## 5 Results and Discussion  
The performance of the GPR model was evaluated using root-mean-square error (_RMSE_), coefficient of determination (\(R^{2}\)), the ratio of the root-mean-square error to the standard deviation of measured data (_RSR_), and Nash-Sutcliffe coefficient (_NSE_) and mean bias error (_MBE_). The results were compared with some popular classical methods suggested in the literature (i.e., [PERSON] [4]; [PERSON] [3]) for determining the UBC of shallow foundation on cohesionless soil. The GPR model in Section 3.3 was developed by learning from the 78 data, and the results of the training performance including \(R^{2}\), _RSR_, _NSE_, and _MBE_ are shown in Table 4. In comparison to the theoretical equations, the output of the GPR model on the testing phase has substantially higher values of coefficient of determination (\(R^{2}\)), Nash-Sutcliffe coefficient (_NSE_), and lower values of _MBE_ and _RSR_. In addition, the equation proposed by [PERSON] [50] shows the best performance in comparison to the [PERSON] [3] theoretical formulas. Figure 2 shows the best-fit line of estimated versus measured UBC and the associated coefficient of determination (\(R^{2}\)). The outputs of the used theoretical formulas are more scattered than the GPR-based model, as shown in this Figure 2. From Table 4, it can be inferred that the \(R^{2}\) and _NSE_ are the highest (0.9404 and 0.8420), and the other criteria such as _RSR_ and _MBE_ are the least (0.8420 and \(-\)69.3268) for the GPR-based modeling in the testing phase. _MBE_ ([PERSON] [4]: \(-\)105.2996 and [PERSON] [3]: \(-\)161.2911) has the largest negative score (see Table 5), indicating that it is the most conservative approach. It is clear that the GPR model is more accurate than analytical  
Figure 1: The flowchart associated with the prediction of ultimate bearing capacity using a data-driven approach. formulas. The model performs better when \(R^{2}\) and \(NSE\) values are greater than 0.85. Thus, it depicts the accuracy of the finalized model. Figure 3 shows the comparison results of the training and testing sets to characterize the difference between the actual and predicted UBC of shallow foundations on cohesionless soil. Good agreements can be also observed between the comparison results, except for the few noise points, but these results are acceptable for the proposed GPR model to predict the UBC of shallow foundation on cohesionless soil. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Index** & **Training** & **Testing** \\ \hline \(R^{2}\) & 0.9552 & 0.9404 \\ \(RSR\) & 0.3428 & 0.3975 \\ \(NSE\) & 0.8825 & 0.8420 \\ \(MBE\) (kPa) & \(-10.9661\) & \(-69.3268\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance statistics of proposed model. Figure 2: Scatter plot presenting the actual UBC values versus the predicted UBC on the testing dataset: (**a**) in GPR model (this study); (**b**) in [PERSON] [4]; (**c**) in [PERSON] [3]. The proposed model was developed and validated using data inside the data range (see Table 2), but it was not tested outside of this range; hence, it is assumed that the suggested model will be valid for data inside within the data range. The GPR model's sensitivity results were evaluated using [PERSON]'s [51] technique to assess the influence of input parameters on \(q_{u}\). This method has been utilized in a number of studies [52; 53; 54; 14], and it is stated as follows:  
\[r_{ij}=\frac{\sum_{m=1}^{n}(y_{im}\times y_{om})}{\sqrt{\sum_{m=1}^{n}y_{im} \sum_{m=1}^{n}y_{om}{}^{2}}} \tag{11}\]  
where \(n\) is the number of data values (this study used 78 data values); \(y_{im}\) and \(y_{om}\) are the input and output parameters. The \(r_{ij}\) value ranges from zero to one for each input parameter, and the highest \(r_{ij}\) values suggested the most efficient output parameter (which was \(q_{u}\) in this study).