## 2 Material and Methods
### Kernel Tree Methods  
Decision trees as applied to continuous data are known as regression trees (RT) [17; 18; 19]. Given that documentation on the methodologies of kernel tree methods is extensive, only a summary is presented. As [PERSON] et al. [20] noted, \"linear regression is a global model, where there is a single predictive formula holding over the entire data-space. An alternative approach is to sub-divide, or partition, the space into smaller regions, where the interactions are more manageable.\" The'regression tree' approach to modeling identifies a hierarchy of interactions that were previously unknown. The regression tree approach creates recursive partitions or cells within the entire data space (i.e., terminal nodes or leaves) and the cells are modeled separately. The cells of regression trees are typically 'pruned' during the model validation phase in identifying the best predictive model. One strength of this method is that data do not need to be imputed, given that partitions will be made in the presence of missing data ([PERSON], 2009). RTs are represented as two-dimensional graphics, which makes it easily understood and interpreted [6; 7]. RTs are quite popular as an exploratory modeling technique, and are commonly associated with data mining techniques [21; 22; 23; 24; 25; 26; 27; 28]. RTs are very resistant to irrelevant regressors given that the recursive tree-building algorithm estimates the optimal variable on which to split at each step, regressors unrelated to the response are not chosen for splitting [29] (pp. 199-215). In theory, a regression tree partitions the data space of all joint regressor values \(\mathbf{X}\) into \(J\)-disjoint regions \(\left\{R_{j}\right\}_{1}^{J}\)[6]. For a given set of joint regressor values \(\mathbf{X}\), the tree prediction \(\hat{Y}=T_{j}(X)\) assigns as the response estimate, the value assigned to each region containing \(\mathbf{X}\):  
\[X\in R_{j}\ \Rightarrow T_{j}(\mathbf{X})=\hat{y}_{j} \tag{1}\]  
Given a set of regions, the optimal response values associated with each region minimize the prediction error in that region:  
\[\hat{Y}_{j}=\arg min_{y^{\prime}}E_{y}[L(y,y^{\prime})\big{|}\mathbf{X}\in\mathbf{R}_{ j}] \tag{2}\]  
Unlike some kernel methods, RTs use the data to estimate a good partition instead of relying on a predefined model by the analyst. 'Boosted trees' (BT) relies on the philosophy that a small number of simple trees of weak learners that are combined as one model outperform the predictions of one large RT [6; 30; 31]. 'Boosting' builds trees sequentially such that each new tree improves the predictive power of the ensemble [32; 33; 34]. The result grows new trees specifically aimed at accommodating observations that an existing ensemble predicts poorly, i.e., overall improved predictive performance of the final BRT model. BRT approximates a solution to the problem of fitting a sum-of-trees by adding new trees one at a time, while keeping all existing trees unchanged [35]. As stated by [PERSON] [31] and [PERSON] et al. [35], BT is a model to enhance the model accuracy and the key step in boosting is to consecutively apply the algorithm to constantly modified data, i.e., it minimizes the loss function through adding a regression tree in each iteration step [35]. 'Random forests' (RF) developed by [PERSON] [36] and as summarized by [PERSON] et al. [37] combines [PERSON]'s bagging sampling approach, and the random selection of features, introduced independently [38; 39] in order to construct a collection of decision trees with controlled variation. Each tree in the ensemble acts as a base classifier to determine the class label of an unlabeled instance.\" Key advantages of RF over BT are robustness to noise and less overfitting [35; 38; 39; 40; 41; 42; 43]. There were five key parameters used for model calibration in this study. A minimum size split of 10 was used for both the boosted tree and random forests to prevent the program from splitting any node with below a specified number of cases [44].