# 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
The number of data sets for which FDUS is superior to RBU is 20, 19, and 18 with respect to the AUC, F-measure, and G-mean, respectively. NB-Tomek is mainly designed for the complex situation in which majority-class samples overlap with minority-class samples, while RBU uses Gaussian kernel functions to deal with the complex distribution of samples in the majority-class sample space. We argue that FDUS alleviates the problem of sample overlap by undersampling in each partition after clustering in the majority class. At the same time, undersampling in the determined majority-class sample space enables some of the minor errors introduced by the NB-Tomek method to be avoided. Therefore, FDUS is superior to NB-Tomek. For RBU, some of the sample distribution features that are captured by Gaussian kernel functions are far from the classification boundary, while those captured by FDUS are closer. Therefore, FDUS exhibits better performance after undersampling. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Data Set** & **FDUS** & **Tomek** & **US** & **RBU** & **Centers** & **CBIS** & **UA-KF** \\ \hline abalone-17\_vs\_7-8-9-10 & 0.8823 & 0.8758 & 0.8123 & 0.8142 & 0.8183 & 0.8554 & 0.8592 & 0.8446 \\ car-vgood & 0.9813 & 0.9642 & 0.6241 & 0.9118 & 0.9446 & 0.9678 & 0.9757 & 0.9446 \\ dermatology-6 & 1 & 0.9849 & 0.9588 & 0.9575 & 0.9985 & 0.9903 & 0.9875 & 0.9985 \\ ecoli & 0.9531 & 0.9211 & 0.8529 & 0.8439 & 0.9038 & 0.8975 & 0.9186 & 0.9141 \\ glass-0-1-6\_vs\_4-5-6 & 0.9958 & 0.9803 & 0.7808 & 0.9785 & 0.9914 & 0.9372 & 0.9547 & 0.9914 \\ glass-0-1-6\_vs\_5 & 1 & 0.8985 & 0.8502 & 0.8262 & 0.8779 & 0.8612 & 0.8922 & 0.8528 \\ iris0 & 0.9961 & 0.9978 & 0.9901 & 0.9878 & 0.9754 & 0.9979 & 0.9824 & 0.9489 \\ kr-vs\_k-three\_vs\_eleven & 0.9972 & 0.9953 & 0.7693 & 0.9863 & 0.981 & 0.9860 & 0.9652 & 0.9881 \\ led/digit-0-2-4-5-6-7-8-9\_vs\_1 & 0.9869 & 0.9209 & 0.6875 & 0.9841 & 0.9862 & 0.9355 & 0.9556 & 0.9862 \\ new-thyroid1 & 0.9875 & 0.9851 & 0.9486 & 0.9522 & 0.9567 & 0.9554 & 0.9741 & 0.9284 \\ page-blocks0 & 0.9659 & 0.9007 & 0.8396 & 0.8276 & 0.9132 & 0.8932 & 0.9607 & 0.8818 \\ segment0 & 0.9993 & 0.9905 & 0.9167 & 0.9064 & 0.9756 & 0.9818 & 0.9015 & 0.9029 \\ shuttle-c0-vs-c4 & 1 & 0.9997 & 0.8757 & 0.9199 & 0.9846 & 0.9978 & 0.9464 & 0.9334 \\ vehicle0 & 0.9777 & 0.9457 & 0.8742 & 0.8866 & 0.9437 & 0.9038 & 0.9567 & 0.9296 \\ vehicle2 & 0.9868 & 0.9598 & 0.6504 & 0.9686 & 0.9738 & 0.9187 & 0.9715 & 0.9738 \\ vowel0 & 0.9857 & 0.9719 & 0.8405 & 0.8279 & 0.9401 & 0.9557 & 0.9782 & 0.9518 \\ winequality-white-3\_vs\_7 & 0.9600 & 0.8058 & 0.7587 & 0.7845 & 0.7968 & 0.7794 & 0.7968 & 0.7903 \\ wisconsin & 0.9794 & 0.9686 & 0.8938 & 0.9092 & 0.9429 & 0.9753 & 0.9532 & 0.9063 \\ yeast-0-5-6-7-9\_vs\_4 & 0.8484 & 0.8049 & 0.7884 & 0.8142 & 0.8315 & 0.8925 & 0.8103 & 0.7935 \\ yeast5 & 0.9922 & 0.9852 & 0.8978 & 0.8278 & 0.9651 & 0.9552 & 0.9773 & 0.8661 \\
**Average** & 0.9738 & 0.9419 & 0.8305 & 0.8958 & 0.9354 & 0.9287 & 0.9394 & 0.9164 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: AUC results using various comparison methods. Abbreviations: Tomek, NB-Tomek; Centers, Centers_nn. **The case in which the sample features sensitivity and noise**. We selected USS and UA-KF to conduct comparisons. Across each data set, FDUS significantly outperforms UA-KF when evaluated using the AUC metric. In the evaluation of the F-measure and G-mean, FDUS outperforms UA-KF on 18 and 16 data sets, respectively, demonstrating its superior performance. Compared with UA-KF, FDUS is 5.74% superior to the AUC, 6.98% superior to F-measure, and 7.31% superior to G-mean with regard to the average of the evaluation indicators for all data sets. For USS, in terms of the average values for the AUC, F-measure, and G-mean evaluations, FDUS exhibits significant priority in each data set, with values of 14.32%, 10.21%, and 12.93%, respectively. Due to the sensitivity of USS for all samples in a majority class, samples near the boundary are also included in the calculation.