## 6 Results and Discussion
Second, the language(s) that was selected allows one to draw some considerations on the nature of the problem; for example, the fact that, in all cases, the relation during or its inverse (i.e., \(\langle D\rangle\) or \(\langle\overline{D}\rangle\)) was selected indicates that the past interactions between the variables is a key element for modelling this particular phenomenon. Because Temporal J48, in this experiment, was run without pruning, the resulting trees cannot be easily displayed because of their dimensions. Nevertheless, thanks to its intrinsic interpretability, meta-rules can be easily extracted from a regression tree, as, for example:  
\[\text{If Rel.humidity is high while traffic is high}\qquad\text{then}\quad NO_{2}\text{ tends to be high}\] \[\text{If Sol.duration is high while traffic is very low}\quad\text{then}\quad NO_{2}\text{ tends to be low} \tag{20}\]  
\[\dots\]  
which can contribute to designing a real-world theory of the modelled phenomenon. The language selection part performed by the optimizer, in general, reduces the set of used temporal operators of HS when extracting the rules (see Table 6), and this is desirable considering that, among many others, one desideratum for interpretability is to explain the reasoning in an understandable way to humans, which have a strong and specific bias towards simpler descriptions [44]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline _Month_ & \(\mathbf{\alpha}\) & _mae_ & _rmse_ & _rae (\%)_ \\ \hline Jan & 0.75 & 10.47 & 13.38 & 63.61 \\ Feb & 0.73 & 10.67 & 12.86 & 67.86 \\ Mar & 0.65 & 12.66 & 16.04 & 73.62 \\ Apr & 0.68 & 12.05 & 14.62 & 75.87 \\ May & 0.71 & 10.00 & 13.63 & 61.86 \\ Jun & 0.61 & 12.57 & 15.34 & 79.93 \\ Jul & 0.59 & 11.90 & 15.09 & 79.35 \\ Aug & 0.69 & 13.62 & 17.07 & 70.74 \\ Sep & 0.72 & 11.47 & 15.21 & 64.24 \\ Oct & 0.83 & 8.84 & 11.11 & 52.95 \\ Nov & 0.76 & 8.58 & 11.25 & 61.18 \\ Dec & 0.77 & 9.32 & 12.05 & 57.15 \\ \hline _average_ & 0.71 & 11.01 & 12.84 & 67.36 \\ \hline _Month_ & _cc_ & _mae_ & _rmse_ & _rae (\%)_ \\ \hline Jan & 0.77 & 9.35 & 13.21 & 56.84 \\ Feb & 0.75 & 9.89 & 12.92 & 62.91 \\ Mar & 0.67 & 12.71 & 16.59 & 73.90 \\ Apr & 0.76 & 9.86 & 13.41 & 62.09 \\ May & 0.71 & 10.34 & 13.99 & 63.97 \\ Jun & 0.70 & 11.24 & 14.58 & 71.45 \\ Jul & 0.67 & 11.21 & 14.87 & 74.74 \\ Aug & 0.76 & 11.87 & 15.96 & 61.63 \\ Sep & 0.60 & 12.88 & 18.82 & 72.13 \\ Oct & 0.76 & 9.74 & 13.35 & 58.37 \\ Nov & 0.74 & 8.91 & 11.93 & 63.58 \\ Dec & 0.75 & 9.55 & 12.93 & 58.55 \\ \hline _average_ & 0.72 & 9.85 & 13.28 & 60.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test results, non-temporal data: linear regression (top) and decision tree regression (bottom). \begin{table}
\begin{tabular}{c c c c c} \hline \hline _Month_ & _cc_ & _mae_ & _rnse_ & _rae (\%)_ \\ \hline Jan & 0.80 & 9.79 & 12.21 & 59.51 \\ Feb & 0.83 & 8.35 & 10.58 & 53.16 \\ Mar & 0.81 & 9.45 & 12.66 & 54.94 \\ Apr & 0.71 & 11.43 & 14.30 & 71.95 \\ May & 0.73 & 10.67 & 13.86 & 66.02 \\ Jun & 0.72 & 10.61 & 13.63 & 67.41 \\ Jul & 0.75 & 9.94 & 12.57 & 66.29 \\ Aug & 0.77 & 12.80 & 15.45 & 66.44 \\ Sep & 0.78 & 11.31 & 14.56 & 63.34 \\ Oct & 0.82 & 9.00 & 11.58 & 53.96 \\ Nov & 0.80 & 8.22 & 10.60 & 58.61 \\ Dec & 0.84 & 8.08 & 10.42 & 49.51 \\ \hline _average_ & 0.78 & 9.09 & 12.70 & 60.93 \\ \hline _Month_ & _cc_ & _mae_ & _rnse_ & _rae (\%)_ \\ \hline Jan & 0.75 & 9.59 & 13.75 & 58.31 \\ Feb & 0.84 & 7.93 & 10.41 & 50.43 \\ Mar & 0.78 & 10.26 & 13.42 & 59.65 \\ Apr & 0.71 & 10.29 & 14.45 & 64.75 \\ May & 0.77 & 9.36 & 12.65 & 57.91 \\ Jun & 0.70 & 11.08 & 14.59 & 70.45 \\ Jul & 0.65 & 10.96 & 15.21 & 73.09 \\ Aug & 0.75 & 12.10 & 16.19 & 62.84 \\ Sep & 0.78 & 10.09 & 14.01 & 56.49 \\ Oct & 0.75 & 10.12 & 13.98 & 60.67 \\ Nov & 0.79 & 7.83 & 10.74 & 55.85 \\ Dec & 0.87 & 7.11 & 9.54 & 43.61 \\ \hline _average_ & 0.76 & 8.87 & 13.24 & 54.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test results, lagged data; linear regression (top), and decision-tree regression (bottom).