## 3 Data Modeling Methodology  
### Support Vector Regression (SVR)  
SVR is an application of Support Vector Machines to regression problems, its main principle is to map low-dimensional linearly indivisible data into a high-dimensional linearly divisible space and find a hyperplane to divide it and realize the mapping with the help of the kernel function [33]. The ultimate goal of SVR is to find a regression function \(y\) such that the data are distributed as much as possible around \(y\). The formula for \(f(x)\) is given below:  
\[y=f(x)=\omega x+b \tag{4}\]  
where \(x\) is the independent variable, \(\omega\) is the set of vectors to be optimized, and \(b\) is the intercept. Meanwhile, when determining the decision boundary of SVR, the pursuit is to maximize the soft interval while minimizing the loss function. By maximizing the soft interval, it is realized to include as many data points as possible in the interval band of the support vectors, but in reality, it is difficult to artificially determine the support vector interval plane where it is located, so the slack variable \(\zeta\) is introduced, which makes the interval requirement of the function become loose, after the introduction of the slack variable \(\zeta\), the loss function of SVR optimization can be expressed as:  
\[ \begin{split}\underset{\omega,b,\xi_{i}}{min}&\frac{ \left\|\omega\right\|^{2}}{2}+C\sum_{i=1}^{n}\zeta_{i}\\ \text{\emph{subject to }}& y_{i}(\omega\cdot \varphi(x_{i})+b\geq 1-\zeta_{i}),\\ \zeta_{i}&\geq 0,\ i=1,\ldots,N\end{split}  \tag{5}\]  
where \(\zeta_{i}\) is the slack variable, \(C\) is the penalty coefficient used to control the penalty term, and \(\varphi(x_{i})\) is the set of points \(x_{i}\) in low dimensions mapping to the high-dimensional space \(\varphi\) in the mapping function. Therefore, the loss function of the nonlinear SVR is obtained. From Equation (5), it can be seen that the loss function is a convex optimization problem in mathematics, and when solving such a convex optimization problem, the Lagrange multiplier method is usually used to solve it. First, the loss function is transformed into a Lagrangian function with constraints, and the solution of the original problem is equivalent to the solution of the dual function under the satisfaction of the Karlovski-Kuhn-Tucker (KKT) condition, so only the solution of the Lagrange dual function is required. Where the formula for the Lagrange dual function is as follows:  
\[ \begin{split} L_{D}=\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i, j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{i}\varphi(x_{i})\varphi(x_{j})\\ \text{\emph{subject to }}C\geq\alpha_{i}\geq 0\end{split}  \tag{6}\]  
where \(\alpha_{i}\) is the Lagrange operator, and \(\varphi(x_{i})\) is the mapping function that maps the set of points \(x_{i}\) to higher dimensions, but it is not clear exactly what mapping function is used to ensure that the decision boundary can be found in the transformed space, and when the computation of a dot product of the form \(\varphi(x_{i})\varphi(x_{j})\), a kernel function needs to be introduced to solve the above problem. After the introduction of the kernel function, there is no need to know exactly what the mapping function is, and it is much easier to use the kernel function to compute the vector relations in low dimensions than in high dimensions \(\varphi(x_{i})\varphi(x_{j})\). In this paper, according to the linear correlation analysis in the data preprocessing, it is found that there is a moderate degree of linear correlation between the DO content in water and other indicators, but at the same time, there is also a certain degree of nonlinear relationship. Therefore, the Gaussian radial basis partial linear kernel function (RBF) is chosen to transform the data from low latitude to high dimension with the following equation:  
\[K(x,y)=e^{-\gamma\left\|x-y\right\|^{2}},\ \gamma>0 \tag{7}\]  
where \(\gamma\) is the parameter gamma to be optimized. As a result, the penalty coefficient \(C\) and the gamma parameter to be optimized in SVR can be obtained.