## 2 Materials and Methods
### Model Evaluation Metrics  
To enable comparison with similar research, the results of the models are evaluated by two very popular statistical evaluation metrics in regression tasks, namely, the Mean Squared Error (MSE) and the coefficient of determination (R2). The MSE evaluates the relative distance of the predicted and observed variable, where values closer to zero indicate lower bias between the predicted and observed value, indicating better model performance.  
\[MSE=\frac{\sum_{i=1}^{n}(y_{i}-y_{i})^{2}}{n} \tag{11}\]  
where: \(y_{i}\) is the observed variable at time \(t\), \(y_{i}^{\prime}\) is the predicted variable at time \(t\), and \(n\) is the length of the dataset.  
The coefficient of determination R2 describes the square of the sample correlation coefficient, ranging from 0 to 1, and measures the observed variance explained by the model. Values near 0 indicate no correlation, while values near 1 indicate good model performance, meaning that the model can explain the variance in the observed data.  
\[R^{2}=\left[\frac{\sum_{i=1}^{n}(y_{i}-\overline{y})(y_{i}-\overline{y}^{ \prime})}{\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}}}\right]^{2} \tag{12}\]  
where: \(\overline{y}\) i is the mean of the observed data and \(\overline{y}^{\prime}\) is the mean of the predicted data.  
Relative squared error (RSE) is a relative measure, denoting it what it would have been if a simple predictor was used. It takes the total squared error and normalizes it by dividing it by the total error of a simple predictor.  
\[RSE=\frac{\sum_{i=1}^{n}(y_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\hat{y})^{2}} \tag{13}\]  
where:  
\[\hat{y}=\frac{\sum_{i=1}^{n}y_{i}}{n} \tag{14}\]  
The lower the RSE, the better the model, so for a perfect fit model, the values of RSE and \(\hat{y}\) are both equal to zero.  
Mean bias error (MBE) denotes the mean forecast error, due to overcast or undercast, meaning it measures how much does the model overpredict or underpredict the values.  
\[MBE=\frac{\sum_{i=1}^{n}(y_{i}-y_{i})}{n} \tag{15}\]Lower values are desirable, where negative MBE means the model predicts lower than measured, while a positive MBE means the model predicts larger values, respectively. The leverage statistic in regression analysis is often used to quantify the influence of certain data points on the regression line, referred as high leverage points. The studentized residuals are calculated as follows:  
\[\hat{\epsilon}=y-y^{\prime} \tag{16}\]  
The average leverage calculated as:  
\[\overline{h}=\frac{\sum_{i=1}^{n}h_{ii}}{n}=\frac{p}{n} \tag{17}\]  
where: \(p\) is the number of predictor variables.  
The specific leverages of each predicted data point are calculated from the weight matrix, by following:  
\[y^{\prime}=Hy \tag{18}\]  
Or written as linear combination:  
\[y^{\prime}_{i}=h_{ii}y_{1}+h_{i2}y_{2}+\ldots h_{in}y_{n} \tag{19}\]  
where: \(h_{in}\) are the leverages of each data point. Each data point whose leverage is 3 times higher than is flagged as \"high leverage point\". The results of leverage analysis are then plotted as leverage versus studentized-residuals plots.