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A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and, at the same time, a single granular object in the quotient space. The new two-space view leads to more insights into and a deeper understanding of rough set theory. In this paper, we revisit results from rough sets from the two-space perspective and introduce the notions of granular rough sets and probabilistic granular rough sets in the quotient space, as three-way approximations of sets in the ground space. We propose a concept of granular shadowed sets in the quotient space, as three-way approximations of fuzzy sets in the ground space. We formulate a cost-sensitive method to construct a granular shadowed set from a fuzzy set. We show that, when the costs satisfy some conditions, the three granular approximations become the same for the special case where a fuzzy set is in fact a set. [ABSTRACT FROM AUTHOR]

Three-way granular computing is a paradigm of thinking and information processing in three granules. We introduce a framework of three-way granular computing based on a hexagon induced by a trisection of a set. This enables us to unify rough-set concept analysis and formal concept analysis. For an object granule (i.e., a subset of a universal set of objects), we construct three attribute granules. Conversely, for an attribute granule (i.e., a subset of a universal set of attributes), we construct three object granules. Rough-set concept analysis uses one kind of trisections to study four types of disjunctive formal concepts and the associated concept lattices. Formal concept analysis uses another kind of trisections to study four types of conjunctive formal concepts and the associated concept lattices. The eight concept lattices can be grouped into two classes such that four lattices in each class are isomorphic. [ABSTRACT FROM AUTHOR]

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space, where the latter is induced by an equivalence relation on the former. With this two-space understanding, it is possible to lift any concepts and notions from the ground space to the quotient space. The results are granular versions that approximate the original concepts and notions. In this paper, we investigate the problem of lifting a fuzzy set in the ground space to granular fuzzy sets in the quotient space. By applying the principles of three-way decision, we introduce the idea of three-way granular approximations of fuzzy sets in terms of three granular fuzzy sets that represent the two extremes and one middle. The two extremes are given by granular rough fuzzy sets. We present several different ways to interpret and construct a middle. [ABSTRACT FROM AUTHOR]

In rough set theory, a decision class (i.e., a subset of objects) is approximated by three pair-wise disjoint positive, boundary, and negative regions. The concept of three-way decisions is introduced to provide a new interpretation of the three regions. We construct acceptance, non-commitment, and rejection rules, respectively, from the positive, boundary, and negative regions. The notion of class-specific attribute reducts concerns a minimal set of attributes used in constructing such rules. Existing studies on class-specific attribute reducts only consider the positive region and hence only the acceptance rules. In many situations such as medical diagnosis, we are also interested in negative rules or rule-out rules. This motivates the present study on three-way decision perspectives on class-specific attribute reducts. In addition to positive-region based attribute reducts, we study negative-region and positive-and-negative-region based attribute reducts. We investigate relationships among the three types of reducts. Although the three types of reducts are equivalent in consistent decision tables, they are not equivalent in inconsistent decision tables. By extending the framework, we study the three types of class-specific attribute reducts in probabilistic rough set models and their relationships. Finally, we give a general definition of class-specific attribute reducts. [ABSTRACT FROM AUTHOR]

The concept of attribute reducts plays a fundamental role in rough set analysis. There are at least two possibilities to define an attribute reduct. A classification-based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification, namely, the positive regions of all decision classes, in a decision table. A class-specific, class-dependent, or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class. While a classification-based reduct may not work equally well for every decision class, a class-specific attribute reduct is optimally tailored to a particular decision class. However, studies in rough set theory are dominated by classification-based reducts; there is very limited investigation on class-specific reducts. An objective of this paper is to draw attention to class-specific reducts. We systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts. It is possible to derive a class-specific reduct from a classification-based reduct and to derive a classification-based reduct from a family of class-specific reducts. The families of all class-specific reducts provide a pair of lower and upper bounds of the family of all classification-based reducts. Based on a three-way classification of attributes into the pair-wise disjoint sets of core, marginal, and nonuseful attributes, we examine relationships between the corresponding classes of classification-based and class-specific attributes. The union of the sets of class-specific core attributes is the set of classification-based core attributes. It is only possible to obtain an upper bound for the set of classification-based marginal attributes and a lower bound for the set of classification-based nonuseful attributes from the family of the class-specific corresponding sets of attributes. [ABSTRACT FROM AUTHOR]

Based on ideas from formal concept analysis, this paper interprets the notions of RS-definable concepts (i.e., rough-set definable concepts) and the Boolean algebra of RS-definable concepts. We explicitly represent a RS-definable concept as a pair of an extension and an intension, where the extension is a set of objects and the intension is a family of sets of attribute-value pairs called avp-sets. An object in the extension satisfies at least one avp-set in the intension and each avp-set in the intension is satisfied by only objects in the extension. The two-directional connections produce an atomic Boolean algebra of RS-definable concepts, corresponding to the lattice of formal concepts in formal concept analysis. The Boolean algebra of RS-definable concepts is used to define and interpret a subset of objects through a pair of lower and upper approximations. The new formulation emphasizes on an in-depth conceptual understanding of rough-set concept analysis. [ABSTRACT FROM AUTHOR]

