## 3 Results and Discussion
### Explanation of the Model Using the SHAP Value Method  
In the following interpretation part, the selected RF model is applied for SHAP analysis, i.e., TreeSHAP, a specialized method for tree-based models [30,41]. Much research is actively conducted regarding XAI because there is uncertainty in decision-making based on the machine learning results. In this study, the SHAP value method is adopted, enabling granular explanations of the contribution of each feature [24,27]. Figure 4a is the SHAP summary plot, showing how the top parameters affect yield prediction. The plot lists in descending order the average of absolute SHAP values and shows the correlation between input variables and the output. For a numerical feature example, as the \"step_aa_T\" parameter value decreases, the SHAP values decrease to \(-0.06\) where a negative SHAP value means a negative impact on prediction as shown in Equation (4). To examine partial dependency, SHAP value scatterplots describe the effect of changing an individual feature [41,42]. In Figure 5, the SHAP values scatterplot with the \"step_aa_T\" parameter shows the details of the nonlinear relationship, in which the EDS yield is roughly proportional to the parameter values and differs only near the peak value. As shown in the other plot of Figure 5, the \"step_ax_P\" feature implies that low and high sensor values are related to lowering yield relatively. As an example of a categorical parameter, the \"step_s_R_53\" parameter shows its yield-enhancing influence with the SHAP value up to 0.04 on the summary plot, as shown in Figure 4a and on the individual feature's SHAP value scatter plot in Figure 5. The \"step_s_R_52\" parameter is the other category processing condition of \"step_s_R_53\" converted from one-hot encoder, and they show opposite responses in the summary plot, as shown in Figure 4a. As shown in the waterfall chart of Figure 4b, these two categorical parameters share SHAP values, while they mean the wafer is processed with the operating condition presented by \"52\" instead of \"53.\" As shown in Figures 4 and 5, the \"53\" process condition is supposed to increase yield in the prediction model. Practically, these parameters represent process design change, and the influence on yield is consistent with the domain knowledge that \"53\" is the advanced process of the step. Figure 4: _Cont._  
The waterfall charts indicate how each feature influences the expected target prediction for a specific data point as shown in Figure 4b-d. In other words, SHAP waterfall charts illustrate how the explanation model decomposes the model output for an instance (i.e., a wafer) as the summands of plus and minus SHAP values of each feature as Equation (4). The basic prediction yield of the RF model is \(\mathrm{E}\Big{[}\hat{f}(x)\Big{]}\), and the predicted yield according to the summation of SHAP values for each example wafer is \(\hat{f}(x)\)[30; 44]. This approach can help analysts perform investigations on specific low-yield wafers. For example, Figure 4c  
Figure 4: (**a**) SHAP value plot of feature attribution for top parameters by the RF model. The color corresponds to the range of feature values from high (red) to low (blue). A positive SHAP value means a positive impact on prediction, leading the model to predict a high yield for the wafer. Waterfall plots of example wafers (**b**) with mid yield, (**c**) with low yield, and (**d**) with high yield. The color corresponds to the SHAP value: positive (red) and negative (blue). Figure 5: SHAP value scatterplots according to the values of six example features. shows that a specific low-yield wafer is processed by a specific unit, as represented by the plus value for \"step_ac_U_8\" and it has a yield lowering effect of \(-0.04\). The sum of all the SHAP values of 983 parameters for the wafer, \(\hat{f}(x)\) of Figure 4c, is \(-0.694\), and the predicted yield of the wafer by the RF model, \(f(x)\), is \(-0.673\). The SHAP value method decodes the model well for the data point and illustrates the stepwise prediction.