# 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 c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Data Set**} & \multicolumn{3}{c}{**Kmeans**} & \multicolumn{3}{c}{**DBSCAN**} & \multicolumn{3}{c}{**SVC**} \\ \cline{2-11}  & **AUC** & **F** & **G** & **AUC** & **F** & **G** & **AUC** & **F** & **G** \\ \hline abalone-17\_vs\_7-8-9-10 & 0.8823 & 0.7782 & 0.7463 & 0.8812 & 0.7778 & 0.7447 & 0.8792 & 0.7986 & 0.7612 \\ car-vgood & 0.9739 & 0.7935 & 0.7877 & 0.9750 & 0.7871 & 0.7910 & 0.9648 & 0.7861 & 0.7816 \\ dermatology-6 & 1 & 1 & 0.9817 & 0.9978 & 0.9993 & 0.9958 & 1 & 1 & 0.9813 \\ ecoli1 & 0.9308 & 0.8801 & 0.8526 & 0.9285 & 0.8619 & 0.8357 & 0.9291 & 0.8689 & 0.8435 \\ glass-0-1-6\_vs\_4-5-6 & 0.9958 & 0.8923 & 0.8595 & 0.9961 & 0.8982 & 0.8601 & 0.9912 & 0.8829 & 0.8546 \\ glass-0-1-6\_vs\_5 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ iris0 & 0.9961 & 0.9895 & 0.9897 & 0.9836 & 0.9865 & 0.9867 & 0.9924 & 0.9876 & 0.9877 \\ kr-vs-k-three\_vs\_eleven & 0.9972 & 0.9618 & 0.9657 & 0.9981 & 0.9633 & 0.9672 & 0.9961 & 0.9588 & 0.9562 \\ led7 digit-0-2-4-5-6-7-8- & 0.9869 & 0.8633 & 0.8431 & 0.9910 & 0.8639 & 0.8482 & 0.9923 & 0.8702 & 0.8682 \\
9\_vs\_1 & & & & & & & & & & \\ new-thyroid1 & 0.9875 & 0.9867 & 0.9871 & 0.9855 & 0.9843 & 0.9847 & 0.9817 & 0.9831 & 0.9803 \\ page-blocks0 & 0.9659 & 0.9051 & 0.8974 & 0.9583 & 0.8979 & 0.8928 & 0.9706 & 0.9033 & 0.9011 \\ segment0 & 0.9993 & 0.9874 & 0.9879 & 0.9990 & 0.9862 & 0.9865 & 0.9993 & 0.9874 & 0.9879 \\ shuttle-cl\_vs\_c4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ vehicle0 & 0.9777 & 0.9455 & 0.9367 & 0.9815 & 0.9465 & 0.9386 & 0.9818 & 0.9466 & 0.9391 \\ vehicle2 & 0.9868 & 0.9162 & 0.9042 & 0.9837 & 0.9108 & 0.9027 & 0.9842 & 0.9125 & 0.9034 \\ vowel0 & 0.9857 & 0.8989 & 0.9064 & 0.9816 & 0.8959 & 0.9036 & 0.9861 & 0.8994 & 0.9066 \\ winequality-white- & 0.9600 & 0.9333 & 0.9082 & 0.9612 & 0.9345 & 0.9095 & 0.9588 & 0.9311 & 0.9061 \\ ivis0 & 0.9794 & 0.9183 & 0.8974 & 0.9805 & 0.9201 & 0.8996 & 0.9748 & 0.9166 & 0.8967 \\ k-vs\_k-three\_vs\_4 & 0.8484 & 0.7385 & 0.7172 & 0.8429 & 0.7331 & 0.7138 & 0.8490 & 0.7392 & 0.7181 \\ yeast5 & 0.9862 & 0.9581 & 0.9534 & 0.9836 & 0.9569 & 0.9514 & 0.9878 & 0.9518 & 0.9515 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different ratios of AUC, F-measure (’F’ for short), and G-mean (’G’ for short) with different clustering methods. It can be seen from Table 4 that the final results achieved by the different separation hyperplanes obtained by the SVM, linear classification, and logistic classification also have their own advantages and disadvantages. This shows that the FDUS algorithm is closely related to the initial \(H\). Secondly, it can be seen from Table 3 that, according to an evaluation of the data using the AUC, the final result of the FDUS method based on linear classification and the _car-cyood_ data set is the best; it can also be observed that the method based on logistic regression classification achieves the best results for the two data sets of _ecoli1_ and _yeast5_, and that the SVM-based algorithm shows the best results for the remaining data sets. This proves that the optimization of the separated hyperplane can improve the results of the algorithm. ### Comparison of FDUS with Benchmarks

In this section, the proposed FDUS is compared with seven representative algorithms on 20 numerical data sets. Tables 5-7 present the evaluations of the AUC, F-measure, and G-mean. Each of their results is obtained from the average of 20 operations. We will compare and analyze these results from four aspects, while examining the performance of FDUS. **Cases with complex sample distribution**. Here, FDUS exhibits significant advantages compared to NB-temek and RBU. In terms of the average results obtained in the AUC, F-measure and G-mean evaluations, FDUS exceeds NB-Tomek by 3.18%, 8.50%, and 6.58%, respectively. It also exceeds RBU by 3.84%, 4.47%, and 4.31%, respectively. Compared to NB-Tomek, FDUS exhibits good priority in each data set.