## 5 Results and Discussion
### Model Interpretation Using SHAP  
Developing complex black box machine learning models is a widely applied highly efficient technique to predict the target variable in mining and geotechnical engineering, particularly when the response variable is dependent on intricately related non-linear features. However, despite being remarkably proficient, these models suffer a significant constraint of being opaque and uninterpretable. This limitation associated with the black box models underscores the necessity to furnish a clarification of the generated outputs and the ramifications of the diverse input features on the resultant output. To address this, a model explanation framework called SHAP (Shapely Additive Explanations) was employed in this study [55; 56; 57]. SHAP analysis is derived from game theory and it encompasses strategic interactions among factors to allocate gains and costs equitably among collaborators. Although versatile, only a few studies have employed it to elucidate black box models [57; 98; 99]. The shapely value for feature \(i\) is calculated using Equation (35), with\(f\) as the black-box model and \(x\) as an input datapoint corresponding to a single tabular dataset row. Initially, all possible subsets \(z^{\prime}\) of the transformed input \(x^{\prime}\) are iterated to account for an interaction between individual feature values. Subsequently, the black box model output is determined for subsets with and without the feature of interest. The difference of \(f_{x}\left(z^{\prime}\right)-f_{x}\left(z^{\prime}\smallsetminus i\right)\) indicates the marginal contribution of feature \(i\). This process is repeated for each subset permutation, and weighting is calculated based on the total number of features, \(M\). On the other hand, an emphasis is placed on small correlations to observe isolated feature effects on predictions [100; 101; 102]. \[\phi_{i}(f,x)=\sum_{z^{\prime}\subseteq x^{\prime}}\frac{\left|z^{\prime} \right|!\left(M-\left|z^{\prime}\right|-1\right)!}{M!}\Big{[}f_{x}\left(z^{ \prime}\right)-f_{x}\left(z^{\prime}\smallsetminus i\right)\Big{]} \tag{35}\]  
In the conducted modeling experiment, the ERTR-BO model demonstrated exceptional proficiency in regression for the provided dataset. Given its tree-based nature, a tree-based SHAP explainer was employed to assess the impacts of distinct features on the predicted variable (\(v\)) [101]. Figure 15 illustrates SHAP plots depicting global and local contributions through bar, dot, and force plots [102]. Feature names in the plots remain consistent with the study's nomenclature, except for 'rhor,' which denotes \(\rho\). In both the bar and dot plots, the vertical axes present input features based on their contribution to the \(v_{f}\) prediction model. The dot plot's horizontal axis (Figure 15a) showcases actual SHAP values, while the bar plot's horizontal axis (Figure 15b) indicates the mean of absolute SHAP values. Each dot in Figure 15a corresponds to a SHAP value tied to a distinct dataset feature. The color transition from purple to pink within the dots signifies escalating SHAP value magnitudes. On the other hand, Figure 15c displays a force plot exemplifying the local contribution of a test dataset point. This plot revealed an actual prediction of 26.02 m/s, while the base value of 24.05 signified the expected value or mean of the predictions made for the whole dataset. Beneath the force plot, insights are provided into each feature's contribution to the 26.02 m/s prediction. Observing Figure 15, it was evident that the bulk density of rock (\(\rho_{r}\)) stood out as the foremost influential feature impacting the flyrock launch velocity, followed by the energy factor (\(W\)), burden (\(B\)), burden to blasthole diameter ratio (\(B\)/\(d\)), stemming length to burden ratio (\(l_{s}\)/\(B\)), and ratio of the stemming length to the length of blasthole (\(l_{s}\)/\(l_{bh}\)), respectively.