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33
Relational interpretations of neighborhood operators and rough set approximation operators
 Information Sciences
, 1998
"... This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations. A special class of neighborhood systems, called 1neighborhood systems, is introduced. Three extensions of Pawl ..."
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Cited by 37 (16 self)
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This paper presents a framework for the formulation, interpretation, and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations. A special class of neighborhood systems, called 1neighborhood systems, is introduced. Three extensions of Pawlak approximation operators are analyzed. Properties of neighborhood and approximation operators are studied, and their connections are examined.
Information granulation and rough set approximation
 International Journal of Intelligent Systems
, 2001
"... Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarsegrained views of the universe. Approximation of concepts, represented by subsets of the universe, d ..."
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Cited by 32 (15 self)
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Information granulation and concept approximation are some of the fundamental issues of granular computing. Granulation of a universe involves grouping of similar elements into granules to form coarsegrained views of the universe. Approximation of concepts, represented by subsets of the universe, deals with the descriptions of concepts using granules. In the context of rough set theory, this paper examines the two related issues. The granulation structures used by standard rough set theory and the corresponding approximation structures are reviewed. Hierarchical granulation and approximation structures are studied, which results in stratified rough set approximations. A nested sequence of granulations induced by a set of nested equivalence relations leads to a nested sequence of rough set approximations. A multilevel granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored. 1
A Partition Model of Granular Computing
 LNCS Transactions on Rough Sets
, 2004
"... There are two objectives of this chapter. One objective is to examine the basic principles and issues of granular computing. We focus on the tasks of granulation and computing with granules. From semantic and algorithmic perspectives, we study the construction, interpretation, and representation ..."
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Cited by 25 (6 self)
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There are two objectives of this chapter. One objective is to examine the basic principles and issues of granular computing. We focus on the tasks of granulation and computing with granules. From semantic and algorithmic perspectives, we study the construction, interpretation, and representation of granules, as well as principles and operations of computing and reasoning with granules. The other objective is to study a partition model of granular computing in a settheoretic setting. The model is based on the assumption that a finite set of universe is granulated through a family of pairwise disjoint subsets. A hierarchy of granulations is modeled by the notion of the partition lattice.
Granular Computing using Neighborhood Systems
 Advances in Soft Computing: Engineering Design and Manufacturing
, 1999
"... A settheoretic framework is proposed for granular computing. Each element of a universe is associated with a nonempty family of neighborhoods. A neighborhood of an element consists of those elements that are drawn towards that element by indistinguishability, similarity, proximity, or functionality ..."
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Cited by 21 (12 self)
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A settheoretic framework is proposed for granular computing. Each element of a universe is associated with a nonempty family of neighborhoods. A neighborhood of an element consists of those elements that are drawn towards that element by indistinguishability, similarity, proximity, or functionality. It is a granule containing the element. A neighborhood system is a family of granules, which is the available information or knowledge for granular computing. Operations on neighborhood systems, such as complement, intersection, and union, are defined by extending settheoretic operations. They provide a basis of the proposed framework of granular computing. Using this framework, we examine the notions of rough sets and qualitative fuzzy sets.
Constructive and algebraic methods of the theory of rough sets
 Information Sciences
, 1998
"... This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtai ..."
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Cited by 21 (4 self)
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This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtained from different types of binary relations. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Various classes of rough set algebras are characterized by different sets of axioms. Axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators. 1
A comparative study of fuzzy sets and rough sets
 Information Sciences
, 1998
"... This paper reviews and compares theories of fuzzy sets and rough sets. Two approaches for the formulation of fuzzy sets are reviewed, one is based on manyvalued logic and the other is based on modal logic. Two views of rough sets are presented, setoriented view and operatororiented view. Rough se ..."
