93,582 research outputs found
Rough matroids based on coverings
The introduction of covering-based rough sets has made a substantial
contribution to the classical rough sets. However, many vital problems in rough
sets, including attribution reduction, are NP-hard and therefore the algorithms
for solving them are usually greedy. Matroid, as a generalization of linear
independence in vector spaces, it has a variety of applications in many fields
such as algorithm design and combinatorial optimization. An excellent
introduction to the topic of rough matroids is due to Zhu and Wang. On the
basis of their work, we study the rough matroids based on coverings in this
paper. First, we investigate some properties of the definable sets with respect
to a covering. Specifically, it is interesting that the set of all definable
sets with respect to a covering, equipped with the binary relation of inclusion
, constructs a lattice. Second, we propose the rough matroids based
on coverings, which are a generalization of the rough matroids based on
relations. Finally, some properties of rough matroids based on coverings are
explored. Moreover, an equivalent formulation of rough matroids based on
coverings is presented. These interesting and important results exhibit many
potential connections between rough sets and matroids.Comment: 15page
A classification study of rough sets generalization
In the development of rough set theory, many different interpretations and formulations
have been proposed and studied. One can classify the studies of rough
sets into algebraic and constructive approaches. While algebraic studies focus on
the axiomatization of rough set algebras, the constructive studies concern with the
construction of rough set algebras from other well known mathematical concepts and
structures. The constructive approaches are particularly useful in the real applications
of rough set theory. The main objective of this thesis to provide a systematic
review existing works on constructive approaches and to present some additional results.
Both constructive and algebraic approaches are first discussed with respect to
the classical rough set model. In particular, three equivalent constructive definitions
of rough set approximation operators are examined. They are the element based, the
equivalence class based, and the subsystem based definitions. Based on the element
based and subsystem based definitions, generalized rough set models are reviewed and
summarized. One can extend the element based definition by using any binary relations
instead of equivalence relations in the classical rough set model. Many classes
of rough set models can be established based on the properties of binary relations.
The subsystem based definition can be extended in the set-theoretical setting, which
leads to rough set models based on Pawlak approximation space, topological space,
and closure system. Finally, the connections between the algebraic studies, relation
based, and subsystem based formulations are established
Matroidal structure of generalized rough sets based on symmetric and transitive relations
Rough sets are efficient for data pre-process in data mining. Lower and upper
approximations are two core concepts of rough sets. This paper studies
generalized rough sets based on symmetric and transitive relations from the
operator-oriented view by matroidal approaches. We firstly construct a
matroidal structure of generalized rough sets based on symmetric and transitive
relations, and provide an approach to study the matroid induced by a symmetric
and transitive relation. Secondly, this paper establishes a close relationship
between matroids and generalized rough sets. Approximation quality and
roughness of generalized rough sets can be computed by the circuit of matroid
theory. At last, a symmetric and transitive relation can be constructed by a
matroid with some special properties.Comment: 5 page
A comprehensive study of implicator-conjunctor based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis
© 2014 Elsevier B.V. Both rough and fuzzy set theories offer interesting tools for dealing with imperfect data: while the former allows us to work with uncertain and incomplete information, the latter provides a formal setting for vague concepts. The two theories are highly compatible, and since the late 1980s many researchers have studied their hybridization. In this paper, we critically evaluate most relevant fuzzy rough set models proposed in the literature. To this end, we establish a formally correct and unified mathematical framework for them. Both implicator-conjunctor-based definitions and noise-tolerant models are studied. We evaluate these models on two different fronts: firstly, we discuss which properties of the original rough set model can be maintained and secondly, we examine how robust they are against both class and attribute noise. By highlighting the benefits and drawbacks of the different fuzzy rough set models, this study appears a necessary first step to propose and develop new models in future research.Lynn D’eer has been supported by the Ghent University Special Research Fund, Chris Cornelis was partially supported by the Spanish Ministry of Science and Technology under the project TIN2011-28488 and the Andalusian Research Plans P11-TIC-7765 and P10-TIC-6858, and by project PYR-2014-8 of the Genil Program of CEI BioTic GRANADA and Lluis Godo has been partially supported by the Spanish MINECO project EdeTRI TIN2012-39348-C02-01Peer Reviewe
Class Association Rules Mining based Rough Set Method
This paper investigates the mining of class association rules with rough set
approach. In data mining, an association occurs between two set of elements
when one element set happen together with another. A class association rule set
(CARs) is a subset of association rules with classes specified as their
consequences. We present an efficient algorithm for mining the finest class
rule set inspired form Apriori algorithm, where the support and confidence are
computed based on the elementary set of lower approximation included in the
property of rough set theory. Our proposed approach has been shown very
effective, where the rough set approach for class association discovery is much
simpler than the classic association method.Comment: 10 pages, 2 figure
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