## 4 Prediction Model Establishment  
### Least Squares Support Vector Machine Algorithm (LSSVM)  
The support vector machine (SVM) has been proposed as a solver for quadratic programming (QP) and regression problems. The least squares support vector machine (LSSVM) is an enhanced version of the support vector machine. Based on the traditional support vector machine (SVM), the inequality constraints in the quadratic programming problem are transformed into equality constraints. This simplifies the calculation of the function approximation problem and greatly facilitates the solution process. It overcomes the abnormal regression caused by the rough dataset and the large fluctuation of the dataset, and can effectively avoid the local optimization problems in BP neural network and other methods [40]. Compared with fuzzy logic systems and artificial neural networks, LSSVM adopts a low-parameter technology, which enables fast calculation and parameter tuning [41]. The principle is as follows:  
For a given training dataset \(\{(x_{i},y_{i}),i=1,2,\ldots,n\}\), where \(x_{i}=(x_{i1},x_{i2},\ldots,x_{id,})^{T}\) is \(d\)-dimensional input vector, and \(y_{i}\) is the corresponding output result. \(N\) represents the total number of training data points. In order to represent the mapping relationship between input variables and output variables, the following nonlinear function is used to estimate the model:  
\[f(x)=\langle\varphi(x),\ \omega\rangle+b \tag{8}\]  
where \(\omega\) represents the weight vector; \(b\) represents the bias term; and symbol \(\langle\cdot\rangle\) denotes the inner product operation. Based on the principle of structural risk minimization, the error square term is expressed as empirical risk, and the quadratic programming problem in SVM is optimized as follows:  
\[\left\{ \begin{array}{l}\min_{\omega,\varepsilon}J(\omega,e)=\min\left( \frac{1}{2}\|\omega\|^{2}+\frac{1}{2}\gamma\sum_{i=1}^{N}e_{i}^{2}\right)\\ y_{i}=\langle\varphi(x_{i}),\omega\rangle+b+e_{i}i=1,2,\ldots,N;\gamma>0\end{array} \right. \tag{9}\]  
where \(\varphi(x)\) represents the kernel function; \(y_{i}\) stands for the regularization parameter; \(e_{i}\) denotes the error vector; and \(\gamma\) signifies the penalty factor. To solve the optimization problem above, we construct the corresponding Lagrange function. \[L_{LSSVM}=\frac{1}{2}\|\omega\|^{2}+\frac{1}{2}\gamma\sum_{i=1}^{N}e_{i}^{2}- \sum_{i=1}^{N}\alpha_{i}\{\langle\varphi(x_{i}),\omega\rangle+b+e_{i}-y_{i}\} \tag{10}\]  
where \(\alpha_{i}\) represents the Lagrange function. Figure 7: Screening and prediction process of dust concentration index based on RF-GA-LSSVM. By calculating the partial derivatives of Formula (3) with respect to \(\omega,b,e_{i},\alpha_{i}\), the conditions for obtaining the optimal solution of the problem are:  
\[\left\{ \begin{array}{l}\frac{\partial L_{i,\text{SVM}}}{\partial\omega}=0\\ \frac{\partial L_{i,\text{SVM}}}{\partial\omega}=0\\ \frac{\partial L_{i,\text{SVM}}}{\partial e_{i}}=0\\ \frac{\partial L_{i,\text{SVM}}}{\partial e_{i}}=0\\ \frac{\partial L_{i,\text{SVM}}}{\partial a_{i}}=0\end{array} \right.\Rightarrow \left\{ \begin{array}{l}\omega=\sum_{i=1}^{N}\alpha_{i}\varphi(x_{i})\\ \sum_{i=1}^{N}\alpha_{i}=0\\ \alpha=\gamma e_{i}\\ \langle\varphi(x_{i}),\omega\rangle+b+e_{i}-y_{i}=0\end{array} \right. \tag{11}\]  
By eliminating variables \(\omega\) and \(e_{i}\), the above linear problem is simplified. \[ \begin{bmatrix}0&E^{T}\\ E&\Omega+\frac{1}{\gamma}E\end{bmatrix}\begin{bmatrix}b\\ \alpha\end{bmatrix} = \begin{bmatrix}0\\ y\end{bmatrix}  \tag{12}\]  
where \(y=[y_{1},y_{2},\ldots,y_{n}]^{T}\); \(\alpha=[\alpha_{1},\alpha_{2},\ldots,\alpha_{n}]^{T}\); \(E\) is an \(n\)-order unit matrix; \(\Omega\) is an \(n\)-order identity matrix of the kernel function.