Some Algebraic Properties of Multigranulations and an Analysis of Multigranular Approximations of Classifications

Abstract

Ever since the introduction of rough sets by Pawlak as a model to capture uncertainty, it has drawn much attention from both theoretical and application point of view. Classifications of universes play very important roles in several fields of study. The study of rough definability of classifications was initiated by Busse. The properties of approximations of classifications were established in the form of four theorems and were used to define the types of classifications. These results were generalised to develop two theorems of necessary and sufficient type were established by Tripathy et al , from which several results including the four theorems of Busse could be derived as corollaries. Recently, rough sets based on Multigranulation were introduced and studied by Qian et al. Also, it has been extended to include incomplete information systems. Many of these results are extended to the multigranular cases. In this paper, we extend the properties of types of classifications to the multigranular context. Also, we introduce some parameters like the accuracy of approximation and the quality of approximation of classifications with respect to Multigranulations. We have obtained interesting criteria under which both types of Multigranulations reduce to single granulation. Also, some algebraic properties of Multigranulations are derived

Approximate Reasoning through Multigranular Approximate Rough Equalities

“…The notations used for the two types of Multigranulations were different in the original papers. But we follow the notations used in a recent paper by Tripathy et al [6,7,8]. That is we use R+S for optimistic Multigranulat ion and RS  for pessimistic Multigranulation, where R and S are two equivalence relations on U.…”

“…Next we present some properties of mu ltigranular rough sets, which shall be used in the proofs of the results of this paper [7,8]. …”

“…Several papers have been published since then. So me of the notable ones are the study of algebraic properties of M GRS [6,7,8], extensions of MGRS in inco mplete informat ion systems [6] and topological properties of M GRS [6]. Many real life examples have been provided in these papers.…”

Study of Covering Based Multi Granular Rough Sets and Their Topological Properties

“…Another such model called pessimistic multigranular rough set was introduced in [9]. Some applications of multigranular rough sets to approximate equalities is made in [18,19] and applications to approximation of classifications is done in [16,17]. In [8] an extension of MGRS, rough set model based on multi tolerance relations in incomplete information systems is developed.…”

“…The initial version of multi-granulation is now called the optimistic multi-granular rough sets [6] after the introduction of a similar notion called the pessimistic multi-granular rough sets [9]. Several fundamental properties of these types of rough sets have been studied [17,18]. Every crisp set X in a universal set U over which an equivalence relation R is defined is associated with a pair of crisp sets called the lower approximation of X with respect to R and the upper approximation of X with respect to R. The lower approximation comprises of those elements of U, which can be definitely be characterized as belonging to X with the knowledge expressed through R and the upper approximation comprises of those elements of U which can possibly be characterized as belonging to X with respect to the knowledge expressed by R. The complement of the lower approximation of X with respect to R, in the upper approximation of X with respect to R is called the boundary of X with respect to R and comprises of the region of uncertainty of X with respect to R.…”