## 2 Materials and Methods
### Regression Models
#### 2.3.2 Classification and Regression Trees  
CART is a tree-based algorithm, and it can be used for both classification and regression [36]. In other words, CARTs contain classification trees and regression trees [35]. However, we only focused on regression trees because this study aimed to predict continuous values.  
To begin with, a CART is typically constrained to construct binary trees in the consideration of computational cost [35]. Specifically, CARTs will grow by means of binary recursive partitioning, i.e., the parent node will always be split into two child nodes, and then each child node will become a parent node and will be split into two child nodes. This process will continue until the node is a terminal node [37].  
Moreover, CARTs will search all possible features and all possible values (split-point) to find the best split which can achieve the maximum homogeneity of the two child nodes.  
In other words, choosing the best split is equivalent to the maximization of the change in impurity function, which is defined as [38]:  
\[\begin{array}{c}\Delta i(t)=i\big{(}t_{p}\big{)}-P_{l}i(t_{l})-P_{r}i(t_{r})\\ \underset{x_{j}\leq x_{j}^{R},\ j=1,\...,\ M}{\arg\max}\left[i\big{(}t_{p} \big{)}-P_{l}i(t_{l})-P_{r}i(t_{r})\right]\end{array} \tag{4}\]  
where \(\Delta i(t)\) is the maximization of the change in impurity function, \(i()\) is the impurity function, \(t_{p}\), \(t_{l}\), and \(t_{r}\) are the parent node, left node, and right node, respectively, \(P_{l}\) and \(P_{r}\) are probabilities of the left node and right node, \(x_{j}\) is feature \(j\), \(x_{j}^{R}\) is the best split value of feature \(j\), and \(M\) is the number of features [38]. After the best split is chosen, the samples whose value are greater than the split-point will be assigned to the right node, whereas the rest will be assigned to the left node [26].  
In addition, the common impurity functions (splitting criteria) for regression trees are the mean squared error criterion and mean absolute error criterion, which are defined as [31]:  
\[\begin{array}{c}\overline{y}_{m}=\frac{1}{N_{m}}\sum\limits_{y\in Q_{m}}y \\ \text{mean squared error criterion}=H(Q_{m})=\frac{1}{N_{m}}\sum\limits_{y\in Q _{m}}(y-\overline{y}_{m})^{2}\end{array} \tag{5}\]  
\[\begin{array}{c}\text{mean absolute error criterion}\ H(Q_{m})=\frac{1}{N _{m}}\sum\limits_{y\in Q_{m}}|y-median(y)_{m}|\end{array} \tag{6}\]  
where \(Q_{m}\) is the set of samples in node \(m\), \(N_{m}\) is the number of samples in node \(m\), \(H()\) is the impurity function, \(y\) is the target values in vector \(y\), \(\overline{y}_{m}\) is the mean of \(y\), and \(median(y)_{m}\) is the median of \(y\)[31].  
If we take the mean squared error criterion, the predicted value can be computed as the average of all samples in the terminal node. However, the predicted value is computed as the median of all samples in the terminal node when we take the mean absolute error criterion [31]. In summary, the key advantage of the CART algorithm is its robustness to outliers and being able to isolate outliers in an individual node or nodes [38]. Moreover, its great interpretability can fully describe the feature space partition [26].