## Materials and Methods
### Analysis Methods
#### 2.3.3 Model Analysis  
The spectral feature sets extracted by the above methods were used as an input variable to models and the models were constructed using the FLDA algorithm and the SVM algorithm, respectively. The FLDA algorithm reduces the dimensions of high-dimensional features, projects the data into a one-dimensional space, and finds an optimal projection direction that minimizes the within-class scatter and maximizes the between-class scatter [18]. The principle is as follows:  
Suppose that a set \(X\) in the original high-dimensional space contains \(N\) samples \(x_{1}\), \(x_{2},\ldots,x_{n}\), where \(N_{1}\) samples belonging to the class \(W_{1}\) are recorded as subset \(X_{1}\), \(N_{2}\); samples belonging to the class \(W_{2}\) are recorded as subset \(X_{2}\); and \(N_{i}\) samples belonging to the class \(W_{i}\) are recorded as subset \(X_{i}\). In the original high-dimensional space, the pooled within-class scatter matrix is \(S_{w}\):  
\[S_{w}=\sum_{i=1}^{\mathrm{C}}\sum_{X\in X_{i}}\left(x-m_{i}\right)\left(x-m_{ i}\right)^{T},\ \ i=1,\ldots,C \tag{3}\]  
The between-class scatter matrix is \(S_{b}\):  
\[S_{b}=\sum_{i=1}^{\mathrm{C}}\left(m_{i}-m\right)\left(m_{i}-m\right)^{T} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Vegetation Index** & **Calculation Formula** & **Reference** \\ \hline PRI & (R531 – R570)/(R531 + R570) & [27] \\ PHRI & (R550 – R531)/(R531 + R570) & [27] \\ MCARI & [(R700 – R670) – 0.2(R700 – R550)](R700/R670) & [28] \\ TVI & 0.5[120(R750 – R550)-200(R670-R550)] & [29] \\ ARI & (R550–1 (R700) – 1) & [30] \\ NDVI & (R800 – R670)/(R800 – R670) & [31] \\ GNDVI & (R747 – R537)/(R747 + R537) & [32] \\ NBNDVI & (R850 – R680)/(R850 + R680) & [33] \\ NRI & (R570 – R670)/(R570 + R670) & [33] \\ PSRI & (R680 – R500)/(R500) & [34] \\ SIPI & (R800 – R540)/(R800 – R680) & [35] \\ NPCI & (R680 – R430)/(R680 – R430) & [36] \\ TCARI & 3[(R700 – R670) – 0.2(R700 – R550)(R700/R670)] & [37] \\ CARI & ((a670 – R670 + b)/(a\({}^{2}\) + 1)\({}^{1/2}\))(R700/R670) & [38] \\ A & = (R700 – R550)/150, b = R550 – (a x 550) & \\ GI & R554/R677 & [39] \\ MSR & (R800/R670 – 1)/sqrtt(R800/R670 + 1) & [40] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Vegetation indices commonly used in stress studies. In the one-dimensional space Y after projection, the pooled within-class scatter matrix is \(\overline{S_{w}}\):  
\[\overline{S_{w}}=\sum_{i=1}^{C}\sum_{y\in Y_{i}}\left(y-\overline{m_{i}}\right)^ {2},\ \ \ i=1,\,\ldots,C \tag{5}\]  
where \(m_{i}\) is the mean vector of each class of samples in the original high-dimensional space; \(\overline{m_{i}}\) is the mean vector of each class of samples in the one-dimensional space Y after projection; \(m\) is the mean vector of all of the samples. The purpose is to find the best projection direction, so that the various samples of the one-dimensional space are separated as much as possible after projection. That is, the difference between the mean vector of each class of samples is as large as possible and the pooled within-class scatter matrix is as small as possible. The Fisher criterion function is defined as:  
\[J_{F}(\omega)=\frac{\omega^{T}S_{b}\omega}{\omega^{T}S_{w}\omega} \tag{6}\]  
Let the denominator be equal to a non-zero constant, \(\omega^{T}S_{w}\omega=C\
eq 0\), and define the Lagrange function as:  
\[L(\omega,\lambda)=\omega^{T}S_{b}\omega-\lambda(\omega^{T}S_{w}\omega-c) \tag{7}\]  
where \(\lambda\) is the Lagrange multiplier. The Fisher criterion function \(J_{F}(\omega)\) is used to obtain the partial derivative of \(\omega\). The partial derivative is set equal to zero to get the equation:  
\[S_{b}\omega^{\star}=\lambda S_{w}\omega^{\star} \tag{8}\]  
where \(\omega^{\star}\) is the maximum value, which is the best projection method. In contrast, SVM is useful in dealing with linear indivisible cases. First, transform the input space into a high-dimensional space by nonlinear transformation (defining the suitable kernel function), and then find the optimal classification surface in the new space.