## 3 Results and Discussion  
### Influence of Variation/Response Space Complexity on BP-ANN Model Establishment  
The relation between the complexity of variation/response space and the node number in the hidden layer for each response was revealed. Except the 32 experiments acting as boundaries, the remaining 128 experiments were tested by leave-one-out cross-validation. The node number in the hidden layer was varied from 1 to 20 for each response. The assessment of the prediction performance is shown in Figure 3. Lower \(RE\_MAX\) and \(MRE\) and larger \(B\) are preferable. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Response** & **Number of Input Variables** & **Variable Labels** & **Contribution to Response (\%)** \\ \hline \multirow{4}{*}{Compressive stress} & 1 & G & 93 \\  & 2 & G, J & 96 \\  & 3 & G, J, D & 98 \\  & 10 & A–J & 100 \\ \hline \multirow{4}{*}{End temperature} & 3 & A, D, F & 71 \\  & 4 & A, D, F, C & 89 \\  & 5 & A, D, F, C, E & 94 \\  & 10 & A–J & 100 \\ \hline \multirow{4}{*}{Tensile stress} & 4 & F, G, D, J & 71 \\  & 5 & F, G, D, J, C & 78 \\ \cline{1-1}  & 6 & F, G, D, J, C, H & 85 \\ \cline{1-1}  & 7 & F, G, D, J, C, H, I & 91 \\ \cline{1-1}  & 10 & A–J & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variable combinations of each response for backpropagation (BP)-ANN models. Figure 3 shows that the performance is significantly improved by increasing the node number in the hidden layer to seven. However, the performance is oscillatory with further increasing node number; the larger number of nodes may lead to over-fitting and affect the generalization capability. For instance, when the node number is 20, the mean relative errors increase for all three responses; the coefficients of determination decrease; the maximum relative error for end temperature is quite high. Furthermore, it shows that each response has different optimal ranges. Figure 3: Performance assessment (**a**) _RE_MAX_—maximum relative error of all testing results, (**b**) _MRE_—mean relative error, and (**c**) _B_—coefficient of determination of the BP-ANN models with different node numbers in the hidden layer. Considering the particular behavior of each response, error criteria were specifically defined for each response and a preferable range of the node number in the hidden layer was proposed for each response (Table 4). The range was 4-6 for the maximum compressive stress at the hot face of the working lining, 5-7 for the end temperature of the steel shell, and 10-12 for the maximum tensile stress at the cold end of the steel shell. These ranges can be correlated with the node numbers in the input and output layers, as shown in Table 5. The number of variables that contribute more than 90% to the response was used to calculate the _PF_ value, which represents the variation/response complexity. This number was divided by ten (the total number of variables) and multiplied by 100, which gave a _PF_ value in percent. Table 5 shows that the _PF_ equaled 10, 50, and 70 for the maximum compressive stress, end temperature, and maximum tensile stress, respectively. Therefore, the relation between the complexity of the variation/response space and the node number in the hidden layer can be associated with the _PF_ and the node numbers in the input and output layers explicitly. Two equations were deduced from Table 5.  
Lower boundary: \(N_{h}=AN_{i}+N_{o}\)  
Upper boundary: \(N_{h}=(A+0.2)\,N_{i}+N_{o}\)  
where \(A\) is a function of the _PF_, \(N_{i}\), \(N_{h}\), and \(N_{o}\) are the node numbers in the input, hidden, and output layers, respectively. The relation between \(A\) and _PF_ was fitted by an exponential equation (Equation (13)) as shown in Figure 4. Equations (11)-(13) provide guidelines to define node number in the hidden layer for a steel ladle system based on _PF_ and node numbers in the input and output layers.