By integrating granular computing with rough set theory, granular rough sets enhance the semantics and effectiveness of decision-making through granule-based representations. Existing research has not thoroughly explored the issues of inducing three-way decision rules with granular rough sets, partly due to the challenge of meaningfully describing granules. To address these gaps, this paper proposes a unified framework for threeway decision models based on granular rough sets. Additionally, we introduce a generalized formulation for granule descriptions. It extends traditional representations to include all possible descriptions within a given domain. Through the lens of the proposed framework and granular descriptions, we formulate a three-way decision model in generalized granular rough sets and further demonstrate its instantiation potential across three specific types of granular spaces: quotient spaces, neighborhood-induced granular spaces, and maximal-clique-induced granular spaces. The effectiveness of the proposed models is illustrated through examples using set-valued information tables and experiments on real-world datasets. The results show that the proposed models have good performance and practical applicability.

Three-way clustering assigns highly uncertain samples to the boundary domains, effectively addressing the problem of misclassification caused by data uncertainty. In numerical attribute information systems, neighborhood rough sets can effectively capture the indiscernibility relations between objects. However, the conventional neighborhood relation suffers from a one-size-fits-all issue due to the fixed neighborhood radius. To solve these issues, we propose an intuitionistic neighborhood and construct a corresponding three-way clustering model. Specifically, we first capture the dual nature and uncertainty of neighborhood relations through the construction of the intuitionistic neighborhood. Then we construct a three-way clustering model with dual and single evaluation functions based on intuitionistic neighborhoods. Finally, we adaptively obtain the optimal threshold pairs by maximizing the clustering effectiveness index. Experiments conducted on twelve datasets demonstrate that our proposed method outperforms baseline methods, showing superior capability in handling the inherent uncertainty in information systems.

We propose a trilevel framework for studying rough sets and granular rough sets by applying the principles of three-way decision as thinking in threes. The framework builds and interprets any model of rough sets at three levels: the binary relations level concerning the relationships between objects, the granular space level concerning granules of objects, namely, sets of objects called granular objects, and the approximation level concerning the approximations of sets of objects by granular objects. We identify and characterize eight classes of rough set models, including Pawlak, covering-based, and granular rough sets. By reviewing the existing studies within the framework, we find that there is a lack of investigations on three classes. To fill in these gaps, we investigate two types of granular spaces induced by any binary relations: neighborhood-induced granular spaces and maximal-clique-induced granular spaces. We examine the properties of the two types of granular space and the properties of rough set approximations in the corresponding two classes of models. We also consider a third class of models of granular rough sets based on granular spaces without referencing a binary relation.

As effective tools for dealing with uncertainty,three-way decision models have been widely applied in the study of conflict analysis.However,existing three-way conflict analysis models are mostly based on single-type conflict information systems,which fail to address the challenges of practical situations where agents may have missing or multiple types of ratings.Furthermore,the existing definitions of alliance sets are typically based on a given agent.Particularly,agents in an alliance set are allied to the given agent,but they are not necessarily allied with each other.To address these issues,this paper proposes a three-way conflict analysis model in an incomplete heterogeneous conflict information system.Additionally,this paper presents the definition and algorithm of a family of maximal consistent alliance interval sets,where agents are allied with each other.Firstly,the incomplete heteroge-neous conflict information system with single-dimensional multi-type ratings is transformed into a two-dimensional fuzzy incomplete conflict information system with two-dimensional single-type ratings.This is achieved by defining the support and opposition degrees for different rating types of agents.Secondly,the agent-based alliance,conflict,and neutral interval sets using optimistic and pessimistic distance functions between agents are defined.Finally,a family of maximal consistent alliance interval sets is defined and obtained by adopting the algorithm of maximal clique enumeration.

Tri-level attribute reduction is an interesting topic that aims to reduce the data dimensionality from different levels and granularity perspectives. However, existing research exhibits limitations, mainly in handling symbolic data, lack of effective reduction algorithms, and scarcity of data experiments and performance evaluations, which would be an obstacle to the further development of tri-level attribute reduction in theory and application. Hence, we systematically investigate tri-level attribute reduction based on neighborhood rough sets (NRSs) for numerical data. We first give the class-specific and object-specific attribute reduction conditions based on NRS, respectively. Furthermore, we explore and analyze relationships of tri-level reducts. From the perspective of forward and backward reduction, we propose algorithms of class-specific attribute reduction based on dependency degree, and object-specific reduction algorithms based on inconsistency degree. Finally, we introduce a novel metric to validate the efficiency of specific class and specific object attribute reductions. The results of data experiments show the feasibility and effectiveness of tri-level attribute reduction based on NRS in data analysis.

