## 3 Simulation Results
### Prediction Based on Dynamic Data
On the other hand, in the test phase on ten meltings, the highest accuracy was achieved by the piecewise-linear MARS model. The average relative error from the ten meltings was 0.73%, and the average absolute error was 12.19 \({}^{\circ}\)C. The SVR-based model achieved the lowest accuracy in endpoint temperature prediction with the Gaussian kernel, i.e., the average relative error was 3.68%, and the average absolute error was 61.68 \({}^{\circ}\)C (see Table 15). \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Method**} & \multicolumn{6}{c}{**Training**} \\ \hline  & **Time (s)** & \(r_{\text{{yy}}^{2}}\) & \(r_{\text{{yy}}^{2}}\) & **MSE** & **RMSE** & **RRMSE (\%)** & **MAPE (\%)** & **PI** \\ \hline SVR (polynomial kernel) & 1.0183 & 0.9842 & 0.9686 & 969.8928 & 31.1431 & 2.0959 & 1.5849 & 1.0563 \\ SVR (Gaussian kernel) & 0.2528 & 0.9883 & 0.9767 & 723.7140 & 26.9019 & 1.8104 & 1.3153 & 0.9105 \\ NN & 1.6471 & 0.9757 & 0.9520 & 1491.0892 & 38.6146 & 2.5987 & 1.6227 & 1.6227 \\ MARS (pieeewise-linear) & 2.4166 & 0.9850 & 0.9702 & 919.6555 & 30.3528 & 2.0409 & 1.4773 & 1.0281 \\ MARS (pieeewise-cubic) & 2.7178 & 0.9837 & 0.9677 & 1002.3017 & 31.6591 & 2.1306 & 1.6113 & 1.0741 \\ \(k\)-NN & 0.0015 & 0.9877 & 0.9756 & 754.7641 & 27.4730 & 1.8489 & 1.4295 & 0.9301 \\ RF & 59.0626 & 0.9918 & 0.9837 & **506.5556** & 22.5068 & 1.5147 & 1.1175 & **0.7604** \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Performance of the machine learning in temperature prediction based on dynamic data. Figure 19: Dynamic input observations of the melting #5. Table 16 compares the performance of individual models for melt carbon concentration. In this case, the individual models were also trained on static and dynamic data, and carbon concentration predictions were tested on ten meltings. The results show that in the training phase, the measured data were best approximated by the RF-based model (i.e., PI = 4.40). On the second side, this model was the slowest in training. Unfortunately, MARS-type models completely failed in the training phase due to the nature of the input data. In the training of the MARS model, numerical issues occurred due to a low correlation between input observations and the target variable. Therefore, testing of MARS models could not be performed. The NN-based model showed the lowest performance in training (i.e., PI = 7.77). In the test phase, i.e., prediction of the BOS endpoint, the best results were achieved by the \(k\)-NN model (i.e., the average relative error from the ten meltings was 12.62%, and the average absolute error was approximately 0.01 vol.%) (see Table 17). The SVR model with Gaussian kernel was, on average, the least accurate (i.e., the average relative error from the ten meltings was up to 1376.47%, and the average absolute error was approximately 0.58 vol.%). In general, it can be stated that modeling and carbon prediction were less successful because the models were less accurate than in the case of temperature, and some models failed. Due to the most precise model for predicting endpoint melt temperature on unknown dynamic data being the piecewise-linear MARS model, its equation is presented in this paper (see Equation (33)). This regression model can be programmed in the monitoring system of the BOS plant and used as a soft sensor to estimate the melt temperature from dynamic measured data. The list of corresponding basis functions for this model is shown in Table 18. On the other hand, the most precise prediction of endpoint carbon content was achieved by the \(k\)-NN model. Unfortunately, this machine model cannot be presented in the form of a single regression equation.