19,948 research outputs found
Covering rough sets based on neighborhoods: An approach without using neighborhoods
Rough set theory, a mathematical tool to deal with inexact or uncertain
knowledge in information systems, has originally described the indiscernibility
of elements by equivalence relations. Covering rough sets are a natural
extension of classical rough sets by relaxing the partitions arising from
equivalence relations to coverings. Recently, some topological concepts such as
neighborhood have been applied to covering rough sets. In this paper, we
further investigate the covering rough sets based on neighborhoods by
approximation operations. We show that the upper approximation based on
neighborhoods can be defined equivalently without using neighborhoods. To
analyze the coverings themselves, we introduce unary and composition operations
on coverings. A notion of homomorphismis provided to relate two covering
approximation spaces. We also examine the properties of approximations
preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin
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
Geometric lattice structure of covering and its application to attribute reduction through matroids
The reduction of covering decision systems is an important problem in data
mining, and covering-based rough sets serve as an efficient technique to
process the problem. Geometric lattices have been widely used in many fields,
especially greedy algorithm design which plays an important role in the
reduction problems. Therefore, it is meaningful to combine coverings with
geometric lattices to solve the optimization problems. In this paper, we obtain
geometric lattices from coverings through matroids and then apply them to the
issue of attribute reduction. First, a geometric lattice structure of a
covering is constructed through transversal matroids. Then its atoms are
studied and used to describe the lattice. Second, considering that all the
closed sets of a finite matroid form a geometric lattice, we propose a
dependence space through matroids and study the attribute reduction issues of
the space, which realizes the application of geometric lattices to attribute
reduction. Furthermore, a special type of information system is taken as an
example to illustrate the application. In a word, this work points out an
interesting view, namely, geometric lattice to study the attribute reduction
issues of information systems
A framework for modelling linear surface waves on shear currents in slowly varying waters
We present a theoretical and numerical framework -- which we dub the Direct
Integration Method (DIM) -- for simple, efficient and accurate evaluation of
surface wave models allowing presence of a current of arbitrary depth
dependence, and where bathymetry and ambient currents may vary slowly in
horizontal directions. On horizontally constant water depth and shear current
the DIM numerically evaluates the dispersion relation of linear surface waves
to arbitrary accuracy, and we argue that for this purpose it is superior to two
existing numerical procedures: the piecewise-linear approximation and a method
due to \textit{Dong \& Kirby} [2012]. The DIM moreover yields the full
linearized flow field at little extra cost. We implement the DIM numerically
with iterations of standard numerical methods. The wide applicability of the
DIM in an oceanographic setting in four aspects is shown. Firstly, we show how
the DIM allows practical implementation of the wave action conservation
equation recently derived by \textit{Quinn et al.} [2017]. Secondly, we
demonstrate how the DIM handles with ease cases where existing methods
struggle, i.e.\ velocity profiles changing direction with
vertical coordinate , and strongly sheared profiles. Thirdly, we use the DIM
to calculate and analyse the full linear flow field beneath a 2D ring wave upon
a near--surface wind--driven exponential shear current, revealing striking
qualitative differences compared to no shear. Finally we demonstrate that the
DIM can be a real competitor to analytical dispersion relation approximations
such as that of \textit{Kirby \& Chen} [1989] even for wave/ocean modelling.Comment: 25 pages, 8 figures, 1 table, submitted to J. Geophys. Res.: Ocean
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