Fuzzy Rough Approximation Research Articles

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.

Rough Fuzzy Set in Incomplete Fuzzy Information System Based on Similarity Dominance Relation

The purpose of this paper is to introduce the concept of dominance-based rough set into the incomplete fuzzy decision system. In such information system, all unknown values are considered as lost, the similarity dominance relation is then used to construct information granules. The lower and upper rough fuzzy approximations in terms of the similarity dominance relation are presented, from which one can derive all at least and at most decision rules from the incomplete fuzzy decision system. Moreover, to obtain the optimal decision rules, we propose two types of knowledge reductions, relative lower and upper approximate reducts. These two reducts are minimal subsets of the condition attributes, which preserve the lower and upper approximate memberships for an object respectively. Some numerical examples are employed to substantiate the conceptual arguments and related patents are also reviewed in the paper.

An Interval-Valued Intuitionistic Fuzzy Rough Set Model

Given a widespread interest in rough sets as being applied to various tasks of data analysis it is not surprising at all that we have witnessed a wave of further generalizations and algorithmic enhancements of this original concept. This paper proposes an interval-valued intuitionistic fuzzy rough model by means of integrating the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory. Firstly, some concepts and properties of interval-valued intuitionistic fuzzy set and interval-valued intuitionistic fuzzy relation are introduced. Secondly, a pair of lower and upper interval-valued intuitionistic fuzzy rough approximation operators induced from an interval-valued intuitionistic fuzzy relation is defined, and some properties of approximation operators are investigated in detail. Furthermore, by introducing cut sets of interval-valued intuitionistic fuzzy sets, classical representations of interval-valued intuitionistic fuzzy rough approximation operators are presented. Finally, the connections between special interval-valued intuitionistic fuzzy relations and interval-valued intuitionistic fuzzy rough approximation operators are constructed, and the relationships of this model and the others rough set models are also examined.

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based ( I , T ) -intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T , lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of ( I , T ) -intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of ( I , T ) -intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.

Rough approximation operators on two universes of discourse and their fuzzy extensions

Based on a crisp binary relation between two universes of discourse, one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper. Based on these induced notions, four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse. Furthermore, basic properties of the new approximation operators are investigated, and the relationships among the operators are also examined. Subsequently, conditions under which some or all of these approximation operators are equivalent are obtained. On the other hand, the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator I and a triangular norm T . Consequently, four pairs of fuzzy rough approximation operators are constructed. Basic properties and comparison of the fuzzy rough approximation operators are discussed. If I is an R-implicator with respect to a t-norm T , then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy T -partition.

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.

On generalized intuitionistic fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak’s rough approximations have been made in the literature over the years. This paper proposes a general framework for the study of relation-based intuitionistic fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation are defined. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established. Finally, an operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators.

Fuzzy rough approximations of process data

This paper concerns the variable precision fuzzy rough set (VPFRS) model with asymmetric bounds. The discussion of the presented approach is preceded by a comparison of the original crisp rough set paradigm to the variable precision crisp rough set model. As a new aspect, a unified form of expressing the lower and upper crisp approximations is considered. It can be applied to defining new fuzzy rough set models. Crucial notions of the VPFRS model are redefined and explained. A new way of determining the upper variable precision fuzzy rough approximation is proposed. The VPFRS model is used for describing and analyzing the control actions which are accomplished by a human operator, who controls a complex dynamic system. The decision model is expressed by means of a decision table with fuzzy attributes. Decision tables are generated by the fuzzification of crisp data based on a set of fuzzy linguistic values of the attributes. A T-similarity relation is chosen for comparing elements of the universe. In an illustrative example, the task of stabilization of the aircraft’s bank angle during a turn maneuver is analyzed.

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.

Set approximations in fuzzy formal concept analysis

Formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems. The purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment. Properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined. Based on both lattice-theoretic and fuzzy set-theoretic operators, two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined. Finally, properties of the rough fuzzy set approximation operators are presented in detail.

( ϑ, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring

In this paper, we consider a ring as a universal set and study ( ϑ, T)-fuzzy rough approximation operators with respect to a TL-fuzzy ideal of a ring. First, some new properties of generalized ( ϑ, T)-fuzzy rough approximation operators are obtained. Then, a new fuzzy algebraic structure – TL-fuzzy rough ideal is defined and its properties investigated. And finally, the homomorphism of ( ϑ, T)-fuzzy rough approximation operators is studied.

