## 2 Theoretical Background of Applied Method
### Observations and Targets in BOS
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Observations** & **Ranges** & **Units** & **Targets** & **Ranges** & **Units** \\ \hline \(x_{1}\)—Steel quality class & 444–959 & (-) & Endpoint melt temperature & 1580–1720 & (\({}^{\circ}\)C) \\ \(x_{2}\)—Amount of blown oxygen & 7000–8900 & (Nm\({}^{3}\)) & Endpoint carbon concentration in melt & 0.02–0.08 & (\%) \\ \(x_{3}\)—Duration of oxygen blowing & 900–1100 & (s) & Melting duration & 25–80 & (min) \\ \(x_{4}\)—Pig iron temperature & 1200–1400 & (\({}^{\circ}\)C) & (Optional) & & \\ \(x_{5}\)—Weight of pig iron & 1,300,000–170,000 & (kg) & & & \\ \(x_{6}\)—Carbon concentration in pig iron & 4.0–4.6 & (\%) & & & \\ \(x_{7}\)—Silicon concentration in pig iron & 0.1–1.5 & (\%) & & & \\ \(x_{8}\)—Manganese concentration in pig iron & 0.1–0.8 & (\%) & & & \\ \(x_{9}\)—Phosphorus concentration in pig iron & 0.04–0.08 & (\%) & & & \\ \(x_{10}\)—Sulfur concentration in pig iron & 0.002–0.02 & (\%) & & & \\ \(x_{11}\)—Titanium concentration in pig iron & 0.005–0.05 & (\%) & & & \\ \(x_{12}\)—Scrap weight added to pig iron & 23,000–5500 & (kg) & & & \\ \(x_{13}\)—Amount of added magneste to the melt & 0–2000 & (kg) & & & \\ \(x_{14}\)—Amount of Fe in pig iron & 140,000–170,000 & (kg) & & & \\ \(x_{15}\)—Amount of after-blown oxygen & 0–1000 & (Nm\({}^{3}\)) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Static observations and targets with measurement ranges. This paper also compares machine regression models in predicting melt temperature or carbon concentration based on four dynamic variables of the BOS. Given that the waste gas flow was not measured in the several melting, it was decided not to use this observation for model training and subsequent prediction. Dynamic data recorded for a period of 1 s were taken from each melting. These are the composition of the waste gas, the waste gas temperature, and the flow rate of the blown oxygen (see Table 2). The accumulated amount of blown oxygen was calculated from the oxygen flow rate. The following data tensor can be defined when modeling from static observation to transform inputs to output of BOS process by the machine model:  
\[S=\{\textbf{x}_{i},y_{i}\}_{i=1}^{N}=\{xs_{1i},...,xs_{ni},ys_{i}\}_{i=1}^{N} \tag{1}\]  
where \(\textbf{x}_{i}\in\mathbb{R}^{n}\) are vectors of independent static variables of the BOS process according to Table 1. Parameter \(n\) represents the number of values in the vectors given by the total number of meltings intended for training. Parameter \(N\) represents the total number of observations used for training and prediction. Dependent variable \(y_{i}\in\mathbb{R}\) represents the target variable (i.e., temperature or carbon) that is modeled by the machine learning method and predicted by the learned machine model. Tensor (1) should be created for each target variable individually. In this study, 17 static input observations were used to predict the selected target. In modeling from merged static and dynamic data and finally prediction from dynamic data, the training data tensor can be defined as the following:  
\[D=\{\textbf{x}_{i},y_{i}\}_{i=1}^{N}=\{xd_{1i},...,xd_{ni},ys_{i}\}_{i=1}^{N} \tag{2}\]  
where \(xd_{1i},...,xd_{ni}\) are vectors of independent dynamic variables of the BOS process according to Table 2. Parameter \(n\) represents the number of data calculated as \((n=m*r)\), where parameter \(m\) represents the total number of meltings in the raw dataset and parameter \(r\) represents the number of repeated hardware-based measurements of the given target variable (i.e., initial,..., endpoint measurement) recorded as static data. Parameter \(N\) represents the total number of dynamic observations used for training and prediction (i.e., four in basic). The dependent variable \(ys_{i}\) represents the target variable (i.e., melt temperature or carbon concentration in the melt) that is modeled by the machine learning method and predicted by the learned machine model.