## 5 Experimental Results
### Influence of Model Parameters on Classification Performance
#### 5.5.2 Influence of the Resampling Rate  
This section aims to study the influence of the resampling rate \(a\) on margin-based bagging performance in imbalanced classification. We first employ the following example to illustrate our experimental design. The maximum value of the resampling rate \(a\) should be equal to or less than 100. When the size of \(A\), the associated set of \(a\) values, is set to 5, the elements of \(A\) are \(\{20,40,60,80,100\}\), i.e., the range of \(a\) is 20-100. When \(A=\{100\}\), our margin based method becomes similar to _UnderBagging_. In this experiment, the size \(T\) of the bagging ensemble is set to 100 and the tested number of elements in \(A\) is set from 1 to 40. Figures 6-8 exhibits the optimal range of \(a\) which respectively lead to the best _average accuracy, F-measure_ and _minimum accuracy per class_ for each of the four margin definitions, on all the data sets. Almost all the classification results are improved compared with those of Tables 3-5. The best increase in average accuracy is about 1.5% for most data. The best increase in minimum accuracy per class is about 10% for datasets _Covtype_, _Statlog_ and _Vehicle_. Hence, it is interesting to further optimize our algorithm by the selection of an optimal resampling range. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Data** & **Bagging** & **Under-Bagging** & **SMOTE-Bagging** & **Max-Margin** & **Unsupervised Max-Margin** & **Sum-Margin** & **Unsupervised Max-Margin** \\ \hline Car & 0.0162 & 0.0787 & 0.0243 & 0.1151 & **0.1153** & 0.1134 & 0.1144 \\ Cleveland & 0.1195 & 0.1642 & 0.1172 & 0.1770 & **0.1793** & 0.1773 & 0.1789 \\ Cotype-data & 0.0155 & 0.0859 &  
Tables 7-9 respectively present the average accuracy, F-measure and minimum accuracy per class, achieved by our margin-based bagging algorithm using respectively max-margin, unsupervised max-margin, sum-margin and unsupervised sum-margin with optimal resampling ranges, on all the data sets. The exhibited results correspond to the classification results presented in Figures 6-8. From these tables, we can see that sum margins obtain slightly better results compared with max margins in terms of the three metrics. Figure 5: Evolution of the average accuracy, F-measure and minimum accuracy per class according to the ensemble size. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Max-Margin** & **Unsupervised Max-Margin** & **Sum-Margin** & **Unsupervised Sum-Margin** \\ \hline Car & **94.1** & 93.7 & 93.8 & 93.9 \\ Cleveland & 30.6 & 28.0 & **31.0** & 29.4 \\ Covtype.data & 67.8 & **68.1** & 67.9 & **68.1** \\ Glass & 94.3 & 94.3 & 94.3 & 94.3 \\ Hayes-roth & **84.3** & 83.8 & **84.3** & 83.4 \\ Newthyroid & 95.8 & 95.7 & **95.9** & 95.1 \\ Optdigit & 90.5 & **90.9** & **90.9** & **90.9** \\ Page-blocks & 94.9 & **95.0** & **95.0** & **95.0** \\ Penbased & 93.0 & 93.2 & 93.3 & **93.6** \\ Pendigit & **91.8** & 91.5 & **91.8** & 91.3 \\ Segment & 94.1 & 94.2 & **94.4** & 94.1 \\ Statlog & 82.7 & 82.8 & **83.0** & 82.8 \\ Urbanlandcover & 75.7 & 76.1 & 75.7 & **77.0** \\ Vehicle & 77.1 & **77.5** & 77.0 & **77.5** \\ Wilt & **96.0** & **96.0** & 95.6 & **96.0** \\ Wine & 99.0 & 99.0 & 99.1 & **99.4** \\ Winequality-red & 34.0 & 34.4 & 34.5 & **34.8** \\ Winequality-white & 41.3 & 40.6 & **42.4** & 42.0 \\ \hline Mean accuracy & 79.8 & 79.7 & **80.0** & 79.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average accuracy of margin-based bagging involving four margins with optimal resampling range.