# Extension of PAR Models under Local All-Sky Conditions to Different Climatic Zones
## 4 Extension of the Models to Other Locations  
The final objective of this work was to verify whether the models obtained with the experimental values obtained in Burgos (Spain) were valid for other climatic zones, for which purpose all seven SURFRAD stations in the USA were selected and the multilinear regression and neural network models were applied with the coefficients obtained for Burgos to check their fit. The seven American stations are at the geographical locations shown in Table 6. The type of climate at each station was also included in this table, according to the Koppen climatic classification [38]: hot desert climate (Bwh); hot-summer humid continental climate (Dfa); warm-summer humid continental climate (Dfb); humid subtropical climate (Cfa); cold semi-arid climate (BSk). The models based on the data collected at Burgos were applied to the seven American meteorological stations and compared with their own primary data. For this purpose, the \(R^{2}\) coefficient and the \(nRSME\) and \(nMBE\) errors were obtained for all stations, both with the MLR and the ANN models for all sky types and for each particular sky type, as shown in Table 7.  
Observing Table 7, it can be seen that the fit values were also very good with an \(R^{2}\) greater than 0.99 for all stations and both the MLR and ANN models. The \(nRMSE\) was always less than 7%, and all \(nMBE\) values showed that all the models overestimated the \(PAR\) values, albeit with minimally higher values. The station with the lowest \(nRMSE\) and the best fit was Penn State, for both the MLR and ANN results. If the results for clear skies are analyzed (Table 8), it can be observed that the fit was also very good with an \(R^{2}\) higher than 0.98 (although slightly lower than the study carried out for all types of sky conditions). In this case, the \(nRMSE\) was lower, and at no station did it exceed 6%. When the study was centred upon partial skies (Table 9), the \(R^{2}\) values were greater than 0.99 and the errors were less than 9.5%. Finally, when the study was performed with overcast skies (Table 10), the \(R^{2}\) value was greater than 0.98 and the errors with MLR were less than 8%. In this case, the \(nRMSE\) at the Desert Rock station where the climate is hot and arid was slightly higher than the rest (11.07%) in the ANN study. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Latitude (\({}^{\circ}\)N)** & **Latitude (\({}^{\circ}\)W)** & **Altitude (m.a.s.l.)** & **Climate** \\ \hline Bondville, Illinois & 40.05192 & 88.37309 & 230 & Dfa \\ Table Mountain, Boulder, Colorado & 40.12498 & 105.2368 & 1689 & Bsk \\ Desert Rock, Nevada & 36.62373 & 116.01947 & 1007 & Bwh \\ Fort Peck, Montana & 48.30783 & 105.1017 & 634 & Bsk \\ Goodwin Creek, Mississippi & 34.2547 & 89.8729 & 6 & Cfa \\ Penn State, Univ. Pennsylvania & 40.72012 & 77.93085 & 376 & Dfb \\ Sioux Falls, South Dakota & 43.73403 & 96.62328 & 473 & Dfa \\ \hline \hline \end{tabular} (m.a.s.l.: meters above sea level).  
\end{table} 
Table 6: Geographical data of SURFRAD weather stations. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \begin{tabular}{c} **Sky** \\ **Conditions** \\ \end{tabular} & \(R^{2}\) & \(nRMSE\) **(\%)** & \(nMBE\) **(\%)** & \begin{tabular}{c} \(Sky\) \\ _Conditions_ \\ \end{tabular} & \(R^{2}\) & \(nRMSE\) **(\%)** & \(nMBE\) **(\%)** \\ \hline All skies & & & & & & \\ (MLR1) & 0.994 & 4.48 & \(-5.20\times 10^{-3}\) & All skies & & & \\ (MLR1) & 0.994 & 4.48 & \(-5.20\times 10^{-3}\) & All skies & & & \\ (MLR1) & & & & & & \\ (MLR2) & 0.992 & 3.00 & \(1.21\times 10^{-1}\) & Clear skies & & & \\ Partial skies & & & & & & \\ (MLR3) & 0.978 & 6.72 & \(-6.02\times 10^{-2}\) & \
\begin{tabular}{c} Partial skies \\ (ANN3) \\ \end{tabular} & 0.978 & 6.71 & \(-5.26\times 10^{-2}\) \\ Overcast skies (MLR4) & 0.982 & 6.63 & \(-4.84\times 10^{-2}\) &
\begin{tabular}{c} Overcast \\ skies (ANN4) \\ \end{tabular} & 0.982 & 6.59 & \(-5.10\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Validation of MLR and ANN models.