## 3 Artificial Neural Network Models  
A topology consisting of an input layer, a hidden layer, and an output layer was considered for the ANN, and the Levenberg-Marquardt algorithm was used to adjust both the weights and the bias of the network, in accordance with Equation (8). The transfer function in the hidden layer, f\({}_{1}\), was sigmoidal, and f\({}_{2}\) in the output layer was a linear function. W\({}_{1}\) and b\({}_{1}\) were the weights and biases of the hidden layer, and W\({}_{2}\) and b\({}_{2}\) were the weights and biases of the output layer, respectively. \[\mathrm{I}_{\mathrm{UV}}=\mathrm{f}_{2}(\mathrm{W}_{2}*\mathrm{f}_{1}( \mathrm{W}_{1}*\mathrm{Inputs}+\mathrm{b}_{1})+\mathrm{b}_{2}) \tag{8}\]  
All the meteorological variables indicated in Table 1 may be used to develop ANN-based models. As indicated above, I\({}_{\mathrm{GH}}\) is the variable with the greatest influence on I\({}_{\mathrm{UV}}\). The Pearson correlation coefficient is 0.99, as shown in Figure 2. Therefore, I\({}_{\mathrm{GH}}\) must be included as a first input of the model. However, some of the variables are correlated between each other and their inclusion together in the ANN model increased the complexity of the model without significantly improving accuracy, both for the nRMSE and the nMBE values. As can be seen in Table 2, the Pearson coefficient r(I\({}_{\mathrm{GH}}\)variable) highlighted a high correlation between I\({}_{\mathrm{GH}}\) and k\({}_{\mathrm{t}}\). On the other hand, as can be observed in Figure 2, a high correlation exits between I\({}_{\mathrm{BH}}\), and D. To determine the optimal neural network, the goodness of fit of the models obtained with the combination of inputs shown in Table 3 was analyzed, using the previously defined indicators, nRMSE, and nMBE. In each combination, one of the variables was eliminated or substituted. For the analysis, the experimental data set was divided into three groups: 70% was used as a training set, 15% as a validation set, and the remaining 15% was used to test the model. The three (training, validation and test) data sets were the same in all the models that were developed. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Variable:** & \(\mathbf{k_{t}}\) & \(\mathbf{\alpha}\) & \(\mathbf{I_{BH}}\) & \(\mathbf{D}\) & \(\mathbf{\varepsilon}\) & T & **Cc** \\ \hline r(I\({}_{\mathrm{GH}}\),Variable): & 0.83 & 0.75 & 0.74 & \(-\)0.71 & 0.66 & 0.56 & \(-\)0.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Pearson correlation coefficient r(I\({}_{\mathrm{GH}}\), variable) for detecting redundant information. Figure 3: The Pearson correlation coefficient, r(I\({}_{\mathrm{UV}}\), variable), between I\({}_{\mathrm{UV}}\) and the meteorological variables considered in the study. Table 3 shows the 18 ANN models developed with the nRMSE, nMBE, and the corresponding R\({}^{2}\) values obtained for the data used in both the training and in the test models. The results shown in Table 3 confirmed that the inclusion of meteorological variables with a high Pearson correlation coefficient in no way improved the accuracy of the ANN. On the one hand, considering I\({}_{\rm GH}\), \(\alpha\), \(\varepsilon\), T, Cc, ANN model number 5 might be more appropriate than models 1 to 4 listed in Table 3. On the other hand, eliminating solar height had a negative influence on the values of the statistics under analysis. Therefore, solar height was left as an independent variable in the models. Likewise, worse results were observed after removing cloud cover and leaving temperature. Based on the above, the model to be used was model number 9, from Table 3, as that model presented a reduced number of inputs (I\({}_{\rm GH}\), \(\alpha\), Cc) and yielded adequate nRMSE and nMBE values. In addition, it was observed that model 9, which uses I\({}_{\rm GH}\), yielded a somewhat lower nRMSE than model 13, obtained using k\({}_{\rm t}\) instead of I\({}_{\rm GH}\) and with the two other above-mentioned variables (\(\alpha\), Cc).