## 4 Model Results  
### K-Fold Cross-Validation  
The training performance of each ML model is depicted in the boxplots in Figure 3 using the \(R^{2}\) and RMSE metrics. The boxplots show the variability and fluctuations of the performance measures by using 5-fold cross-validation on 197 actual values from the training data sets. The five-number summary of a dataset, consisting of the minimum, first quartile, median, third quartile, and maximum values, is graphically displayed in a box plot, also known as a box-and-whisker plot. These values are represented in the box plot by a box that spans from the first quartile to the third quartile and a line inside the box that represents the median. The box plot also includes \"whiskers\" that extend from the box to the minimum and maximum values and diamonds that represent any values that fall outside of the whiskers. Specifically, outliers are any data points that are higher than 1.5 times the Interquartile Range (IQR) and are below the first quartile (Q1) or above the third quartile (Q3). Box plots are commonly used in statistics and data analysis to quickly and easily visualize the distribution of a dataset. The minimum, maximum, median, and average for RMSE and \(R^{2}\) for three machine learning models using fivefold cross-validation are presented in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Data** & **Statistics** & \(q_{u}\) **(kPa)** & \(t\)**/**D** & \(d\)**/**D** & \(LId_{sc}\)** & \(hId_{sc}\)** & \(q_{rs}\) **(kPa)** \\ \hline Training data & Standard deviation & 15.05 & 0.08 & 1.42 & 0.96 & 2.02 & 118.33 \\  & Mean & 38.89 & 0.16 & 1.64 & 5.88 & 5.01 & 197.26 \\  & Median & 45.02 & 0.15 & 2.00 & 6.00 & 6.00 & 178.81 \\  & Maximum & 53.82 & 0.40 & 4.00 & 8.00 & 8.00 & 452.21 \\  & Minimum & 3.86 & 0.00 & 0.00 & 2.00 & 0.00 & 15.44 \\  & Kurtosis & \(-\)0.52 & 1.97 & \(-\)1.13 & 8.58 & 0.80 & \(-\)0.87 \\ \hline Testing data & Standard deviation & 17.35 & 0.09 & 1.38 & 1.26 & 1.93 & 120.11 \\  & Mean & 36.66 & 0.19 & 1.85 & 5.67 & 4.89 & 195.44 \\  & Median & 44.24 & 0.15 & 2.00 & 6.00 & 6.00 & 178.40 \\  & Maximum & 53.82 & 0.40 & 4.00 & 8.00 & 8.00 & 467.01 \\  & Minimum & 4.99 & 0.00 & 0.00 & 2.00 & 0.00 & 14.11 \\  & Kurtosis & \(-\)1.03 & 0.63 & \(-\)0.89 & 4.44 & \(-\)0.17 & \(-\)0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model-variable descriptive statistics. From Figure 3 and Table 2, the MPR model exhibits better performance than LR and RF in terms of the median and mean values of the two metrics. This superiority is due to the non-linear associations between the bearing capacity and the input variables that nonlinear models can effectively capture. Further investigation is needed to ensure the stability and dependability of the model, in addition to evaluating its prediction accuracy. The random error should have a normal distribution with a mean of 0 and a symmetric shape, with most errors being close to the center. Figure 4 displays the residuals and Gaussian functions for the first three models, which all have a normal distribution pattern. The MPR model seems to have a higher level of prediction accuracy when compared to the RF and LR algorithms, as 80% of the residuals for the MPR fall within the range of [\(-\)17.267, 17.267]. Moreover, the mean value of the fitted normal distribution of residuals for the MPR model is 0.703. It is much closer to 0; that is, the MPR model has better error randomness. So, it can be stated that the MPR algorithm is more advantageous.