# Numerical Association Rule Mining from a Defined Schema Using the VMO Algorithm
## 3 Related Work
### Techniques for Numerical Association Rule Mining
#### 3.1.1 Discretization of Numerical Attributes  
In 1996, [PERSON] and [PERSON] [11] included numerical attributes in the problem of association rules. The data collected to deal with everyday problems are not always discrete elements. Numerical attributes have different characteristics from those of categorical attributes, so they require other methods for their study. The discretization of numerical variables was the theoretical foundation used to develop the first contributions in this area. However, possible missing information is a persistent limitation. The APACS2 [24] technique is based on the partitioning principle. It stands out because it eliminates user intervention when defining parameters and generating both positive and negative rules. Unlike others of its kind, this technique uses a metric called \"adjusted difference\" to select the rules of most significant interest. Experiments showed that the algorithm was capable of outperforming others of its time. The valuable information lost by partitioning became notable. Thus, in their experiments, studies corroborated that, for equal support, the partitioning technique does not significantly minimize the information lost in all its variants. Therefore, more efficient optimization methods are needed [25]. The large number of rules generated with the primitive techniques is another problem derived from partitioning. Studies have been conducted to deal with this problem, and they highlighted a simultaneous multi-attribute discretization procedure supported by the use of clustering techniques [26]. The numerical association rule discovery algorithm called NAR-DISCOVERY [27] can be summarized in two phases. The first adopts a strategy of \"divide and conquer\", which allows it to approach a set of initial rules obtained through a basic partitioning that defines a small number of intervals with a high amplitude. The initial rules are evaluated with classical mechanisms of frequent patterns, and in the second phase, successive partitions are applied to this set of initial rules. In addition, a search based on a tree structure is implemented to manage rule derivations, and through an optimization mechanism based on temporary tables, it is applied in order to select exciting rules. The selection of the discretization method is a problem for the partitioning technique. One criterion used is the a priori knowledge of the post-discretization application. Thus, a supervised technique may be an option for classification purposes. This technique could also not help in identifying associations. With this theoretical basis, the authors of [28] proposed an unsupervised technique that does not require thresholds in order to generate the intervals. The procedure uses the standard deviation as a measuring instrument for the analysis and definition of the method. The experiments carried out showed that the technique was better than other existing ones. The statistical analysis of the variables is a strategy used to determine patterns in the numerical association rules. The consequent of a rule may correspond to the distribution of the numerical variables of the antecedent. Some researchers considered that the distribution of the values in the numerical attributes can have a generalized representation of a categorical definition when the appropriate statistical measures are used [29]. Then, they were reconverted into quantitative association rules in a post-processing stage [30]. Finally, the researchers introduced a definition for domain partitioning, and they proposed bi-partitioning techniques based on the mean, median, and minimization of the standard deviation based on this concept. The clustering theory, which is usually attributed to data-mining classification, has been extended to discretization procedures. In this context, the authors of [31] proposed an efficient hierarchical clustering algorithm based on density variation to solve the interval partitioning problem, which was their main contribution.