3,111 research outputs found
Geometric lattice structure of covering-based rough sets through matroids
Covering-based rough set theory is a useful tool to deal with inexact,
uncertain or vague knowledge in information systems. Geometric lattice has
widely used in diverse fields, especially search algorithm design which plays
important role in covering reductions. In this paper, we construct four
geometric lattice structures of covering-based rough sets through matroids, and
compare their relationships. First, a geometric lattice structure of
covering-based rough sets is established through the transversal matroid
induced by the covering, and its characteristics including atoms, modular
elements and modular pairs are studied. We also construct a one-to-one
correspondence between this type of geometric lattices and transversal matroids
in the context of covering-based rough sets. Second, sufficient and necessary
conditions for three types of covering upper approximation operators to be
closure operators of matroids are presented. We exhibit three types of matroids
through closure axioms, and then obtain three geometric lattice structures of
covering-based rough sets. Third, these four geometric lattice structures are
compared. Some core concepts such as reducible elements in covering-based rough
sets are investigated with geometric lattices. In a word, this work points out
an interesting view, namely geometric lattice, to study covering-based rough
sets
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
NMGRS: Neighborhood-based multigranulation rough sets
AbstractRecently, a multigranulation rough set (MGRS) has become a new direction in rough set theory, which is based on multiple binary relations on the universe. However, it is worth noticing that the original MGRS can not be used to discover knowledge from information systems with various domains of attributes. In order to extend the theory of MGRS, the objective of this study is to develop a so-called neighborhood-based multigranulation rough set (NMGRS) in the framework of multigranulation rough sets. Furthermore, by using two different approximating strategies, i.e., seeking common reserving difference and seeking common rejecting difference, we first present optimistic and pessimistic 1-type neighborhood-based multigranulation rough sets and optimistic and pessimistic 2-type neighborhood-based multigranulation rough sets, respectively. Through analyzing several important properties of neighborhood-based multigranulation rough sets, we find that the new rough sets degenerate to the original MGRS when the size of neighborhood equals zero. To obtain covering reducts under neighborhood-based multigranulation rough sets, we then propose a new definition of covering reduct to describe the smallest attribute subset that preserves the consistency of the neighborhood decision system, which can be calculated by Chen’s discernibility matrix approach. These results show that the proposed NMGRS largely extends the theory and application of classical MGRS in the context of multiple granulations
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