## 4 Results
### Data-Driven Modeling Results  
#### 4.2.1 Multiple Linear Regression (MLR)  
Table 4 presents the Pearson correlation matrix, showcasing the linear relationships between input and output parameters. This table assesses linear relationships between various parameters. The Pearson correlation matrix is a statistical instrument employed to assess the linear connection between variable pairs within a dataset. It encompasses correlation coefficients, which gauge the magnitude and orientation of the linear link between these variables. These matrix elements span from \(-1\) to \(1\), furnishing valuable insights into the interrelationship among variables. When the correlation coefficient is positive, it signifies a direct positive linear association between the two variables. In other words, as one variable increases, the other also tends to increase. The closer the coefficient approaches \(+1\), the more robust the positive relationship. Conversely, when the correlation coefficient is negative, it indicates an inverse linear relationship between the two variables. In this scenario, as one variable increases, the other typically decreases. The closer the coefficient nears \(-1\), the more pronounced the negative relationship. When the correlation coefficient stands precisely at zero, it denotes the absence of a linear relationship between the variables. Changes in one variable do not provide predictive information about changes in the other. The findings reveal that there are no input parameters that exhibit a strong linear relationship with the output parameter. The highest correlation value observed is 0.594. The findings from Table 4 indicate that the MLR model is unlikely to yield strong predictions for the output parameter. However, the MLR model was thoroughly examined, and the best performing model was selected. Figure 5 illustrates the comparison between the predicted and actual values generated by the numerical model. Despite the efforts, the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **B** & **D** & **L/B** & \(\gamma\) & \(\phi\) & \(\mathbf{q_{u}}\) \\ \hline Footing Width, B & **1** & 0.149 & \(-0.346\) & \(-0.906\) & \(-0.183\) & 0.269 \\ Footing Depth, D & 0.149 & **1** & \(-0.132\) & \(-0.081\) & \(-0.111\) & 0.594 \\ Footing Geometry, L/B & \(-0.346\) & \(-0.132\) & **1** & 0.388 & 0.202 & \(-0.105\) \\ Sand Unit Weight, \(\gamma_{\text{d}}\) or \(\gamma^{\prime}\) & \(-0.906\) & \(-0.081\) & 0.388 & **1** & 0.384 & \(-0.103\) \\ Internal Friction Angle, \(\phi\) & \(-0.183\) & \(-0.111\) & 0.202 & 0.384 & **1** & 0.532 \\ Footing Bearing Capacity, \(\text{q}_{u}\) & 0.269 & 0.594 & \(-0.105\) & \(-0.103\) & 0.532 & **1** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Pearson matrix for input and output parameters (darkness level represents the degree of strength in the linear relationship). Furthermore, the results reveal that the suggested MLR model performed better in predicting lower q\({}_{\text{u}}\) values compared to higher q\({}_{\text{u}}\) values. This observation suggests that, in cases of lower q\({}_{\text{u}}\), the relationship between inputs and output parameters tends to exhibit closer adherence to linear behavior. Table 5 summarizes the overall performance of the best multiple linear regression (MLR) model in predicting the ultimate bearing capacity of shallow foundations for both the training and testing databases. Various performance metrics are employed to assess the accuracy and precision of the model's predictions. In the training database, the model exhibits an average Mean Absolute Error (MAE) of 33.938 kPa, indicating the average absolute difference between the predicted and actual values. The Mean Squared Error (MSE) is 2189.854, representing the average squared difference between predictions and actual values. The Root Mean Squared Error (RMSE) is 46.796, which indicates the average magnitude of the error.