## 2 Preliminaries and Background
### Discretization  
The discretization process aims to transform a set of continuous attributes into discrete ones. Although there is a substantial number of discretization methods in the literature, [PERSON] et al. (2018) recently carried out extensive testing of the 30 most representative and newest discretization techniques in supervised classification. Amongst the best performing algorithms, FUSINTER, ChiMerge, CAIM, and Modified Chi2 obtained the highest average accuracies; it is possible to add Zeta and MDLP to this list if the Cohen's kappa metric is considered. In the authors' taxonomy, the evaluation measures for comparing solutions were broken down into five families: information, statistics, rough set, wrapper, and binning. Subsequently, we review few evolutionary approaches to solve discretization problems and succeeding methods of CAIM.  
In [46], a supervised method called Evolutionary Cut Points Selection for Discretization (ECPSD) was introduced. The technique exploits the fact that boundary points are suitable candidates for partitioning numerical attributes. Hence, a complete set of boundary points for each attribute is first generated. A CHC model [47] then searches the optimal subset of cut points while minimizing the inconsistency. Later on, the evolutionary multivariate discretizer (EMD) was proposed on the same basis [27]. The inconsistency was substituted for the aggregate classification error of an unpruned version of C4.5 and a Naive Bayes. Additionally, a chromosome length reduction algorithm was added to overcome large numbers of attributes and instances in datasets. However, the selection of the most appropriate discretization scheme relies on the weighted-sum of each objective functions, where a user-defined parameter is provided. This approach is thus limited even though varying parameters of a parametric scalarizing approach may produce multiple different Pareto-optimal solutions. In [25], a multivariate evolutionary multi-objective discretization (MEMOD) algorithm is proposed. It is an enhanced version of EMD, where the CHC has been replaced by the well-known NSGA-II, and the chromosome length reduction algorithm hereafter exploits all Pareto solutions instead of the best one. The following objective functions have been considered: the number of cut points currently selected, the average classification error produced by a CART and Naive Bayes, and the frequency of the selected cut points.  
As previously exposed, CAIM stands out due to its performance amongst the classical techniques. Some extensions have been proposed, such as Class-Attribute Contingency Coefficient [48], Autonomous Discretization Algorithm (Ameva) [49], and ur-CAIM [30]. Ameva has been successfully applied in activity recognition [50] and fall detection for people who are older [51]. The technique is designed for achieving a lower number of discretization intervals without prior user specifications and maximizes a contingency coefficient based on the \(\chi^{2}\) statistics. The Ameva criterion is formulated as follows:  
\[Ameva(k)=\frac{\chi^{2}}{k(l-1)} \tag{4}\]  
where \(k\) and \(l\) are the number of discrete intervals and the number of classes, respectively. The ur-CAIM discretization algorithm enhances CAIM for both balanced and imbalanced classification problems. It combines three class-attribute interdependence criteria in the following manner:  
\[\text{ur-CAIM}=\text{CAIM}_{N}*\text{CAIR}*(1-\text{CAIU}) \tag{5}\]  
where \(\text{CAIM}_{N}\) denotes the CAIM criterion scaled into the range [0,1]. CAIR and CAIU stand for Class-Attribute Interdependence Redundancy and Class-Attribute Interdependence Uncertainty, respectively. In the ur-CAIM criterion, the CAIR factor has been adapted to handle unbalanced data.