## 3 Simulation Results
### Prediction Based on Dynamic Data
\begin{table}
\begin{tabular}{l l l l} \hline
**BF** & **Equation** & **BF** & **Equation** \\ \hline BF1 & max(0, 1687\(-\)\(x_{18}\)) & BF14 & max(0, 1333.9\(-\)\(x_{4}\)) \(\times\) max(0, \(x_{15}\)) \(\times\) max(0, 8151.8\(-\)\(x_{2}\)) \\ BF2 & max(0, \(x_{8}\)\(-\)\(\backslash\)58643) & BF15 & BF14 \(\times\) max(0, \(x_{3}\)\(-\)971) \\ BF3 & max(0, 0.58643\(-\)\(x_{8}\)) & BF16 & BF1 \(\times\) max(0, \(x_{17}\)\(-\)647.5) \\ BF4 & max(0, \(x_{4}\)\(-\)1333.9) & BF17 & BF4 \(\times\) max(0, \(x_{1}\)\(-\)875) \\ BF5 & max(0, \(x_{5}\)\(-\)146,000) & BF18 & BF4 \(\times\) max(0, 875\(-\)\(x_{1}\)) \\ BF6 & max(0, 146,000\(-\)\(x_{5}\)) & BF19 & max(0, 1333.9\(-\)\(x_{4}\)) \(\times\) max(0, \(x_{2}\)\(-\)8044.3) \\ BF7 & max(0, 0.028\(-\)\(x_{11}\)) & BF20 & max(0, 1333.9\(-\)\(x_{4}\)) \(\times\) max(0, 0.8044.3\(-\)\(x_{2}\)) \\ BF8 & max(0, 922\(-\)\(x_{3}\)) & BF21 & max(0, \(x_{3}\)\(-\)922) \(\times\) max(0, \(x_{17}\)\(-\)406.53) \\ BF9 & max(0, \(x_{13}\)\(-\)9272) & BF22 & max(0, \(x_{3}\)\(-\)922) \(\times\) max(0, 406.53\(-\)\(x_{17}\)) \\ BF10 & max(0, 9722\(-\)\(x_{13}\)) & BF23 & BF19 \(\times\) max(0, \(x_{5}\)\(-\)150100) \\ BF11 & max(0, \(x_{12}\)\(-\)50300) & BF24 & BF19 \(\times\) max(0, 150100\(-\)\(x_{5}\)) \\ BF12 & max(0, 50300\(-\)\(x_{12}\)) & BF25 & max(0, 1333.9\(-\)\(x_{4}\)) \(\times\) max(0, \(x_{15}\)) \(\times\) max(0, 8151.8\(-\)\(x_{2}\)) \(\times\) max(0, 971\(-\)\(x_{3}\)) \(\times\) max(0, 596\(-\)\(x_{1}\)) \\ BF13 & BF5 \(\times\) max(0, \(x_{2}\)\(-\)8503) & \\ \hline \end{tabular}
\begin{tabular}{l l}  & Carbon \((\%)=0.050403+0.00012192\times\) BF1 \(-0.030111\times\) BF2 \\  & \(-0.025931\times\) BF3 \(+0.00037055\times\) BF4 \(+0.000013054\times\) BF5 \\  & \(-0.00000071876\times\) BF6 \(-0.50589\times\) BF7 \(+0.000020491\times\) BF8 \\  & \(+0.0000068037\times\) BF9 \(+0.0000014624\times\) BF10 \(-0.0000076596\times\) BF11 \\  & \(-0.00000077033\times\) BF12 \(-0.000000073973\times\) BF13 \\  & \(+0.0000000021832\times\) BF14 \(-0.000000000097109\times\) BF15 \(+0.0000012\times\) BF16 \\  & \(-0.0000067856\times\) BF17 \(-0.00000068003\times\) BF18 \(-0.00000028498\times\) BF19 \\  & \(-0.00000037196\times\) BF20 \(-0.00000029068\times\) BF21 \(-0.00000013186\times\) BF22 \\  & \(+0.000000000037285\times\) BF23 \(+0.000000000044636\times\) BF24 \\  & \(+0.0000000000081557\times\) BF25. \\ \end{tabular}
\end{table} 
Table 13: Basis functions of the piecewise-linear MARS model of carbon content (lime and dolomitic lime added as supplementary observations). Dynamic data (see Table 2) from ten selected meltings that were not part of the training were used to test the trained models. For illustration, Figure 2 shows the time behavior of dynamic data recorded during melting #5. In practice, dynamic data are continuously measured during melting and can be used as the input to the machine model to make online predictions of temperature or carbon concentration. The performances of individual machine models from the training and testing phase are shown in Table 14. In the training phase, the statistical measures were calculated to indicate the performance level of the individual machine learning methods. The performance index (PI) was considered the primary indicator for performance assessment. In addition, the time required to train individual models was determined. Table 15 presents the performance of the models in the testing phase on dynamic data from melts that were not the subject of their training. Since the dynamic course of temperature and carbon was not measured, only the simulated (i.e., predicted) and actual measured value (i.e., temperature/carbon) in the BOS endpoint were compared. To determine the model's accuracy, the absolute and relative error at this endpoint was calculated. The results show that in the training phase, the RF-based model was best (i.e., PI = 0.76), although this model was the slowest in training. The NN-based model approximated the measured data the least satisfactorily (PI = 1.32). The number of neurons in one hidden layer was set to 9.