# An Undersampling Method Approaching the Ideal Classification Boundary for Imbalance Problems
## 3 Proposed Method
### Undersampling That Enables Close-Fitting Data Distribution at the Ideal Classification Boundary  
During undersampling, a small number of samples are extracted from a large number of data objects in the majority class, thus causing some information loss. Therefore, multiple undersampling is generally used to obtain more classifiers during ensemble learning, which enhances the uniformity of the samples at the classification boundary. The questions  
Figure 5: Diameter of the cluster facing the separating hyperplane. considered in this process are thus as follows: How can the number of classifiers \(m\) in ensemble learning be determined? Does the sample distribution that was obtained after determination conform to the ideal data distribution? Herein, one-time undersampling gains a classifier. **Definition 2**. : \(\phi(x,f(x_{1}))\) _is the sample distribution of the majority class sampling, as determined by the classification boundary function. As shown in Figure 3, there is a linear separation function \(f(x_{1})\) between the majority class and the minority class, and \(C_{i}\) is obtained by clustering in the majority class. At the boundary of the majority class, \(f(x_{2})\) is parallel to \(f(x_{1})\). Then, the space formed by \(f(x_{2})\) and \(C_{i}\) facing \(f(x_{1})\) is the undersampled sampling space. We call the sample distribution \(\phi(x,f(x_{1}))\)._  
\(t\) is the number of classifiers in ensemble learning. When \(m\to t\), the data distribution of ensemble learning \(\phi(x,t)\rightarrow\phi(x,f(x_{1}))\). Since \(f(x_{1})\) is obtained via the performance rough classification, it generally does not coincide with the ideal classification \(f(x_{ideal})\). As such, when \(\phi(x)\rightarrow\phi(x,f(x_{1}))\), the classification results obtained according to this distribution \(\phi(x)\) deviate. As is evident from area \(B\) in Figure 5, the sample distribution in the sampling space is biased to the left. In a situation with the ideal sampling distribution \(\phi(x,f(x_{ideal}))\), where the sampling distribution is mostly uniform, this may work if there are fewer samples on the left side. That is, if \(m\) takes a value from 0 to \(t\), the total sampling distribution of \(t_{opt}\) in the ensemble learning will approach the ideal state. When \(m>t\), \(\phi(x,m)\) will gradually deviate from \(\phi(x,f(x_{1}))\), and then begin to approach \(\phi(x,f(x_{1}))\). After analyzing the above situation, we believe that at the initial stage of ensemble learning, the result of the undersampling classification gradually becomes better; then, when it reaches \(m\), the optimal value is obtained. After this, it gradually decreases. Based on this information, we argue that the sample distribution is uniform for \(f(x_{ideal})\) when the number of undersampling is taken as \(m\). In order to further investigate the improvement in the algorithm according to the number of classifiers that were built using the sampling data obtained in the ensemble learning, we tested 20 data sets. For each data set, we assumed that the number of classifiers in ensemble learning ranged from 1 to 2000, and then created a graph. The graph for each data set is very similar to Figure 6. As the number of classifiers increased, the AUC of the classification result exhibited a sawtooth shape, as shown in Figure 6. Therefore, we speculate that the AUC may exhibit periodicity, accompanied by an increase in the number of classifiers in ensemble learning. We then conducted a time series periodicity test on the results of all 20 data sets after performing a Fourier transform and found that they did not show periodicity. Figure 6: AUC value obtained from the _number of classifiers_/undersampling times_ in ensemble learning for the range 1–2000 in the _Yast5_ data set. In this experiment, we found that each data set exhibits a cyclic phenomenon, in which the accuracy rate increases from small to large and then decreases. The sub-picture shown in Figure 6 is an enlargement of the 1-30 samples obtained from the Yeast5 data set.