There exist several approaches to rough set approximations in a multigranulation space, namely, a family of equivalence relations. In this paper, we propose a unified framework to classify and compare existing studies. An underlying principle is to explain rough sets in a multigranulation space through rough sets derived by using individual equivalence relations. Two basic models are suggested. One model is based on a combination of a family of equivalence relations into an equivalence relation and the construction of approximations with respect to the combined relation. By combining equivalence relations through set intersection and union, respectively, we construct two sub-models. The other model is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations. By using set intersection and union to combine a family of approximations, respectively, we again build two sub-models. As a result, we have a total of four models. We examine these models and give conditions under which some of them become the same. [ABSTRACT FROM AUTHOR]

Abstract: Subsethood measures, also known as set-inclusion measures, inclusion degrees, rough inclusions, and rough-inclusion functions, are generalizations of the set-inclusion relation for representing graded inclusion. This paper proposes a framework of quantitative rough sets based on subsethood measures. A specific quantitative rough set model is defined by a particular class of subsethood measures satisfying a set of axioms. Consequently, the framework enables us to classify and unify existing generalized rough set models (e.g., decision-theoretic rough sets, probabilistic rough sets, and variable precision rough sets), to investigate limitations of existing models, and to develop new models. Various models of quantitative rough sets are constructed from different classes of subsethood measures. Since subsethood measures play a fundamental role in the proposed framework, we review existing methods and introduce new methods for constructing and interpreting subsethood measures. [Copyright &y& Elsevier]

Abstract: We propose a framework for the study of covering based rough set approximations. Three equivalent formulations of the classical rough sets are examined by using equivalence relations, partitions, and σ-algebras, respectively. They suggest the element based, the granule based and the subsystem based definitions of approximation operators. Covering based rough sets are systematically investigated by generalizing these formulations and definitions. A covering of universe of objects is used to generate different neighborhood operators, neighborhood systems, coverings, and subsystems of the power set of the universe. They are in turn used to define different types of generalized approximation operators. Within the proposed framework, we review and discuss covering based approximation operators according to the element, granule, and subsystem based definitions. [Copyright &y& Elsevier]

Abstract: Three-way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier. This paper compares probabilistic three-way decisions, probabilistic two-way decisions, and qualitative three-way decisions of the standard rough set model. It is shown that, under certain conditions when considering the costs of different types of miss-classifications, probabilistic three-way decisions are superior to the other two. [ABSTRACT FROM AUTHOR]

Abstract: The role of topological De Morgan algebra in the theory of rough sets is investigated. The rough implication operator is introduced in strong topological rough algebra that is a generalization of classical rough algebra and a topological De Morgan algebra. Several related issues are discussed. First, the two application directions of topological De Morgan algebras in rough set theory are described, a uniform algebraic depiction of various rough set models are given. Secondly, based on interior and closure operators of a strong topological rough algebra, an implication operator (called rough implication) is introduced, and its important properties are proved. Thirdly, a rough set interpretation of classical logic is analyzed, and a new semantic interpretation of Łukasiewicz continuous-valued logic system Łuk is constructed based on rough implication. Finally, strong topological rough implication algebra (STRI-algebra for short) is introduced. The connections among STRI-algebras, regular double Stone algebras and RSL-algebras are established, and the completeness theorem of rough logic system RSL is discussed based on STRI-algebras. [Copyright &y& Elsevier]

Abstract: The rough set theory approximates a concept by three regions, namely, the positive, boundary and negative regions. Rules constructed from the three regions are associated with different actions and decisions, which immediately leads to the notion of three-way decision rules. A positive rule makes a decision of acceptance, a negative rule makes a decision of rejection, and a boundary rule makes a decision of abstaining. This paper provides an analysis of three-way decision rules in the classical rough set model and the decision-theoretic rough set model. The results enrich the rough set theory by ideas from Bayesian decision theory and hypothesis testing in statistics. The connections established between the levels of tolerance for errors and costs of incorrect decisions make the rough set theory practical in applications. [Copyright &y& Elsevier]

Abstract: This paper proposes a reduct construction method based on discernibility matrix simplification. The method works in a similar way to the classical Gaussian elimination method for solving a system of linear equations. Elementary matrix simplification operations are introduced. Each operation transforms a matrix into a simpler form. By applying these operations a finite number of times, one can transform a discernibility matrix into one of its minimum (i.e., the simplest) forms. Elements of a minimum discernibility matrix are either the empty set or singleton subsets, in which the union derives a reduct. With respect to an ordering of attributes, which is either computed based on a certain measure of attributes or directly given by a user, two heuristic reduct construction algorithms are presented. One algorithm attempts to exclude unimportant attributes from a reduct, and the other attempts to include important attributes in a reduct. [Copyright &y& Elsevier]

Abstract: Probabilistic approaches have been applied to the theory of rough set in several forms, including decision-theoretic analysis, variable precision analysis, and information-theoretic analysis. Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough set approximation operators and present a critical review of existing studies. Intuitively, they are defined based on a pair of thresholds representing the desired levels of precision. Formally, the Bayesian decision-theoretic analysis is adopted to provide a systematic method for determining the precision parameters by using more familiar notions of costs and risks. Results from existing studies are reviewed, synthesized and critically analyzed, and new results on the decision-theoretic rough set model are reported. [Copyright &y& Elsevier]

