## 3 Results and Discussion
### Estimation of Bearing Capacity Using Multiple Regression Analysis  
In the existing models for ultimate bearing capacity, \(s_{u}\) and \(a_{r}\) considerably influenced the prediction of the ultimate bearing capacity. Therefore, two parameters and their combinations are considered as input variables, and the construction conditions and shape factor of the aggregate pier were additionally considered as input variables. To consider the nonlinear relationship between the independent and the dependent variables in the multiple regression analysis, various types of input values were considered given frequently used parameter types, as listed in Table 2.  
Although a slight difference was observed in prediction accuracy depending on the number of input variables, most of them agreed well with the observed ultimate bearing capacity, and the lowest MAE was obtained when four input variables were used. Therefore, the final regression equation used four selected input variables (\(\frac{1}{a_{r}}\), \(\sqrt{s_{u}a_{r}}\), \(d_{f}^{2}\), and \(\frac{1}{S_{r}}\)) and all load test data were used to estimate the coefficients of the regression equation. The functional form of the best-fitting bearing capacity model is expressed as  
\[q_{ult}=67.8\frac{1}{a_{r}}+169.3\sqrt{s_{u}a_{r}}+271.4d_{f}^{2}-626.5\frac{ 1}{S_{r}}-256.8 \tag{5}\]  
where the units of \(s_{u}\) and \(d_{f}\) correspond to kPa and m, and the units of \(a_{r}\) and \(S_{r}\) are the number, which is a dimensionless derived unit. If a high correlation occurs between the independent variables in the multiple regression calculation, then multicollinearity problems may occur, thereby causing problems in the reliability of the regression coefficient estimation. Therefore, the multicollinearity between the parameters was evaluated through a variance inflation factor (VIF), where VIF = \(1\) / \((1-R^{2})\), and \(R^{2}\) is the coefficient of determination. A VIF that is greater than approximately 5 indicates the presence of potential interdependence [37]. The standard error and VIF for each parameter are summarized in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Number of Input Variable**} & \multirow{2}{*}{**Rank**} & \multirow{2}{*}{**Input Variables**} & \multicolumn{1}{c}{**MAE**} & \multirow{2}{*}{**R\({}^{2}\)**} & \multicolumn{2}{c}{**Bias, \(\lambda\)**} \\ \cline{5-8} \cline{7-9}  & & & & **Mean** & & **COV (\%)** \\ \hline \multirow{4}{*}{3} & 1 & \(\frac{1}{a_{r}}\), \(\sqrt{s_{u}a_{r}}\), \(\frac{1}{S_{r}}\) & 77.0 & 0.91 & 1.000 & 13.9 \\ \cline{2-9}  & 2 & \(\frac{1}{a_{r}}\), \(\sqrt{s_{u}  
The VIF value for all variables was less than 3. Therefore, the estimates of the fitted coefficients were considered appropriate. Figure 7 shows the comparison between the observed and predicted values of bearing capacity using the proposed MLR equation. The MAE, bias, and coefficient of variation (COV) in bias for the proposed MLR equation were 61.4 kPa, 1.000, and 12.4%, respectively, and the predictions of the proposed MLR model coincide well with the observed ultimate bearing capacity. Sensitivity analysis was performed to evaluate the influence of each input variable on the bearing capacity prediction. There are generally two types of sensitivity analysis: global sensitivity analysis and local sensitivity analysis, and the global sensitivity analysis was performed through Monte Carlo simulations because this helps gain an overall vision of the system, which is especially useful for distinguishing a significant parameter from insignificant input parameters [38]. The range of each variable was estimated through the database in Table 1; \(s_{u}\) ranged from 12 to 100 kPa, \(a_{r}\) ranged from 0.16 to 1.22, \(d_{r}\) ranged from 0 to 0.61 m, and \(S_{r}\) ranged from 2.0 to 26.3, and their distributions were assumed to be uniform.