# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 2 Information Granules and Minkowski Distance
### The Distance Measure and Merging Method between Information Granules  
In the field of fuzzy granular classification, it is very necessary to study the location relationship of the constructed information granules in the Minkowski space. It is easy to know that the center and the radius determine the position and size (space occupation) of an information granule. Also, there is an important indicator to measure the location relationship of information granules, that is, the distance between information granules. In the Minkowski space, we anticipate the distance between two information granules, such as \(\Omega_{a}=(\mathbf{v}_{a},\rho_{a})_{p}\) and \(\Omega_{b}=(\mathbf{v}_{b},\rho_{b})_{p}\), can be calculated by  
\[d(\Omega_{a},\Omega_{b})_{p}=\|\mathbf{v}_{a}-\mathbf{v}_{b}\|_{p}-\rho_{a}- \rho_{b}, \tag{6}\]  
where \(\mathbf{v}_{a}\) and \(\mathbf{v}_{b}\) are the centers, \(\rho_{a}\) and \(\rho_{b}\) are the corresponding radii. In our proposed granular classification method, a point-to-granule and bottom-up model is constructed, which involves a process of merging method between information  
Figure 3: The geometric shapes of 2-dimensional and 3-dimensional information granules with three Minkowski distances: (**a**) a circle and a sphere with Euclidean distance (\(p=2\)), (**b**) a diamond and a octahedron with Manhattan distance (\(p=1\)), (**c**) a square and a cube with Chebyshev distance (\(p=\infty\)). granules. For two information granules \(\Omega_{a}=(\mathbf{v}_{a},\rho_{a})_{p}\) and \(\Omega_{b}=(\mathbf{v}_{b},\rho_{b})_{p}\), they can be merged into a bigger information granule \(\Omega_{ab}=(\mathbf{v}_{ab},\rho_{ab})\), where  
\[\mathbf{v}_{ab}=(\mathbf{s}_{a}+\mathbf{s}_{b})/2, \tag{7}\]  
\[\rho_{ab}=\frac{1}{2}\|\mathbf{s}_{1}-\mathbf{s}_{2}\|_{p}. \tag{8}\]  
\(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\) are two points from the surface of both \(\Omega_{a}\) and \(\Omega_{b}\) owning the greatest Minkowski distance among others, which are calculated by  
\[\mathbf{s}_{1}=\mathbf{v}_{a}-\rho_{a}\times\vec{u}_{ab}, \tag{9}\]  
\[\mathbf{s}_{2}=\mathbf{v}_{b}+\rho_{b}\times\vec{u}_{ab}, \tag{10}\]  
where \(\vec{u}_{ab}\) is the unit vector in Minkowski space which records the direction from \(\mathbf{v}_{a}\) to \(\mathbf{v}_{b}\), viz.,  
\[\vec{u}_{ab}=\frac{\mathbf{v}_{b}-\mathbf{v}_{a}}{\|\mathbf{v}_{b}-\mathbf{v }_{a}\|_{p}}. \tag{11}\]  
For convenience, we take the information granules in \(\mathbb{R}^{2}\) space as an example and set the value of \(p\) as \(2\), \(1\) and \(\infty\), respectively. Three couples of information granules are shown in Figure 4. Through the above-mentioned distance calculating and merging methods, for \(\Omega_{1}=((0.1,0.1),0.1)_{2}\) and \(\Omega_{2}=((0.2,0.2),0.15)_{2}\) in Figure 4a, the distance between them is \(d(\Omega_{1},\Omega_{2})_{2}=-0.1086\) and the information granule obtained by merging them is \(\Omega_{12}=((0.167,0.167),0.196)_{2}\). For \(\Omega_{3}=((0.1,0.1),0.05)_{1}\) and \(\Omega_{4}=((0.2,0.2),0.075)_{1}\) in Figure 4b, the distance between them is \(d(\Omega_{3},\Omega_{4})_{1}=0.075\) and the information granule obtained by merging them is \(\Omega_{34}=((0.15625,0.0.15625),0.1625)_{1}\). For \(\Omega_{5}=((0.1,0.1),0.05)_{\infty}\) and \(\Omega_{6}=((0.25,0.2),0.1)_{\infty}\) in Figure 4c, the distance between them is \(d(\Omega_{5},\Omega_{6})_{\infty}=0\) and the information granule obtained by merging them is \(\Omega_{56}=((0.2,0.167),0.15)_{\infty}\). The Proposed Fuzzy Granular Classification Methods for Imbalanced Datasets Based on Minkowski Distance  
In this section, the proposed Minkowski distance-based fuzzy granular classification method is detailed. To better demonstrate the modeling process, a normalized \(n\)-dimensional imbalanced dataset \(\mathbf{D}\) is used.