# Crop Growth Stage GPP-Driven Spectral Model for Evaluation of Cultivated Land Quality Using GA-BPNN
## 2 Materials and Methods
### Methods
#### 2.3.4 Support Vector Regression  
The SVR is a widely used supervised learning method for solving the problem of regression fitting. Different from the traditional process from induction to deduction, SVR greatly simplifies the usual regression process by making efficient \"transductive inference\" from training samples to prediction [41; 42; 43]. Traditional regression algorithms use training samples to generate a model (a global trend) that is used to predict values of the dependent variable at unknown locations. In SVR, the values of 2011-2015 MODIS-GPPs as predictors \(x\) in this study are first plotted onto an m-dimensional feature space (m-number of predictors) and a linear model with its epsilon band-acceptable prediction surface is constructed in the feature space.  
\[f(x)=\omega\varnothing(x)+b \tag{7}\]  
where \(f(x)\) presents CLQ estimates, \(\varnothing(x)\) denotes the GPP set of nonlinear transformations, \(b\) is the error term, and \(\omega\) is the weight coefficient. After preprocessing of data, the error term often has a zero mean and can be dropped. According to structural risk maximization principles, \(\omega\) and \(b\) are calculated by following the objective function \(R(x)\).  
\[R(x)=\frac{1}{2}\|\omega\|^{2}+\frac{1}{l}\sum_{i=1}^{l}\left|f(x_{i})-y_{i} \right|_{\varepsilon} \tag{8}\]  
where \(\left|f(x_{i})-y_{i}\right|_{\varepsilon}\) is the insensitive loss function, \(\varepsilon\) is the error tolerance of the insensitive loss function, \(l\) denotes the number of samples, \(\|\omega\|^{2}\) reflects the flatness of in the m-dimensional space.  
The model complexity can be simplified by minimizing \(\|\omega\|^{2}\). By introducing (non-negative) slack variables \(\xi_{i}\), \(\xi_{i}^{*},i=1,\ldots,n\) and penalty factor (C) to derive the deviation of training samples outside the insensitive loss function, SVR is formulated by minimizing the following objective function:  
\[\begin{split}&\text{R}\Big{(}\omega,\xi_{i},\xi_{i}^{*}\Big{)}= \frac{1}{2}\|\omega\|^{2}+\text{C}\sum\limits_{i=1}^{1}\left(\xi_{i}+\xi_{i}^{ *}\right)\\ &\text{s.t.}\left\{\begin{array}{l}\text{y}_{i}-\varnothing( \text{x})-b\leq\varepsilon+\xi_{i}^{*}\\ \omega\varnothing(\text{x})+b-\text{y}_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0,i=1,\ldots,n\end{array}\right.\\ \end{split}\right.\end{split} \tag{9}\]  
The above problem can be translated into the following dual problem:  
\[\begin{split}\min_{\alpha_{1}^{(*)}}\varepsilon\text{R}^{21}& \frac{1}{2}\sum\limits_{i,j=1}^{1}\left(\alpha_{i}-\alpha_{i}^{*}\right) \Big{(}\alpha_{i}-\alpha_{j}^{*}\Big{)}\Big{(}\varnothing(\text{x}_{i}) \varnothing(\text{x}_{j})\Big{)}+\varepsilon\sum\limits_{i=1}^{1}\left(\alpha_ {i}^{*}+\alpha_{i}\right)-\sum\limits_{i=1}^{1}\text{y}_{i}\Big{(}\alpha_{i}^ {*}-\alpha_{i}\Big{)}\\ &\text{s.t.}\left\{\begin{array}{l}\sum\limits_{i=1}^{1} \left(\alpha_{i}-\alpha_{i}^{*}\right)=0\\ 0\leq\alpha_{i}^{(*)}\leq C,i=1,\ldots,1.\end{array}\right.\\ \end{split} \tag{10}\]  
where \(\alpha_{i}-\alpha_{i}^{*}\) is the transformation of the \(\omega\) variable after introducing the Lagrange factor. The SVR function is obtained by solving the above problems:  
\[f(x)=\omega\varnothing(x)+b=\sum\limits_{i=1}^{1}\Big{(}\alpha_{i}^{*}-\alpha_{ i}\Big{)}\text{K}(\text{x}_{i},\text{x})+b \tag{11}\]  
where \(K(x_{i},x)=\varnothing(x_{i})\varnothing(x)\) is kernel function.