The theory of three-way decision has contributed to data science in many topics, such as three-way classification, three-way clustering, and three-way feature selection. Most three-way decision models are formulated based on evaluation functions that commonly consider two opposite aspects of positive and negative, alliance and conflict, etc. This idea coincides with the concept of bipolarity, which studies the two opposite poles of positive and negative. However, the connections between bipolarity and three-way decision models have not been well investigated, despite the fact that they share many common features. Therefore, this work explores their relationships and proposes a new bipolar three-way decision model. Firstly, we examine the connections between the polarity theory, including the concepts of unipolarity and bipolarity, and decision models, including the two-way and three-way decision models. The examination suggests a lack of a three-way decision model corresponding to certain types of bipolarity. Thus, secondly, we propose a new bipolar three-way decision model 3WD-(ep,en)2, which considers a pair of a positive evaluation function ep and a negative evaluation function en and applies a pair of thresholds on each function. Finally, we illustrate the usefulness and effectiveness of 3WD-(ep,en)2 in data science through its application in analyzing incomplete data. In particular, we present a computational formulation based on similarity classes and a conceptual formulation based on a conjunctive description language. The two formulations together provide a multi-view understanding of the approach.

In the open-world environment, the incremental updating approaches to attribute reduction based on rough sets are efficient and effective to evaluate and search an optimal subset of attributes from two-dimensionally time-evolving data, which can be interpreted as the complex changes of dynamic data, i.e., four types of combinations induced by the insertion/deletion of objects and the addition/remove of attributes. To avoid the time-consuming and repetitive computation from scratch in such dynamic data, this paper mainly focuses on constructing a unified incremental framework to attribute reduction by the matrix-based accelerated updating strategies. We systematically discuss and present a series of incremental updating mechanisms and algorithms of approximation quality in the neighborhood-based probabilistic rough sets. Besides, a unified framework of dynamic attribute reduction in four situations of changes is proposed to develop the performance of updating reduct. Finally, we report the comparative experiments between the non-incremental and incremental algorithms of reduct to demonstrate the feasibility and efficiency of proposed approaches.

Trisecting agents, issues, and agent pairs are essential topics of three-way conflict analysis. They have been commonly studied based on either a rating or an auxiliary function. A rating function defines the positive, negative, or neutral ratings of agents on issues. An auxiliary function defines the alliance, conflict, and neutrality relations between agents. These functions measure two opposite aspects in a single function, leading to challenges in interpreting their aggregations over a group of issues or agents. For example, when studying agent relations regarding a set of issues, a standard aggregation takes the average of an auxiliary function concerning single issues. Therefore, a pair of alliance +1 and conflict -1 relations will produce the same result as a pair of neutrality 0 relations, although the attitudes represented by the two pairs are very different. To clarify semantics, we separate the two opposite aspects in an auxiliary function into a pair of alliance and conflict functions. Accordingly, we trisect the agents, issues, and agent pairs and investigate their applications in solving a few crucial questions in conflict analysis. Particularly, we explore the concepts of alliance sets and strategies. A real-world application is given to illustrate the proposed models.

针对网络中异常数据类别分布的不平衡性和异常用户检测代价的敏感性,在序贯三支决策框架下,提出了一种基于生成式对抗网络和集成学习模型的异常用户检测方法.利用生成式对抗网络(generative adversarial nets,GANs)模型对异常/非异常数据进行类别平衡,并在多层次多粒度的特征空间下训练LightGBM模型,持续地处理不确定域的样本以识别异常用户.实验结果表明,与传统的机器学习算法相比,该方法在异常用户检测中具有较高的AUC值和较低的检测代价.

The Pawlak model of conflict analysis uses three-valued ratings (i.e., positive, neutral, and negative) of a set of agents on a set of issues. Several extensions to the Pawlak model, namely, rough sets based qualitative and quantitative models, formal concept analysis based quantitative models, and three-way conflict analysis models, have been proposed in recent years. The main objective of this paper is to propose a more general model that unifies these existing models in an evaluation-based framework of three-way decision. The proposed model uses a pair of evaluations, one for support and the other for opposition, for trisecting the set of agents. By considering qualitative and quantitative evaluations, we derive a qualitative model and a quantitative model of three-way conflict analysis, respectively. The corresponding two models built based on rough sets and the corresponding two models built based on formal concept analysis are special cases. A unification of existing models provides insights into a common structure in formulating three-way conflict analysis with different choices of evaluations. We illustrate an application of the three-way conflict analysis model in making development plans for Gansu Province in China.

Although incomplete information is a well studied topic in rough set theory, there still does not exist a general agreement on the semantics of various types of incomplete information. This has led to some confusions and many definitions of similarity or tolerance relations on a set of objects, without a sound of semantical justification. The main objective of this paper is to address semantics issues related to incomplete information. We present a four-step model of Pawlak rough set analysis, in order to gain insights on how an indiscernibility relation (i.e., an equivalence relation) is defined and used under complete information. The results enable us to propose a conceptual framework for studying the similarity of objects under incomplete information. The framework is based on a classification of four types of incomplete information (i.e., "do-not-care value", "partially-known value", "class-specific value", and "non-applicable value") and two groups of methods (i.e., relation-based and granule-based methods) for modeling similarity. We examine existing studies on similarity and their relationships. In spite of their semantics differences, all four types of incomplete information can be uniformly represented in a set-valued table. We are therefore able to have a common conceptual possible-world semantics. Finally, to demonstrate the value of the proposed framework, we examine three-way decisions under incomplete information.

