Fuzziness in rough sets

Chakrabarty, Kankana; Biswas, Ranjit; Nanda, Sudarsan

doi:10.1016/s0165-0114(97)00414-4

Cited by 122 publications

(46 citation statements)

References 10 publications

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“…Because the second step has been discussed in the previous two sections, only the description of rough sets based on their fuzziness is dealt with here. The later problem has been addressed in [20][21][22][23][24][26][27][28], for example, Wang [22] discussed the relationship between the fuzzy measure and granularity in detail…”

Section: Fuzzy Measures Of Rough Setsmentioning

confidence: 99%

“…In summary, the works noted above extended the measures of common sets to the family of fuzzy sets from various viewpoints. From the concept of information entropy, Chakrabarty [20] presented the measure of the fuzziness of fuzzy sets by using the relationship between fuzzy sets and their nearby crisp sets. He [21] defined the concept of fuzzy entropy using the linear fuzzy measure.…”

Section: Introductionmentioning

confidence: 99%

“…)), and Mao [33] showed that all fuzzy measures (roughness, rough entropy, or fuzzy entropy) defined in [20][21][22][23][24][26][27][28] can be represented by the above formula as long as the functions f and g are chosen properly.…”

Section: Introductionmentioning

confidence: 99%

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The structural analysis of fuzzy measures

Zhang

2011

Sci. China Inf. Sci.

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From quotient space based granular computing theory we explore the fuzzy measures of fuzzy sets. Based on the structural analysis of fuzzy sets, we present an "isotropism" assumption. Under this assumption we achieved the following insights: (1) the uniqueness of a fuzzy measure being isotropic on a finite complete semi-ordered set; (2) the necessary and sufficient condition of the isomorphism for fuzzy measure functions in fuzzy mathematics; (3) the necessary and sufficient condition that fuzzy measures have fuzzy and granular monotonicity; (4) under a certain assumption, the analytic expression for fuzzy measures. The results clarify the relationship between fuzzy measures and granular computing, reveal the essence of fuzzy measures, and provide a simple way of constructing fuzzy measures.

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Section: Fuzzy Measures Of Rough Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The structural analysis of fuzzy measures

Zhang

2011

Sci. China Inf. Sci.

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“…We have utilised it in order to identify the decision model of a military pilot [11]. The idea of VPRS is based on a changed relation of set inclusion given in (1) and (2) [18], defined for any nonempty crisp subsets A and B of the universe X. We say that the set A is included in the set B with an inclusion error β:…”

Section: Variable Precision Rough Sets Modelmentioning

confidence: 99%

Variable Precision Fuzzy Rough Sets

Mieszkowicz-Rolka

Rolka

2004

Transactions on Rough Sets I

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Abstract. In this paper the variable precision fuzzy rough sets (VPFRS) concept will be considered. The notions of the fuzzy inclusion set and the α-inclusion error based on the residual implicators will be introduced. The level of misclassification will be expressed by means of α-cuts of the fuzzy inclusion set. Next, the use of the mean fuzzy rough approximations will be postulated and discussed. The concept of VPFRS will be defined using the extended version of the variable precision rough sets (VPRS) model, which utilises a general allowance for levels of misclassification expressed by two parameters: lower (l) and upper (u) limit. Remarks concerning the variable precision rough fuzzy sets (VPRFS) idea will be given. An example will illustrate the proposed VPFRS model.

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“…Extensive applications of the fuzzy set theory have been found in various fields. There have been many papers studying the connections and difference of fuzzy set theory and rough set theory (Banerjee and Pal 1996;Chakrabarty et al 2000;Davvaz 2006a;Dubois and Prade 1990;Mi and Zhang 2004;Radzikowska and Kerre 2002). Dubois and Prade (1990) were one of the first who investigated the problem of fuzzification of rough sets.…”

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confidence: 99%

Approximations in n-ary algebraic systems

Davvaz

2007

Soft Comput

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Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.Keywords Rough set · Lower approximation · Upper approximation · Algebraic system · n-ary semigroup · Fuzzy set A survey of related worksIn this section, we describe the motivation and a survey of related works.The rough set theory was introduced by Pawlak in 1981. And Pawlak is acknowledged to be a "father" of rough sets. Both rough set theory and theory of evidence deal with uncertainty. They are strongly connected, but they are different approach to uncertainty. The main difference between the rough set theory (Pawlak 1982) and Dempster Shafer theory of evidence (Shafer 1976) is that Dempster Shafer theory uses B. Davvaz (B) belief functions as a main tool, while rough set theory makes use of sets lower and upper approximations, for example see Skowron and Grzymala-Busse (1994).Rough set theory, a new mathematical approach to deal with inexact, uncertain or vague knowledge, has recently received wide attention on the research areas in both of the real-life applications and the theory itself. Rough set theory is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. There are at least two methods for the development of this theory, the constructive and axiomatic approaches. In constructive methods, lower and upper approximations are constructed from the primitive notions, such as equivalence relations on a universe Quafafou 2000) and neighborhood systems (Wong and Pin Nie 1994;Yao 1998). In Pawlak rough sets, the equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a nonempty intersection with the set. It is well known that a partition induces an equivalence relation on a set and vice versa. The properties of rough sets can thus be examined via either partition or equivalence classes. Rough sets are a suitable mathematical model of vague concepts, i.e., concepts without sharp boundaries. Rough set theory is emerging as a powerful theory dealing with imperfect data (Pawlak 1991). It is an expendin...

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Fuzziness in rough sets

Cited by 122 publications

References 10 publications

The structural analysis of fuzzy measures

The structural analysis of fuzzy measures

Variable Precision Fuzzy Rough Sets

Approximations in n-ary algebraic systems

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