# 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

Table 6: F-measure results using various comparison methods. Abbreviations: Tomek, NB-Tomek; Centers, Centers_nn. set to \(m\). Furthermore, because of these \(m\) samples, they have \(m\) fewer samples near the boundary compared to FDUS. Consequently, FDUS achieves a closer approximation to the ideal classification boundary than the boundaries generated by these methods. Hence, FDUS demonstrates superior performance. Based on the above analysis of the four scenarios, FDUS offers the following advantages when evaluated against the three indicators. Firstly, it effectively handles data contaminated by noise and outliers. Secondly, it successfully captures the features of majority-class samples with complex distributions, leading to a model that more closely aligns with the ideal classification boundary. Thirdly, as demonstrated by the data sets used in the experiment, FDUS is capable of handling highly imbalanced data sets. ### Stability Comparison of Algorithms

We used 20 results pertaining to each algorithm to draw box plots and examine the stability of these algorithms. As is evident in Figure 7, the FDUS method proposed in this paper is the most stable of all the algorithms. In the box plots of the 20 data sets, the area in which the FDUS results are scattered is very small, and almost all of them form a straight line; this shows that the stability of the algorithm is excellent. Other algorithms, however, mostly show the characteristics of larger scattered areas in these data sets; this is because these algorithms all use random sampling methods that lead to unstable results. Although our method also uses random sampling, the result of each experiment remains immensely stable. This shows that an optimal value for FDUS exists, and that the algorithm can find the optimal value or the approximate optimal value. This phenomenon coincides with the assumption that an optimal value exists in FDUS ensemble learning, which we have proved by testing our hypothesis. \begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Data Set** & **FDUS** & **Tomek** & **USS** & **RUS** & **RBU** & **Centers** & **CBIS** & **UA-KF** \\ \hline abalone-17\_vs\_7-8-9-10 & 0.7463 & 0.7193 & 0.6571 & 0.7087 & 0.7091 & 0.6939 & 0.7347 & 0.7047 \\ car-vgood & 0.8001 & 0.3788 & 0.4310 & 0.8912 & 0.8634 & 0.7716 & 0.7812 & 0.7156 \\ dermatology-6 & 1 & 0.9713 & 0.9702 & 0.9021 & 0.9013 & 0.9675 & 0.9321 & 0.9693 \\ eco1i & 0.7983 & 0.8396 & 0.7393 & 0.8312 & 0.8387 & 0.8279 & 0.8633 & 0.8083 \\ glass-0-1-6\_vs\_4-5-6 & 0.8595 & 0.8845 & 0.7536 & 0.8762 & 0.8186 & 0.7977 & 0.9116 & 0.8882 \\ glass-0-1-6\_vs\_5 & 1 & 0.8294 & 0.7706 & 0.8121 & 0.8438 & 0.8559 & 0.8473 & 0.8173 \\ iris0 & 0.9897 & 0.9425 & 0.9369 & 0.9418 & 0.9033 & 0.9566 & 0.9159 & 0.9159 \\ kr-vs-k-three\_vs\_eleven & 0.9657 & 0.8799 & 0.6561 & 0.9191 & 0.9012 & 0.9411 & 0.8991 & 0.9631 \\ led?digit-0-2-4-5-6-7-8-9\_vs\_1 & 0.8431 & 0.8263 & 0.6445 & 0.8661 & 0.7516 & 0.8390 & 0.8659 & 0.9001 \\ new-thyroid1 & 0.9871 & 0.9255 & 0.8466 & 0.9163 & 0.9503 & 0.9431 & 0.9497 & 0.8097 \\ page-blocks0 & 0.8974 & 0.8145 & 0.8657 & 0.8081 & 0.8985 & 0.8795 & 0.9002 & 0.8146 \\ segment0 & 0.9879 & 0.9288 & 0.8583 & 0.9015 & 0.9427 & 0.9405 & 0.9589 & 0.8889 \\ shuttle-c0-vs-c4 & 0.9617 & 0.9047 & 0.8619 & 0.8432 & 0.9501 & 0.9572 & 0.9144 & 0.8444 \\ vehicle0 & 0.9367 & 0.9286 & 0.8101 & 0.8334 & 0.8987 & 0.9068 & 0.9278 & 0.8778 \\ vehicle2 & 0.9042 & 0.8978 & 0.6048 & 0.8971 & 0.8421 & 0.8525 & 0.9401 & 0.9073 \\ vowel0 & 0.9064 & 0.8822 & 0.8684 & 0.7561 & 0.8872 & 0.8777 & 0.8728 & 0.8028 \\ winequality-white-3\_vs\_7 & 0.9082 & 0.7208 & 0.7379 & 0.7427 & 0.7686 & 0.6905 & 0.7462 & 0.6962 \\ wisconsin & 0.8974 & 0.8712 & 0.8681 & 0.8341 & 0.8762 & 0.8248 & 0.8208 & 0.7508 \\ yeast-0-5-6-7-9\_vs\_4 & 0.7172 & 0.7014 & 0.6801 & 0.7055 & 0.7126 & 0.6898 & 0.7074 & 0.7002 \\ yeast5 & 0.9573 & 0.8815 & 0.8976 & 0.7839 & 0.9242 & 0.9101 & 0.9076 & 0.8076 \\
**Average** & 0.9022 & 0.8364 & 0.7729 & 0.8385 & 0.8591 & 0.8562 & 0.8699 & 0.8291 \\ \hline \end{tabular}
\end{table}