# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 1 Introduction
The above-mentioned methods aim at improving the classification performance by adding some auxiliary methods, such as data sampling methods. However, interpretability is also a significant requirement when constructing classification models. As we all know, fuzzy rules or fuzzy sets constructed by fuzzy algorithms are usually highly interpretable which can reveal the structure of data. For example, the Takagi-Sugeno-Kang fuzzy classifier [26], a famous two-stage fuzzy classification model, obtains the antecedent parameters or initial structure characteristics through clustering algorithms [27; 28]. However, when dealing with imbalanced datasets, the samples in the minority class may be treated as outliers or noise points or be completely ignored when applying clustering methods. This obviously affects the classification performance of imbalanced datasets. Thus, a fuzzy method considering both classification performance and interpretability should be constructed. Information granules are a concept worth considering. Information granules [29] are entities composed of data with similar properties or functional adjacency which are the core concept in the field of granular computing (GrC) [30; 31]. To a certain degree, an information granule is the indistinguishable minimum entity with explicit semantics which can depict data. This implies that data with similar numerical values can be arranged together and abstracted into an information granule with specific fuzzy semantics. This is consistent with the abstract process of building fuzzy sets around data. Since information granules can be built in different formalisms such as one-dimensional intervals, fuzzy sets, rough sets, and so on. Thus, in the view of high dimensionality and the special geometric structure of imbalanced datasets, we can construct two different collections of information granules to depict the characteristics of the majority class and the minority class [32], respectively. For instance, we can see clearly in Figure 1 that the hollow square-shape samples can be abstracted into a fuzzy set composing six cube-shaped information granules and the solid dot-shape ones can be abstracted into a fuzzy set containing only one cube-shaped information granule. Intuitively, the data is divided into two parts, i.e., the majority class is represented by a fuzzy set composed of six cubes and the minority one is represented by a fuzzy set composed of only one cube. This intuitive classification process demonstrates the principle of using information granules to classify imbalanced data. Therefore, the objective of this paper is to generate some information granules and then use them to form the fuzzy rules for the classification of imbalanced data. To achieve the objective, we proposed a Minkowski distance-based granular classification method. In our proposed method, the information granules in different Minkowski spaces are constructed based on a spectrum of Minkowski distance, which can well reveal the geometric structure of both the majority class and minority class of data. In other words, our approach aims at improving classification performance by exploring and understanding the geometric characteristics of imbalanced datasets, rather than just using some auxiliary methods. At the first stage of our Minkowski distance-based granular classification method, the imbalanced dataset is divided into two partitions in light of their class labels, viz., the majority class and the minority class. Each sample in each partition is considered a \"spot\" information granule. At the second stage, a series of bigger union information granules is constructed in each partition through using a Minkowski distance-based merging mechanism. The last stage aims at adjusting the radius of the overlapping information granules and building the granular description for each class of data by uniting the refined information granules contained by each corresponding partition.