## 6 Results and Discussion  
All results can be seen in the tables from Tables 3-6, in which we report, per each experiment, not only the correlation coefficient (cc) between the ground truth \(b\in dom(B)\) and the predicted value \(\hat{b}\in Dom(B)\)[20; 42; 43], but also the mean average error (mae), the root mean squared error (rmse), and the relative absolute error (rae). The first group of results concerns non-lagged data and standard approaches. As we can see, the correlation coefficient ranges from 0.59 to 0.83, with an average of 0.71, in the linear regression models, and from 0.60 to 0.72, with an average of 0.72, in the decision tree models. The fact that the latter show a slightly better behaviour than the former may indicate that the underlying process is not (strongly) linear and that a stepwise function may approximate this reality in a better way. The fact that the average correlation is not too high in both cases and that in both case, there is at least one month in which it is particularly low may indicate that non-lagged data probably do not capture the underlying phenomenon in its full complexity. As much as lagged data are concerned, in linear regression models, the correlation coefficients range from 0.71 to 0.84, with an average of 0.78, while in decision tree models from 0.65 to 0.87, with an average of 0.76, presented in Table 4. As we can see, the situation reversed itself, the linear models being more precise than the decision tree ones. A possible explanation is that, while lagged data, in general, offer more information about the underlying process, reasoning with more variables (i.e., 60 vs. six) allows finding very complex regression hyperplanes, which adapt to the data in a natural way; unfortunately, this is a recipe for non-interpretability, as having such a complex regression function, with different coefficients for the same independent variable at different lags, makes it very difficult for the expert to create an explanatory physical theory. To give one example, we consider the linear model extracted from \(\mathcal{A}_{lan}\) and, in particular, the coefficients of each variable, as shown in Table 5. As can be observed, the alleged influence of every variable seems to have some erratic behaviour, with coefficients with different signs and absolute values at different lags. It could be argued that such a matrix of values is no different from a weight matrix of a neural network, in some sense. Finally, in Table 6, we can see the results of Temporal J48, in which case the correlation coefficient ranges from 0.78 to 0.87, with an average of 0.83. As can be noticed, in exchange for a higher computational experimental complexity, this method returns clearly better results. This is to be expected, as, by its nature, it combines the benefits of the lagged variables with those of symbolic regression. One can observe not only the improvement in the average, but also in the stability among the twelve months: in the worst case, the correlation index is 0.78, which is to be compared, for example, with the worst case of simple linear regression (0.59). Moreover, it seems that Temporal J48 behaves in a particularly good way on difficult cases: if the case of \(\mathcal{A}_{jul}\), for example, we have a correlation coefficient of 0.67 with non-lagged data and decision trees, 0.65 with lagged data and decision trees, and 0.78 with serialized data. In addition to the statistical performances of these models, the following aspects should be noticed. First, these models were extracted in a feature selection context; however, in all cases, the evolutionary algorithm found that all variables had some degree of importance, and no variable was eliminated.