Wen-Xiu Zhang

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.

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.

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.

Generalized fuzzy rough approximation operators based on fuzzy coverings

This paper focuses on the generalization of covering-based rough set models via the concept of fuzzy covering. Based on a fuzzy covering of a universe of discourse, two pairs of generalized lower and upper fuzzy rough approximation operators are constructed by means of an implicator I and a triangular norm T. Basic properties of the generalized fuzzy rough approximation operators are investigated. Topological properties of the generalized fuzzy rough approximation operators and characterizations of the fuzzy T-partition by the generalized upper fuzzy rough approximation operators are further established. When fuzzy coverings are a family of R-foresets or R-aftersets of all elements of a universe of discourse with respect to a fuzzy binary relation R, the corresponding generalized fuzzy rough approximation operators degenerate into the fuzzy-neighborhood-oriented fuzzy rough approximation operators. Combining with the fuzzy-neighborhood-operator-oriented fuzzy rough approximation operators, conditions under which some or all of these approximation operators are equivalent are subsequently determined.

Connections between rough set theory and Dempster-Shafer theory of evidence

In rough set theory there exists a pair of approximation operators, the upper and lower approximations, whereas in Dempster-Shafer theory of evidence there exists a dual pair of uncertainty measures, the plausibility and belief functions. It seems that there is some kind of natural connection between the two theories. The purpose of this paper is to establish the relationship between rough set theory and Dempster-Shafer theory of evidence. Various generalizations of the Dempster-Shafer belief structure and their induced uncertainty measures, the plausibility and belief functions, are first reviewed and examined. Generalizations of Pawlak approximation space and their induced approximation operators, the upper and lower approximations, are then summarized. Concepts of random rough sets, which include the mechanisms of numeric and non-numeric aspects of uncertain knowledge, are then proposed. Notions of the Dempster-Shafer theory of evidence within the framework of rough set theory are subsequently formed and interpreted. It is demonstrated that various belief structures are associated with various rough approximation spaces such that different dual pairs of upper and lower approximation operators induced by the rough approximation spaces may be used to interpret the corresponding dual pairs of plausibility and belief functions induced by the belief structures.

Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.

Variable threshold concept lattices

In this paper, the definition of a variable threshold concept lattice is introduced. Based on a Galois connection, three kinds of variable threshold concept lattices, in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold, are defined. The number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice. The three kinds of variable threshold concept lattices are constructed between two crisp sets, between a crisp set and a fuzzy set, and between a fuzzy set and a crisp set. Their properties are analogous to that of the classical concept lattices, and can be induced by the fuzzy concept lattice.

On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of (I,T)-interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T-similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I-lower and T-upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I-lower and T-upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.

Attribute reduction theory of concept lattice based on decision formal contexts

The theory of concept lattices is an efficient tool for knowledge representation and knowledge discovery, and is applied to many fields successfully. One focus of knowledge discovery is knowledge reduction. Based on the reduction theory of classical formal context, this paper proposes the definition of decision formal context and its reduction theory, which extends the reduction theory of concept lattices. In this paper, strong consistence and weak consistence of decision formal context are defined respectively. For strongly consistent decision formal context, the judgment theorems of consistent sets are examined, and approaches to reduction are given. For weakly consistent decision formal context, implication mapping is defined, and its reduction is studied. Finally, the relation between reducts of weakly consistent decision formal context and reducts of implication mapping is discussed.