# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 4 Experiment Studies and Discussion
### Synthetic Datasets  
Two synthetic datasets showing imbalanced characteristics and unique geometrical structures are used, see Figure 6. They are generated in the following way. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Positive (Minority) Prediction** & **Negative (Majority) Prediction** \\ \hline Positive (minority) class & True positive (TP) & False negative (FN) \\ Negative (majority) class & False positive (FP) & True negative (TN) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Confusion matrix for a two-class problem. 1. Moon-Blob dataset. This dataset contains 420 samples and two classes. One class, as the majority class, shows a shape of a moon and contains 400 samples. It is governed as follows: \[ \begin{cases}x_{1}=cos(\theta)+\delta\\ x_{2}=cos(\theta)+\delta\end{cases} \] (18) where \(\theta\) ranges in \([0,\pi]\) and \(\delta\) is a noise variable in a normal distribution \(N(0,0.15)\). The samples of this Moon class are marked in red in Figure 6a. Another class, the minority class, shows a shape of a blob and contains 20 samples. The samples are randomly generated with regard to the normal distribution with the mean vector \(\mu=[0,0.42]\) and the covariance matrix \(\Sigma= \begin{bmatrix}0.20&0\\ 0&0.20\end{bmatrix} \), which are marked in blue in Figure 6b. 2. Circles dataset. It is a two-dimensional dataset containing 420 samples with two classes. The majority class contains 400 samples, and governed as follows: \[ \begin{cases}x_{1}=\alpha_{1}cos(\theta)+\delta\\ x_{2}=\alpha_{1}cos(\theta)+\delta\end{cases} \] (19) The minority class contains 20 samples, and governed as follows: \[ \begin{cases}x_{1}=\alpha_{2}cos(\theta)+\delta\\ x_{2}=\alpha_{2}cos(\theta)+\delta\end{cases} .\] (20)  
\(\theta\) ranges in \([0,2\pi]\), \(\alpha_{1}=0.5\) and \(\alpha_{2}=1.0\) are scale factors and \(\delta\) is s noise following a normal distribution \(N(0,0.15)\). In Figure 6b, the majority class is represented in red points and the minority class is in blue. The above two imbalanced datasets are generated to validate the feasibility of the Minkowski distance-based method. The experimental results on the two synthetic imbalanced datasets are presented in Tables 2 and 3, where \(eps\) stands for the radius threshold, \(p\) stands for the parameter for calculating Minkowski distances, and the symbol \(GM\) contains the average value and the standard deviation of the geometric mean, referring to (17), delivered by the models on the testing sets. In individual tables, the results with the highest value of average value are highlighted in bold-face. Figure 6: The visualization of synthetic datasets involved in the experiments: (**a**) Moon-Blob dataset, (**b**) Circles dataset. For Moon-Blob dataset, when the value of Minkowski parameter \(p\) equals 1 and the radius threshold \(eps\) is set as 0.10, the value of \(GM\) reaches its maximum, say, 94.54%. Refer to Figure 7a, we can see clearly that the majority class are exactly covered by the red diamond-shape information granules. The union information granule represents a moon shape just matching the original distribution of the samples in the majority class. In contrast, much less blue information granules are constructed for capturing the shape of the minority class. The corresponding classification decision boundary with \(p=1\) and \(eps=0.1\) is presented in Figure 7b. The darker the color of an area in Figure 7b, the greater probability the samples in this area belong to the corresponding class. However, when \(p=2\) and \(eps=0.06\), the generated round-shape information granules do not cover the samples in majority class and minority class better than those with \(p=1\) and \(eps=0.1\), see Figure 7c. For Circles dataset, when the value of Minkowski parameter \(p\) is set as \(\infty\) and the one of the radius threshold \(eps\) is set as 0.04, the value of \(GM\) reaches the maximum, say, 95.92%.