## 3 Results and Discussion
### Model Interpretation  
Figure 11 shows the variable importance score for different inputs for the four datasets. As illustrated, \(T/2R\) was the most significant variable for all stiffness coefficients, with an average variable importance score of around 0.58. This suggested that soil layering played an important role in foundation stiffness. Furthermore, it can be seen that the importance of the \(T/2R\) on the four stiffness coefficients was comparable, which means that the soil layering affected the foundation's kinematic behavior under all loading conditions. The \(G_{\rm sand}/G_{\rm clay}\) also contributed greatly to the output, and its average variable importance score was 0.30. Unlike the nearly equal influence of \(T/2R\) on all stiffness coefficients, the importance of \(G_{\rm sand}/G_{\rm clay}\) was most significant for the vertical foundation stiffness coefficient. The above results were in good agreement with the DT visualization results in Figure 10. Comparatively speaking, the influence of other input variables on output variables was relatively small, but it cannot be ignored. For example, embedment conditions  
Figure 9: Scatter plot of the actual and predicted dimensionless elastic stiffness coefficients.  
had an obvious influence on the kinematic behavior of circular foundations under lateral load, and the importance of Case 2 reached 0.2388.  
Figure 11: Variable importance score for four datasets.  
Figure 10: The visualization of a DT for \(K_{\rm c}\). Note that X[4] and X[5] represent T/2R and \(G_{\rm sand}\)/ \(G_{\rm clay}\), respectively.  
Figure 12 analyzes how the features \(T/2R\), \(G_{\rm sand}/G_{\rm clay}\), and \(Z_{\rm D}/2R\) affected the prediction for the target variables \(K_{\rm c}\), \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\). \(T/2R\) had the greatest influence on the stiffness coefficient and was negatively correlated with \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\). Within a certain \(T/2R\) range (normalized \(T/2R<0.2\)), the values of \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\) decreased rapidly with the increase in \(T/2R\). When the value of \(T/2R\) was greater than 0.2, \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\) tended to remain unchanged. While \(K_{\rm c}\) first increased and then decreased with the increase in the \(T/2R\) value, 0.1 was the critical value of \(K_{\rm c}\) change. Furthermore, we also found that \(G_{\rm sand}/G_{\rm clay}\) was positively correlated with \(K_{\rm c}\), \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\); with the gradual increase in the feature value of \(Z_{\rm D}/2R\), \(K_{\rm c}\), \(K_{\rm h}\), \(K_{\rm m}\), and \(K_{\rm v}\) changed little, indicating that \(Z_{\rm D}/2R\) was relatively less important for the stiffness coefficient.  
Figure 13 further illustrates the interaction influence of input variables on the output, taking the \(K_{\rm c}\) dataset as an example. There is a strong correlation among \(T/2R\), \(G_{\rm sand}/G_{\rm clay}\), and \(Z_{\rm D}/2R\), and they affect the stiffness coefficient in a coupled way. For example, when \(T/2R\) was less than 0.25 and \(G_{\rm sand}/G_{\rm clay}\) was more than 20, the model tended to predict a large \(K_{\rm c}\). A large \(K_{\rm c}\) was also observed when \(G_{\rm sand}/G_{\rm clay}\) was greater than 30 and the \(Z_{\rm D}/2R\) was in the range of 0-0.5. The above analysis provided a certain reference for improving the stiffness of the circular foundation.  
Figure 12: PDPs of \(T/2R\), \(G_{\rm sand}/G_{\rm clay}\), and \(Z_{\rm D}/2R\) for all output variables. Note that the features were normalized so that their influences on the output could be better compared.