# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 4 Experiment Studies and Discussion
### Synthetic Datasets
It can be seen clearly that union information granules composed of red and blue cubes cover the samples of both majority and minority classes perfectly. Apparently, the configuration of Minkowski parameter \(p\) and the radius threshold \(eps\) dramatically affects the classification performance of the Minkowski distance-based granular models constructed by our proposed method. The main reason behind this is that \(p\) directly determines the geometric shape of the constructed information granules and \(eps\) determines the sizes. In detail, different imbalanced datasets have different sample distributions, which leads to the diversification of the geometric structure of datasets. The Minkowski parameter \(p\) used in our method enriches the geometric shapes of the constructed information granules. This enables the constructed granular classification models to explore the geometric structure of data from multiple perspectives and as accurately as possible, which helps to improve the classification performance. The radius threshold \(eps\) is the key parameter of this method to deal with imbalanced data. Since the minority class occupy tiny spaces, such as three blue diamonds in Figure 7a, \(eps\) transform the spaces occupied by the majority class into a union of some tiny space (similar to the ones occupied by the minority class). The proposed [PERSON] distance-based granular classification method broadens the perspective of classification modeling, and also deftly solves the problem of too few samples in minority class in imbalanced data classification. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\boldsymbol{\mathit{GM}}\) (\%) & \(\boldsymbol{\mathit{eps}}\) & \(\boldsymbol{\mathit{0.02}}\) & \(\boldsymbol{\mathit{0.04}}\) & \(\boldsymbol{\mathit{0.06}}\) & \(\boldsymbol{\mathit{0.08}}\) & \(\boldsymbol{\mathit{0.10}}\) & \(\boldsymbol{\mathit{0.12}}\) \\ \hline \(1\) & \(94.39\pm 6.36\) & \(94.45\pm 6.41\) & \(94.47\pm 6.43\) & \(94.52\pm 6.47\) & \(94.54\pm 6.48\) & \(94.52\pm 6.47\) \\ \(\sqrt{2}\) & \(94.39\pm 6.36\) & \(94.39\pm 6.36\) & \(94.43\pm 6.40\) & \(94.45\pm 6.41\) & \(94.49\pm 6.44\) & \(94.51\pm 6.47\) \\ \(2\) & \(94.39\pm 6.36\) & \(94.39\pm 6.36\) & \(94.43\pm 6.40\) & \(94.45\pm 6.41\) & \(94.39\pm 6.36\) & \(94.43\pm 6.40\) \\ \(2\sqrt{2}\) & \(94.39\pm 6.36\) & \(94.39\pm 6.36\) & \(94.39\pm 6.36\) & \(94.39\pm 6.36\) & \(94.42\pm 6.38\) & \(93.90\pm 7.57\) \\ \(\infty\) & \(94.52\pm 6.47\) & \(94.52\pm 6.47\) & \(93.46\pm 8.61\) & \(93.75\pm 8.17\) & \(93.93\pm 7.88\) & \(94.05\pm 7.68\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results (\(GM\)) obtained by the corresponding classification models with different values of \(p\) and \(eps\) established on Moon-Blob dataset. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\boldsymbol{\mathit{GM}}\) (\%) & \(\boldsymbol{\mathit{eps}}\) & \(\boldsymbol{\mathit{0.02}}\) & \(\boldsymbol{\mathit{0.04}}\) & \(\boldsymbol{\mathit{0.06}}\) & \(\boldsymbol{\mathit{0.08}}\) & \(\boldsymbol{\mathit{0.10}}\) & \(\boldsymbol{\mathit{0.12}}\) \\ \hline \(1\) & \(94.14\pm 11.71\) & \(95.73\pm 9.24\) & \(94.27\pm 10.26\) & \(89.33\pm 22.55\) & \(89.12\pm 21.17\) & \(90.13\pm 10.57\) \\ \(\sqrt{2}\) & \(91.35\pm 11.65\) & \(92.94\pm 9.63\) & \(92.61\pm 8.74\) & \(90.86\pm 10.46\) & \(88.54\pm 12.32\) & \(88.55\pm 12.33\) \\ \(2\) & \(94.64\pm 6.56\) & \(93.00\pm 9.60\) & \(95.55\pm 8.73\) & \(92.45\pm 8.26\) & \(90.55\pm 9.68\) & \(90.34\pm 9.11\) \\ \(2\sqrt{2}\) & \(91.64\pm 6.56\) & \(93.30\pm 6.70\) & \(94.40\pm 8.64\) & \(92.96\pm 9.53\) & \(92.20\pm 8.99\) & \(91.15\pm 9.34\) \\ \(\infty\) & \(91.96\pm 6.56\) & \(95.92\pm 6.10\) & \(90.90\pm 8.38\) & \(91.02\pm 9.34\) & \(90.02\pm 9.54\) & \(90.32\pm 9.15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental results (\(GM\)) obtained by the corresponding classification models with different values of \(p\) and \(eps\) established on Circles dataset.