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Cited by 16 (2 self)
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This paper reviews and compares theories of fuzzy sets and rough sets. Two approaches for the formulation of fuzzy sets are reviewed, one is based on manyvalued logic and the other is based on modal logic. Two views of rough sets are presented, setoriented view and operatororiented view. Rough sets under setoriented view are closely related to fuzzy sets, which leads to nontruthfunctional fuzzy set operators. Both of them may be considered as deviations of classical set algebra. In contrast, rough sets under operatororiented view are different from fuzzy sets, and may be regarded as an extension of classical set algebra. Key words: approximation operators, fuzzy sets, interval fuzzy sets, modal logic, manyvalued logic, possibleworld semantics, product systems, rough sets. 1
Probabilistic approaches to rough sets
 Expert Systems
, 2003
"... This paper reviews probabilistic approaches to rough sets in granulation, approximation, and rule induction. The Shannon entropy function is used to quantitatively characterize partitions of a universe. Both algebraic and probabilistic rough set approximations are studied. The probabilistic approxim ..."
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Cited by 16 (8 self)
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This paper reviews probabilistic approaches to rough sets in granulation, approximation, and rule induction. The Shannon entropy function is used to quantitatively characterize partitions of a universe. Both algebraic and probabilistic rough set approximations are studied. The probabilistic approximations are defined in a decisiontheoretic framework. The problem of rule induction, a major application of rough set theory, is studied in probabilistic and informationtheoretic terms. Two types of rules are analyzed, the local, low order rules, and the global, high order rules. 1
A comparative study of formal concept analysis and rough set theory in data analysis
 Proceedings of 3rd International Conference on Rough Sets and Current Trends in Computing, RSCTCâ€™04
, 2004
"... Abstract. The theory of rough sets and formal concept analysis are compared in a common framework based on formal contexts. Different concept lattices can be constructed. Formal concept analysis focuses on concepts that are definable by conjuctions of properties, rough set theory focuses on concepts ..."
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Cited by 15 (7 self)
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Abstract. The theory of rough sets and formal concept analysis are compared in a common framework based on formal contexts. Different concept lattices can be constructed. Formal concept analysis focuses on concepts that are definable by conjuctions of properties, rough set theory focuses on concepts that are definable by disjunctions of properties. They produce different types of rules summarizing knowledge embedded in data. 1
Information granulation and approximation in a decisiontheoretical model of rough sets
, 2003
"... Summary. Granulation of the universe and approximation of concepts in the granulated universe are two related fundamental issues in the theory of rough sets. Many proposals dealing with the two issues have been made and studied extensively. We present a critical review of results from existing studi ..."
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Cited by 12 (8 self)
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Summary. Granulation of the universe and approximation of concepts in the granulated universe are two related fundamental issues in the theory of rough sets. Many proposals dealing with the two issues have been made and studied extensively. We present a critical review of results from existing studies that are relevant to a decisiontheoretic modeling of rough sets. Two granulation structures are studied, one is a partition induced by an equivalence relation and the other is a covering induced by a reflexive relation. With respect to the two granulated views of the universe, element oriented and granule oriented definitions and interpretations of lower and upper approximation operators are examined. The structures of the families of fixed points of approximation operators are investigated. We start with the notions of rough membership functions and graded set inclusion defined by conditional probability. This enables us to examine different granulation structures and the induced approximations in a decisiontheoretic setting. By reviewing and combining results from existing studies, we attempt to establish a solid foundation for rough sets and to provide a systematic way for determining the required parameters in defining approximation operators. 1
On generalizing Pawlak approximation operators
 Proceedings of the First International Conference, RSCTCâ€™98, LNAI 1424
, 1998
"... Abstract. This paper reviews and discusses generalizations of Pawlak rough set approximation operators in mathematical systems, such as topological spaces, closure systems, lattices, and posets. The structures of generalized approximation spaces and the properties of approximation operators are anal ..."
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Cited by 12 (4 self)
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Abstract. This paper reviews and discusses generalizations of Pawlak rough set approximation operators in mathematical systems, such as topological spaces, closure systems, lattices, and posets. The structures of generalized approximation spaces and the properties of approximation operators are analyzed. 1