# Novel Nature-Inspired Hybrids of Neural Computing for Estimating Soil Shear Strength
## 4 Results and Discussion
### Accuracy Assessment of Predictive Models  
The results are evaluated in this part by comparing the predicted values of SSS with actual values. In this sense, the error of the performance is calculated by \(RMSE\) and \(MAE\) error criteria. Besides, the R\({}^{2}\) index is used to measure the correlation between the actual and predicted SSSs. Needless to say, the results of the training phase denote the capability of the models for pattern recognition, while the testing results indicate the prediction power. In the training phase, all four models achieved a good understanding of the relationship between the SSS and independent parameters. In this phase, the calculated \(RMSE\) values of 0.0328, 0.0272, 0.0238, and 0.0222, respectively for the ANN, DA-ANN, WOA-ANN, and IWO-ANN demonstrate that the learning error of the unreinforced ANN experienced a considerable decrease as a result of being coupled with the mentioned algorithms. The values of the \(MAE\) also support the mentioned claim as it decreased from 0.0258 to 0.0200, 0.0186, and 0.0171. Furthermore, the calculated values of R\({}^{2}\) proved that the correlation of the ANN products with actual SSSs increased from 0.9010, 0.9300, 0.9465, and 0.9534. As for testing data, the trained networks were applied to some unseen soil condition to estimate the SSS for the testing phase. Figure 9 shows the results based on the computed error (i.e., the difference between the output and target values) of each sample, alongside the histogram of errors. Note that the current histogram charts depict the frequency of each error extents. The higher the frequency of errors that are close to zero, the greater the accuracy of the prediction. According to these figures, the prediction reliability of the ANN (\(RMSE\) = 0.0515 and \(MAE\) = 0.0408) increased by applying DA (\(RMSE\) = 0.0425 and \(MAE\) = 0.0329), WOA (\(RMSE\) = 0.0436 and \(MAE\) = 0.0328), and IWO (\(RMSE\) = 0.0432 and \(MAE\) = 0.0322) metaheuristic techniques. Moreover, the histogram chart of the ANN results yields the highest standard error (0.0522). Moreover, Figure 10 shows the correlation between the actual values of SSS with ones predicted by each model. Referring to the respective R\({}^{2}\)s of 0.6329, 0.7647, 0.7442, and 0.7519, it can be deduced that the outputs of the hybrid ensembles are more consistent than the typical MLP. In this phase, the actual values of SSS range from 0.197308 to 0.522631, while the outputs of the ANN, DA-ANN, WOA-ANN, and IWO-ANN are in the ranges of [0.225142, 0.519525], [0.203952, 0.522346], [0.212126, 0.528062], and [0.197868, 0.538494], respectively. Figure 8: The computation time and the best root mean square error (\(RMSE\)) of the best-fitted ensembles. Figure 9: The testing results obtained for (**a**) ANN, (**b**) ANN frequency, (**c**) DA-ANN, (**d**) DA-ANN frequency (**e**) WOA-ANN, (**f**) WOA-ANN frequency, (**g**) IWO-ANN (**h**) IWO-ANN frequency predictions. The findings of this study revealed that applying the named evolutionary algorithms can effectively help ANN to adjust the weights and biases more properly, and consequently present a more accurate estimation of the SSS. Considering the obtained results, a score-based ranking system is developed to have a relative evaluation of the performance of the used models. To do so, scores are attributed to each model based on the calculated values of the accuracy criteria. In this way, the higher reliability of the model, the larger the score assigned to it. From a comparison point of view, it can be said that the IWO-ANN (training score = 12) performed more efficiently in terms of all three _RMSE_, _MAE_, and R\({}^{2}\) in comparison with DA- and WOA-based neural ensembles. After that, all three criteria demonstrated  
Figure 10: The testing results obtained for (**a**) ANN, (**b**) DA-ANN, (**c**) WOA-ANN, and (**d**) IWO-ANN. more reliability of the WOA than DA for training the ANN.