# Impact of Imbalanced Datasets Preprocessing in the Performance of Associative Classifiers
## 3 Materials and Methods
### Associative Classifiers
#### 3.2.2 Extended Gamma Classifier  
The Gamma Associative Classifier was proposed by [PERSON] as a prediction model [43], and has been used successfully in supervised classification tasks [44]. In its original version, this algorithm was unable to handle qualitative data, or absence of information in the data. That is why an extension of this classifier was made, to overcome these limitations [45]. This extension modifies the way in which the similarity between instances is calculated, and allows the direct application of this classification model to databases with mixed and incomplete attributes, very common in the financial field. Let \(X\) and \(P\) be the training and testing sets, respectively, from a universe \(U\), where each instance \(x\in X,\ p\in P\) is described by a set of attributes \(\mathrm{A}=\{A_{1},A_{2},\cdots,A_{m}\}\); each attribute \(A_{i}\) has a definition domain \(dom(A_{i})\), which can be numeric or categorical. As a particular case, if the value of the attribute \(A_{i}\) in an instance \(x\) is unknown, it is considered to be a missing data, and denoted as \(x_{i}=\ ^{\prime}?^{\prime}\). It is assumed that there is a set of classes \(K=\{K_{1},\ldots,K_{c}\}\) associated with the training instances. The Extended Gamma Associative Classifier (EG) consists of two phases: training and classification. The training phase of this classifier begins with the storage of the training set, and includes the subsequent calculation of various parameters (Table 2). In the classification phase, EG uses an iterative process, based on the calculation of the average similarity to each class of the instance to be classified. To analyze the similarity between the test instances and training instances, the extended generalized similarity \(\gamma_{ext}\) is used. After obtaining the similarities, the average of the generalized similarity of said test pattern for each class \(k_{l}\in K\) (Equation (1)) is calculated. Let \(p\in P\) be an instance to classify and let \(p_{j}\) be the value corresponding to the j-th attribute. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Meaning** & **Recommendation** \\ \hline \(w\) & It is the vector of weights of the attributes, which & Computed by Differential \\  & indicates the importance of each attribute. & Evolution [40] \\  & It is the value that will initially take \(\theta\) and indicates & \\ \(\theta\) & how different two numerical values can be and that the & \\  & extended generalized similarity operator considers & them similar. \\  & It is the stop parameter and refers to the maximum & \\  & value allowed to \(\theta\), which allows to continue looking & \\ \(\rho\) & for the disambiguation of patterns near the border; & when \(\theta=\rho\), the CAG will stop iterating and \\  & & disambiguate the class. \\  & It is the pause parameter. In this pause an evaluation of & \\  & the pattern to be classified is carried out, in order to & \\ \(\rho_{0}\) & determine whether or not it belongs to the unknown & \\  & class: it depends on whether the normal operation of & \\  & the algorithm is continued. & \\  & It is the threshold to decide if the pattern to be & \\ \(u\) & classified belongs to the unknown class or to any of the & \\  & known classes. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Empirical values for the parameters of the Gamma classifier. The number of instances belonging to the class \(k_{l}\) in the training set is given by \(n\), and \(x_{j}^{i}\) represents the value of the j-th attribute of the i-th instance of the class \(k_{l}\), and \(w_{j}\) represents the weight of the j-th attribute. \[c_{k_{l}}=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{j}\gamma^{\prime} \alpha\left( \begin{subarray}{c}i\\ j\end{subarray} ,p_{j}\right)}{n} \tag{1}\] \[\gamma_{ext}\Big{(}x_{j},y_{j},\theta\Big{)}=