Minimization of axiom sets on fuzzy approximation operators

Axiomatic characterization of approximation operators is an important aspect in the study of rough set theory. In this paper, we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators.

A novel approach to fuzzy rough sets based on a fuzzy covering

This paper proposes an approach to fuzzy rough sets in the framework of lattice theory. The new model for fuzzy rough sets is based on the concepts of both fuzzy covering and binary fuzzy logical operators (fuzzy conjunction and fuzzy implication). The conjunction and implication are connected by using the complete lattice-based adjunction theory. With this theory, fuzzy rough approximation operators are generalized and fundamental properties of these operators are investigated. Particularly, comparative studies of the generalized fuzzy rough sets to the classical fuzzy rough sets and Pawlak rough set are carried out. It is shown that the generalized fuzzy rough sets are an extension of the classical fuzzy rough sets as well as a fuzzification of the Pawlak rough set within the framework of complete lattices. A link between the generalized fuzzy rough approximation operators and fundamental morphological operators is presented in a translation-invariant additive group.

Fuzzy rough sets and multiple-premise gradual decision rules

We propose a new fuzzy rough set approach which, differently from most known fuzzy set extensions of rough set theory, does not use any fuzzy logical connectives ( t-norm, t-conorm, fuzzy implication). As there is no rationale for a particular choice of these connectives, avoiding this choice permits to reduce the part of arbitrary in the fuzzy rough approximation. Another advantage of the new approach is that it is based on the ordinal properties of fuzzy membership degrees only. The concepts of fuzzy lower and upper approximations are thus proposed, creating a base for induction of fuzzy decision rules having syntax and semantics of gradual rules. The proposed approach to rule induction is also interesting from the viewpoint of philosophy supporting data mining and knowledge discovery, because it is concordant with the method of concomitant variations by John Stuart Mill. The decision rules are induced from lower and upper approximations defined for positive and negative relationships between credibility degrees of multiple premises, on one hand, and conclusion, on the other hand.

On characterizations of [formula omitted]-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.

Knowledge acquisition in incomplete fuzzy information systems via the rough set approach

Abstract: Machine learning can extract desired knowledge from training examples and ease the development bottleneck in building expert systems. Most learning approaches derive rules from complete and incomplete data sets. If attribute values are known as possibility distributions on the domain of the attributes, the system is called an incomplete fuzzy information system. Learning from incomplete fuzzy data sets is usually more difficult than learning from complete data sets and incomplete data sets. In this paper, we deal with the problem of producing a set of certain and possible rules from incomplete fuzzy data sets based on rough sets. The notions of lower and upper generalized fuzzy rough approximations are introduced. By using the fuzzy rough upper approximation operator, we transform each fuzzy subset of the domain of every attribute in an incomplete fuzzy information system into a fuzzy subset of the universe, from which fuzzy similarity neighbourhoods of objects in the system are derived. The fuzzy lower and upper approximations for any subset of the universe are then calculated and the knowledge hidden in the information system is unravelled and expressed in the form of decision rules.

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.

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.

GENERALIZED FUZZY ROUGH SETS BY CONDITIONAL PROBABILITY RELATIONS

In 1982, Pawlak proposed the concept of rough sets with a practical purpose of representing indiscernibility of elements or objects in the presence of information systems. Even if it is easy to analyze, the rough set theory built on a partition induced by equivalence relation may not provide a realistic view of relationships between elements in real-world applications. Here, coverings of, or nonequivalence relations on, the universe can be considered to represent a more realistic model instead of a partition in which a generalized model of rough sets was proposed. In this paper, first a weak fuzzy similarity relation is introduced as a more realistic relation in representing the relationship between two elements of data in real-world applications. Fuzzy conditional probability relation is considered as a concrete example of the weak fuzzy similarity relation. Coverings of the universe is provided by fuzzy conditional probability relations. Generalized concepts of rough approximations and rough membership functions are proposed and defined based on coverings of the universe. Such generalization is considered as a kind of fuzzy rough set. A more generalized fuzzy rough set approximation of a given fuzzy set is proposed and discussed as an alternative to provide interval-value fuzzy sets. Their properties are examined.