## 5 Performance Assessment
### Statistical Indicators  
The models' accuracy was evaluated in terms of three statistical indicators: the determination coefficient \(R^{2}\), the normalized root mean square error \(nRMSE\), and the normalized mean bias error \(nMBE\). The indicators are defined in Appendix B.  
Figure 3: Density plot of diffuse fraction \(k_{d}\) vs. the clearness index \(k_{t}\) at the station in Payerne, Switzerland, in 2020. Measured and estimated values of \(k_{d}\) with the tested separation models are displayed. First, we look closer at the performance of the M1 and M2 models in estimating DNI. For M1, the determination coefficient \(R^{2}\) falls between 0.745 and 0.949. The minimum value of \(R^{2}\) is obtained at the arid climate station in Tamanrasset, Algeria. The maximum value of \(R^{2}\) is achieved at Payerne, Switzerland, with a temperate continental climate. This was expected, with Payerne being the origin location of M1. Similarly, for M2, the determination coefficient \(R^{2}\) falls between 0.720 and 0.937. The minimum and maximum values were also achieved at Tamanrasset and Payerne, respectively. This means that the reduction by half of M1's predictor number roughly keeps the fraction of variance in D_TEST explained by M1. The models' performance in estimating _DNI_ from the diffuse fraction estimates through Equations (2) and (3) is displayed in Figure 4a in terms of \(nRMSE\). M1 achieves the best performance at the arid station in Gobabeb, Namib Desert, Namibia (\(nRMSE=14.0\%\)), while the worst performance is reached at Lindenberg, Germany, a station with a temperate climate (\(nRMSE=41.2\%\)). At the different stations in D_TEST, M1 exhibits both positive and negative biases. The minimum \(nMBE\) was achieved at the tropical station on Cocos Island (\(nMBE=0.19\%\)). The highest positive bias is experienced at the continental station in Regina, Canada (\(nMBE=15.3\%\)), while the highest negative bias is experienced at the temperate continental station in Boulder, USA (\(nMBE=-13.9\%\)). M2 achieves the best performance at Gobabeb (\(nRMSE=14.8\%\)) and the worst at Lindenberg (\(nRMSE=48.1\%\)). In terms of \(nMBE\), M2 achieves the best result at the temperate station in Tateno, Japan (\(nMBE=0.40\%\)). Similarly to M1, M2 experiences the highest positive bias at Regina (\(nMBE=19.0\%\)) and the highest negative bias at the station in Boulder (\(nMBE=-12.0\%\)). At first glance, the accuracy of M1 and M2 seems to be relatively low. However, this is not the case. With all the progress registered by the separation models, there is plenty of room for improvement. Visual inspection of Figure 4 reveals high values of \(nRMSE\) when the reference models were applied to estimate _DNI_: \(nRMSE\) ranges between 15.2% and  
Figure 4: Models’ accuracy is evaluated in terms of \(nRMSE\) at every station from D_TEST when the solar irradiance components (**a**) direct normal _DNI_ and (**b**) diffuse DHI are estimated. 49.4%, both values being reached by S3. A large bias is also present, with \(nMBE\) taking values between \(-19.0\%\) (reached by E2) and \(+18.2\%\) (reached by PB). Secondly, the models' performance in estimating \(DHI\) from \(k_{d}\) estimates through Equation (2) was tested. The results are summarized in Figure 4b in terms of \(nRMSE\). Overall, the \(nRMSE\) range is wider than in the case of \(DNI\), making the distinction between the models more visible. The determination coefficient \(R^{2}\) of M1 falls between 0.428 and 0.904. The minimum value is reached at the arid climate station in Solar Village, Saudi Arabia. The maximum value is achieved at the continental climate station in Toravere, Estonia. The determination coefficient \(R^{2}\) of M2 falls between 0.427 and 0.892. The minimum value is reached at the arid climate station in Tamanrasset, Algeria. The maximum value is also achieved at Toravere. In terms of \(nRMSE\), M1 achieves the best result at the temperate station in Cabauw, Netherlands (\(nRMSE=23.1\%\)), while the worst is obtained at Tamanrasset (\(nRMSE=51.9\%\)).