# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 2 Information Granules and Minkowski Distance
### The Construction of Information Granules for Each Class  
Considering the large difference between the quantities of samples in the majority class and the minority one for imbalanced datasets, it is unrealistic to use the same amount of information granules to capture the geometric characteristics of both majority and minority classes. It is a feasible option to construct two collections containing different quantities of granules to depict the structure of both two classes. Thus, the task of this section is to elaborate the process of constructing the corresponding collection of information granules on \(\mathbf{D}^{maj}\) and \(\mathbf{D}^{min}\). Take the majority class \(\mathbf{D}^{maj}=\big{\{}\mathbf{x}_{i}=(x_{i1},x_{i2},\cdots,x_{in})\ |\ \mathbf{x}_{i}\in[0,1]^{n},\ i=1,2,\cdots,N_{1}\big{\}}\) as an example. Through the merging mechanism aforementioned in Section 2, a collection of information granules can be constructed by the following steps. At the beginning, each point in \(\mathbf{D}^{maj}\) is regarded as a \"spot\" information granule whose radius equals zero and center is the point itself. Thus, we obtain the initial collection of \(\mathbf{D}^{maj}\), says, \(IG^{maj}=\{\Omega_{1}^{maj},\Omega_{2}^{maj},\cdots,\Omega_{N_{1}}^{maj}\}= \{(\mathbf{x}_{1},0)_{p},(\mathbf{x}_{2},0)_{p},\cdots,(\mathbf{x}_{N_{1}},0) _{p}\}\), where \(p\) is the Minkowski parameter. Next, the merging mechanism is conducted among these \"spot\" information granules in \(IG^{maj}\). Specially, the distances between any two information granules in collection \(IG^{maj}\) are first calculated with (6) and the nearest pair of information granules \(\Omega_{2}^{maj}\) and \(\Omega_{b}^{maj}\), \(a\
eq b,a,b\in 1,2,\cdots,N_{1}\) are obtained. Then, we suppose that \(\Omega_{a}^{maj}\) and \(\Omega_{b}^{maj}\) are merged into a bigger information granule \(\Omega_{ab}^{maj}\). Its radius, i.e., \(\rho_{ab}^{maj}\) can be calculated by (8). Here, a key parameter is imported, that is, the radius threshold \(eps\) to adjust the size of the generated information granules. If this new radius is greater than the predefined \(eps\), i.e., \(\rho_{ab}^{maj}>eps\), the merging mechanism will not be executed and the two information granules \(\Omega_{a}^{maj}\) and \(\Omega_{b}^{maj}\) are maintained in collection \(IG^{maj}\). Otherwise, \(\Omega_{a}^{maj}\) and \(\Omega_{b}^{maj}\) are merged into \(\Omega_{ab}^{maj}=(\mathbf{v}_{ab}^{maj},\rho_{ab}^{maj})_{p}\) with (7) and (8). Usually, when designing a granular model, a right value of \(eps\) is usually selected which makes \(\rho_{ab}^{maj}<=eps\). In this situation, the contents of the collection \(IG^{maj}\) is updated by removing \(\Omega_{a}^{maj}\) and \(\Omega_{b}^{maj}\) and adding \(\Omega_{ab}^{maj}\). So far, two \"spot\" information granules are merged into one new bigger one. After this, the above merging process can be repeated until the radius of the information granule obtained by merging any two information granules in \(IG^{maj}\) is less than \(eps\), which is represented in Algorithm 1. Figure 5: The blueprint of the proposed fuzzy granular classification method based on Minkowski distance when \(p\) is set as \(\infty\). Once the merging process is accomplished, a collection containing \(M_{1}\) merged information granules is obtained, viz. \(IG^{maj}=\{\Omega_{1}^{maj},\Omega_{2}^{maj},\cdots,\Omega_{M_{1}}^{maj}\}\). Through uniting all the elements in the collection \(IG^{maj}\), we obtain a \"union information granule\", says, \(\mathbf{\Omega}^{maj}=\Omega_{1}^{maj}\cup\Omega_{2}^{maj}\cup\cdots\cup \Omega_{M_{1}}^{maj}\). For the minority subset \(\mathbf{D}^{min}\), the corresponding union can also be obtained in the same way, says, \(\mathbf{\Omega}^{min}=\Omega_{1}^{min}\cup\Omega_{1}^{min}\cup\cdots\cup \Omega_{M_{2}}^{min}\).