# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 2 Information Granules and Minkowski Distance
### The Emergence and Evaluation of the Minkowski Distance-Based Fuzzy Granular Classification Model  
Through the processing at the above two stages, the two union information granules, i.e., \(\mathbf{\Omega}^{maj}\) and \(\mathbf{\Omega}^{min}\), are produced to describe the key features of the majority class \(\mathbf{D}^{maj}\) and minority class \(\mathbf{D}^{min}\). Note that both \(\mathbf{\Omega}^{maj}\) and \(\mathbf{\Omega}^{min}\) can depict the distribution and location of samples belonging to their corresponding classes, such as \(\mathbf{\Omega}^{maj}\) may occupy much more Minkowski space than \(\mathbf{\Omega}^{min}\), which indicates that the two unions are non-overlapping with each other. Therefore, before establishing the Minkowski distance-based classification model, the overlap between the two union information granules \(\mathbf{\Omega}^{maj}\) and \(\mathbf{\Omega}^{min}\) should be eliminated. If there is an overlap between \(\mathbf{\Omega}^{maj}\) and \(\mathbf{\Omega}^{min}\), there must be their element information granules are overlapping with each other. For instance, for \(\Omega_{k}^{maj}=(\mathbf{v}_{k}^{maj},\rho_{k}^{maj})_{p}\) from \(\mathbf{\Omega}^{maj}\) and \(\Omega_{h}^{min}=(\mathbf{v}_{h}^{min},\rho_{h}^{min})_{p}\) from \(\mathbf{\Omega}^{min}\), if the Minkowski distance between their centers is lee than the sum of their radii, viz.,\(\|\mathbf{v}_{k}^{maj}-\mathbf{v}_{h}^{min}\|_{p}<\rho_{k}^{maj}+\rho_{h}^{min}\), they are overlapped. In order to eliminate the overlap, we let \(\Omega_{k}^{maj}\) and \(\Omega_{h}^{min}\) tangent to each other through scaling their radii into the half of the Minkowski distance between their centers,  
\[\rho_{opt}^{maj}=\rho_{opt}^{min}=\frac{\|\mathbf{v}_{k}^{maj}-\mathbf{v}_{h}^ {min}\|_{p}}{2}. \tag{12}\]In this way, we can eliminate all overlaps between two union information granules \(\mathbf{\Omega}^{maj}\) and \(\mathbf{\Omega}^{min}\) and obtain two optimized ones, i.e., \(\mathbf{\Omega}^{maj}_{opt}=\Omega^{maj}_{opt\,1}\cup\Omega^{maj}_{opt\,2}\cup\cdots \cup\Omega^{maj}_{opt\,M_{1}}\) and \(\mathbf{\Omega}^{min}_{opt}=\Omega^{min}_{opt\,1}\cup\Omega^{min}_{opt\,2}\cup \cdots\cup\Omega^{min}_{opt\,M_{2}}\). Thus they can be tagged with the corresponding majority class and minority class with \"If-Then\" rules. A [PERSON] distance-based granular classification model containing two fuzzy rules are emerged, i.e.,  
\[\text{majority class: If }\mathbf{x}\text{ is }\mathbf{\Omega}^{maj}_{opt}=\Omega^{maj}_{ opt\,1}\cup\Omega^{maj}_{opt\,2}\cup\cdots\cup\Omega^{maj}_{opt\,M_{1}},\text{ then }\mathbf{x}\text{ is }\textit{majority class}; \tag{13}\] \[\text{minority class: If }\mathbf{x}\text{ is }\mathbf{\Omega}^{min}_{opt}=\Omega^{min}_{ opt\,1}\cup\Omega^{min}_{opt\,2}\cup\cdots\cup\Omega^{min}_{opt\,M_{2}},\text{ then }\mathbf{x}\text{ is }\textit{minority class}.\]  
For a given test sample \(\mathbf{x}_{t}\in[0,1]^{n}\), we can calculate its activation levels versus the rule of majority class and the one of minority class in (13). Since the activation levels are usually determined by distances, those can be obtained by judging the position relation between the sample and union information granules, which can be considered by discussing the following three situations:  
1. \(\mathbf{x}_{t}\in\mathbf{\Omega}^{maj}_{opt}\) : In this situation, the sample \(\mathbf{x}_{t}\) is positioned within the boundary of the union information granule \(\mathbf{\Omega}^{maj}_{opt}\) of majority class. We intuitively decide that the distance between the \(\mathbf{x}_{t}\) and \(\mathbf{\Omega}^{maj}_{opt}\) is zero, viz., \(d_{t-maj}=0\)
2. \(\mathbf{x}_{t}\in\mathbf{\Omega}^{min}_{opt}\) : In this situation, the sample \(\mathbf{x}_{t}\) is positioned within the boundary of union information granule \(\mathbf{\Omega}^{min}_{opt}\) of minority class. We intuitively decide that the distance between the \(\mathbf{x}_{t}\) and \(\mathbf{\Omega}^{min}_{opt}\) is zero, viz., \(d_{t-min}=0\)
3.