## 4 Model Result
### K-Fold Cross-Validation Approach  
It is important to achieve the required accuracy of the models that are being used for the prediction of the models. This is a crucial component for ensuring the effectiveness of anticipated models, and validation is critical in this regard. The process of validation is used to evaluate the accuracy of data by mixing it randomly, and the K-fold validation test was used in this research to evaluate model accuracy. This strategy is used to eliminate the bias associated with randomly picking the training database from the training database. It separates the samples of data obtained from the experimental approach into ten equal subgroups using a random number generator. It absorbs nine out of ten subsets to shape the effective learner, with the remaining subset being used to test the model's predictions. The process is repeated approximately ten times to achieve the average accuracy of the repetitions performed. Many researchers consider that the use of this ten-fold cross-validation approach is effective in demonstrating both the generality and reliability of a model's performance [81].  
Figure 13 depicts the error measurement-based validation test of the nonlinear model and ensemble model (RF and GBR) for error estimation. The model validation is checked by different error indicators, including mean absolute error (MAE), root mean square error (RMSE), and root mean square log error (RMSLE), mean absolute error (MAE), and root mean square log error (RMSE), as presented in Figure 13. Even though there were fluctuations reported in the results, the accuracy remained high, as demonstrated in Figure 13a-f. The result of K-fold cross-validation for the decision tree model via MAE, RMSE, and RMSLE produced an error below 10 MP, as shown in Figure 13a-f and Table 10, respectively. Similarly, the MLPNN with bagging and Adaboost showed the same trend by showing minimal errors (see Table 11). This was due to their lower bias in predicting a model. Moreover, random forest regression and gradient boosting regression showed  
Figure 12: (**a**) Prediction vs targets of GB; (**b**) Distribution of errors of GB model.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Statistical Analysis** & **GB** \\ \hline Average EMLPNN & 3.59 \\ Minimum EMLPNN & 0.00 \\ Maximum EMLPNN & 23.97 \\ Entries lies below 10 MPa & 88 \\ Entries lies between 10 MPa and 15 MPa & 4 \\ Entries lies between 15 MPa and 20 MPa & 1 \\ Entries lies between 20 MPa and 25 MPa & 1 \\ Total data in testing data & 94 \\ Average below 10 MPa & 93.61 \\ Average between 10 MPa and 15 MPa & 4.25 \\ Average between 15 MPa and 20 MPa & 1.06 \\ Average between 20 MPa and 25 MPa & 1.06 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Statistical analysis of GB.  
a strong correlation in terms of k-fold cross-validation. It can be seen in Figure 13e-f and Table 12 that the error indicators for model validation showed less error than 10 MPa.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Folds**} & \multicolumn{2}{c}{**DT-Bagging**} & \multicolumn{4}{c}{**DT-Boosting**} \\ \hline  & MAE & RMSLE & RMSE & MAE & RMSLE & RMSE \\ \hline
1 & 7.812 & 0.173 & 8.822 & 7.259 & 0.159 & 8.665 \\
2 & 7.156 & 0.134 & 7.950 & 8.502 & 0.117 & 9.150 \\
3 & 3.970 & 0.033 & 5.746 & 4.854 & 0.0429 & 7.342 \\
4 & 5.316 & 0.075 & 8.306 & 3.620 & 0.0422 & 4.943 \\
5 & 4.850 & 0.058 & 7.441 & 4.865 & 0.0519 & 5.970 \\
6 & 5.308 & 0.079 & 7.289 & 4.761 & 0.0413 & 6.305 \\
7 & 2.924 & 0.034 & 4.906 & 2.667 & 0.0328 & 3.946 \\
8 & 9.307 & 0.103 & 11.095 & 6.800 & 0.0526 & 8.433 \\
9 & 7.684 & 0.124 & 11.759 & 5.761 & 0.0878 & 8.926 \\
10 & 6.622 & 0.085 & 9.458 & 6.495 & 0.0659 & 8.018 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Result of K-Fold cross-validation for ensemble models.  
Figure 13: Result of validation; (**a**,**b**) DT-bagging and DT-boosting; (**c**,**d**) MLPNN-bagging and MLPNN-boosting; (**e**) random forest; (**f**) gradient boosting.