## 3 Methods
### VNIR-Based Prediction Methods
#### 3.3.1. PLS-MARS Method  
The PLS-MARS method is a combination of PLS and MARS methods that constructs a nonlinear prediction model with local regression. The flow of PLS-MARS comprises two parts (Figure 3). Part 1 removes the multicollinearity of the explanatory variables by obtaining PCs using a PLS regression method. Part 2 constructs the MARS model based on the predictors and PCs obtained in Part 1. The details of PLS-MARS are described as follows. PCs were abstracted in Part 1. The explanatory variables and predictors were labeled as \(E_{0}\) and \(F_{0}\), respectively. The procedure of obtaining PCs was to abstract the PCs first. Thereafter, a series of PLS regression models based on the obtained PCs and predictors was built. Lastly, the accuracy of the models was evaluated. The model with the highest accuracy was selected, and the corresponding PCs were output as the eventual PCs. The steps of this procedure are described in detail as follows. Step 1: The PC \(t_{1}\) of the explanatory variables \(E_{0}\) and PC \(u_{1}\) of the predictors \(F_{0}\) were extracted. The estimated values of PCs can be expressed in accordance with Equation (1). The abstracted PCs should represent sufficient information of the variables, and the PCs  
Figure 3: Procedure of constructing the partial least squares-based multivariate adaptive regression spline (PLS-MARS) model. and \(u_{1}\) should be highly correlated. The preceding calculation targets imply that \(t_{1}\) and \(u_{1}\) can be calculated based on the conditions in Equation (2) by using the Lagrange multipliers. \[\left\{ \begin{array}{c}t_{1}=E_{0}w_{1}\\ u_{1}=F_{0}v_{1}\end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{c}\langle t_{1},u_{1}\rangle=\langle E_{0}w_{1},F_{0}v_ {1}\rangle=w_{1}^{T}E_{0}^{T}r_{0}\Rightarrow Max\\ w_{1}^{T}w_{1}=1,\;v_{1}^{T}v_{1}=1\end{array} \right. \tag{2}\]  
Step 2: The regression (i.e., Equation (3)) was constructed based on \(t_{1}\) and \(u_{1}\); the model is expressed as follows:  
\[\left\{ \begin{array}{c}E_{0}=t_{1}\alpha_{1}^{T}+E_{1}\\ F_{0}=u_{1}\beta_{1}^{T}+F_{1}\end{array} \right. \tag{3}\]  
where \(\alpha_{1}\) and \(\beta_{1}\) are the model parameters to ensure the high correlation between \(t_{1}\) and \(u_{1}\), which can be estimated using least squares criterion; \(E_{1}\) and \(F_{1}\) are the residual matrixes. Step 3: The residual matrixes \(E_{1}\) and \(F_{1}\) were set as the explanatory variables and predictors, respectively. Step 1 was iteratively repeated until the count of the selected PCs reached \(h\) (i.e., the matrix rank of \(E_{0}\)) (Equation (4)). The PCs \(t\) of the explanatory variables can be calculated using Equation (2). \[\left\{ \begin{array}{c}E_{0}=t_{1}\alpha_{1}^{T}+\cdots+t_{h}\alpha_{h}^{T}+ E_{h}\\ F_{0}=u_{1}\beta_{1}^{T}+\cdots u_{h}\beta_{h}^{T}+F_{h}\end{array} \right. \tag{4}\]  
Step 4: The first \(\delta\) principal component PCs (\(0<\delta\leq h\)) were successively selected. In addition, the corresponding PLS regression model between the predictors and extracted PCs of the explanatory variables was constructed. Step 5: The precision of the PLS model was evaluated using the root-mean-square error (RMSE) calculated by the cross-validation method (leave-one-out sample). High precision of the PLS regression model led to a small value of RMSEV. The PCs \(t=\left\{t_{1},\cdots,t_{p}\right\}\) for constructing the PLS regression model with the highest precision were output as PCs for constructing the following the MARS model. In Part 2, the MARS prediction model was built based on the explanatory variables \(\left\{t_{1},\cdots,t_{p}\right\}\) and predictors. The MARS model was established based on several BFs in the form of an expansion of splines.