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Approximation-oriented Fuzzy Rough Set Approaches

Inuiguchi M.

Fundamenta Informaticae

2015

Vol. 142, nr 1-4

21--51

In this paper we focus on generalizations of the classical rough set approach to fuzzy environments. There are two aspects of rough set approaches: classification and approximation. In the classification aspect, by rough set approaches we can classify objects into positive and negative examples of a class. On the other hand, in the approximation aspect, by rough set approaches we obtain the lower and upper approximations of a class. The former model works better in the attribute reduction while the latter model works better in the rule induction. In the setting of the classical rough set approach, the lower approximation is nothing but the set of positive examples and the upper approximation is the complementary set of negative examples. However, these equalities do not always hold in the generalized settings. Most of fuzzy rough set models proposed earlier are defined in the classification aspect. The approaches based on those models do not always work well in approximating fuzzy subsets. In this paper we define the fuzzy rough set models in the approximation aspect. We investigate their fundamental properties and demonstrate the advantages of fuzzy set approximation. Finally we consider attribute reduction based on the proposed fuzzy rough set models.

Crisp and Fuzzy Topological Interior and Closure Operators with Inclusion Degree : Theory and Applications

Fashandi H., Peters J.F.

Fundamenta Informaticae

2013

Vol. 122, nr 3

207--225

This article introduces interior and closure operators with inclusion degree considered within a crisp or fuzzy topological framework. First, inclusion degree is introduced in an extension of the interior and closure operators in crisp topology. This idea is then introduced in fuzzy topology by incorporating a relaxed version of fuzzy subsethood. The introduction of inclusion degree leads to a means of dealing with imperfections and small errors, especially in cases such as digital images where boundaries of subsets of an image are not crisp. The properties of the new operators are presented. Applications of the proposed operators are given in terms of rough sets and mathematical morphology.

A Development of Inclusion-degree-based Rough Fuzzy Random Sets

Ha M., Pedrycz W., Zhang A., Fan Y.

Fundamenta Informaticae

2008

Vol. 86, nr 4

481-502

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. In this study, we investigate an idea of rough fuzzy random sets. This construct provides us with a certain generalization of rough sets by introducing the concept of inclusion degree. The underlying objective behind this development is to address the problems which involve co-existing factors of fuzziness and randomness thus giving rise to a notion of the fuzzy random approximation space based on inclusion degree. Some essential properties of rough approximation operators of such rough fuzzy random sets are discussed. Further theoretical foundations for the formation of rules constructed on a basis of available decision tables are offered as well.