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

Rough sets, their extensions and applications

Rough set theory provides a useful mathematical foundation for developing automated computational systems that can help understand and make use of imperfect knowledge. Despite its recency, the theory and its extensions have been widely applied to many problems, including decision analysis, data-mining, intelligent control and pattern recognition. This paper presents an outline of the basic concepts of rough sets and their major extensions, covering variable precision, tolerance and fuzzy rough sets. It also shows the diversity of successful applications these theories have entailed, ranging from financial and business, through biological and medicine, to physical, art, and meteorological

Quantification of R-Fuzzy sets

The main aim of this paper is to connect R-Fuzzy sets and type-2 fuzzy sets, so as to provide a practical means to express complex uncertainty without the associated difficulty of a type-2 fuzzy set. The paper puts forward a significance measure, to provide a means for understanding the importance of the membership values contained within an R-fuzzy set. The pairing of an R-fuzzy set and the significance measure allows for an intermediary approach to that of a type-2 fuzzy set. By inspecting the returned significance degree of a particular membership value, one is able to ascertain its true significance in relation, relative to other encapsulated membership values. An R-fuzzy set coupled with the proposed significance measure allows for a type-2 fuzzy equivalence, an intermediary, all the while retaining the underlying sentiment of individual and general perspectives, and with the adage of a significantly reduced computational burden. Several human based perception examples are presented, wherein the significance degree is implemented, from which a higher level of detail can be garnered. The results demonstrate that the proposed research method combines the high capacity in uncertainty representation of type-2 fuzzy sets, together with the simplicity and objectiveness of type-1 fuzzy sets. This in turn provides a practical means for problem domains where a type-2 fuzzy set is preferred but difficult to construct due to the subjective type-2 fuzzy membership

Rough Neutrosophic Sets

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties

Rough Neutrosophic Sets

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties

The Quantification of Perception Based Uncertainty Using R-fuzzy Sets and Grey Analysis

The nature of uncertainty cannot be generically defined as it is domain and context specific. With that being the case, there have been several proposed models, all of which have their own associated benefits and shortcomings. From these models, it was decided that an R-fuzzy approach would provide for the most ideal foundation from which to enhance and expand upon. An R-fuzzy set can be seen as a relatively new model, one which itself is an extension to fuzzy set theory. It makes use of a lower and upper approximation bounding from rough set theory, which allows for the membership function of an R-fuzzy set to be that of a rough set. An R-fuzzy approach provides the means for one to encapsulate uncertain fuzzy membership values, based on a given abstract concept. If using the voting method, any fuzzy membership value contained within the lower approximation can be treated as an absolute truth. The fuzzy membership values which are contained within the upper approximation, may be the result of a singleton, or the vast majority, but absolutely not all. This thesis has brought about the creation of a significance measure, based on a variation of Bayes' theorem. One which enables the quantification of any contained fuzzy membership value within an R-fuzzy set. Such is the pairing of the significance measure and an R-fuzzy set, an intermediary bridge linking to that of a generalised type-2 fuzzy set can be achieved. Simply by inferencing from the returned degrees of significance, one is able to ascertain the true significance of any uncertain fuzzy membership value, relative to other encapsulated uncertain values. As an extension to this enhancement, the thesis has also brought about the novel introduction of grey analysis. By utilising the absolute degree of grey incidence, it provides one with the means to measure and quantify the metric spaces between sequences, generated based on the returned degrees of significance for any given R-fuzzy set. As it will be shown, this framework is ideally suited to domains where perceptions are being modelled, which may also contain several varying clusters of cohorts based on any number of correlations. These clusters can then be compared and contrasted to allow for a more detailed understanding of the abstractions being modelled