In this chapter we mainly focus on the study of some topological aspects of rough sets and approximations of classifications. The topological classification of rough sets deals with their types. We find out types of intersection and union of rough sets, New concepts of rough equivalence (top, bottom and total) are defined, which capture approximate equality of sets at a higher level than rough equality (top, bottom and total) of sets introduced and studied by Novotny and Pawlak [23,24,25] and is also more realistic. Properties are established when top and bottom rough equalities are interchanged. Also, parallel properties for rough equivalences are established. We study approximation of classifications (introduced and studied by Busse [12]) and find the different types of classifications of an universe completely. We find out properties of rules generated from information systems and observations on the structure of such rules. The algebraic properties which hold for crisp sets and deal with equalities loose their meaning when crisp sets are replaced with rough sets. We analyze the validity of such properties with respect to rough equivalences. © 2009 Springer-Verlag Berlin Heidelberg.

Balakrushna Tripathy

Department of Analytics

School of Computer Science and Engineering

Vellore Campus

Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

VIT is one of the top ranked private universities in India according to NIRF, THE and QS Rankings.&nbsp;Govt. of India has recognized&nbsp;VIT, Vellore as an&nbsp;Institution of Eminence. This has allowed VIT to take independent quality initiatives and move up in world ranking.

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VIT University

On Approximation of Classifications, Rough Equalities and Rough Equivalences

Studies in Computational Intelligence Rough Set Theory: A True Landmark in Data Analysis

The multigranular rough set (MGRS) models of Qian et al. were extended to put forth covering-based multigranular rough sets (CBMGRS) by Liu et al. in 2012. The equality of sets, which is restrictive and redundant, was extended first in Pawlak (Rough sets: theoretical aspects of reasoning about data. Kluwer, London, 1991) and subsequently in Tripathy (Int J Adv Sci Technol 31:23–36, 2011) to propose four types of rough set-based approximate equalities. These basic concepts of rough equalities have been extended to several generalized rough set models. In this paper, covering-based pessimistic multigranular (CBPMG) approximate rough equivalence is introduced and several of their properties are established. Real life examples are taken for constructing counter examples and also for illustration. We have also discussed how these equalities can be applied in approximate reasoning and our latest proposal is no exception. © Springer Nature Singapore Pte Ltd. 2018.

Smart Computing and Informatics Smart Innovation, Systems and Technologies

Covering-Based Pessimistic Multigranular Approximate Rough Equivalences and Approximate Reasoning

As the notion of equality in mathematics is too stringent and less applicable in real life situations, Novotny and Pawlak introduced approximate equalities through rough sets. Three more types of such equalities were introduced by Tripathy et al. as further generalisations of these equalities. As rough set introduced by Pawlak is unigranular from the granular computing point of view, two types of multigranulations rough sets called the optimistic and the pessimistic multigranular rough sets have been introduced. Three of the above approximate equalities were extended to the multigranular context by Tripathy et al. recently. In this paper, we extend the last but the most general of these approximate equalities to the multigranular context. We establish several direct and replacement properties of this type of approximate equalities. Also, we illustrate the properties as well as provide counter examples by taking a real life example. © Springer India 2015.

Computational Intelligence in Data Mining - Volume 2 Smart Innovation, Systems and Technologies

On Multigranular Approximate Rough Equivalence of Sets and Approximate Reasoning

The notion of rough sets [4] was used by Novotny and Pawlak [1, 2, 3] to introduce the concept of rough equality, in order to incorporate user knowledge in determining the equality of sets, which is a natural phenomenon in day to day life. Even this notion was shown to be deficient and was extended by Tripathy et al [6, 9] to define the notion of rough equivalence of sets, which is more suitable in real life situations. Tripathy [7] added two more approximate equalities to this list in order to complete the kinds of approximate equalities. Algebraic properties of sets with mathematical equality have very little meaning when sets are replaced with rough sets. As a consequence, replacing equality by rough equivalence the algebraic properties were studied by Tripathy et al [10]. In our present work, we have made an analysis of the validity of the algebraic properties with respect to the four kinds of approximate equalities and provide a comparative study to establish that rough equivalence is the best and rough equality is the worst among these approximate equalities with respect to the algebraic properties involving rough sets. The topological characterisations of rough sets leading to four types of rough sets and operations on them [8] are being used in the sequel. We support our descriptions with a running real life example of shares. © 2013 IEEE.

2013 IEEE International Conference on Computational Intelligence and Computing Research

Algebric properties of rough sets using topological characterisations and approximate equalities

Rough sets have been a successful model to capture impreciseness in data. The basic notion rough set introduced by Pawlak is a single granulation model from the granular computing point of view. Recently, this has been extended to multigranular rough set models. Pawlak and Novotny introduced the notions of rough set equalities which is called approximate equalities. These notions of equalities use the user knowledge to decide the equality of sets and hence generate approximate reasoning. In this article, we introduce the concepts of multigranular approximate equalities and establish their properties. Also, the replacement properties, which are obtained by interchanging the bottom equalities with the top equalities, have been established. We provide a real life example for both types of multigranulation and illustrate the interpretation of the approximate equalities through the example. © 2013 IEEE.

2013 Fourth International Conference on Computing, Communications and Networking Technologies (ICCCNT)

On the approximate equalities of multigranular rough sets and approximate reasoning

In an attempt to incorporate user knowledge in order to decide about the equality of sets, the concepts of approximate equalities using rough sets were introduced. These notions have been generalised in several ways and very recently [1] extended four types of approximate equalities using rough fuzzy sets instead of only rough sets. To be precise, a concept of leveled approximate equality was introduced and properties were studied. In this paper we extend this work with case studies to illustrate the applications of the concepts and compare them respectively. We also introduce and discuss the rough measures of basic sets, fuzzy sets and interpret four types of approximate equalities in terms of the accuracy measure as well as rough measures. The analysis had provided a clear distinguish notion in terms of the measures. © 2013 IEEE.

Journal	Data powered by TypesetStudies in Computational Intelligence Rough Set Theory: A True Landmark in Data Analysis
Publisher	Data powered by TypesetSpringer Berlin Heidelberg
ISSN	1860-949X
Open Access	0