ \end{tabular} &
\begin{tabular}{c} **Size of Subensemble** \\ **Model** \\ \end{tabular} & **RMSE** & \(\mathbf{R^{2}}\) \\ \hline ALL & MLV-VSEPLS & - & 30 & 30 & 0.01223 & 0.09994 \\ SEL & SE-MLV-VSPLS & Accuracy & 30 & 9 & 0.01220 & 0.09992 \\ SEL & SE-MLV-VSPLS & Diversity & 30 & 8 & 0.01271 & 0.09990 \\ SEL & MOSE-MLV-VSPLS & Accuracy and diversity & **30** & **10** & **0.01063** & **0.09998** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of selective ensemble soft sensor models under ensemble pruning with different indicators.