## 4 Results and Discussion  
The remaining 15% of the data, not used previously for generating the MLR and ANN models, were used as a validation set for both MLR and ANN models. The results are shown in Table 6, with MLR outcomes on the left and ANN results on the right. The best results were obtained for clear skies (R\({}^{2}>0.97\), and nRMSE \(<12.4\%\)). Both nRMSE and nMBE values obtained through multilinear regressions exceeded 20% in all cases, results \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sky Types** & **MLR Model** & **R\({}^{2}\)** & **nRSME(\%)** & **nMBE (\%)** \\ \hline All skies & \(\rm GHUVE=1.90\cdot 10^{-2}+235\cdot 10^{-4}GHI-1.74\cdot 10^{-2}\theta_{z}-9.85\)- & 0.937 & 25.60 & \(-2.05\cdot 10^{-1212}\) \\  & \(\rm 10^{-2}k_{x}+7.44\cdot 10^{-4}T\) & 0.926 & 29.80 & 6.92\(\cdot 10^{-13}\) \\ Overcast & \(\rm GHUVE=2.98\cdot 10^{-2}+1.61\cdot 10^{-4}GHI-2.71\cdot 10^{-2}\theta_{z}\) & 0.926 & 29.80 & 6.92\(\cdot 10^{-13}\) \\  & \(\rm+1.30k_{d}-9.89\cdot 10^{-4}DHI\) & 0.934 & 22.49 & 3.23\(\cdot 10^{-12}\) \\ Intermediate & \(\rm GHUVE=8.54\cdot 10^{-2}+2.59\cdot 10^{-4}GHI-6.98\cdot 10^{-2}\theta_{z}-1.80\cdot \\  & \(\rm 10^{-4}DHI-9.91\cdot 10^{-5}DNI\) & 0.940 & 21.10 & 9.87\(\cdot 10^{-13}\) \\ Clear & \(\rm GHUVE=1.75\cdot 10^{-1}+5.84\cdot 10^{-5}GHI-1.52\cdot 10^{-1}\theta_{z}+6.89\cdot \\  & \(\rm 10^{-4}T\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Multilinear regression (MLR) models of \(\rm GHUVE\) and goodness of fit (based on 85% of data). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sky Condition** & **R\({}^{2}\)** & **nRMSE** (\%) & **nMBE** (\%) \\ \hline All skies & & & \\ (MLR1) & 0.988 & 14.36 & 6.69 \(\cdot 10^{-2}\) \\ Overcast & 0.962 & 21.32 & 6.69 \(\cdot 10^{-2}\) \\ Intermediate & 0.955 & 18.68 & 9.09 \(\cdot 10^{-2}\) \\ Clear & 0.981 & 11.92 & 7.59 \(\cdot 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Goodness of fit of the ANN models (based on 85% of data). close to those obtained by other authors [2] who performed models using second-degree polynomials, obtaining nRMSE values between 20% and 54%. Previous works modeled hourly [31] and daily [34; 35; 36; 37] GHUVE values through ANN obtaining nRMSE ranging from 14 and 21%. Considerable improvement in performances occurred when the highly influential parameter TOC was introduced as ANN input. Table 6 comparison revealed that ANN models performed better than MLR models. For all skies and clear skies, the nRMSE value improved significantly, decreasing over 40% with respect to the MLR models. The nMBE value was overestimated for intermediate and clear skies. Figures 8 and 9 compare the modeled GHUVE values from derived MLR models and ANN models, respectively, and the corresponding experimental GHUVE measurements obtained from the Burgos meteorological station using the referred validation data set. A good determination coefficient was obtained (R\({}^{2}>\) 0.92, MRL models, R\({}^{2}>\) 0.95 ANN models) both for all skies, overcast, intermediate, and clear sky conditions. To improve the previous results, a new approach was introduced to optimize the number of variables for minimizing nRMSE using ANN models for all-sky conditions. The number of neurons in the hidden layer can be adjusted accordingly to accommodate the complexity and dimensionality of the input data, from one neuron to the number of input variables of the ANN. The two variables identified as strongly correlated to GHUVE based on [PERSON]'s coefficient, as shown in Table 3, \(\mathrm{GHI}\) and \(\theta_{x}\), were used as reference in this study. The ANN with two neurons in the hidden layer, with \(\mathrm{GHI}\) and \(\theta_{x}\) as input variables presented nRMSE of 17.02%, as shown in Table 7. By retaining \(\mathrm{GHI}\) and \(\theta_{x}\) as input variables, new ANN models were generated, increasing one by one the number of variables included in Table 2 and, consequently, the number of neurons in the hidden layer, from one neuron to as many neurons as the number of variables in each case.