## 5 Results and Discussions  
As described above earlier, the whole dataset of soil samples includes 441 data points. To construct the ABC-LM-ANN model used for CC prediction, the original dataset is divided into a training set (80%), a validating set (10%), and a testing set (10%). The first two datasets are used in the phase of model construction and the third dataset acts as novel soil samples to verify the predictive performance of the proposed model in real-world applications of CC estimation. It is worth noting that before the model construction and testing phases, the original dataset had been normalized with the use of the \(Z\)-score transformation. This data transformation is to prevent the undesired situation in which variables with large magnitude dominate the ones with small magnitude. To evaluate the performance of the ABC-LM-ANN model, root mean square error (RMSE), the mean absolute percentage error (MAPE), the mean absolute error (MAE), and the coefficient of determination (R\({}^{2}\)) are employed. The equations used to compute these indices are provided as follows [68]:  
\[\text{RMSE}=\text{sqrt}(\sum_{t=1}^{N}\frac{\left(Y_{\text{A},i}-Y_{\text{P},i} \right)^{2}}{N}) \tag{11}\]  
Figure 6: The training flow of the ABC-LM algorithm. \[\text{MAPE}=\frac{100\%}{\text{N}}\sum_{\text{t=1}}^{\text{N}}\frac{\left|\text{ Y}_{\text{A,i}}-\text{Y}_{\text{P,i}}\right|}{\text{Y}_{\text{A,i}}} \tag{12}\]  
\[\text{MAE}=\frac{1}{\text{N}}\sum_{\text{t=1}}^{\text{N}}\left|\text{Y}_{\text{A, i}}-\text{Y}_{\text{P,i}}\right| \tag{13}\]  
\[\text{R}^{2}=\frac{\text{SS}_{\text{yy}}-\text{SSE}}{\text{SS}_{\text{yy}}} \tag{14}\]  
where YA,i and YP,i is the actual and the predicted values of soil CC of the ith data instance. N is the number of data instances in the set of interest. SS\({}_{\text{yy}}\) and SEE are calculated as follows:  
\[\text{SS}_{\text{yy}}=\sum_{\text{i=1}}^{\text{N}}\left(\text{Y}_{\text{A,i}} -\text{Y}_{\text{A,m}}\right)^{2} \tag{15}\]  
\[\text{SSE}=\sum_{\text{i=1}}^{\text{N}}\left(\text{Y}_{\text{A,i}}-\text{Y}_ {\text{P,i}}\right)^{2} \tag{16}\]  
where YA,m is the mean value of actual soil CC. The optimization process of the proposed ABC-LM-ANN is illustrated in Figure 7. It can be seen from this figure that the hybridization of ABC and LM can help the searching process to converge into a good solution of ANN structure within 63 iterations. Beyond this point, no better solution of the ANN structure can be found. The ANN optimized by the LM-ANN can attain a promising predictive result with RMSE = 0.009, MAPE = 10.705%, MAE = 0.007, and R\({}^{2}\) = 0.840. With R\({}^{2}\) = 0.840 (see Figure 8), the constructed ANN can explain 84% of the variation of the actual CC values. This is a good outcome since the problem of CC estimation, like other problems in the field of geotechnical engineering, is known to be complex. In addition, Figure 9 demonstrates the collected CC values accompanied by the corresponding CC results predicted by the proposed approach and Figure 10 presents the error distribution of the ABC-LM-ANN model in both training and testing phases. To confirm the performance of the ABC-LM-ANN, its outcome is compared to those of the LM backpropagation artificial neural network (LM-ANN), the ABC trained ANN (ABC-ANN), and the regression tree (RegTree). The two models of LM-ANN and RegTree are implemented in MATLAB  
Figure 8: The performance of the proposed ABC-LM-ANN model used for predicting CC. Figure 10: The error distribution of ABC-LM-ANN. Figure 6:0 The actual CC vs. the value of CC predicted by the proposed ABC-LM-ANN model. environment with the help of the Statistics and Machine Learning Toolbox [66]. To ensure a fair comparison, the hyper-parameters of LM-ANN are selected as follows: the number of neurons in the hidden layer = 10, the learning rate = 0.01, and the maximum number of training epochs = 5000. Moreover, the number of minimum number of leaf node observations is set to be 1 as suggested in [66].