## 6 Experiments and Results
### Inertia Tank Modeling  
Table 10 lists the five top models sorted by their average R\({}^{2}\) values (R\({}^{2}_{\text{avg}}\)). For comparison purposes, standalone data-driven models were trained using the same grid search of  
Figure 14: Best forecast temperatures of the laboratory’s model. (**a**) Best R\({}^{2}\) of the best-performing model. (**b**) Second-best R\({}^{2}\) of the best-performing model.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Metric for Each System** & **Day 1** & **Day 2** & **Day 3** & **Day 4** & **Day 5** & **Day 6** & **Day 7** & **Day 8** & **Day 9** & **Day 10** & **Day 11** \\ \hline R\({}^{2}_{\text{lab}}\) & 0.66 & 0.86 & 0.69 & 0.91 & 0.60 & 0.63 & **0.96** & \(-\)0.27 & 0.88 & 0.77 & \(-\)0.69 \\ MAPE\({}_{\text{zip}}\) (\%) & 1.51 & 0.83 & 0.78 & 0.61 & 0.84 & 0.90 & **0.44** & 2.22 & 0.64 & 2.1 & 2.63 \\ MAE\({}_{\text{zip}}\) (\%) & 0.35 & 0.19 & 0.18 & 0.14 & 0.20 & 0.22 & **0.10** & 0.54 & 0.15 & 0.46 & 0.65 \\ MSE\({}_{\text{zip}}\) (\%\({}^{\text{C}}\)) & 0.17 & 0.05 & 0.04 & 0.04 & 0.05 & 0.07 & **0.02** & 0.38 & 0.04 & 0.30 & 0.52 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Metrics table for each test day. Lab stands for laboratory.  
Figure 13: Worst forecast temperatures of the laboratory’s model. (**a**) Worst R\({}^{2}\) of the best-performing model. (**b**) Second-worst R\({}^{2}\) of the best-performing model.  
hyperparameters. The model with the best performance achieved an R\({}_{avg}^{2}\) of 0.16, indicating a weak correlation between the predicted and observed values.  
For both R\({}_{avg}^{2}\) and MAPE\({}_{avg}\), the best hyperparameters are kernel = \(rbf\), C = 50, \(\varepsilon\) = 0.05, and \(\gamma=scale\).  
The observed deviations in the laboratory temperature cannot be extrapolated to the inertia tank temperature. This is because the inertia tank is more insulated than the laboratory and less susceptible to external temperature fluctuations. Additionally, the variables influencing the behavior of the inertia tank are not subject to randomness or hard-to-predict factors, such as laboratory occupancy. Consequently, the models achieve higher R\({}^{2}\) values and lower MAPE scores for each day tested, as shown in Table 11. Figure 15a,b illustrate the two worst-performing days.  
Even though the last figures were the worst predicted, the values taken by their score metrics indicate a good fit, with high accuracy also shown by the prediction profile of the temperature. Figure 16a,b show the best predicted values of the test set.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**R\({}_{avg}^{2}\)** & **MAPE\({}_{avg}\) (\%)** & **Kernel** & **C** & \(\varepsilon\) & \(\gamma\) \\ \hline
0.93 & 1.31 & rbf & 50 & 0.05 & auto \\
0.92 & 1.44 & rbf & 10 & 0.05 & auto \\
0.90 & 1.54 & rbf & 10 & 0.1 & auto \\
0.90 & 1.46 & rbf & 5 & 0.1 & auto \\
0.90 & 1.37 & rbf & 100 & 0.05 & auto \\ \hline \hline \end{tabular}
\end{table}
Table 10: Five top-performing models sorted by R\({}^{2}\).  
Figure 15: Worst forecast temperatures of the inertia tank’s model. (**a**) Worst R\({}^{2}\) of the best-performing model. (**b**) Second-worst R\({}^{2}\) of the best-performing model.