## 4 Case Study
### Fermentation Process for Penicillin Production
#### 4.1.2 Performance Test  
In this study, to demonstrate the effectiveness of the diversity perturbation mechanism in the base model generation stage, we use an ensemble of all base models generated on different diversity mechanisms at first. At this stage of the experiment, the number of base models under each method is the same, which is set to 12, 30, 45, and 60. In the EPLS, VSEPLS, and LV-VSEPLS methods, the number of base models equals the resampling times. In MLV-VSEPLS, three kinds of kernel functions are used for each resampled sample space, and three different base models are constructed; the number of base models is three times the resampling times. The performance of the penicillin fermentation process under different diversity perturbation mechanisms in the framework of the full ensemble soft sensor is given in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Variables** & **Variable Description** & **Units** & **Input Variables** & **Variable Description** & **Units** \\ \hline x\({}_{1}\) & Aeration rate & L/h & x\({}_{8}\) & Substrate feed temperature & K \\ x\({}_{2}\) & Agitator power & W & x\({}_{9}\) & Dissolved oxygen concentrate & g/L \\ x\({}_{3}\) & Substrate feed rate & L/H & x\({}_{10}\) & Culture volume & L \\ x\({}_{4}\) & Acid flow rate & L/H & x\({}_{11}\) & Carbon dioxide concentrate & g/L \\ x\({}_{5}\) & Base flow rate & L/H & x\({}_{12}\) & Fermenter temperature & K \\ x\({}_{6}\) & Cooling water flow rate & L/H & x\({}_{13}\) & pH & - \\ x\({}_{7}\) & Hot water flow rate & L/H & x\({}_{14}\) & Generated heat & kcal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input variables in the penicillin fermentation process. \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} & \
\begin{tabular}{c} **RMSE** \\ \end{tabular} &
\begin{tabular}{c} **R2** \\ \end{tabular} \\ \hline \multirow{4}{*}{EPLS} & & 12 & 12 & 0.04977 & 0.9975 \\  & & 30 & 30 & 0.04891 & 0.9976 \\  & & 45 & 45 & 0.04890 & 0.9976 \\  & & **60** & **60** & **0.04889** & **0.9976** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of fully ensemble soft sensor models in penicillin fermentation under different diversity perturbation mechanisms. According to the RMSE criteria, the optimal set of data in each method is bold. From this, the EPLS method incorporates only input data perturbations in the base model generation and shows the worst prediction accuracy on the test set. The VSEPLS method adds an input variable selection mechanism, which improves the prediction performance of the ensemble model to some extent compared with EPLS, but the accuracy of the two methods is not high. LV-VSEPLS combines the perturbation input features with the perturbation kernel function parameters on the basis of perturbed input data and improves the accuracy of the prediction to a certain extent. By comparison, in the MLV-VSEPLS method, the latent data relationships of the samples are fully explored from different perspectives, and a diversity of latent variable modeling subspaces are constructed. In the full ensemble framework, the number of resampling is reduced, and the optimal performance is achieved with the same number of base models. It is also clear from the data that the rate of improvement in the model accuracy is higher when the number of modeling sample spaces is increased from 15 to 30. However, the subsample set of the base model for constructing the ensemble learning algorithms in this study was obtained through bootstrap resampling, and the base learners were homogeneous. Due to the limited data samples, the base learners were not independent of each other, so when the number of base models was increased from 45 to 60, the improvement rate of prediction accuracy of ensemble learning soft sensor models showed a decreasing and stabilizing trend.