## 2 Theoretical Background
### Comparison with Existing Modeling Techniques  
Previously, several studies have been conducted for predicting bearing capacity using MLR or AI. Although the same data set is used, the performance of the model may differ depending on the modeling technique. This study has some differences in modeling approaches from previous studies in order to improve the accuracy and robustness of the prediction model, and they are depicted in Figure 5.  
The accuracy of the prediction model can be different depending on the selection of input variables, and its effect should be evaluated to establish the optimal prediction model. Additionally, if the independent variables have a nonlinear relationship with the dependent variable, a suitable relationship must be identified considering the nonlinear function, such as second- or third-order polynomial, exponential, and logarithm functions. However, these effects were rarely considered in MLR modeling in most previous studies. The commonly used independent variable selection methods include forward selection, backward elimination, and stepwise methods. However, even if the input variables are selected by these variable selection methods, the optimal input variables for MLR are not always selected. In addition, if there is a multicollinearity among the independent variables, another model may be selected without analyzing the optimal prediction model. [PERSON] and [PERSON] [33] reviewed the existing variable selection methods, and pointed out their disadvantages. They noted that there are many advantages in selecting a suitable model by evaluating all combinations of independent variables after making all possible models. Therefore, in this study, MLR was used for the basic regression equation, and nonlinearity with dependent variables is considered by adding  
Figure 4: Leave-one-out cross-validation.  
Figure 5: Comparison of modeling approaches between previous and present studies.  
various variable forms, such as \(x^{2}\), \(1/x\), \(\ln(x)\), and \(\sqrt{x}\). In particular, to select the appropriate number of independent variables, the number of independent variables in the MLR equation was set to three, four, and five, and the accuracy of the model was evaluated considering all combinations of independent variables. The number of input variables and their equation types were selected through the results of cross-validation, and the coefficients of the final model were re-estimated using all data. Finally, the multicollinearity between the input variables has been evaluated to confirm the suitability of the selected variables.  
As an artificial intelligence technique, a DNN was applied to overcome the shortcomings of the conventional ANN. In DNN training, how to deal with underfitting or overfitting is very important. However, the performance of a model may differ depending on the data splitting, especially when the data size is small. Unfortunately, there is no standard way to determine the structure of a model (in terms of the number of hidden layers and nodes, batch size, etc.). Therefore, various training conditions were evaluated to determine an optimal structure, and the structure with the minimum cross-validation error was selected as the optimal structure.