Rough set theory is one of important tools of soft computing, and rough approximations are the essential elements in rough set models. However, the existing fuzzy rough set model for set-valued data, which is directly constructed based on a kind of similarity relation, fail to explicitly define fuzzy rough approximations. To solve this issue, in this paper, we propose two types of fuzzy rough a...

In this paper, a different approach to extract the threshold value β of Variable Precision Rough Set (VPRS) applied to continuous information systems is presented. This study combines the Fuzzy Set and Rough Fuzzy Set (RFS) theories to determine the β value of VPRS. The β value was determined by the Fuzzy C-means and relevant Fuzzy theories, for the reason that errors of system classification c...

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 approxim...

In this paper, we discuss rough inclusions defined in Rough Mereology – a paradigm for approximate reasoning introduced by Polkowski and Skowron – as a basis for common models for rough as well as fuzzy set theories. We justify the point of view that tolerance (or, similarity) is the motif common to both theories. To this end, we demonstrate in Sect. 6 that rough inclusions (which represent a h...

Fuzzy sets and rough sets are known as uncertainty models. They are proposed to treat different aspects of uncertainty. Therefore, it is natural to combine them to build more powerful mathematical tools for treating problems under uncertainty. In this chapter, we describe the state of the art in the combinations of fuzzy and rough sets dividing into three parts. In the first part, we first desc...

Urban fringe is the transition zone fine grained with urban and non-urban land cover types. The complex landscape mosaic in this area challenges the land cover classification based on the remote-sensing data. Spectral signatures are not efficient to discriminate all pixels into classes. To improve the recognition and handle the uncertainty, this paper provides a novel integrated approach, based...

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough sets and rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a class of revised...

In this paper, two new operators are introduced for the rough set theory. Using them, two inequalities well known in the rough set theory can now be modified to become equalities. With this change, no information will be lost in the new expressions. Hence, many properties in rough set theory can be improved and in particular, the union, the intersection, and the complement operations can be red...

This paper proposes an approach to deal with the problem of producing a set of cross-level fuzzy certain and possible rules from examples with hierarchical and quantitative attributes. The proposed approach combines the rough-set theory and the fuzzy-set theory to learn. Some pruning heuristics are adopted in the proposed algorithm to avoid unnecessary search. A simple example is also given to ...

As medical images contain uncertainties, there are difficulties in classification of images into homogeneous regions. Fuzzy sets, rough sets and the combination of fuzzy and rough sets plays a prominent role in formalizing uncertainty, vagueness, and incompleteness in diagnosis. Development of hybrid approaches for the segmentation of the magnetic resonance imaging (MRI) with the ability of com...