## 3 Methods
### Gradient Boosting Regression Tree (GBRT)  
GBRT, also known as gradient boosting decision tree (GBDT) or multiple additive regression tree (MART), is one of the most widely used ML algorithms for the model's great generalization ability and highly robust performance in practical applications, and this GBRT was introduced by [PERSON] (1973); [PERSON] (1974). As one of the ensemble learning methods that combines different weak learners to generate strong learners, GBRT uses boosting techniques that aim to reduce bias, rather than bagging (e.g., RF algorithm), which aims to reduce variance. To evaluate the accuracy of the model, a variety of loss functions can be used during boosting, including least squares, least absolute deviation, Huber and quantile for regression, binomial deviance, multinomial deviance, and exponential loss for classification. GBRT considers additive models of the following form:  
\[F(x)=\sum_{m=1}^{M}\gamma_{m}h_{m}(x) \tag{9}\]  
where \(h_{m}(x)\) represents the basis learners in boosting. Then, GBRT builds the additive model in a forward stagewise fashion:  
\[F_{m}(x)=F_{m-1}(x)+\gamma_{m}h_{m}(x) \tag{10}\]  
At each stage, the decision tree \(h_{m}(x)\) is chosen to minimize the loss function \(L\), given the current model \(F_{m-1}\) and its fitting of \(F_{m-1}(x_{i})\):  
\[F_{m}(x)=F_{m-1}(x)+arg\underset{h}{min}\sum_{i=1}^{n}L(y_{i},F_{m-1}(x_{i})+h (x)) \tag{11}\]  
The initial model \(F_{0}\) is problem specific. Gradient boosting attempts to solve this minimization problem numerically via the steepest descent. The steepest descent direction is the negative gradient of the loss function evaluated at the current model \(F_{m-1}\), which can be calculated for any differentiable loss function as follows:  
\[F_{m}(x)=F_{m-1}(x)-\gamma_{m}\sum_{i=1}^{n}\
abla_{F}L(y_{i},F_{m-1}(x_{i})) \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Description** & **GridSearch Values** & **Searching Results** \\ \hline n\_estimators & The number of trees in the forest. & 1, 5, 10, 20, 40, 60, 80, 120 & 40 \\ \hline max\_depth & The maximum depth of the tree. & 2, 5, 8, 15, 25, 30, 50, & 15 \\ \hline min\_samples\_split & The minimum number of samples required to split an internal node. & 2, 5, 10, 15, 100 & 2 \\ \hline min\_samples\_leaf & The minimum number of samples required to be at a leaf node; a split point at any depth will only be considered if it leaves at least min\_samples\_leaf training samples in each of the left and right branches; this may have the effect of smoothing the model, especially in regression. & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameter settings to determine the optimal hyperparameters for the random forest (RF) model.  
where the step length \(\gamma_{m}\) is chosen using a line search:  
\[\gamma_{m}=\underset{\gamma}{arg\ min}\sum_{i=1}^{n}L\left(y_{i},F_{m-1}(x_{i})- \gamma\frac{\partial L(y_{i},F_{m-1}(x_{i}))}{\partial F_{m-1}(x_{i})}\right) \tag{13}\]  
The GBRT algorithm can provide feature relevance information and partial dependence, where this partial dependence shows the dependence among the target response and the most important features (not shown in this study). However, despite the GBRT's demonstrated satisfactory accuracy for versatile domains of regression problems [29; 75; 76], to our knowledge, this algorithm has not been previously applied to crop LAI retrievals with remote sensing. GBRT should be studied as it might be a promising alternative in crop LAI retrieval with remote sensing.  
With regard to GBRT model hyperparameters, loss, n_estimators, max_depth, min_samples_split and min_samples_leaf are selected for hyperparameter tuning using GridSearch and 5-fold cross-validation, the results are exhibited in Table 8. The remainder of the GBRT model hyperparameters are set as defaults.