## 3 Methodology
### Other Machine Learning Methods  
We replaced the GBRT in the AGB upscaling model by other machine learning algorithms, including random forest (RF) or extremely randomized tree (ERT) to compare the performance of models.  
(1) RF: RF is a combination of tree predictors that every tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. [66; 67]. The single tree predictor in the entirety is produced by stochastically choosing independent variables with the bootstrap sampling approach [68]. The RF method usually ensures optimal predictions through three main steps. First, original data are divided into subdivisions. Second, every subdivision is used to develop an individual decision tree representing a subregression model that provides its own results. Finally, outputs derived from the trees are averaged to estimate the optimal results [69]. The randomness added to the forests when coming into being decision trees may decouple estimated errors. Some errors can be removed by having an average of those estimations [70]. RF for regression are formed by growing trees depending on a random vector \(\Theta\) such that the tree predictor \(h(x,\Theta)\) takes on numerical values as opposed to class labels. The output values are numerical, and we assume that the training set is independently drawn from the distribution of the random vector \(Y\), \(X\). The mean-squared generalization error for any numerical predictor \(h(X)\) is:  
\[E_{X,Y}(Y\ -\ h(X))^{2} \tag{3}\]  
The random forest predictor is formed by taking the average over k of the trees \(\{h(x,\Theta_{k})\}\)[66].  
(2) ERT: ERT, which was raised by [PERSON], is a tree-based machine learning ensemble model for supervised learning of classification and regression [71]. It consists of strongly randomizing both attribute and cut-point choices while splitting a tree node essentially. In the extreme case, it builds totally randomized trees with structures that are independent of the output values of the learning sample. The strength of randomization can be tuned to problem specifics by the appropriate choice of a parameter. ERT has three important parameters: \(K\), \(n_{\min}\), and \(M\)[71]. The parameter \(K\) is the number of random splits at every node to build extra trees. It may be selected from one to \(n\), where \(n\) is the number of attributions. According to a given situation, the smaller the \(K\) is, the stronger the randomization of the trees and the weaker the dependence of their structure on the output values of the learning sample will be. Previous studies have demonstrated that the best value of \(K\) is \(K=n\) for regression [71]. The parameter \(n_{\min}\) of original samples asked for splitting a node. Smaller trees, higher bias, and smaller variance result from the larger values of \(n_{\min}\). Therefore, optimal value of \(n_{\min}\) depends on the principle under the level of output noise in the dataset [71]. The parameter \(M\) refers to the amount of ensemble trees. It is well known that the behavior of estimation error is a monotonically decreasing function of \(M\) for randomization methods [66]. Relative variance reduction is used in regression. \(S_{I}\) and \(S_{r}\) refer to two subsets of cases from \(S\) corresponding to the two outcomes of a split \(s\), then the score is defined as follows [71]:  
\[Score_{R}(s,S)=\frac{\mathrm{var}\{y|S\}-\frac{|S_{i}|}{|S|}\mathrm{var}\{y|S_ {i}\}-\frac{|S_{i}|}{|S|}\mathrm{var}\{y|S_{r}\}}{\mathrm{var}\{y|S\}} \tag{4}\]  
where \(\mathrm{var}\{y|S\}\) is the variance of the output \(y\) in the sample \(S\).