Yiyu Yao

Relational interpretations of neighborhood operators and rough set approximation operators

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 1-neighborhood systems, is introduced. Three extensions of Pawlak approximation operators are analyzed. Properties of neighborhood and approximation operators are studied, and their connections are examined.

Constructive and algebraic methods of the theory of rough sets

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.

Three-way decisions with probabilistic rough sets

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.

Two views of the theory of rough sets in finite universes

Abstract This paper presents and compares two views of the theory of rough sets. The operator-oriented view interprets rough set theory as an extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are related to the interior and closure operators in topological spaces, the necessity and possibility operators in modal logic, and lower and upper approximations in interval structures. The set-oriented view focuses on the interpretation and characterization of members of rough sets. Iwinski type rough sets are formed by pairs of definable (composed) sets, which are related to the notion of interval sets. Pawlak type rough sets are defined based on equivalence classes of an equivalence relation on the power set. The relation is defined by the lower and upper approximations. In both cases, rough sets may be interpreted by, or related to, families of subsets of the universe, i.e., elements of a rough set are subsets of the universe. Alternatively, rough sets may be interpreted using elements of the universe based on the notion of rough membership functions. Both operator-oriented and set-oriented views are useful in the understanding and application of the theory of rough sets.

A decision theoretic framework for approximating concepts

This paper explores the implications of approximating a concept based on the Bayesian decision procedure, which provides a plausible unification of the fuzzy set and rough set approaches for approximating a concept. We show that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained. On the other hand, if a given concept is approximated by two sets, we can derive both the algebraic and probabilistic rough set approximations. Moreover, based on the well known principle of maximum (minimum) entropy, we give a useful interpretation of fuzzy intersection and union. Our results enhance the understanding and broaden the applications of both fuzzy and rough sets.

Attribute reduction in decision-theoretic rough set models

Rough set theory can be applied to rule induction. There are two different types of classification rules, positive and boundary rules, leading to different decisions and consequences. They can be distinguished not only from the syntax measures such as confidence, coverage and generality, but also the semantic measures such as decision-monotocity, cost and risk. The classification rules can be evaluated locally for each individual rule, or globally for a set of rules. Both the two types of classification rules can be generated from, and interpreted by, a decision-theoretic model, which is a probabilistic extension of the Pawlak rough set model. As an important concept of rough set theory, an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table. This paper addresses attribute reduction in decision-theoretic rough set models regarding different classification properties, such as: decision-monotocity, confidence, coverage, generality and cost. It is important to note that many of these properties can be truthfully reflected by a single measure @c in the Pawlak rough set model. On the other hand, they need to be considered separately in probabilistic models. A straightforward extension of the @c measure is unable to evaluate these properties. This study provides a new insight into the problem of attribute reduction.

Information Granulation and Rough Set Approximation

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 coarse‐grained 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 multi‐level granulation, characterized by a special class of equivalence relations, leads to a more general approximation structure. The notion of neighborhood systems is also explored. © 2001 John Wiley & Sons, Inc.

Generalization of Rough Sets using Modal Logics

ABSTRACTThe theory of rough sets is an extension of set theory with two additional unary set-theoretic operators defined based on a binary relation on the universe. These two operators are related to the modal operators in modal logics. By exploring the relationship between rough sets and modal logics, this paper proposes and examines a number of extended rough set models. By the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, various classes of algebraic rough set models can be derived. They correspond to different modal logic systems. With respect to graded and probabilistic modal logics, graded and probabilistic rough set models are also discussed.

Decision-theoretic rough set models

Decision-theoretic rough set models are a probabilistic extension of the algebraic rough set model. The required parameters for defining probabilistic lower and upper approximations are calculated based on more familiar notions of costs (risks) through the well-known Bayesian decision procedure. We review and revisit the decision-theoretic models and present new results. It is shown that we need to consider additional issues in probabilistic rough set models.

The superiority of three-way decisions in probabilistic rough set models

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.