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

Fuzzy covering based rough sets revisited

In this paper we review four fuzzy extensions of the so-called tight pair of covering based rough set approximation operators. Furthermore, we propose two new extensions of the tight pair: for the first model, we apply the technique of representation by levels to define the approximation operators, while the second model is an intuitive extension of the crisp operators. For the six models, we study which theoretical properties they satisfy. Moreover, we discuss interrelationships between the models

Implicator-Conjunctor Based Models of Fuzzy Rough Sets: Definitions and Properties

Ever since the first hybrid fuzzy rough set model was pro- posed in the early 1990¿s, many researchers have focused on the definition of the lower and upper approximation of a fuzzy set by means of a fuzzy relation. In this paper, we review those proposals which generalize the logical connectives and quantifiers present in the rough set approxima- tions by means of corresponding fuzzy logic operations. We introduce a general model which encapsulates all of these proposals, evaluate it w.r.t. a number of desirable properties, and refine the existing axiomatic approach to characterize lower and upper approximation operators. © 2013 Springer-Verlag.This work was partially supported by the Spanish Ministry of Science and Technology under Project TIN2011-28488. Lluis Godo has been partially supported by the MINECO Project TIN2012-39348-C02-01.Peer Reviewe

Modelos de Conjuntos Rugosos Difusos Tolerantes al Ruido: Definiciones y Propiedades

Desde que a principios de los años noventa se propuso el primer modelo híbrido difuso-rugoso, muchos investigadores se han interesado en la generalizacioón de la nocioón de aproximacioón inferior y superior de un conjunto difuso a partir de una relación difusa. Un tipo específico de estos nuevos modelos se centran en proponer aproximaciones robustas al posible ruido en los datos. En este artículo, repasamos las propuestas más prominentes, y comprobamos cuáles de las propiedades deseables del modelo estándar de los conjuntos rugosos se mantienen en estas propuestas.Este trabajo ha sido financiado parcialmente por el Ministerio de Ciencias y Tecnología bajo el Proyecto con referencia TIN2011-28488, y por el Fondo Especial de Investigación (BOF) de la Universidad de Gante. Lluis Godo reconoce financiación parcial por parte del Proyecto MINECO con referencia TIN2012-39348-C02-01.Peer Reviewe

Decision reducts and bireducts in a covering approximation space and their relationship to set definability

In this paper, we discuss the relationship between different types of reduction and set definability. We recall the definition of a decision reduct, a gamma-decision reduct, a decision bireduct and a gamma-decision bireduct in a Pawlak approximation space and the notion of set definability both in a Pawlak and a covering approximation space. We extend the notion of discernibility between objects in a Pawlak approximation space to a covering approximation space. Moreover, we introduce the definition of a decision reduct, a gamma-decision reduct, a decision bireduct and a gamma-decision bireduct in a covering approximation space. In addition, we study interrelationships between the four types of reduction, the correspondence with positive regions and the relationship to set definability in Pawlak and covering approximation spaces. (C) 2019 Elsevier Inc. All rights reserved

Fuzzy neighborhood operators based on fuzzy coverings

In many data mining processes, neighborhood operators play an important role as they are generalizations of equivalence classes which were used in the original rough set model of Pawlak. In this article, we introduce the notion of fuzzy neighborhood system of an object based on a given fuzzy covering, as well as the notion of the fuzzy minimal and maximal descriptions of an object. Moreover, we extend the definition of four covering-based neighborhood operators as well as six derived coverings discussed by Yao and Yao to the fuzzy setting. We combine these fuzzy neighborhood operators and fuzzy coverings and prove that only sixteen different fuzzy neighborhood operators are obtained. Moreover, we study the partial order relations between those sixteen operators. © 2016 Elsevier B.V.Lynn D'eer was supported by the Ghent University Special Research Fund. Chris Cornelis was partially supported by the Spanish Ministry of Science and Technology under the Project TIN2014-57251-P and the Andalusian Research Plans P10-TIC-6858, P11-TIC-7765 and P12-TIC-2958, and by Project PYR-2014-8 of the Genil Program of CEI BioTic GRANADA. Lluís Godo was partially supported by the Spanish MINECO project RASO TIN2015-71799-C2-1-P and the network TIN2014-56381-REDTLODISCO, as well as the Catalan government grant 2014SGR-118.Peer Reviewe