## 4 Results  
### Database Preparation  
In the database, individual variables were measured in their respective units. Data normalization is employed to decrease network error and enhance network training speed. The normalization linear function utilized in this study is presented below.  
\[X_{norm}=\frac{X-X_{min}}{X_{max}-X_{min}} \tag{1}\]  
This equation has four terms, \(X_{max}\), \(X_{min}\), \(X\) and \(X_{norm}\), which are the maximum, minimum, actual and normalized values, respectively.  
In all mathematical models, the training and testing databases were randomly divided. A total of 80% of the main database was used for training, while the remaining 20% was used for testing.  
Evaluating statistical parameters for both databases is crucial, as it directly impacts the final model's accuracy. Based on Tables 2 and 3, the training and testing databases exhibited similar statistical parameters, including minimum, maximum, mean, and standard deviation values.  
The performance of a network can be evaluated through a variety of parameters such as coefficient of determination (\(R^{2}\)) and mean absolute error (MAE). Equations (2)-(7) show the definitions of Mean Absolute Error (MAE), Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Squared Logarithmic Error (MSLE), Root Mean Squared Logarithmic Error (RMSLE), and Coefficient of Determination (\(R^{2}\)) [81].  
\[MAE=\frac{\sum_{N}\bigl{(}X_{m}-X_{p}\bigr{)}}{N} \tag{2}\]  
\[MSE=\frac{\sum_{N}\bigl{(}X_{m}-X_{p}\bigr{)}^{2}}{N} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Variable** & **Datasets** & **Minimum** & **Maximum** & **Mean** & **Std. Deviation** \\ \hline \multirow{3}{*}{Output} & Footing Bearing & \multirow{3}{*}{60} & \multirow{3}{*}{67.700} & \multirow{3}{*}{464.000} & \multirow{3}{*}{208.524} & \multirow{3}{*}{108.113} \\  & Capacity, q\({}_{\text{u}}\) (kPa) & & & & & \\ \cline{1-1}  & Footing Width, B (m) & & & & & \\ \cline{1-1}  & Footing Depth, D (m) & & & & & \\ \cline{1-1}  & Footing Geometry, L/B & & & & & \\ \cline{1-1}  & Sand Unit Weight, \(\gamma_{\text{d}}\) or \(\gamma^{\prime}\) & & & & \\ \cline{1-1}  & (kN/m\({}^{3}\)) & & & & & \\ \cline{1-1}  & Internal Friction Angle, \(\phi\) & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Distribution of training database.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Variable** & **Datasets** & **Minimum** & **Maximum** & **Mean** & **Std. Deviation** \\ \hline \multirow{3}{*}{Output} & Footing Bearing Capacity, & \multirow{3}{*}{15} & \multirow{3}{*}{58.500} & \multirow{3}{*}{425.000} & \multirow{3}{*}{222.273} & \multirow{3}{*}{115.466} \\  & q\({}_{\text{u}}\) (kPa) & & & & & \\ \cline{1-1}  & Footing Width, B (m) & & & & & \\ \cline{1-1}  & Footing Depth, D (m) & & & & & \\ \cline{1-1}  & Footing Geometry, L/B & & & & & \\ \cline{1-1}  & Sand Unit Weight, \(\gamma_{\text{d}}\) or \(\gamma^{\prime}\) & & & & \\ \cline{1-1}  & (kN/m\({}^{3}\)) & & & & & \\ \cline{1-1}  & Internal Friction Angle, \(\phi\) & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Distribution of testing database.  
\[RMSE=\sqrt{\frac{\sum_{N}\left(X_{m}-X_{p}\right)^{2}}{N}} \tag{4}\]  
\[MSLE=\frac{\sum_{N}\left(\log\left(X_{m}+1\right)-\log\left(X_{p}+1\right) \right)^{2}}{N} \tag{5}\]  
\[RMSLE=\sqrt{\frac{\sum_{N}\left(\log\left(X_{m}+1\right)-\log\left(X_{p}+1 \right)\right)^{2}}{N}} \tag{6}\]  
\[R^{2}=\left[\frac{\sum_{i=1}^{N}\left(X_{m}-\overline{X_{m}}\right)\left(X_{p} -\overline{X_{p}}\right)}{\sum_{i=1}^{N}\left(X_{m}-\overline{X_{m}}\right)^{ 2}\sum_{i=1}^{n}\left(X_{p}-\overline{X_{p}}\right)^{2}}\right]^{2} \tag{7}\]  
where \(N\) is the number of datasets, \(X_{m}\) and \(X_{p}\) are actual and predicted values, and \(\overline{X_{m}}\) and \(\overline{X_{p}}\) are the average of actual and predicted values, respectively. Ideally, the model should have an \(R^{2}\) value of 1 and a MAE, MSE, RMSE, MSLE, RMSLE value of 0.