# Optimum Design of Cylindrical Walls Using Ensemble Learning Methods
## 3 Results and Discussion
### Interpretation of the ML Models  
The SHAP algorithm and Shapley values of input variables are commonly used for the interpretation of machine learning models and to quantify the contribution of a variable to the machine learning model output. The Shapley values of a variable indicate the effect of its variable on the predicted outcome of a machine learning model. These values are calculated using Equation (2), where \(\phi_{i}\) are the Shapley values, M is the number of input variables, x is a vector of input variables, \(\chi^{\prime}\) is a simplified input vector that is related to x with a mapping function, and \(f\) is an explanatory model. Further details of the SHAP algorithm can be found in [PERSON] and [PERSON] (2013), [PERSON] et al. (2014), and [PERSON] et al. (2015).  
\[\phi_{i}(f,x)=\sum_{x^{\prime}\subseteq\ \chi^{\prime}}\frac{|\ x^{\prime}|!(M-|\ x^{ \prime}|-1)!}{M!}\big{[}f_{x}\big{(}\ x^{\prime}\big{)}-f_{x}\big{(}\ x^{\prime }\big{)}i\big{)}\big{]} \tag{2}\]  
Figures 9-12 show the Shapley values of all three input variables in this study for different ensemble machine learning models. In these plots, every data sample is represented by a dot, and the color of a dot shows whether a variable has a high or low value in its range. In these plots, a positive SHAP value indicates a positive effect on the model outcome, whereas a negative SHAP value indicates a negative effect on the output. The order of the variables along the vertical axis indicates their impact on the model outcome such that H has the greatest impact, followed by r and C/S. It follows that high values of H and r lead to high values of optimum wall thickness. On the other hand  
The feature dependence plots in Figure 13 were obtained by plotting the SHAP value of each input variable for its actual value using four different ensemble learning models. In each of these plots, the colors of the dots represent the values of the most strongly dependent variable. A common denominator of all plots in Figure 13 is that increasing the values of H and r has an increasing effect on the predicted output in all models. This indicates a positive correlation between H and r and the corresponding optimum wall thickness. On the other hand, increasing the C/S value leads to reduced SHAP values associated with this variable up to the point where C/S is equal to about 0.15. For greater values of C/S, the SHAP value stays constant. The variation in the SHAP value associated with C/S is affected by the model used. In all models, r was the variable most strongly dependent on C/S. Furthermore, in all models, the variable most dependent on H was r, and the variable most dependent on r was H. It could be observed that increasing the values of both r and H have an increasing effect on the optimum wall thickness. The feature dependence plots of all models show that at H = 10 m and r = 10 m, the SHAP value changes its sign. For H \(\leq\) 10 the r value and the SHAP value are inversely proportional, whereas the opposite is true for H > 10. A similar relationship can be observed in the dependence plots of r too.  
Figure 12: Shapley values of the Random Forest model.  
Figure 13: Feature dependence plots for XGBoost, LightGBM, Random Forest and CatBoost. (**a**) XGBoost dependence plot for r, (**b**) XGBoost dependence plot for H, (**c**) XGBoost dependence plot for C/S, (**d**) LightGBM dependence plot for r, (**e**) LightGBM dependence plot for H, (**f**) LightGBM dependence plot for C/S, (**g**) RF dependence plot for r, (**h**) RF dependence plot for H, (**i**) RF dependence plot for C/S, (**j**) CatBoost dependence plot for r, (**k**) CatBoost dependence plot for H, (**l**) CatBoost dependence plot for C/S.