## 3 Results and Discussions
### Modelling of Tensile Strength by ANN  
The initially tested feedforward ANN configurations with training and testing results are presented in Table 9. Training procedures and transfer functions were tested in eight different configurations. The training procedures included the Levenberg-Marquardt (\"trainlm\"), scaled conjugate gradient (\"trainscg\"), gradient descent with adaptive learning rate (\"trainingda\"), and Bayesian regularization backpropagation (\"trainbr\"). The adopted transfer functions were tangential (\"tansig\") and logarithmic (\"logsig\"). The best results for test samples (MAE test = 0.2100, MSE test = 0.0674) were obtained with the Bayesian regularization learning method and the tangential transfer function; the specified configuration was adopted further on. To confirm the previous statement and to complement the prediction accuracy, the ratio of prediction to deviation (RPD) was calculated by dividing the standard deviation of the reference test data by the standard error of the prediction [40]. The highest value was obtained with the 7 th ANN configuration (for MAE and MSE test). Feedforward ANNs were tested with nine different topologies, namely, ANN 4, ANN 5, ANN 6, ANN 2_4, ANN 4_6, ANN 6_4, ANN 2\(4\)6, ANN 4\(6\)4, and ANN 6\(4\)2, comprising one to three hidden layers. For example, the ANN 4 topology contains one hidden layer with four neurons, the ANN 2\(4\)6 contains three hidden layers with two neurons in the first hidden layer, four in the second, and six in the third hidden layer. As mentioned earlier, all neurons in the hidden layers adopted the tangential (\"tansig\") transfer function and the Bayesian regularization backpropagation learning method (\"trainbr\"). The LOSO k-fold cross-validation with 25 folds was used for the ANN's validation, wherein each fold contained 24 training data points and 1 test data point. The learning procedure lasted for six different epoch durations (10, 50, 100, 200, 400 and 800 iterations). After the training procedure, the performance of ANN was calculated as the average values of MAE and MSE for both test and training data of all 25 folds. The results for MAE test and MAE train are presented in Figures 9 and 10, and for MSE test and MSE train, are presented in Figures 11 and 12. The ANN 6 configuration achieved the best result for the test set after 400 epochs (MAE test = 0.1745, MSE test = 0.0469). A higher number of epochs was not necessary, since the MAE and MSE converged after 800 epochs in all the adopted ANN configurations. The graphs above indicate that a higher number of hidden layers in ANNs is not suitable for this application. ANNs with two or three hidden layers are prone to overfitting, since the results do not change with a higher number of epochs. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**ANN Information** & **Config 1** & **Config 2** & **Config 3** & **Config 4** & **Config 5** & **Config 6** & **Config 7** & **Config 8** \\ \hline Training procedure & trainlm & trainlm & trainscg & trainscg & traingda & trainspd & trainbr & trainbr \\ Learning epochs & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\ Transfer function & tansig & logsig & tansig & logsig & tansig & logsig & tansig & logsig \\ Architecture & ANN 6 & ANN 6 & ANN 6 & ANN 6 & ANN 6 & ANN 6 & ANN 6 & ANN 6 \\ Cross validation & 25-fold & 25-fold & 25-fold & 25-fold & 25-fold & 25-fold & 25-fold & 25-fold \\ MAE test & 0.3243 & 0.316 & 0.2129 & 0.2157 & 0.2662 & 0.2618 & 0.21 & 0.2267 \\ RPD (MAE test) & 1.7213 & 1.7665 & 2.6217 & 2.5877 & 2.097 & 2.1321 & 2.6584 & 2.4624 \\ MAE train & 0.0001 & 0.0001 & 0.0255 & 0.0476 & 0.1578 & 0.2355 & 0.0549 & 0.0619 \\ MSE test & 0.1341 & 0.1854 & 0.0869 & 0.0725 & 0.1115 & 0.124 & 0.0674 & 0.0928 \\ RPD (MSE test) & 4.1631 & 3.0102 & 6.4213 & 7.7042 & 5.0072 & 4.5016 & 8.2797 & 6.0144 \\ MSE train & 0.0001 & 0.0001 & 0.0016 & 0.0043 & 0.0389 & 0.09 & 0.0075 & 0.0087 \\ Time [s] & 3.6 & 3.2 & 3.9 & 3.9 & 3.6 & 3.7 & 5.4 & 5.3 \\ \hline \hline \end{tabular}
\end{table}