# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 1 Introduction
For example, in a two-classes (denoted as ClassA and ClassB) classification problem, the uncertain region is formed by data points of both not-ClassA and not-ClassB, namely the negative regions of ClassA and ClassB. To describe these uncertainty regions, fuzzy or rough sets are applied in three-way decision methods. Then, probability and fuzzy membership are utilized as two common indicators expressing uncertainty in the final decision-making. Compared with fuzzy-/rough-set-based methods, granules via GrC seem more natural to describe the uncertainty regions [18], especially in decision-making with multiple-level mechanisms [19]. For example, in a classification problem, the generic two-way decision requires finding an optimal threshold on decision indicators to divide the output space into two regions representing two classes. For three-way decisions, the indicator value of uncertain data would lie around the threshold of two-way decisions, e.g., 0.5 of probability or membership. By making use of GrC, we can easily extend the threshold to a granular interval for three regions' descriptions. By optimizing this granular interval to have a high coverage of uncertain data, the performance of three-way decisions would be improved. According to the above descriptions, in this paper, we develop a granular method for three-way decisions. Firstly, to enhance the data description ability, this paper proposed utilizing information granules (IGs) instead of fuzzy or rough sets [20] in the description of certain and uncertain data in data space directly. Meanwhile, to express the uncertainty of data, information entropy is introduced in the process of GrC to construct granules with better information description ability. Then, based on these constructed granular descriptors, fuzzy rules are formed and used to guide the final three-way decision-making in the following way. For example, a rule via the uncertain descriptor is formed to distinguish uncertain data for \"clerical investigation\". Moreover, certain data is also described by some granules and recognized into different classes via the formed rules in a generic two-way decision process. With the help of the proposed method, three contributions are obtained in this paper. 1. A three-way decision method based on granular classification rules is proposed for decision-making in this paper. By making use of granules' advantages on data description [21,22], this method combines GrC and FCM (Fuzzy C-Means) [23] to construct uncertain data descriptors and to guide the formation of classification rules in three-way decision-making. As we know, FCM is a commonly used algorithm for data clustering. Here, it is a novelty developed for decision-making with the help of granules for data description and validated feasible to solve the uncertainty investigation in classification problems;
2. Three-way decisions in various scenarios are studied via the proposed method. For example, three-way decisions in one-dimensional and high-dimensional data spaces are separately discussed, including the granular descriptors construction, rules formation and the final decision-making process. All implementations are demonstrated feasible and effective through experiments on synthetic and public data;
3. Good performances are verified in uncertainty data descriptions and three-way decisions. In this paper, firstly, several useful evaluation metrics are developed for uncertainty description. Then, both synthetic data and publicly available data are applied to evaluate the performance of the proposed three-way decision method. By considering different parameters in constructing uncertainty granules, the performance of decision-making in classification problems is discussed. The feasibility and effectiveness of the proposed method are validly demonstrated. Besides the introduction part, Section 2 describes the problem of uncertain data in the decision-making process of classification problems. Section 3 describes the processes of how to construct information granules for describing uncertainty.