Wei-Zhi Wu

Generalized fuzzy rough sets

This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized approximation operators is defined. The connections between fuzzy relations and fuzzy rough approximation operators are examined. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms. Axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators.

Constructive and axiomatic approaches of fuzzy approximation operators

This paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized fuzzy rough (and rough fuzzy, respectively) approximation operators is first defined. The representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented. The connections between fuzzy (and crisp, respectively) relations and fuzzy rough (and rough fuzzy, respectively) approximation operators are further established. In axiomatic approach, various classes of fuzzy approximation operators are characterized by different sets of axioms. The minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators.

Approaches to knowledge reduction based on variable precision rough set model

This paper deals with approaches to knowledge reduction based on variable precision rough set model. The concepts of β lower distribution reduct and β upper distribution reduct based on variable precision rough sets (VPRS) are first introduced. Their equivalent definitions are then given, and the relationships among β lower and β upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated. It is proved that for some special thresholds, β lower distribution reduct is equivalent to the maximum distribution reduct, whereas β upper distribution reduct is equivalent to the possible reduct. The judgement theorems and discernibility matrices associated with the β lower and β upper distribution reducts are also established, from which we can obtain the approaches to knowledge reduction in VPRS.

Knowledge acquisition in incomplete information systems: A rough set approach

Abstract This paper deals with knowledge acquisition in incomplete information systems using rough set theory. The concept of similarity classes in incomplete information systems is first proposed. Two kinds of partitions, lower and upper approximations, are then formed for the mining of certain and association rules in incomplete decision tables. One type of “optimal certain” and two types of “optimal association” decision rules are generated. Two new quantitative measures, “random certainty factor” and “random coverage factor”, associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables. The reduction of descriptors and induction of optimal rules in such tables are also examined.

Granular Computing and Knowledge Reduction in Formal Contexts

Granular computing and knowledge reduction are two basic issues in knowledge representation and data mining. Granular structure of concept lattices with application in knowledge reduction in formal concept analysis is examined in this paper. Information granules and their properties in a formal context are first discussed. Concepts of a granular consistent set and a granular reduct in the formal context are then introduced. Discernibility matrices and Boolean functions are, respectively, employed to determine granular consistent sets and calculate granular reducts in formal contexts. Methods of knowledge reduction in a consistent formal decision context are also explored. Finally, knowledge hidden in such a context is unraveled in the form of compact implication rules.

Knowledge reduction in random information systems via Dempster-Shafer theory of evidence

Knowledge reduction is one of the main problems in the study of rough set theory. This paper deals with knowledge reduction in (random) information systems based on Dempster-Shafer theory of evidence. The concepts of belief and plausibility reducts in (random) information systems are first introduced. It is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in (random) information systems. The relative belief and plausibility reducts in consistent and inconsistent (random) decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined.

On characterizations of ( I,T)-fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Various fuzzy generalizations of rough approximations have been made in the literature. This paper proposes a general framework for the study of (I,T)-fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper generalized fuzzy rough approximation operators, determined by an implicator I and a triangular norm T, is first defined. Basic properties of (I,T)-fuzzy rough approximation operators are investigated. The connections between fuzzy relations and fuzzy rough approximation operators are further established. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, (I,T)-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and I-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, an open problem proposed by Radzikowska and Kerre in (Fuzzy Sets and Systems 126 (2002) 137) is solved.

Approaches to knowledge reductions in inconsistent systems

This article deals with approaches to knowledge reductions in inconsistent information systems (ISs). The main objective of this work was to introduce a new kind of knowledge reduction called a maximum distribution reduct, which preserves all maximum decision classes. This type of reduction eliminates the harsh requirements of the distribution reduct and overcomes the drawback of the possible reduct that the derived decision rules may be incompatible with the ones derived from the original system. Then, the relationships among the maximum distribution reduct, the distribution reduct, and the possible reduct were discussed. The judgement theorems and discernibility matrices associated with the three reductions were examined, from which we can obtain approaches to knowledge reductions in rough set theory (RST).

Neighborhood operator systems and approximations

This paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k-step neighborhood systems. Based on a binary relation on a finite universe, six families of binary relations are obtained, and the corresponding six classes of k-step neighborhood systems are derived. Extensions of Pawlaks rough set approximation operators based on such neighborhood systems are proposed. Properties of neighborhood operator systems and rough set approximation operators are investigated, and their connections are examined.

A rough set approach for the discovery of classification rules in interval-valued information systems

A novel rough set approach is proposed in this paper to discover classification rules through a process of knowledge induction which selects decision rules with a minimal set of features for classification of real-valued data. A rough set knowledge discovery framework is formulated for the analysis of interval-valued information systems converted from real-valued raw decision tables. The minimal feature selection method for information systems with interval-valued features obtains all classification rules hidden in a system through a knowledge induction process. Numerical examples are employed to substantiate the conceptual arguments.