## 2 Methods and Materials
### Variable Selection and Parameter Estimation  
#### 2.4.1 Variable Selection  
The variable selection has a vital role in the model building process because in some cases modelling will result in several candidate independent variables hence there is a need for variable selection. Variables are selected to avoid over-fitting, problems of multi-collinearity, and this makes it easier to interpret the model and reduction in computational time. Lasso, Ridge and elasticnet are the types of regularisation techniques that can be used for variable selection. The method adopted for the selection of variables for this research is called Lasso which was first developed by [PERSON] [28]. It performs both shrinkage and variable selection which is used to find the subset of variables that will minimise prediction error. Lasso is a powerful shrinkage method that performs feature selection and regularisation and this tackles the problem of over-fitting, which enhances its predictive accuracy. [PERSON] [28] even goes on to state that Lasso provides probabilities of conducting statistical estimation. Lasso does eliminate the coefficients thus it automatically selects the models. It implements a constraint on model parameters that affects regression coefficients for the variables so that they will shrink towards zero.  
In this study, the variables will be selected using Lasso via hierarchical interactions ([29]). Considering a regression model with pairwise interactions between the predictors \(x_{1},\ldots,x_{P}\) (i.e., temperature, wind speed, barometric pressure, etc.) for a response variable \(Y\) (i.e.,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Description** & **Measuring Units** \\ \hline Air Temperature & Temp & \({}^{\circ}\)C \\ Relative Humidity & RH & \% \\ Wind Speed & WS & m/s \\ Wind Speed Maximum & WS\_Max & m/s \\ Barometer Pressure & BP & mbar \\ Wind Direction & WD & \({}^{\circ}\) \\ Wind Direction Standard Deviation & WD\_Stv & \({}^{\circ}\) \\ Rain Total & Rain\_Tot & mm \\ Wind Vector & WVec & m/s \\ \hline \hline \end{tabular}
\end{table}
Table 1: Independent variables: VEN station.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Description** & **Measuring Units** \\ \hline Temperature & AirTC\_Avg & \({}^{\circ}\)C \\ Relative Humidity & RH & \% \\ Wind Speed & WS\_ms\_S\_WVT & m/s \\ Wind Direction & WindDir\_D1\_WVT & \({}^{\circ}\) \\ Wind direction standard deviation & WindDir\_SD1\_WVT & \({}^{\circ}\) \\ Barometric pressure & BP\_mB\_Avg & mbar \\ \hline \hline \end{tabular}
\end{table}
Table 2: Independent variables: SUN station.  
hourly GHI). The pairwise interaction model is given in Equation (11) which follows a standard GP and discussed in detail in ([29]).  
\[y_{t,h}=\beta_{0}+\sum_{j}\beta_{j}x_{j}+\frac{1}{2}\sum_{j\
eq k}\Theta_{jk}x_{ j}x_{k}+\epsilon_{t,h}, \tag{11}\]  
where \(y_{t,h}\) is GHI on day \(t\) at hour \(h\), \(x_{j}\) and \(x_{k}\) are predictor variables, \(\beta_{0}\) is the intercept and \(\epsilon_{t,h}\) is the error term, \(\beta_{0}\) is the intercept, \(\beta_{j}\), \(\Theta_{jk}\), are the parameters, respectively. The second term after the intercept \(\beta_{0}\) is referred to as the main effects and the third as the interaction terms.