# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 1 Introduction  
In the current information era, decision-making in industrial systems becomes more and more reliable to data, that is to say, a confident decision relies much on the confidence of data [1]. However, due to various reasons in the real-world context, it is commonly found that collected data contains noise, faults, anomalies, mismatches or other problematical samples [2]. When considering data with these kinds of issues in data-driven modeling, the learned systems would not be justified for decision-making analysis. On the other hand, the basic decision in system analysis is commonly known in a binary way. A complex decision-making process can be regarded as the combination of several binary decisions, e.g., the multi-classes classification [3]. However, the decision results are not always confident for customers due to some data quality problems, such as adversarial data, boundary data, etc. [4]. For example, in a binary classification problem, there could be a situation where the prediction probability of two classes is very close, e.g., 0.49 vs. 0.51. In this case, the general two standard alternatives are not suitable, and they should be included with a warning for the third option, e.g., unclear scenarios in autonomous driving decision systems [5]. Therefore, further investigation from humans is required to guarantee confidence in the final decisions. In a word, considering data quality, a three-way decision would be more meaningful and reliable than a two-way decision in the decision-making of real-world industrial systems. The essence of the three-way decision process is actually extending two-valued logic to three-valued logic, namely considering acceptance, rejection and uncertainty in decisions [6]. This logic is attractive for developing multiple-valued logic models; for example, some interesting applications related to machine learning and software programming [7, 8]. To realize this kind of decision mechanism, fuzzy and rough sets are the two most commonoptions in three-way decisions [9] in the literature review. For example, three-valued logics were described by rough sets directly as well as their extensions in [10]. In [11], an advanced three-way decision model was developed based on the combination of rough sets and game theory. This method proposed to set up optimal parameters to construct probabilistic rough sets for describing the related logic in the three-way decision. Furthermore, the game theory was introduced in the formulation of an objective function aiming to minimize the uncertainty of the probabilistic rough sets in the three-way decision; then, the determined parameters were used for the distinction of certain and uncertain logics [12]. Moreover, the shadow set is a kind of development on fuzzy sets which can be used to generalize the decision-making into a three-valued determination process. For instance, in [13] they were effectively leveraged to support the three-way decision. Moreover, some advanced three-way decision methods were modeled based on the methodology of Granular Computing (GrC) [14,15], which can make use of different granularity to describe the uncertainty in three-valued logics, so they can have more flexibility than generic fuzzy-sets- or rough-sets-based methods. In [16], a granular method was proposed for the problem of record linkage which is regarded as a three-way decision. On the other hand, by summarizing three-way methods in the literature, it is found that uncertainty is the key factor affecting the final performance of decision-making. If more and deeper investigation on uncertainty itself is available, the negative influence of uncertainty in three-way decisions can be reduced. Therefore, describing the uncertain data along with certain data is a necessary technique in uncertainty investigation for three-way decisions. According to existing studies reported in [17], uncertainty is generally taken from the negative parts of the two-way decision.