## 4 Algorithm
: _The function \(p\) is defined as the distribution of the subgroup target s.target with respect to the instances \(i_{x}\) from a dataset \(d\) covered by the subgroup description s.description. Formally:_

Figure 8:5 (A19)

Figure 8:6 (A20)

**Definition A22** (Function \(p_{0}\)). : _The function \(p_{0}\) is defined as the distribution of the subgroup target s.target with respect to all instances \(i_{x}\) from a dataset \(d\). Formally:_

Figure 8:7 (A21)

Figure 8:8 (A22)

**Definition A23** (Function \(tn\) (true negatives)). : _The function \(tn\) is defined as the number of instances \(i_{x}\) from the dataset \(d\) that are covered by neither the subgroup description \(s.\)description nor the subgroup target \(s.\)target. Formally:_

Figure 8:9 (A23)

Figure 9:0 (A24)

**Definition A24** (Function \(fn\) (false negatives)). : _The function \(fn\) is defined as the number of instances \(i_{x}\) from the dataset \(d\) that are not covered by the subgroup description \(s.\)description, but are covered by the subgroup target \(s.\)target. Formally:_

Figure 9:1 (A25)

Figure 9:2 (A26)

Table 1 shows the confusion matrix of a subgroup \(s\) with respect to a dataset \(d\). This matrix summarizes the functions describe above. With regard to the functions defined previously, some equivalences are defined in Section 3. Moreover, some popular quality measures for SD presented in the literature are described in that section. **Definition A25** (Subgroup Discovery problem). : _Given a dataset \(d\), a quality measure \(q\) and a numeric value quality_threshold, the subgroup discovery problem consists of exploring the search space of \(d\) in order to enumerate the subgroups that have a quality measure value above the selected threshold. Formally:_

Figure 9:3 (A27)

The search space of a problem (i.e., of a dataset \(d\)) can be visually illustrated as a lattice [24] (see Figure 1). According to this comparison, the first level of the search space of a problem contains all those subgroups \(s\) whose descriptions have a size of 1 (i.e., \(|s.\)description) is equal to 1), the second level of the search space of a problem contains all those subgroups \(s\) whose descriptions have a size of 2 (i.e., \(|s.\)description\(|\) is equal to 2) and, in general, the level \(n\) of the search space of a problem contains all those subgroups \(s\) whose descriptions have the size \(n\) (i.e., \(|s.\)description\(|\) is equal to \(n\)). ## References

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