# 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
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Data Set** & **Dimension** & **Examples** & **ir** \\ \hline abalone-17\_vs\_7-8-9-10 & 8 & 2338 & 39.31 \\ car-vgood & 6 & 1728 & 25.58 \\ dermatology-6 & 34 & 358 & 16.9 \\ ecoli1 & 7 & 336 & 3.36 \\ glass-0-1-6\_vs\_4-5-6 & 9 & 214 & 3.2 \\ glass-0-1-6\_vs\_5 & 9 & 184 & 19.44 \\ iris & 4 & 150 & 2.0 \\ kr-vs\_k-three\_vs\_eleven & 6 & 2935 & 35.23 \\ led7 digit-0-2-4-5-6-7-8\_9\_vs\_1 & 7 & 443 & 10.97 \\ new-thyroid1 & 5 & 215 & 5.14 \\ page-blocks0 & 10 & 5472 & 8.79 \\ segment0 & 19 & 2308 & 6.02 \\ shuttle-c0-vs-c4 & 9 & 1829 & 13.87 \\ vehicle0 & 18 & 846 & 3.25 \\ vehicle2 & 18 & 846 & 2.88 \\ vowel0 & 13 & 988 & 9.98 \\ winequality-white-3\_vs\_7 & 11 & 900 & 44 \\ wisconsin & 9 & 683 & 1.86 \\ yeast-0-5-6-7\_9\_vs\_4 & 8 & 528 & 9.35 \\ yeast5 & 8 & 1484 & 32.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data set statistics. \[F\text{-}measure = \frac{2\times Precision\times Recall}{Precision+Recall} \tag{10}\] \[G\text{-}mean = \sqrt{Precision\times Recall} \tag{11}\]

### Hypothesis Test to Determine If There Is an Optimal Value in a Cycle of Ensemble Learning

The results of the ensemble learning performed by the method proposed in this paper are presented. The number of classifiers is set to a range of 1 to 2000. We then select data in the first cycle of the AUC that are similar to those presented in Figure 6.

The selected data generally remove the first data, and then the remaining values are used to perform a normal distribution hypothesis test. The detection results of the 20 data sets used in this paper are recorded in Table 2. **stat** is a statistical measure that is used to measure the degree of fit between the sample data and the normal distribution, and **P** is a probability value; when P is less than the significance level of 0.05, it is considered to not follow a normal distribution. As shown in Table 2, the \(p\) values are greater than 0.05%, indicating that the confidence level is 95%. This shows that all the data have passed the hypothesis test, and therefore follow a normal distribution. Therefore, they all hold an optimal value in a cycle. ### Effect of Different Clustering Methods on Research Results

In order to discuss the impact of clustering algorithms for large-class partitioning on the entire method, we selected two classic methods, namely DBSCAN and SVR, and compared them with Kmeans. Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [39] is a data-clustering algorithm that is used to discover clusters of arbitrary shapes and is based on the concept of density reachability and density connectivity. Support Vector Clustering (SVC) [40] is used to cluster data points in a high-dimensional space. From Table 3, it can be seen that the results of the three methods are very similar, with Kmeans performing slightly better than DBSCAN and SVC. According to the principle of Occam's razor, we chose to use Kmeans in our method. \begin{table}
\begin{tabular}{l l l l} \hline
**Data Set** & **Stat** & \(p\) & **Number of Classifiers** \\ \hline abalone-17\_vs\_7-8-9-10 & 0.954 & 0.766 & 0–30 \\ car-vgood & 0.795 & 0.074 & 1–10 \\ dermatology-6 & 1 & 1 & 1–10 \\ ecoli1 & 0.949 & 0.156 & 0–30 \\ glass-0-1-6\_vs\_4-5-6 & 0.964 & 0.636 & 0–20 \\ glass-0-1-6\_vs\_5 & 0.938 & 0.081 & 0–30 \\ iris0 & 1 & 1 & 0–5 \\ kr-vs\_k-three\_vs\_eleven & 0.917 & 0.508 & 0–5 \\ led7 digit-0-2-4-5-6-7-8-9\_vs\_1 & 0.929 & 0.442 & 0–10 \\ new-thyroid1 & 0.835 & 0.05 & 1–10 \\ page-blocks0 & 0.776 & 0.051 & 0–5 \\ segment0 & 0.858 & 0.222 & 0–5 \\ shuttle-c0-vs-c4 & 1 & 1 & 0–30 \\ vehicle0 & 0.959 & 0.315 & 1–30 \\ vehicle2 & 0.904 & 0.429 & 1–10 \\ vowel0 & 0.872 & 0.273 & 0–5 \\ winequality-white-3\_vs\_7 & 0.959 & 0.284 & 0–30 \\ wisconsin & 0.806 & 0.091 & 0–56 \\ yeast-0-5-6-7-9\_vs\_4 & 8 & 528 & 9.35 \\ yeast5 & 0.847 & 0.186 & 0–5 \\ \hline \end{tabular}
\end{table} 
Table 2: Normal distribution test of the data sets.