# Extension of PAR Models under Local All-Sky Conditions to Different Climatic Zones
## 3 Methodology
### Artificial Neural Network Model  
Another \(\mathit{PAR}\) estimation method used an ANN trained with the Levenberg-Marquardt Back-Propagation (LMBP) algorithm [37]. In this case, a 3-layer configuration was chosen: the input layer, with each MI for each model, the hidden layer with the information processing centres (neurons), and the output layer, with the results. The explanation of the iterative process and the fitting are explained elsewhere [32].  
For this purpose, four networks were developed and tested in this work, one for all skies and another three for each sky type, based on each selected MI with its previously determined Pearson's correlation coefficient shown in Table 2. The R\({}^{2}\), the \(nMBE\), and the \(nRMSE\) statistics, previously defined above, were used for the validation of the models. Table 4 summarizes the statistical results of each ANN. It can be seen that a very good fit was obtained for each ANN with an \(R^{2}>0.97\) and \(nRMSE<8\%\).  
A check was performed with 15% of the values not used in the generation of the models to validate the results of both the MLR and ANN models, which confirmed the good fit of the models for all sky types, both for the regressions and the neural networks. The results, presented below in Table 5, show the MLR results on the left and ANN results on the right.  
As can be seen from the data in the table, both \"data treatment\" methods are optimal, as the \(R^{2}\) is greater than 0.97 and the errors are less than 7%. The best-fitting model in both cases is the one applied to clear skies with fit values of 0.99 and mean square errors of less than 3%.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sky Conditions** & \(\mathit{R}^{2}\) & \(\mathit{nRMSE\%}\) & \(\mathit{nMBE\%}\) \\ \hline All skies (ANN1) & 0.994 & 4.22 & \(2.84\times 10^{-3}\) \\ Clear skies (ANN2) & 0.992 & 3.01 & \(-4.68\times 10^{-2}\) \\ Partial skies (ANN3) & 0.977 & 6.80 & \(4.06\times 10^{-3}\) \\ Overcast skies (ANN4) & 0.978 & 7.28 & \(-3.50\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical results of the ANN models.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sky Conditions** & **Multilinear Regression Model** & \(\mathit{R}^{2}\) & \(\mathit{nRMSE\%}\) & \(\mathit{nMBE\%}\) \\ \hline All skies (MLR1) & \(\mathit{PAR}=12.12+0.40\cdot\mathit{RaGH}+\) & 0.994 & 4.37 & \(-2.74\times 10^{-13}\) \\  & \(15.74\cdot\mathit{cosZ}-114.4\cdot\mathit{kt}-10.64\cdot\mathit{kd}-0.47\cdot \varepsilon\) & & & \\ \hline Clear skies (MLR2) & \(\mathit{PAR}=-18.12+0.33\cdot\mathit{RaGH}+\) & & & \\  & \(83.15\cdot\mathit{cosZ}+24.19\cdot\mathit{kt}+0.71\ \varepsilon\) & & & \\ \hline Partial skies (MLR3) & \(\mathit{PAR}=-1.81+0.40\cdot\mathit{RaGH}+13.75\cdot\mathit{cosZ}\) & 0.977 & 6.80 & \(-5.32\times 10^{-13}\) \\ \hline Overcast skies (MLR4) & \(\mathit{PAR}=-0.03+0.42\cdot\mathit{RaGH}+6.88\cdot\mathit{cosZ}+\) & & & \\  & \(1.58\cdot\mathit{kt}-6.12\cdot\Delta\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Multilinear regression models of \(\mathit{PAR}\).