## 2 Methods and Materials
### Variables, Data and Software  
Let \(y_{th}\), the dependent variable denote global horizontal irradiance (GHI) measured in \(\mathrm{W}/\mathrm{m}^{2}\), where \(t\) represents day and \(h\) the hour of the day, having \(t=1,\ldots,n\) and \(h=1,\ldots,24\). The covariate are denoted by \(\mathbf{x}=x_{th1},x_{th2},\ldots,x_{thp}\), where \(p\) is the number of the covariates. In this study, we considered solar hours or sunlight hours only. The data used in this study were hourly-averaged for the period 1 March 2018 to 31 March 2020, for both stations. Data for the period 1 March 2018 to 26 October 2019 were used for training the models, while the remaining data (27 October 2019 to 31 March 2020) were used for testing the models.  
Figure 2 shows the map of the distribution of GHI in South Africa. The information on the map is from Solargis website [27].  
Figure 2: World bank, Global Solar Atlas 2.0, solar resource data, Solargis. Source: [[https://solargis.com/maps-and-gsi-data/download/south-africa](https://solargis.com/maps-and-gsi-data/download/south-africa)]([https://solargis.com/maps-and-gsi-data/download/south-africa](https://solargis.com/maps-and-gsi-data/download/south-africa)), accessed on 23 October 2020.  
The data used in this study are from (Southern African Universities Radiometric Network (SAURAN) website [[https://sauran.ac.za/](https://sauran.ac.za/)]([https://sauran.ac.za/](https://sauran.ac.za/)), accessed on 17 June 2020), one from a coastal area (Stellenbosch University) is referred to as SUN, Western Cape latitude: \(-33.9281^{\circ}\) (E), longitude: \(18.8654^{\circ}\) (S) and elevation: \(119\) m. This study focuses on two radiometric stations due to different weather conditions at coastal and inland locations. The instruments are positioned where there is good solar exposure. The pyranometer of SUN which is shown in Figure 3 is on top of a roof of an Engineering building at Stellenbosch University.  
Another one is from an inland area USAid Venda referred to as VEN, Vuwani, latitude: \(-23.1310^{\circ}\) (E), longitude: \(30.4239^{\circ}\) (S), elevation: \(628\) m. The pyranometer of VEN which is shown in Figure 4 is at Vuwani.  
The variable of interest is GHI and the independent variables with their respective abbreviations, as used in the dataset, are given in Table 1 for VEN data and Table 2 for SUN data:  
Figure 4: Weather instruments at UNV station. Source: [[https://sauran.ac.za](https://sauran.ac.za)]([https://sauran.ac.za](https://sauran.ac.za)), accessed on 17 June 2020.  
Figure 3: Weather instruments at SUN station. Source: [[https://sauran.ac.za](https://sauran.ac.za)]([https://sauran.ac.za](https://sauran.ac.za)), accessed on 17 June 2020.  
R version 4.0.4 and Python version 3.7.6 are the statistical packages that were used in this study. The package GaussianProcessRegressor from sklearn.gaussian_process was used for Gaussian process regression when Python was used. Missing data points were imputed using multiple iterative imputations.  
GHI in this research is the variable of interest that is used as a measure of solar irradiance. It is defined as the aggregate sum of short wave radiation that is gotten from over a surface which is even to the ground and is made up of two forms of irradiances called the Direct Normal Irradiance (DNI) and Diffuse Horizontal Irradiance (DHI).