## 2 Materials and Methods
### Explainable Tree-Based Models
#### 2.2.1 Model Evaluation  
We integrated the Shapley additive explanation (SHAP) method, developed by [PERSON] and [PERSON] [44], with the optimized ML to interpret the prediction criteria of the models and the level of contribution of the individual features that strongly correlate in organic matter prediction. Feature importance provides a crucial reference for feature selection [45]. A higher feature importance value, compared with another, implies the greater importance of the feature for generating a prediction. Shapley values quantify the contribution of each explanatory variable in each instance. The term \"additive\" signifies that the corresponding Shapley value for each explanatory variable of an instance can be additively combined [46]. This approach provides quantitative information to explain how individual explanatory variables either positively or negatively impact the target variable of interest in the model. The following formula represents the Shapley value, \(\phi_{i}(v)\):  
\[\phi_{i}(v)=\frac{1}{|\mathbf{N}|!}\sum_{S\subseteq[\mathbf{N}]\setminus\{i\}}|S|!(|\mathbf{ N}|-|S|-1)![v(S\cup\{i\})-v(S)],\]  
where \(v(S)\) represents the output of the ML model being explained with a set \(S\) of features, and \(N\) refers to the entire set of available features. The Shapley value or contribution of a specific feature \(i\) (\(\phi_{i}\)) is calculated as the average contribution across all possible permutations of the feature set [47], akin to determining the marginal contribution (\([v(S\cup\{i\})-v(S)]\)). This method evaluates the unique impact of feature \(i\) by assessing how the inclusion or exclusion of this feature alters the model's output, mirroring the process of assessing a feature's extra value in the model. Additionally, it applies a weighting factor (\(|\mathbf{S}|! (|\mathbf{N}|-|\mathbf{S}|-1)!\)) to account for every potential combination of features, ensuring that the influence of each feature is measured fairly and equitably. Lastly, summing over all coalitions and normalizing with denominator \(|\mathbf{N}|!\) aggregates the contributions of each feature to recognize the significance of individual features in proportion to their effect on the model's performance. The Shapley-based feature importance method, however, does not provide insights into two crucial aspects: (1) how changes in these important features affect predictions and (2) whether a specific threshold for these key features can enhance the accuracy of predicting the target variable. To address these concerns, we employed partial dependence (_PD_) plotting, which enabled us to analyze how specific input features contribute to variations in the expected response of the target variable. One-dimensional _PD_ plots illustrate how changes in the chosen independent variables impact the expected value of their dependent variable while other independent variables remain constant. Furthermore, two-dimensional _PD_ plots are used to assess predicted values for their dependent variable as both independent variables simultaneously vary. _PD_ plots are a tool for comprehending the impacts of features in any ML model [48], such as \(\hat{f}(\mathbf{x})\). The explanatory variables \(x=(x_{s}\), \(x_{c})\) can be divided into two subsets to emphasize the concept and effectively convey this relationship and visualization: \(x_{s}\) and \(x_{c}\). \(x_{s}\) represents the \"chosen\" independent variable, and \(x_{c}\) comprises the set of other independent variables. The function that represents _PD_ is defined as follows:  
\[PD=\frac{1}{n}\!\sum_{i=1}^{n}\hat{f}(x_{s},x_{ic}).\]  
_PD_ represents the partial dependence of a subset of features \(x_{s}\) on the model's predictions, illustrating how the average prediction changes when the values of \(x_{s}\) are altered while all other variables (\(x_{ic}\)) remain constant.