## 2 Methods
### KNN-Is and SMOTE in Big Data
\[TPR=\frac{TP}{TP+FN}\] (5)
* **The True Negative Rate** or Recall, which represents the proportion of negative examples correctly classified. \[TNR=\frac{TN}{TN+FP}\] (6)
* **The False Negative Rate**, which is the proportion of positive examples misclassified. \[FNR=\frac{FN}{TP+FN}\] (7)
* **The False Positive Rate**, which is the proportion of negative examples misclassified. \[FPR=\frac{FP}{TN+FP}\] (8)

A metric encompassing individual rates per class is the geometric mean,

\[G-mean=\sqrt{TPR\times TNR}, \tag{9}\]

which maximizes the effectiveness of each class while maintaining balance. The classifiers can also be evaluated using graphical methods, which allow for the visualization of their effectiveness [46]. One of the most popular strategies is the ROC graph (Receiver Operating Characteristic curve). In this two-dimensional space, curves represent the trade-off between \(TPR\) (\(Y\)-axis) and \(FPR\) (\(X\)-axis). A scalar value can be derived from the ROC space by calculating the area under the curve, known as the \(AUC\) (Area Under the ROC curve). In the literature, there are several ways of computing the \(AUC\), here, we have adopted the version that captures a single point on the ROC curve, also called balanced accuracy, computed as [47]

\[AUC=\frac{TPR+TNR}{2}. \tag{10}\]

## 4 Results and Discussion

The experiments were conducted to evaluate three strategies for dealing with imbalanced and high-dimensional Big Data datasets: (1) SMOTE, (2) dissimilarity with fractional norms (\(D_{d_{p}}\)), and (3) the joint use of dissimilarity and SMOTE (\(D_{d_{p}}\)+ SMOTE). For clarity and simplicity, in strategies (2) and (3), we only selected the two best results in terms of AUC and G-mean. Table 4 presents the experimental results for each strategy used on the nine datasets. It includes the values of \(p\) in the Minkowsky distance metric used in constructing the \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Positive Prediction** & **Negative Prediction** \\ \hline Actual Positive & True Positives (\(TP\)) & False Negatives (\(FN\)) \\ Actual Negative & False Positives (\(FP\)) & True Negatives (\(TN\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Confusion matrix for a two-class problem. dissimilarity matrix and the resulting dimensionality (number of features) obtained when applying this mapping. Additionally, the classification results on the original imbalanced dataset and with SMOTE are provided for comparison purposes. The experimental results using SMOTE align well with the general conclusions from numerous studies on standard problems, indicating the effectiveness of dealing with the class imbalance problem. Despite the high dimensionality of the data, the use of SMOTE consistently improves both the AUC and G-mean across all cases when compared with the baseline strategy (without the preprocessing technique). However, this enhancement comes at the cost of increasing the size of the dataset. When the imbalanced dataset is transformed into a dissimilarity representation with fractional norms, the results generally exhibit lower performance than those obtained with the original dataset. However, it is important to note that the dataset continues presenting the class imbalance problem in this new representation. Despite this, the decrease in performance is not always drastic, even though the dimensionality is reduced by 92% and 98% in some cases. Similarly, while it is challenging to recommend a specific value of \(p\), in most situations, better results are observed when \(p<1\).