Abstract: Rough set theory models similarities and differences of objects based on the notions of indiscernibility and discernibility. With respect to any subset of attributes, one can define two pairs of dual relations: the strong indiscernibility and weak discernibility relations, and the weak indiscernibility and strong discernibility relations. The similarities of objects are examined by the indiscernibility relations, and the differences by the discernibility relations, respectively. Alternatively, one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations. One also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations. The consideration of the matrix-counterpart of relations, and the relation-counterpart of matrices, brings more insights into rough set theory. Based on indiscernibility and discernibility, three different types of reducts can be constructed, keeping the indiscernibility, discernibility, and indiscernibility-and-discernibility relations, respectively. Although the indiscernibility reducts have been intensively studied in the literature, the other two types of reducts are relatively new and require more attention. The existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts. An empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts. [Copyright &y& Elsevier]

This paper proposes a reflexive fuzzy β -neighborhood operator by modifying Ma's fuzzy β -neighborhood operators. Then, we use such an operator to build a covering-based variable precision fuzzy rough set (CVPFRS) model that can deal with the issue of misclassifications and perturbations in decision-making problems. By combining the CVPFRS model with two traditional decision-making methods (the PROMETHE method and the DEAS method), we introduce a novel method for addressing multi-attribute decision-making (MADM) problems. An illustrative example is utilized to demonstrate the practicality of the proposed method. The effectiveness of the proposed method is validated by comparing it with existing methods. By virtue of the cross-validation and hypothesis testing, we give an experimental analysis to interpret the validity and stability of the proposed method. [ABSTRACT FROM AUTHOR]

In order to obtain the weights of a set of criteria by means of real-world data, an effective method based on the covering-based variable precision intuitionistic fuzzy rough set (CVPIFRS) models is presented. By combining the CVPIFRS models with the idea of TOPSIS, we propose a decision-making method to effectively settle the complex and changeable bone transplant selections, which is one of typical multi-attribute decision-making (MADM) problems. The sensitivity analysis of the proposed method shows that the approach is highly flexible and can be applied to a wide range of environments by adjusting the values of the intuitionistic fuzzy (IF) variable precision, together with the choice of different IF logical operators. Through a comparison of the proposed method and some existing MADM methods, it is shown that our method is more effective in dealing with these complex and changeable bone transplant selections issues. [ABSTRACT FROM AUTHOR]

Data in real-world applications are typically changing with time and are often incomplete. To address the challenge of processing such dynamic and incomplete data, we propose a model of dynamic probabilistic rough sets with incomplete data. We introduce incremental methods for estimating the conditional probability and present principles for updating probabilistic approximations when adding and removing objects, respectively. Based on the proposed updating strategies, algorithms are designed for dynamically updating probabilistic approximations with incomplete data. We report experimental evaluations of the efficiency and effectiveness of the proposed incremental algorithms for constructing probabilistic rough set approximations in terms of the size of data and updating ratio by comparing with a non-incremental algorithm. The results show that the new algorithms can effectively utilize the previously acquired knowledge, leading to significantly improved performance over a non-incremental algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a semantically sound approach to covering-based rough sets. We recall and elaborate on a conceptual approach to Pawlak's rough set model, in which we consider a two-part descriptive language. The first part of the language is used to describe conjunctive concepts, while in the second part disjunctions are allowed as well. Given the language, we discuss its elementary and definable sets, and we study how the approximation operators can be seen as derived notions of the family of definable sets, which is represented by a Boolean algebra over a partition. Furthermore, we generalize the two parts of the language in order to describe concepts of covering-based rough sets. Unfortunately, the family of definable sets will no longer be represented by a Boolean algebra over a partition, but by the union-closure of a covering. Therefore, only the derived covering-based lower approximations of sets are definable for the generalized language. In addition, it is discussed how the two-part languages are used to construct decision rules, which are used in data mining and machine learning. [ABSTRACT FROM AUTHOR]

Abstract: The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of sets described by a single binary relation on the universe. In the view of granular computing, the classical rough set theory is established through a single granulation. This paper extends Pawlak’s rough set model to a multi-granulation rough set model (MGRS), where the set approximations are defined by using multi equivalence relations on the universe. A number of important properties of MGRS are obtained. It is shown that some of the properties of Pawlak’s rough set theory are special instances of those of MGRS. Moreover, several important measures, such as accuracy measure , quality of approximation and precision of approximation , are presented, which are re-interpreted in terms of a classic measure based on sets, the Marczewski–Steinhaus metric and the inclusion degree measure. A concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in MGRS as well. Finally, we discuss how to extract decision rules using MGRS. Unlike the decision rules (“AND” rules) from Pawlak’s rough set model, the form of decision rules in MGRS is “OR”. Several pivotal algorithms are also designed, which are helpful for applying this theory to practical issues. The multi-granulation rough set model provides an effective approach for problem solving in the context of multi granulations. [Copyright &y& Elsevier]