## 3 Symbolic Temporal Regression
The value \(\gamma\) allows us a certain degree of uncertainty: we interpret the decision \(A\bowtie a\) on an interval \([x,y]\) with a certain value \(\gamma\) as true if and only if the ratio of points between \(x\) and \(y\) satisfying \(A\bowtie a\) is at least \(\gamma\). A temporal regression tree is obtained by the following grammar:  
\[\tau::=(S\wedge\tau)\vee(\
eg S\wedge\tau)\mid\hat{b}, \tag{14}\]  
where \(S\) is a (temporal or atemporal) decision and \(\hat{b}\in Dom(B)\), in full analogy with non-temporal trees. The idea that drives the extraction of a regression tree is the same in the propositional and the temporal case, and it is based on the concept of splitting by variance. The result is a staircase function, with the additional characteristic that each leaf of the tree, which represents such a function, can be read as a formula of HS. Therefore, if a propositional tree for regression gives rise to tree rules of the type if \(A_{1}<3\) two units before now and \(A_{2}>5\) one unit before now, then, on average, \(B=3.2\) when used on lagged data, Temporal J48 gives rise to rules of the type if mostly \(A_{1}<3\) during an interval before now and mostly \(A_{2}>5\) in an interval during it, then, on average, \(B=3.2\). It should be clear, then, that Temporal J48 presents a superior expressive power that allows one to capture complex behaviours. It is natural to compare the statistical behaviour of regression trees over lagged data and that of Temporal J48 using the same temporal window. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**HS Modality** & **Definition w.r.t. the Interval Structure** & **Example** \\ \hline \(\langle A\rangle\) (after) & \([x,y]R_{A}[w,z]\) & \(\Leftrightarrow\) & \(y=w\) \\ \(\langle L\rangle\) (later) & \([x,y]R_{L}[w,z]\) & \(\Leftrightarrow\) & \(y<w\) \\ \(\langle B\rangle\) (begins) & \([x,y]R_{B}[w,z]\) & \(\Leftrightarrow\) & \(x=w\wedge z<y\) \\ \(\langle E\rangle\) (ends) & \([x,y]R_{E}[w,z]\) & \(\Leftrightarrow\) & \(y=z\wedge x<w\) \\ \(\langle D\rangle\) (during) & \([x,y]R_{D}[w,z]\) & \(\Leftrightarrow\) & \(x<w\wedge z<y\) \\ \(\langle O\rangle\) (overlaps) & \([x,y]R_{O}[w,z]\) & \(\Leftrightarrow\) & \(x<w<y<z\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: [PERSON]’s relations and their logical notation. HS, [PERSON] and [PERSON]. A temporal regression tree such as Temporal J48 is extracted from a temporal data set following the greedy approach of splitting by variance as in the propositional case. Being sub-optimal, a worse local choice may, in general, produce better global ones. This is the idea behind feature selection: different subsets of attributes lead to different local choices, in search for global optima. In the case of temporal regression trees, however, the actual set of interval relations that are used for splitting behaves in a similar way: given a subset of all possible relations, a greedy algorithm for temporal regression trees' extraction may make worse local choices that may lead to better global results. Therefore, we can define a generalization of (11):  
\[\left\{ \begin{array}{ll}\max~{}Performance(\bar{U},\bar{V})\\ \min~{}Cardinality(\bar{U}),\end{array} \right. \tag{15}\]  
in which \(\bar{U}\) represents a selection of features and \(\bar{V}\) represents a selection of interval relations to be used during the extraction. This is a multi-objective optimization problem that generalizes the feature selection problem, and we can call it the feature and language selection problem. Observe that there is, in general, an interaction between the two choices: different subsets of features may require different subsets of relations for a regression tree to perform well. The number of interval relations that are actually chosen, however, does not affect the interpretability of the result, and therefore, it is not optimized (in the other objective function).