# A New Index Assessing the Viability of PAR Application Projects Used to Validate PAR Models
## 2 Materials and Methods
Modified Akima's piecewise cubic Hermite interpolation [55; 56] helped to produce hourly estimates based on MODIS data. Then, this estimated hourly dataset served to train different types of models. The proposed methodology is focused on developing regional models. With that purpose, the use of a clustering of the field of study contributed to constructing models for a whole region. This methodology employs a clustering of \(k_{PAR}\), defined as the ratio between global extraterrestrial \(PAR\) (\(PAR_{0}\)) and \(PAR\) reaching the Earth's surface, to separate the territory of study into two areas with similar characteristics. A clustering served to establish the regions with similar behavior of \(PAR\). The clustering was applied to \(k_{PAR}\), defined in Equation (5). \[k_{PAR}=\frac{PAR}{PAR_{0}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Station** & **Abbreviation** & **Latitude (\({}^{\circ}\), +N)** & **Longitude (\({}^{\circ}\), +E)** \\ \hline Albacete—ITAP & ALB & 39.04 & \(-\)2.08 \\ Córdoba—IFAPA & COR & 37.86 & \(-\)4.80 \\ Lugo—USC & LUG & 43.00 & \(-\)7.54 \\ Salamanc—CIALE & SAL & 40.98 & \(-\)5.72 \\ Villaviciosa—SERIIDA & VLL & 43.48 & \(-\)5.44 \\ Vitoria—NEIKER & VIT & 42.85 & \(-\)2.62 \\ Zaragoza—AULA DEI & ZAR & 41.73 & \(-\)0.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Localization of \(PAR\) stations. where the \(PAR_{0}\) value was estimated as 39.8% of \(G_{0}\) according to [57; 58]. The clustering technique employed was the k-means method, which consists of minimizing the sum of squared distances between the centroid of the group and each object of this group. There are many clustering methods, but k-means represents a widely used method, and its results over mainland Spain fitted well with the orographic and climatological characteristics of the territory. The clustering divided the field of study into two regions with similar behavior of \(k_{PAR}\). Once the regions with similar behavior were set, the development of both multilinear regression and ANN models for each region begun. The models were prepared using data provided by MODIS and ECMWF from all the points of the grid belonging to each region and subsequently validated with data provided by the stations located in the corresponding region. The set of variables used in each model was the same for both multilinear regression and ANN models, as Table 2 explains. Combinations of sets of variables that are commonly used in the bibliography were selected as \(GHI\), \(RH\), \(T\), \(k_{t}\), \(\cos\theta\), and the product \(k_{t}\)-\(\cos\theta\). Examples of models that utilize these variables can be found in [25; 26; 32; 33; 34; 38; 39; 40; 41; 59; 60; 61; 62; 63]. The aim making different combinations of them is to produce different models so that we can compare them and choose which model is the best and which variables are more interesting for \(PAR\) modeling. In the first five models, combinations of these variables always including \(GHI\) were utilized, since the presence of this variable is the key to obtaining good results according to [25]. However, in the last four models, the combination of variables always included the product of \(k_{t}\) and \(\cos\theta\), as this product obtained promising results in [41]. Afterward, a site-adaptation technique, used normally to correct models using local data, was utilized to improve the results [25; 64; 65; 66]. To perform a site-adaptation correction, Equation (6) was applied. \[y_{new\ mod}=y_{\text{mod}}-[(a-1)x_{local}+b] \tag{6}\]  
where \(y_{new\ mod}\) is the corrected model value, \(y_{mod}\) is the value of the original model, \(x_{local}\) is a local measured value, and the coefficients \(a\) and \(b\) are the coefficients obtained in a linear fitting between measured and modeled values.