## 4 Conclusions  
In this study, a multiple regression analysis and DNN were applied to predict the ultimate bearing capacity of aggregate pier-reinforced clay. For a robust prediction of the ultimate bearing capacity of aggregate pier-reinforced clay in the MLR, the most efficient input variables were selected through leave-one-out cross-validation. Based on the results of this study, the following conclusions were drawn:  
1. To select the effective input variables in the MLR, the prediction errors according to the number of input variables and their various equation forms were evaluated through the leave-one-out cross-validation. Accordingly, the ultimate bearing capacity was effectively predicted by using only four input variables (\(\frac{1}{a}\), \(\sqrt{s_{u}a_{r}}\), \(d_{f}^{2}\), and \(\frac{1}{s}\)) with an MAE of 70.5 kPa and bias of 1.003 in cross-validation;
2. The final MLR equation was proposed using all load test data, and the MAE, R\({}^{2}\), bias, and COV in bias were 61.4 kPa, 0.93, 1.000, and 12.4%, respectively. Therefore, the effective prediction of the ultimate bearing capacity was feasible using the proposed MLR equation, thereby resulting in improved prediction accuracy and reduced variability;
3. Global sensitivity analysis was performed to evaluate the influence of each input variable, and \(s_{u}\) has the highest influence on the bearing capacity prediction. The other variables were found to influence bearing capacity in the order of \(a_{r}\), \(S_{r}\), and \(d_{f}\). In addition, four input variables demonstrated a positive correlation with bearing capacity;
4. A DNN was applied to estimate the ultimate bearing capacity, and various training conditions were examined for accuracy to identify the optimal DNN structure. The optimal DNN model was suggested through the cross-validation error evaluation and showed favorable performance in predicting the ultimate bearing capacity with an MAE of 62.1 kPa and bias of 0.999;
5. The proposed MLR equation showed the best performance in the three models and could be applied to single and group aggregate piers. Thus, the proposed MLR equation could be recommended as a robust model for predicting the ultimate bearing capacity of aggregate pier-reinforced clay. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Prediction Method**} & \multirow{2}{*}{**Data**} & \multicolumn{2}{c}{**MAE (kPa)**} & \multicolumn{2}{c}{**Bias, \(\lambda\)**} \\ \cline{3-6}  & & **Mean** & **COV (\%)** & \multirow{2}{*}{**\(R^{2}\)**} & **Mean** & **COV (\%)** \\ \hline \multirow{2}{*}{[PERSON] and [PERSON] (2013)} & Cross-validation & 89.7 & 73.5 & 0.88 & 1.023 & 19.0 \\  & All data & 74.7 & 75.7 & 0.92 & 1.008 & 13.0 \\ \hline \multirow{2}{*}{Proposed MLR in this study} & Cross-validation & 70.5 & 89.1 & 0.91 & 1.003 & 14.2 \\  & All data & 61.4 & 91.6 & 0.93 & 1.000 & 12.4 \\ \hline \multirow{2}{*}{DNN} & Cross-validation & 74.9 & 86.9 & 0.91 & 0.999 & 16.3 \\  & All data & 62.1 & 104.9 & 0.92 & 0.999 & 13.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance of prediction methods for ultimate bearing capacity. Author Contributions: Funding acquisition, T.B. ; methodology, T.B. ; validation, S.-R.K. and B.-I.K. ; writing--original draft preparation, T.B. and S.-R.K.; writing--review and editing, B.-I.K. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2019R1C1C1010053). Conflicts of Interest: The authors declare no conflict of interest. * _[PERSON] et al. (2015)_ [PERSON]; [PERSON] Estimation of Bearing Capacity and Settlement of Spread Footing over Stone Column Reinforced clay Using Fuzzy Models and Artificial Neural Networks. _Indian J. Fundam. Appl. Life Sci._**2015**, _S2_, 3038-3050. * [PERSON] and [PERSON] (2018) [PERSON]; [PERSON] Prediction of Bearing Capacity of Stone Columns Placed in Soft Clay Using ANN Model. _Geotech. Geol. Eng._**2018**, _36_, 1845-1861.