# A New Index Assessing the Viability of PAR Application Projects Used to Validate PAR Models
## 2 Materials and Methods
The statistics used to estimate the goodness of the models developed were the slope and the intercept of the linear regression between measured and modeled \(PAR\), the mean bias error (MBE), the root mean square error (RMSE), and the determination coefficient (R\({}^{2}\)), whose expressions are shown from Equation (7) to Equation (11). \[Slope=\frac{\sum\limits_{i=1}^{n}{(PAR_{Model}-\overline{PAR}_{Model})\left( PAR_{Measured}-\overline{PAR}_{Measured}\right)}}{\sum\limits_{i=1}^{n}{ (PAR_{Measured}-\overline{PAR}_{Measured})}^{2}} \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Multilinear Regression** & **ANN** \\ \hline Model 1 & \(GHI\) & \(GHI\) \\ Model 2 & \(GHI\), \(k_{t}\) & \(GHI\), \(k_{t}\) \\ Model 3 & \(GHI\), \(k_{t}\), \(\cos\theta\) & \(GHI\), \(k_{t}\), \(\cos\theta\) \\ Model 4 & \(GHI\), \(RH\), \(T\) & \(GHI\), \(RH\), \(T\) \\ Model 5 & \(GHI\), \(RH\), \(T\), \(k_{t}\), \(\cos\theta\) & \(GHI\), \(RH\), \(T\), \(k_{t}\), \(\cos\theta\) \\ Model 6 & \(k_{t}\)-\(\cos\theta\) & \(k_{t}\)-\(\cos\theta\) \\ Model 7 & \(k_{t}\)-\(\cos\theta\), \(GHI\) & \(k_{t}\)-\(\cos\theta\), \(GHI\) \\ Model 8 & \(k_{t}\)-\(\cos\theta\), \(RH\), \(T\) & \(k_{t}\)-\(\cos\theta\), \(RH\), \(T\) \\ Model 9 & \(k_{t}\)-\(\cos\theta\), \(GHI\), \(RH\), \(T\) & \(k_{t}\)-\(\cos\theta\), \(GHI\), \(RH\), \(T\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Set of variables used in each model. \[Intercept=\overline{PAR}_{Model}-Slope\cdot\overline{PAR}_{Measured} \tag{8}\]  
\[MBE=\frac{1}{n}\sum_{i=1}^{n}(PAR_{Model}-PAR_{Measured}) \tag{9}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(PAR_{Model}-PAR_{Measured})^{2}} \tag{10}\]  
\[R^{2}=\frac{\sigma_{PAR-Mod\ PAR-Maa}^{2}}{\sigma_{PAR-Mod}^{2}\sigma_{PAR-Maa} ^{2}} \tag{11}\]  
where \(n\) is the number of recordings, \(\sigma_{PAR-Mod}\) is the variance of the modeled _PAR_, \(\sigma_{PAR-Max}\) is the variance of the measured _PAR_, and \(\sigma_{PAR-Mod\ PAR-Maa}\) is the covariance of both modeled and measured _PAR_. Additionally, to estimate the goodness of the models from the point of view of their representativeness when assessing the viability of _PAR_ projects, a new index was defined. Indeed, the new index--called the Exceedance Probability Index (EPI)--is based on the exceedance probabilities, which are widely utilized to evaluate the viability of solar power plant projects. The proposed EPI estimates the difference between the exceedance probabilities for the modeled and measured variable (_PAR_ could be replaced by any other modeled variable) as Equation (12) indicates. This difference is normalized dividing by the exceedance probability of the measured variable. According to this expression, when the exceedance probabilities of the measured and modeled values are close, EPI tends to zero. \[EPI=\frac{\int\limits_{ \begin{subarray}{c}PAR_{T}=PAR_{\min}\end{subarray} } ^{PAR_{T}=PAR_{\max}}abs(P(PAR_{M}>PAR_{T})-P(PAR_{O}>PAR_{T}))}{\int\limits_{ \begin{subarray}{c}PAR_{T}=PAR_{\max}\end{subarray} }abs(P(PAR_{O}>PAR_{T}))} \tag{12}\]  
where \(PAR_{T}\) is a threshold _PAR_ value, \(PAR_{\min}\) is the lower _PAR_ value, \(PAR_{\max}\) is the higher _PAR_ value, \(PAR_{M}\) is the modeled _PAR_, and \(PAR_{O}\) is the measured _PAR_.