## 3 Results
### SARS-CoV-2 Time Series Results  
The research also focuses on analyzing the daily time series data of SARS-CoV-2 cases in three different countries and making predictions specifically for the first quarter of 2021.  
In the case of Spain, two neural networks, NAR and NAR-SVR, were utilized with a structure consisting of nine inputs in the input layer and six nodes in the hidden layer. The backpropagation algorithm was employed, using both sigmoidal and linear activation functions with a learning coefficient \(\eta\) of 0.1. For the NAR-SVR model, an SVR with a Gaussian activation function was employed, with the following parameters: \(\gamma=0.143\), \(\epsilon=0.1\), and \(C=1\).  
For the dataset of daily SARS-CoV-2 cases in Italy, two neural networks were employed with a structure of 8 inputs in the input layer and 10 nodes in the hidden layer. Similar to the previous case, sigmoidal and linear activation functions were used with a learning coefficient \(\eta\) of 0.1. The ML part of the NAR-SVR model utilized an SVR with a Gaussian activation function, along with the parameters \(\gamma=0.143\), \(\epsilon=0.1\), and \(C=1\).  
Lastly, for the dataset of daily cases in Turkey, the neural networks had a structure of 13 inputs and 11 nodes in the hidden layer. Sigmoidal and linear activation functions were employed, along with a learning coefficient \(\eta\) of 0.1 in both neural networks. The NAR-SVR model utilized an SVR with a Gaussian activation function and the following parameters: \(\gamma=0.5\), \(\epsilon=0.1\), and \(C=1\).  
The results obtained from the NAR and NAR-SVR networks were compared with other prediction techniques, including AR (1), ARIMA (1,1,1), and SVR models with Gaussian activation functions and varying numbers of inputs. The prediction results for all the SARS-CoV-2 time series data are presented in Table 2. It is evident that the corrected NAR-SVR network outperforms the NAR network and the other prediction techniques. It demonstrates superior adaptation to the observed variable, resulting in a significant decrease in RMSE and MAE values. Notably, the traditional models employed in the series from Spain and Italy tend to yield unsatisfactory results due to their limited ability to adapt to the real values of the time series.  
The comparison between the NAR and NAR-SVR models is represented in Figure 3.  
\begin{table}
\begin{tabular}{c c c c} \hline
**COVID-19 Cases in Spain** & \(R^{2}\) & **MAE** & **RMSE** \\ \hline NAR & 0.297 & 11,039.31 & 16,740.69 \\ NAR-SVR & 0.648 & 7852.59 & 11,840.78 \\ AR(1) & 0.000 & 12,444.86 & 20,378.32 \\ ARIMA(1,1,1) & 0.155 & 12,752.84 & 18,358.76 \\ SVR & 0.242 & 10,911.04 & 17,388.00 \\ \hline
**COVID-19 Cases in Italy** & \(R^{2}\) & **MAE** & **RMSE** \\ \hline NAR & 0.583 & 2614.85 & 3235.67 \\ NAR-SVR & 0.727 & 1787.28 & 2619.95 \\ AR(1) & 0.585 & 2618.05 & 3229.73 \\ ARIMA(1,1,1) & 0.583 & 2608.97 & 3235.73 \\ SVR & 0.688 & 2223.86 & 2800.11 \\ \hline
**COVID-19 Cases in Turkey** & \(R^{2}\) & **MAE** & **RMSE** \\ \hline NAR & 0.966 & 971.63 & 1380.42 \\ NAR-SVR & 0.970 & 939.97 & 1332.71 \\ AR(1) & 0.953 & 1079.75 & 1637.19 \\ ARIMA(1,1,1) & 0.953 & 1146.95 & 1631.25 \\ SVR & 0.944 & 1382.90 & 1775.86 \\ \hline \end{tabular}
\end{table}
Table 2: Results for COVID-19 cases time series, 1 January 2021/31 March 2021.