## 4 Regression Models
 \tag{13}\]A simpler model is shown in Equation (14), in which cloud cover is removed as an independent variable, which may be useful if cloud-cover data are unavailable. \[ \begin{array}{l}\mathrm{I}_{\mathrm{UV}}=-1.1625+0.0238\cdot\mathrm{I}_{\mathrm{ GH}}+10.2243\cdot\alpha-1.27\cdot 10^{-5}\cdot\mathrm{I}_{\mathrm{GH}}^{2}+0.0253\cdot \mathrm{I}_{\mathrm{GH}}\cdot\alpha-8.8361\cdot\alpha^{2}\\ \left[\mathrm{R}_{\mathrm{Training}}^{2}=99.30\%,\,\mathrm{nRMSE}_{\mathrm{ Training}}=5.63\%,\,\mathrm{nMBE}_{\mathrm{Training}}=0.00\%\right]\\ \left[\mathrm{R}_{\mathrm{Test}}^{2}=99.32\%,\,\mathrm{nRMSE}_{\mathrm{Test} }=5.60\%,\,\mathrm{nMBE}_{\mathrm{Test}}=0.07\%\right]\end{array}  \tag{14}\]  
A study was also conducted on the use of \(\mathrm{k}_{\mathrm{t}}\) instead of \(\mathrm{I}_{\mathrm{GH}}\). In this case, the values are shown below in Equations (15)-(18). \[\mathrm{I}_{\mathrm{UV}}=-11.3956+22.8218\cdot\mathrm{k}_{\mathrm{t}}+24.3899 \cdot+0.0021\cdot\mathrm{Cc} \tag{15}\]  
\[ \begin{array}{l}\left[\mathrm{R}_{\mathrm{Training}}^{2}=93.18\%,\,\mathrm{ nRMSE}_{\mathrm{Training}}=17.57\%,\,\mathrm{nMBE}_{\mathrm{Training}}=0.00\%\right]\\ \left[\mathrm{R}_{\mathrm{Test}}^{2}=93.01\%,\,\mathrm{nRMSE}_{\mathrm{ Test}}=17.95\%,\,\mathrm{nMBE}_{\mathrm{Test}}=0.11\%\right]\\ \mathrm{I}_{\mathrm{UV}}=-11.1581+22.5424\cdot\mathrm{k}_{\mathrm{t}}+24.4465 \cdot\alpha\\ \left[\mathrm{R}_{\mathrm{Training}}^{2}=93.18\%,\,\mathrm{nRMSE}_{\mathrm{ Training}}=17.57\%,\,\mathrm{nMBE}_{\mathrm{Training}}=0.00\%\right]\\ \left[\mathrm{R}_{\mathrm{Test}}^{2}=93.01\%,\,\mathrm{nRMSE}_{\mathrm{ Test}}=17.95\%,\,\mathrm{nMBE}_{\mathrm{Test}}=0.11\%\right]\\ \mathrm{I}_{\mathrm{UV}}=-0.3544+2.0833\cdot\mathrm{k}_{\mathrm{t}}+14.0893\cdot \alpha-0.0165\cdot\mathrm{Cc}-9.0813\cdot\mathrm{k}_{\mathrm{t}}^{2}+40.7916 \cdot\mathrm{k}_{\mathrm{t}}\cdot\alpha\\ +0.0405\cdot\mathrm{k}_{\mathrm{t}}\cdot\mathrm{Cc}-7.9650\cdot\alpha^{2}-0.0170 \cdot\alpha\cdot\mathrm{Cc}-8.8310^{-5}\cdot\mathrm{Cc}^{2}\\ \left[\mathrm{R}_{\mathrm{Training}}^{2}=99.39\%,\,\mathrm{nRMSE}_{ \mathrm{Training}}=5.26\%,\,\mathrm{nMBE}_{\mathrm{Training}}=0.00\%\right]\\ \mathrm{R}_{\mathrm{Test}}^{2}=99.40\%,\,\mathrm{nRMSE}_{\mathrm{Test}}=5.24 \%,\,\mathrm{nMBE}_{\mathrm{Test}}=0.08\%\right]\end{array}  \tag{17}\]  
Analogously to \(\mathrm{I}_{\mathrm{GH}}\), the model in Equation (17), in which cloud cover as an independent variable is removed, is useful if the cloud-cover data are unavailable. \[ \begin{array}{l}\mathrm{I}_{\mathrm{UV}}=-2.7058+7.3586\cdot\mathrm{k}_{ \mathrm{t}}+11.4945\cdot\alpha-10.8880\cdot+42.6023\cdot\mathrm{k}_{\mathrm{ t}}\cdot\alpha-7.7515\\ \cdot\alpha^{2}\\ \left[\mathrm{R}_{\mathrm{Training}}^{2}=99.30\%,\,\mathrm{nRMSE}_{\mathrm{ Training}}=5.63\%,\,\mathrm{nMBE}_{\mathrm{Training}}=0.00\%\right]\\ \left[\mathrm{R}_{\mathrm{Test}}^{2}=99.32\%,\,\mathrm{nRMSE}_{\mathrm{ Test}}=5.62\%,\mathrm{nMBE}_{\mathrm{Test}}=0.08\%\right]\end{array}  \tag{18}\]  
It can be observed that when quadratic terms are included in the regression models, the models containing \(\mathrm{k}_{\mathrm{t}}\) instead of \(\mathrm{I}_{\mathrm{GH}}\) led to similar results, with the models containing \(\mathrm{I}_{\mathrm{GH}}\) yielding somewhat better results. Therefore, in the case of using a second-order model, it may be preferable to use \(\mathrm{k}_{\mathrm{t}}\) in the formulation, as it gives an idea of the clearness of the sky, and the results are similar to using \(\mathrm{I}_{\mathrm{GH}}\). The same is not true when using linear models, where the accuracy of considering \(\mathrm{k}_{\mathrm{t}}\) in the formulation worsens significantly, as can be seen when comparing Equations (15) and (16) with (11) and (12), which correspond to the first order models for \(\mathrm{k}_{\mathrm{t}}\) and \(\mathrm{I}_{\mathrm{GH}}\), respectively. Therefore, the use of first-order models makes it preferable to introduce \(\mathrm{I}_{\mathrm{GH}}\) in the formulation, as the models with \(\mathrm{k}_{\mathrm{t}}\) yielded less accurate results.