# Granular Classification for Imbalanced Datasets: A Minkowski Distance-Based Method
## 1 Introduction  
As one of the key components of machine learning, fuzzy rule-based classifiers [1, 2, 3] explore the features of data by constructing fuzzy sets with strong generalization ability and extracting fuzzy rules with good interpretability. Compared with some traditional classification algorithms, such as decision trees [4, 5], logistic regression [6], naive Bayes [7], neural networks [8], etc., fuzzy classifiers tend to consider both classification performance and interpretability of fuzzy rules when designing models. Fuzzy techniques are also combined with some classic classifiers to deal with human cognitive uncertainties in classification problems. Fuzzy decision trees [9] is a famous one of those, which have been proved as an efficient classification method in many areas [10, 11, 12]. Similarly, there are also some multiple classifier systems, such as fuzzy random forest [13] which are constructed by combining a series of fuzzy decision trees. However, when dealing with the data in practice, such as disease diagnosis [14], protection systems [15], nature disaster predictions [16] and financial problems [17], traditional fuzzy rule-based classifiers could not always extract fuzzy classification rules with good interpretability, which directly leads to the decrease of classification accuracy. This is mainly because the data in the real world often have the characteristic of imbalance [18, 19], that is, the samples of a certain class (called the minority class) is far less than those of other classes (collectively called the majority class). Traditional fuzzy classifiers usually assume that the number of samples contained in each class in the dataset is similar, while the classification for imbalanced data focus on just two classes, viz., the minority class and the majority class. To resolve the issue of classification for imbalanced data, some auxiliary methods have emerged. Data sampling is a common one of them, whose aim is to balance the dataset through increasing the sample size of the minority class (oversampling) or decreasing the sample sizes of majority classes (undersampling) before modeling. A famous oversampling method called \"synthetic minority over-sampling technique (SMOTE)\" is proposed by [20], where the minority class is oversampled through selecting the proper nearest neighbor samples from the minority class. In [21; 22], the SMOTE part is modified before its first stage in which samples weight and k-means are added right before selecting nearest neighbor samples from the minority class of a dataset, respectively. Similarly, there are also a series of studies based on undersampling to help better deal with imbalanced data classification [23]. A number of fuzzy algorithms are combined with these data sampling techniques to handle imbalanced data classification issues. [24] presented a study in which the synergy between three fuzzy rule-based systems and preprocessing techniques (data sampling included) is analyzed in the issue of classification for imbalanced datasets. In [25], some new fuzzy decision tree approaches based on hesitant fuzzy sets are proposed to classify imbalanced datasets. An obvious advantage of the above-mentioned data balancing techniques is that they do not depend on the specific classifier and have good adaptability. However, the data are balanced with compromise, i.e., oversampling would increase the size of training set which may cause overfitting problems and undersampling would remove some useful samples from the training data. This is obviously contrary to the nature of data classification. In addition to the method of data sampling, another commonly used approach focuses on modifying the cost function of classification models so that the penalty weight of the misclassified minority samples in the cost function is greater than the one of the misclassified majority samples. The disadvantage is that there are no universally adequate standards to quantify the penalty weight of the misclassified minority samples.