## 2 Tools and Methods
### Model Assessment Metrics
A value of 1 indicates that the regression model accounts for all the predicted variables, signifying a perfect correlation between the two variables [15; 26]. \[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}- \overline{y}_{i})^{2}}\] (5) where \(y_{i}\) represents the actual value, \(\hat{y}_{i}\) denotes the predicted value and \(\overline{y}_{i}\) signifies the mean value of the actual variable. The following figure (Figure 2) shows the interpretation of the given metric. Metrics derived from metrics or squared errors are called scale-dependent metrics. They correspond to the scale of the original data and present errors in the same unit [3]. Nevertheless, the R\({}^{2}\) value alone is not an adequate measure to assess the validity of the model and the relationship with unexplained variance must be taken into account [24; 27]. Scale-dependent metrics:
* The Sum of Squared Errors (SSE) is a statistical measure employed to assess the quality of a model or the difference between predicted outcomes and actual data. It is often used in regression, modelling, and machine learning to quantify model error. \[SSE=\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2},\] (6)  
Figure 2: Interpretation of R squared [26]. where \(n\) represents the number of values, \(y_{i}\) measured values and \(\hat{y}_{i}\) forecasted values. SSE does not have units comparable to the original data, so it is less intuitive for direct interpretation. To achieve this, a derivative like Mean Squared Error (MSE) or Root Mean Squared Error (RMSE) is frequently utilised. * Root Mean Squared Error (RMSE), \[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}},\] (7) where \(n\) represents the number of values, \(y_{i}\) measured values and \(\hat{y}_{i}\) forecasted values. In contrast to MSE, RMSE shares the same units as the original data, which enhances its interpretability compared to MSE [28]. R\({}^{2}\) is considered more independent than RMSE since it is not reliant on the unit, allowing for comparison among models trained on various datasets. * Mean Squared Error (MSE)--The value of Mean Squared Error (MSE) indicates the proximity of the regression line to a group of points. It calculates the Mean Squared Error between the predicted and actual values, assigning greater importance to larger discrepancies. When the average of the collection of errors is determined, it is referred to as the Mean Squared Error. The smaller the MSE value, the more accurate the prediction becomes. MSE is measured in units that are the square of the target variable. The units of MSE are the square of the target variable. \[MSE=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2},\] (8)
* Mean Absolute Error (MAE)--The Mean Absolute Error reflects the average discrepancy between the measured and forecasted values. The nearer the MAE value is to 0, the more accurate the prediction becomes [7]. \[MAE=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-\hat{y}_{i}|,\] (9)
* MAPE indicates the level of accuracy of the forecasted value compared to the actual measured values [7] (expressed in percent). For the MAPE index, an effective model has a value of less than 14%. \[MAPE=\frac{1}{n}\sum_{i=0}^{n}\frac{|y_{i}-\hat{y}_{i}|}{y_{i}},\] (10) where \(n\) represents the number of values, \(y_{i}\) measured values and \(\hat{y}_{i}\) forecasted values. RMSE, MAE and MAPE are widely used accuracy metrics because they are easy to interpret. They offer a plausible assessment of the typical error anticipated from the predictions at each prediction stage [7]. * SMAPE (symmetric mean absolute percentage error). \[SMAPE=\frac{1}{n}\sum_{i=1}^{n}\frac{|\hat{y}_{i}-y_{i}|}{\frac{(y_{i}+\hat{y }_{i})}{2}},\] (11)  
It conveys the typical variation between the actual values \(y_{i}\) and the estimated values \(\hat{y}_{i}\), normalising the error relative to the average value. A limitation of SMAPE is that if the actual value or predicted value is 0, the error value will increase up to the upper error bound. However, different accuracy metrics rely on different types of measurement.