# A Machine Learning-Based Method for Modeling TEC Regional Temporal-Spatial Map
## 2 Modeling
### Parameter Determination  
#### 2.4.1 Temporal Model Determination  
Figure 8 shows the correlation coefficients between the observed and predicted TECs with \(F10.7_{12}\), \(R_{12}\), and dual-parameter (\(F10.7_{12}\)+\(R_{12}\)) regression for 24 h in the specific year of 2008, 2011, and 2014. It can be seen that the predicted TECs obtained by dual-parameter regression are highly correlated with the observed TECs compared with those obtained by single-parameter regression of \(F10.7_{12}\) and \(R_{12}\). And the correlation coefficients of dual-parameter regression are closer to 1, which further confirms the correctness of introducing the dual-parameter into the model. In particular, the correlation coefficients are higher during periods of high solar activity than during other periods of solar activity, which has also been mentioned in [69; 70]. Figure 9 shows the temporal regression results under different solar activity parameters, and the RMSEs and RRMSEs with \(F10.7_{12}\), \(R_{12}\), and the dual-parameter. It is easy to see that the RRMSE of temporal regression with dual-parameter at the same time is less than that of single-parameter regression under the same conditions. Moreover, the regression results coincide with the TEC observations, in which some outliers were filtered out. Therefore, we simultaneously introduce \(F10.7_{12}\) and \(R_{12}\) into the temporal regression model. Figure 8: The correlation coefficients between the observed and predicted TECs with \(F10.7_{12}\), \(R_{12}\), and dual-parameter (\(F10.7_{12}\)+\(R_{12}\)) regression for 24 h in the specific year of 2008, 2011, and 2014: (**a**) regression result using \(F10.7_{12}\) in 2008; (**b**) regression result using \(R_{12}\) in 2008; (**c**) regression result using \(F10.7_{12}\) and \(R_{12}\) in 2008; (**d**) regression result using \(F10.7_{12}\) in 2011; (**e**) regression result using \(R_{12}\) in 2011; (**f**) regression result using \(F10.7_{12}\) and \(R_{12}\) in 2011; (**g**) regression result using \(F10.7_{12}\) in 2014; (**h**) regression result using \(R_{12}\) in 2014; (**i**) regression result using \(F10.7_{12}\) and \(R_{12}\) in 2014. Figure 10a shows the RRMSE of TEC prediction when \(K\) and \(L\) take different values in the case of dual-parameter is introduced into the temporal map. When \(L\) is fixed, the value of \(K\) goes from 1 to 2 and then to 3, and the regression error decreases significantly, whereas \(K\) goes from 3 to 4, and the regression error does not change significantly. Therefore, we take \(K\) equal to 3 as the upper bound of the time cycle. When \(K\) is 3, the dual-parameter regression error corresponding to different \(L\) and the time required for regression are shown in Figure 10b. \(L\) increased from 3 to 5, the corresponding regression error decreased significantly, and regression time has not increased dramatically. In contrast, \(L\) rose from 5 to 7, the corresponding regression error decreased slightly, and the regression time increased sharply. Therefore, considering the regression error and regression time comprehensively, we take \(L\) equal to 5. Figure 10: Regression analysis results in the case of dual-parameter (\(F101.7_{12}+R_{12}\)): (**a**) regression error against the upper bound of the time cycle; (**b**) regression error and modeling time corresponding to different \(L\) values when \(K\) is 3. Figure 9: Regression results under different solar activity parameters: (**a**) comparison of TEC regression curves of the Roquetes station; (**b**) comparison of TEC regression curves of the Rome station; (**c**) TEC regression error of the Roquetes station; (**d**) TEC regression error of the Rome station. In conclusion, the temporal map (2) of TEC can be written as:  
\[ \begin{array}{l}\text{TEC}=\hat{\text{TEC}}(p,r,m,t)\\ =\sum\limits_{k=0}^{3}\sum\limits_{l=0}^{5}[c_{k,l}^{p}p^{l}\cdot\cos(2\pi km/12 )+s_{k,l}^{p}p^{l}\cdot\sin(2\pi km/12)+,\\ c_{k,l}^{r}r^{l}\cdot\cos(2\pi km/12)+s_{k,l}^{r}r^{l}\cdot\sin(2\pi km/12)]. \end{array}