Three-way decision is widely applied with rough set theory to learn classification or decision rules. The approaches dealing with complete information are well established in the literature, including the two complementary computational and conceptual formulations. The computational formulation uses equivalence relations, and the conceptual formulation uses satisfiability of logic formulas. In this paper, based on a brief review of these two formulations, we generalize both formulations into three-way decision with incomplete information that is more practical in real-world applications. For the computational formulation, we propose a new measure of similarity degree of objects as a generalization of equivalence relations. Based on it, we discuss two approaches to three-way decision using α-similarity classes and approximability of objects, respectively. For the conceptual formulation, we propose a measure of satisfiability degree of formulas as a quantitative generalization of satisfiability with complete information. Based on it, we study two approaches to three-way decision using α-meaning sets of formulas and confidence of formulas, respectively. While using similarity classes is a common method of analyzing incomplete information in the literature, the proposed concept of approximability and the two approaches in conceptual formulation point out new promising directions.

The incremental learning methods based on rough set theory are effective in acquiring knowledge in dynamically changing information tables. In this paper, we focus on the effective acquisition of decision rules by incrementally updating approximations when an incomplete information table changes. First of all, we present a four-step model to obtain three-way decision rules in an incomplete information table based on the graded tolerance relation. The first step presents the graded tolerance relation between objects. The second step calculates the degrees of objects belonging to approximations by using fuzzy logic operators. Besides, we propose a relation matrix to calculate the degrees efficiently. The third step gets three-way approximations by applying a pair of thresholds to the degrees. The fourth step obtains three-way decision rules based on the descriptions of objects. According to the four-step model, we find the notion of approximations plays an essential role in rule acquisition. Incrementally updating approximations are an effective method to obtain decision rules when an incomplete information changes. Accordingly, we study the incrementally updating approximations by incrementally updating the relation matrix when changing attributes, objects, and the attribute value of an object. Finally, experimental results illustrate that the incremental methods are more effective than non-incremental methods.

In this work, the axiomatical definition of similarity measure, distance measure and inclusion measure for interval-valued intuitionistic fuzzy soft set (IV IFSSs) are given.An axiomatical definition of entropy measure for IV IFSSs based on distance is firstly proposed, which is consistent with the axiomatical definition of fuzzy entropy of fuzzy sets introduced by De Luca and Termini.By different compositions of aggregation operators and a fuzzy negation operator, we obtain eight general formulae to calculate the distance measures of IV IFSSs based on fuzzy equivalences.Then we discuss the relationships among entropy measures, distance measures, similarity measures and inclusion measures of IV IFSSs.We prove that the presented entropy measures can be transformed into the similarity measures and the inclusion measures of IV IFSSs based on fuzzy equivalences.

The incremental updating of lower and upper approximations under the variation of information systems is an important issue in rough set theory. Many incremental updating approaches with respect to different kinds of indiscernibility relations have been proposed. The grade indiscernibility relation is a fuzzification of classical Pawlak's indiscernibility relation which can characterize the similarity between objects more precisely. Based on fuzzy rough set model, this paper discusses the approaches for dynamically acquiring of the upper and lower approximations with respect to the grade indiscernibility relation when adding and removing an attribute or an object, and changing the attribute value of the object, respectively. Since the approaches are used in succession, they make the approximations can be updated correctly and effectively when any kind of possible change in the information system. Finally, extensive experiments on data sets from University of California, Irvine (UCI) show that the incremental methods effectively reduce the computing time in comparison with the traditional non-incremental method.

信息系统知识约简与知识发现是粗糙集理论的重要研究方向。通过对不完备信息系统中对象的相似性刻画方式以及相应的粗糙集模型进行对比分析,基于限制容差关系提出了一种新的限制相似关系,建立了基于限制相似关系的粗糙集模型,讨论了模型的基本性质及其与已有模型之间的关系。

Rough set approach for knowledge discovery in incomplete information systems has been extensively studied. This paper conduct a further study of valued tolerance relation based rough approximations. We make an analysis of the existing rough approximabilities and propose a new approach for lower (upper) approximability, which is a generalization of Pawlak approximation operators for complete information system. The approach has also been generalized to fuzzy cases. Some basic properties of the approximation operators are examined.

In this paper, we propose a fuzzy relation, called the grade indiscernibility relation, for incomplete information system to characterize the similarity between objects. The rough set model based on the grade indiscernibility relation is presented with its basic properties being discussed. Furthermore, decision rules acquisitions based on the grade indiscernibility relation are analyzed.