## 2 Theoretical Background of Applied Method
### Model Performance Indicators  
The performance of individual machine learning methods, and the created regression models, respectively, after application to BOS process data, were evaluated using statistical indicators. The performance indicators were calculated for each set \(\left\{y_{i},Y_{i}\right\}_{1}^{N}\), i.e., values of measured target \(y\) and simulated target \(Y\). The following measures were calculated for each method [56]:  
* _Coefficient of correlation_ (\(r_{yY}\))--This coefficient expresses the force of the linear relationship (i.e., degree of dependence) between two variables. The range of this coefficient is (\(-1\),\(1\)), and its formula is as follows: \[r_{yY}=\frac{\sum_{i=1}^{N}(Y_{i}-Y^{avg})(y_{i}-y^{avg})}{\sqrt{\sum_{i=1}^{N }\left(Y_{i}-Y^{avg}\right)^{2}}},\] (21) where \(y^{avg}\) and \(Y^{avg}\) are average values of \(y\) and \(Y\).
* _Coefficient of determination_ (\(r_{yY}^{2}\))--Expresses the degree of causal dependence of two variables. The coefficient gives information about the level of tightness, and the goodness of fit of the model, respectively (e.g., \(r_{yY}^{2}=1\) indicates that the model perfectly fits the measured target data, and \(r_{yY}^{2}<1\) corresponds to lower tightness between \(y\) and \(Y\)). The following formula can calculate the coefficient of determination \(r_{yY}^{2}\): \[r_{yY}^{2}=1-\frac{\frac{1}{N}\sum_{i=1}^{N}\left(Y_{i}-Y^{avg}\right)^{2}}{ \frac{1}{N}\sum_{i=1}^{N}\left(y_{i}-y^{avg}\right)^{2}}.\] (22)* _Root-mean-squared error_ (RMSE)--Represents the square root of mean square error (MSE). The value of RMSE may vary from 0 to positive infinity. The smaller MSE or RMSE, the better the model performance. The calculation formula is as follows: \[\text{RMSE}=\sqrt{\text{MSE}}=\sqrt{\frac{1}{N}\sum_{i=1}^{n}(Y_{i}-y_{i})^{2}}.\] (23)
* _Relative root-mean-squared error_ (RRMSE)--Expresses RMSE divided by the average of measured value \(y_{i}\). The value of RRMSE may vary from 0 to positive infinity. The smaller RRMSE, the better the model performance. The formula to calculate RRMSE is as follows: \[\text{RRMSE}=\frac{\text{RMSE}}{y^{avg}}\times 100\ \ \ \ (\%).\] (24)
* _Mean absolute percentage error_ (MAPE)--This error indicates how accurate a prediction method is. The MAPE expresses this accuracy in a percentage. If values of \(y_{i}\) are very low, then MAPE can exceed 100% extremely. Otherwise, if values \(y_{i}\) are very high (i.e., above \(Y_{i}\)), MAPE will not exceed 100%. Therefore, the MAPE can be calculated according to the following formula: \[\text{MAPE}=\frac{1}{N}\sum_{i=1}^{N}\frac{|y_{i}-Y_{i}|}{|y_{i}|}\times 100\ \ \ \ (\%).\] (25)
* _Performance index_ (PI)--This index expresses the overall performance of a given prediction method. The value of PI may vary from 0 to positive infinity. A lower PI value means better model performance. The PI is calculated as follows [84; 85]: \[\text{PI}=\frac{\text{RRMSE}}{r_{yY}+1}.\] (26)  
An absolute and relative error calculation formula was applied to determine the deviation between the measured and calculated temperature or carbon concentration at the BOS endpoint of selected meltings.  
* _The absolute error_\(\Delta_{x}\) is the difference between the measured value of the process variable \(y\) and the calculated value by the model \(Y\). For example, the following equation can express the absolute error: \[\Delta_{x}=Y-y.\] (27) The value of the absolute error is given unsigned in the evaluations, always as a positive number.
* _The relative error_\(\delta_{x}\) is the ratio of the absolute error \(\Delta_{x}\) to the actual value of the process variable \(y\). The relative error is usually expressed in percentages and can be expressed by the following equation: \[\delta_{x}=\frac{\Delta_{x}}{y}\times 100.\] (28)  
Besides these indicators, the time required to train the model was measured to assess the performance of the individual machine learning techniques.