## 2 Materials and Methods
### Statistical Models for Estimating the FSV  
The MLR is a statistical technique that uses several explanatory variables (independent variables) to predict the outcome of a response variable (dependent variable). The MLR is a widely used method to construct regression models for various applications [72; 84; 85; 86; 87]. The goal of the MLR is to model the linear relationship between the explanatory (independent) variables and response (dependent) variable (FSV). The formula for the MLR is  
\[\text{y}=a+b_{1}x_{1}+b_{2}x_{2}+b_{3}x_{3}+\ldots+b_{i}x_{i}+\varepsilon \tag{1}\]  
where \(i\) is the number of variables, \(\text{y}\) is the dependent variable, \(x_{i}\) is the independent variable, \(a\) is the intercept, \(b_{i}\) is the slope coefficient for the explanatory variable, and \(\varepsilon\) is the error term of the model. The SVR maps the output data to the high-dimensional feature space by defining the kernel function and constructs an optimal classification hyperplane in this space. The SVR can be regarded as a linear algorithm in high-dimensional space [88]. The SVR has been proven to be a popular method for regression modeling [89; 90; 91; 92]. In this study, it was constructed using the following SVR function from the training process:  
\[\text{FSV}=\sum_{i=1}^{n}\alpha_{i}k(x_{i};x)+b \tag{2}\]  
where \(k(x_{i};x)\) is the kernel function, \(\alpha_{i}\) is the Lagrange multiplier, \(x_{i}\) is the training vector, and \(b\) is the bias term in the regression. Using the e1071 package in the R 3.5.3 software, the parameters of the SVR were optimized and cross-validated. The radial basis function (RBF) kernel was selected following previous studies [71; 93]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Characteristic** & **Index** & **Calculation Method** \\
**Variable** & **Short Name** & \\ \hline \multirow{8}{*}{Vegetation indices} & NDVI\_B5 & (B5 \(-\) B4)(B5 \(+\) B4) \\  & NDVI\_B6 & (B6 \(-\) B4)(B6 \(+\) B4) \\  & NDVI\_B7 & (B7 \(-\) B4)(B7 \(+\) B4) \\  & NDVI\_B8 & (B8 \(-\) B4)(B8 \(+\) B4) \\  & NDVI\_B8A & (B8A \(-\) B4)(B8A \(+\) B4) \\  & SAVI & \(1.5^{\text{\text{\textregistered}}}(B8\) \(-\) B4)(B8 \(+\) B4 \(+\) 0.5) \\  & RVI & B8/B4 \\  & MSI & B8/B11 \\  & EVI & \(2.5^{\text{\text{\textregistered}}}(B8\)\(-\) B4)(B8 \(+\) \(6^{\text{\text{\text  
Then, the best regression model was optimized by the tune function (the SVM-type is eps-regression, the SVM-Kernel is the radial, cost is set as 2\({}^{c}\) (c is 2-9 sequence with an interval of 1), epsilon is set as a 0-1 sequence with an interval 0.01, and gamma is default). The RF is a decision tree algorithm and an effective machine learning model for predicting a forest of variables. Based on its powerful modeling capabilities, the RF regression has been widely used in scientific research [94; 95; 96; 97; 98; 99]. The principle of the RF algorithm is to use the bootstrap method to randomly extract multiple samples to generate a group of regression trees (ntree) from the original sample population. To build each tree, a randomly-chosen subset of predictors is used at each splitting node (mtry) by the RF, and there is no need to prune each tree grown. For each tree grown, a procedure named the \"out-of-bag\" (OOB) error was used to independently build each tree based on the training data. The variable importance (VI) and OOB error were calculated by the RF algorithm [100]. The OOB error can be estimated as follows:  
\[OOB_{error}=\ \frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2} \tag{3}\]  
where \(y_{i}\) is the measured FSV, \(\hat{y}_{i}\) is the predicted FSV, and n is the total number of OOB samples. After selecting the variables, the number of grown trees (ntree) and the number of variables (mtry) at the time of node splitting were determined using the minimum mean square error as the criterion. First, the RF model's ntree parameter was set to 500, and the mtry parameter was set to the total number of variables.