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. 1.

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In this paper, we argue that granular computing may have many potential applications in knowledge discovery and data mining. Three related basic operations of granular computing are examined: granulation of the universe, characterization of granules, and relationships between granules. Their connections to the tasks of knowledge discovery and data mining are analyzed.

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 decision-theoretic framework. The problem of rule induction, a major application of rough set theory, is studied in probabilistic and information-theoretic terms. Two types of rules are analyzed, the local, low order rules, and the global, high order rules. 1

One of the challenges a decision maker faces is choosing a suitable rough set model to use for data analysis. The traditional algebraic rough set model classifies objects into three regions, namely, the positive, negative, and boundary regions. Two different probabilistic models, variableprecision and decision-theoretic, modify these regions via l,u user-defined thresholds and α, β values from loss functions respectively. A decision maker whom uses these models must know what type of decisions can be made within these regions. This will allow him or her to conclude which model is best for their decision needs. We present an outline that can be used to select a model and better analyze the consequences and outcomes of those decisions. 1.

. The main objective of this chapter is to discuss different approaches to searching for optimal approximation spaces. Basic notions concerning rough set concept based on generalized approximation spaces are presented. Different constructions of approximation spaces are described. The problems of attribute and object selection are discussed. 1 Introduction Rough set theory was proposed [21, 22] as a new approach to processing of incomplete data. Suppose we are given the finite non-empty set U of objects, called the universe. Each object of U is characterized by a description, for example a set of attribute values. In standard rough sets [21, 22] introduced by Pawlak an equivalence relation (reflexive, symmetric and transitive relation) on the universe of objects is defined based on the attribute values. In particular, this equivalence relation is constructed based on the equality relation on attribute values. Many attempts were made to resolve limitations of this approach and many ...

In this paper, we would like to present some logics with semantics based on rough set theory and related notions. These logics are mainly divided into two classes. One is the class of modal logics and the other is that of quantifier logics. For the former, the approximation space is based on a set of possible worlds, whereas in the latter, we consider the set of variable assignments as the universe of approximation. In addition to surveying some well-known results about the links between logics and rough set notions, we also develop some new applied logics inspired by rough set theory.

. Rough set methodology is based on concept (set) approximations constructed from available background knowledge represented in information systems [14]. In many applications only partial knowledge about approximated concepts is given. Hence quite often first a parametrized family of concept approximations is built and next by tuning of the parameters the best, in a sense, approximation is chosen (see e.g. variable precision rough set model [40]) in approximation spaces. In this paper we follow this approach in generalized approximation spaces. We discuss rough set model based on approximation spaces with uncertainty functions and rough inclusions. Both elements of approximation space are parametrized and for the proper application of such model to a particular data set it is necessary to make optimization of the parameters. We discuss basic properties of the mentioned model and also strategies of parameters optimization. We also present different notions of rough relations....

Abstract. Rough sets have been applied to many areas where multiattribute data is needed to be analyzed to acquire knowledge for decision making. Web-based Support Systems (WSS) are a new research area that aims to support human activities and extend human physical limitations of information processing with Web technologies. The applications of rough set analysis for WSS is looked at in this article. In particular, our focus will be on Web-Based Medical Support Systems (WMSS). A WMSS is a support system that integrates medicine practices (diagnosis and surveillance) with computer science and Web technologies. We will explore some of the challenges of using rough sets in a WMSS and detail some of the applications of rough sets in analyzing medical data. 1