## 2 Methodology  
SVR, proposed in 1995 [21], is an application branch of SVM in the field of regression. It is suitable for handling nonlinear relationships and is excellently robust. The basic idea of SVR is to map nonlinear sample data from a low-dimensional space to a high-dimensional feature space using a nonlinear mapping function. In this high-dimensional space, linear regression is performed on the samples to obtain the regression function in the high-dimensional space. SVR aims to find a function that ensures the error for all training samples does not exceed a predetermined threshold while being as flat as possible. Given the Gaussian kernel's (GK) suitability for nonlinearly separable datasets, ability to avoid the curse of dimensionality, high flexibility, and excellent generalization capabilities, we selected GK as the nonlinear mapping function for SVR in this experiment. Based on the above principles, establishing a prediction model using SVR first involves determining the model's inputs and outputs. Considering the parameters that are highly correlated with the dynamic changes in TEC, the regression function is established as follows:  
\[\text{TEC}^{\prime}(x)=W\times\varphi(x)+b, \tag{1}\]  
where TEC\({}^{\prime}(x)\) represents the regression function, \(x\) denotes the input data, \(W\) is the regression weight, and \(\varphi(x)\) is the mapping function corresponding to the input data. The specific formula for \(\varphi(x)\) needs to be determined based on the characteristics of the data. Based on the training dataset, construct a margin band centered at TEC\({}^{\prime}(x)\) with a width of \(2\delta_{i}^{\prime}\). When the training samples fall within this margin band, the prediction is considered to have no loss; otherwise, the loss is calculated. Introducing the slack variables \(\delta_{i}\) and \(\delta_{i}^{\prime}\), SVR can be represented as:  
\[\min_{\omega,\delta_{i},\delta_{i}^{\prime}}\frac{1}{2}\|W\|^{2}+C\sum_{i=1}^{ n}\big{(}\delta_{i}+\delta_{i}^{\prime}\big{)}, \tag{2}\]  
\[s.t.\left\{ \begin{array}{l}\text{TEC}(x_{i})-\text{TEC}_{i}\leq\zeta+\delta_ {i}\\ \text{TEC}_{i}-\text{TEC}(x_{i})\leq\zeta+\delta_{i}^{\prime}\\ \delta_{i}\geq 0,\delta_{i}^{\prime}\geq 0,i=1,2,\cdots,n\end{array}  \right., \tag{3}\]  
where \(C\) represents the penalty coefficient, \(n\) is the number of data, and TEC\({}_{i}\) is the observed value. The term \"\(s.t.\)\" means \"subject to\", indicating the constraints applied to the optimization problem. By introducing the Lagrangian function and converting Equation (3) into its dual form, and then incorporating the GK to address the dimensionality issue, the SVR function, satisfying the Karush-Kuhn-Tucker conditions, is expressed as follows [22; 23]:  
\[ \begin{array}{rl}\text{TEC}^{\prime}(x)&=\sum_{i=1}^{n}\big{(}a_{i}^{\prime}- a_{i}\big{)}x_{i}^{T}x+b\\ &=\sum_{i=1}^{n}\big{(}a_{i}^{\prime}-a_{i}\big{)}\exp\Big{(}-g\big{\|}x_{i}-x _{j}\big{\|}^{2}\Big{)}+b\end{array} , \tag{4}\]  
where \(a_{i}\) and \(a_{i}^{\prime}\) are the Lagrange multipliers, and \(g\) is the kernel function parameter. In this model, \(C\) and \(g\) influence the accuracy of the SVR model's prediction. Determining the model involves selecting appropriate values for \(C\) and \(g\). In the context of machine learning elements--data, model, method, and strategy--the modeling process can be divided into four stages:  
1. Data Collection and Dataset Splitting: Gather and divide the required data into training and validation datasets. 2. Defining Model Inputs, Outputs, and Hyperparameter Set: In this modeling process, the inputs mainly include solar activity indices, month, hour, season, and the TEC value of the same hour in the previous month. The output is the median TEC value at the corresponding time. The hyperparameters are the values of \(C\) and \(g\) in the model. 3. Training the Model Using the SVR Method: Determine the specific hyperparameters \(C\) and \(g\) values. 4.