# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 4 Experiment Studies and Discussion
### Publicly Available Datasets and Comparison with Other Methods
In other words, the union information granule constructed for the majority class contains much more information granules and occupies more Minkowski space than those for the minority class, which ensures that the geometric characteristics of the two highly different classes can be captured separately. Another reason is that the information granules that make up each union information granule are produced based on Minkowski distance with various values of \(p\), which results in the generated information granules having various geometric shapes. By adjusting the value of parameter \(p\), we can explore the geometric structure of imbalanced data from different perspectives, and then achieve a more accurate capture of data features. In summary, the proposed Minkowski distance-based method shows some unique advantages over the other four classification methods in dealing with imbalanced datasets, which are shown in the following aspects. (i) The condition parts of \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Dataset** & **Ichibushi** & \(\mathbf{E}\)**-Algorithm** & **C4.5** & **Smote-HFRBCS** & **Ours** \\ \hline \multicolumn{6}{l}{Datasets with low imbalance ratio (1.5 to 9)} \\ \multicolumn{6}{l}{Ecoli0 vs1} & \(96.66\pm 2.40\) & \(95.25\pm 4.75\) & \(96.78\pm 2.1\) & \(93.63\pm 6.45\) & \(95.09\pm 4.99\) \\ Iris0 & \(100.00\pm 0.00\) & \(100.00\pm 0.00\) & \(98.97\pm 2.29\) & \(100.00\pm 0.00\) & \(100.00\pm 0.00\) \\ Glass0 & \(69.34\pm 8.12\) & \(0.00\pm 0.00\) & \(78.14\pm 2.21\) & \(76.57\pm 8.05\) & \(82.07\pm 6.93\) \\ Vehicle0 & \(74.27\pm 2.12\) & \(41.22\pm 18.29\) & \(90.01\pm 4.22\) & \(88.92\pm 1.96\) & \(94.45\pm 2.60\) \\ New-thyroid1 & \(89.21\pm 12.66\) & \(84.87\pm 7.56\) & \(97.86\pm 2.84\) & \(98.58\pm 2.48\) & \(98.75\pm 3.44\) \\ Glass6 & \(87.77\pm 7.91\) & \(91.63\pm 3.56\) & \(84.19\pm 10.12\) & \(86.95\pm 10.84\) & \(91.87\pm 8.09\) \\ \hline \multicolumn{6}{l}{Mean} & \(86.21\pm 5.54\) & \(68.83\pm 22.36\) & \(90.99\pm 3.96\) & \(90.78\pm 4.96\) & \(93.71\pm 4.34\) \\ \hline \multicolumn{6}{l}{Datasets with high imbalance ratio (higher than 9)} \\ \multicolumn{6}{l}{Yeast2 vs4} & \(71.15\pm 22.57\) & \(80.22\pm 10.01\) & \(85.09\pm 10.16\) & \(89.32\pm 4.18\) & \(90.24\pm 6.22\) \\ Glass2 & \(43.21\pm 16.12\) & \(9.87\pm 22.07\) & \(33.88\pm 29.14\) & \(54.84\pm 20.57\) & \(61.77\pm 12.62\) \\ Glass4 & \(78.12\pm 16.64\) & \(83.38\pm 19.89\) & \(83.34\pm 10.36\) & \(70.39\pm 40.48\) & \(87.06\pm 14.85\) \\ Page-Blocks13 vs4 & \(94.63\pm 5.26\) & \(93.87\pm 9.43\) & \(98.75\pm 0.67\) & \(98.34\pm 1.85\) & \(97.78\pm 4.43\) \\ Yeast1289 vs7 & \(49.22\pm 14.23\) & \(49.24\pm 12.89\) & \(65.21\pm 8.77\) & \(69.13\pm 5.32\) & \(71.67\pm 4.54\) \\ Ecoli0137 vs26 & \(70.89\pm 40.88\) & \(72.17\pm 45.13\) & \(71.21\pm 41.31\) & \(71.36\pm 42.16\) & \(78.13\pm 39.13\) \\ \hline \multicolumn{6}{l}{Mean} & \(67.87\pm 19.28\) & \(64.79\pm 19.90\) & \(72.91\pm 16.74\) & \(75.56\pm 19.09\) & \(81.11\pm 13.63\) \\ \hline \multicolumn{6}{l}{_Alldatasets_} \\ \hline
**Mean** & \(77.04\pm 12.41\) & \(66.81\pm 21.13\) & \(81.95\pm 10.35\) & \(83.17\pm 12.03\) & \(87.41\pm 8.99\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The comparison results obtained by implementing different classifiers on all datasets in terms of the average and standard deviation of \(GM\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Method** & **Parameters** \\ \hline \multicolumn{2}{l}{Ichibushi} & Number of antecedent attributes of rules: \(r<3\) \\ \multicolumn{2}{l}{number restriction of rules in rule base: \(n=30\)} \\ \multicolumn{2}{l}{E-Algorithm} & Number of antecedent attributes of rules: \(r<3\) \\ \multicolumn{2}{l}{number restriction of rules in rule base: \(n=30\)} \\ \multicolumn{2}{l}{C4.5} & The maximum depth of the tree: 5 \\ \multicolumn{2}{l}{Smote-HFRBCS} & linguistic partition terms selector: 0.1 \\ \multicolumn{2}{l}{Number of evaluations: 10,000} \\ \multicolumn{2}{l}{Population length: 61} \\ \hline \hline \end{tabular}
\end{table}