## 4 Regression Models  
Having determined the ANN-based models, a regression fit was performed with the same selected variables (\(\mathrm{I_{GH}}\), \(\alpha\), and \(\mathrm{Cc}\)) analyzing first and second-order models. A similar  
Figure 4: Ultraviolet irradiance, \(\mathrm{I_{UV}}\), vs. global horizontal irradiance, \(\mathrm{I_{GH}}\). Figure 5: Ultraviolet irradiance, \(\mathrm{I_{UV}}\), vs. solar altitude, \(\alpha\). Figure 6: Ultraviolet irradiance, \(\mathrm{I_{UV}}\), vs. cloud cover, \(\mathrm{Cc}\). Figure 7: Results of \(\mathrm{I_{UV}}=\mathrm{f(I_{GH},\ \alpha,\ \mathrm{Cc})}\), using ANN model 9.  
study was also performed with (\(\rm k_{t}\), \(\rm\alpha\), and \(\rm Cc\)). The models were fitted with 85% of the data to generate the regression models, and the remaining 15% of the data were used to test the models. The data used for fitting and testing the regression models were the same as those used for the ANN models. It should be noted that when using \(\rm I_{BH}\) instead of \(\rm I_{GH}\) or \(\rm k_{t}\), the accuracy of the models worsened significantly, which could be expected a priori in view of the correlation matrix shown in Figure 2. Therefore, these models were not included in the present study. From the data shown in Figure 4 an approximately linear relationship between \(\rm I_{UV}\) and \(\rm I_{GH}\) can be deduced after fitting a first-order model as a function of \(\rm I_{GH}\), \(\rm I_{UV}=0.3676+0.0369\cdot I_{GH}\). It could therefore be considered, as an approximate value, that \(\rm I_{UV}\) was 3.7% of \(\rm I_{GH}\), for the data analyzed within the study period and in the locality of Burgos, Spain. The results with the linear model \(\rm R^{2}=97.32\%\), \(\rm RSME=11.02\%\), and \(\rm nMBE=0.00\%\) were worse than those obtained with more complex models, which included a larger number of variables but had the advantage of simplicity. The following linear regression models given by Equation (9) can be considered as a first option, in the case of preferring greater accuracy, than the linear model with \(\rm I_{GH}\) without using ANNs:  
\[\rm I_{UV}=b_{0}+\sum_{j=1}^{3}b_{j}*x_{j} \tag{9}\]  
Although this means including more independent variables in the model. The use of a second-order regression model, such as the one shown in Equation (10), could also be considered, should greater precision be needed:  
\[\rm I_{UV}=b_{0}+\sum_{j=1}^{3}b_{j}*x_{j}+\sum_{j=1}^{3}b_{j}*x_{j}^{2}+\sum _{k>j}^{3}b_{jk}*x_{j}*x_{k} \tag{10}\]  
Equations (9) and (10) show the first-order and the second-order models to be fitted to the ultraviolet irradiance model, considering \(\rm x_{1}=I_{GH},\ x_{2}=\rm\alpha\) and \(\rm x_{3}=\rm Cc\) both in the first-order and the second-order models. Equation (11) shows the first-order regression model fitted with the above three variables, and Equation (12) also shows the linear regression model fitted only for \(\rm I_{GH}\) and solar altitude, i.e., eliminating cloud cover from the linear-regression model. \[ \begin{array}{l}\rm I_{UV}=-1.9243+0.0323\cdot I_{GH}+6.5892*\alpha+0.0035 \cdot Cc\\ \left[R_{\rm Training}^{2}=98.83\%,\,\rm nRMSE_{\rm Training}=7.29\%,\, \rm nMBE_{\rm Training}=0.00\%\right]\\ \left[R_{\rm Test}^{2}=98.82\%,\,\rm nRMSE_{\rm Test}=7.36\%,\,\rm nMBE_{ \rm Test}=0.11\%\right]\end{array}  \tag{11}\]  
\[\rm I_{UV}=-1.7224+0.0317\cdot I_{GH}+6.9531\cdot\alpha \tag{12}\]  
Equation (13) shows the second-order model, with the three independent variables selected for ANN model 9, fitted to the data that are studied in this work. \[ \begin{array}{l}\rm I_{UV}=-1.8212+0.0167\cdot I_{GH}+19.0462\cdot\alpha+0.0 126\cdot Cc-9.29\cdot 10^{-6}\cdot I_{GH}^{2}+0.0188\cdot I_{GH}\cdot\alpha+8.80\cdot 10^{-5}\\ \cdot I_{GH}\cdot Cc-7.2559\cdot\alpha^{2}-0.0958\cdot\alpha\cdot Cc-6.59\cdot 10^{- 5}\cdot Cc^{2}\\ \left[R_{\rm Training}^{2}=99.43\%,\,\rm nRMSE_{\rm Training}=5.06\%,\, \rm nMBE_{\rm Training}=0.00\%\right]\\ \left[R_{\rm Test}^{2}=99.45\%,\rm nRMSE_{\rm Test}=5.02\%,\,\rm nMBE_{ \rm Test}=0.06\%\right]\end{array}