A fuzzy covering-based rough set model and its generalization over fuzzy lattice
Introduction
Various theories and methods have been proposed to deal with incomplete and insufficient information in classification, concept formation, and data analysis in data mining. For example, rough set theory (RST) [34], [35] and fuzzy set theory (FST) [66] have been developed and applied to real-world problems. RST was originally proposed by Pawlak [34], [35] as a useful tool for dealing with the fuzzy and uncertain problems in information systems and have already been an efficient tool for data pre-process and widely used in fields such as process control, economics, medical diagnosis, conflict analysis, and other fields. This theory can approximately characterize an arbitrary subset of a universe by using two definable subsets called lower and upper approximations [3]. Pawlak’s rough set is based on equivalence relations. However, an equivalence relation imposes restrictions and limitations on many applications [6], [18], [21], [24], [26], [53], [62]. Hence, many extensions have been made in recent years by replacing equivalence relation or partition by notions such as binary relations [12], [25], [43], [44], [61], [62], neighborhood systems and Boolean algebras [1], [54], [60], and coverings of the universe of discourse [2], [37], [38]. Based on the notion of covering, Pomykala [37], [38] obtained two pairs of dual approximation operators. Yao [60] further examined these approximation operators by the concepts of neighborhood and granularity. Such undertaking has stimulated more research in this area [4], [27], [28], [51], [64], [67], [68], [69], [70], [71], [72], [73].
However, as Pawlak’s rough set, covering-based rough set is designed to process discrete data, and it also faces great limitations on the applications for dealing with real-valued data sets [19]. Fuzzy set theory (FST) has been proposed by Zadeh [66] in 1965 as an effective tool for overcoming these limitations and it can deal effectively with vague concepts and graded indiscernibility. Nowadays, RST and FST are two main tools used to process uncertainty and incomplete information in the information systems. The two theories are related but distinct and complementary [36], [63]. In the past twenty years, more and more researchers have focused on the connection between rough sets and fuzzy sets. Intentions on combining RST and FST can be found in different mathematical fields [33], [52], [63]. Dubois and Prade firstly proposed the concept of fuzzy rough sets(FRS) [9], which combined these two theories and influenced numerous authors who used different fuzzy logical connectives and fuzzy relations to define fuzzy rough set models. Then more scholars generalized the fuzzy rough sets by using various methods [7], [13], [14], [15], [17], [30], [31], [32], [39], [40], [48], [55], [56], [57], [65]. The most common FRS is obtained by replacing crisp binary relations and crisp subsets with fuzzy relations and fuzzy subsets on the universe respectively.
As well as RST, some researchers have tried to generalize the fuzzy rough set based on fuzzy relation by using the concept of fuzzy covering. De Cock et al. [5] defined fuzzy rough sets based on the R-foresets of all objects in a universe of discourse with respect to (w.r.t.) a fuzzy binary relation. When R is a fuzzy serial relation, the family of all R-foresets forms a fuzzy covering of the universe of discourse. Analogously, Deng [8] examined the issue with fuzzy relations induced by a fuzzy covering. Li and Ma [22], on the other hand, constructed two pairs of fuzzy rough approximation operators based on fuzzy coverings, the standard min operator and the Kleene-Dienes implicator . It should be noted that fuzzy coverings in the models proposed by Deng [8] and De Cock et al. [5] are induced from fuzzy relations. So, they are not fuzzy coverings in the general sense. Although fuzzy coverings are used by Li and Ma [22] in their models, they only employed two special logical operators i.e. the standard min operator and the Kleene-Dienes implicator. Thus, it is necessary to construct more general fuzzy rough set models based on fuzzy coverings. Recently, an excellent introduction to the topic of fuzzy covering-based rough set is due to some scholars [10], [23], [29], [41], [47], [59], which can be regarded as a bridge linking covering-based rough set theory and fuzzy rough set theory. The original definition of fuzzy covering is defined as follows (see [10], [23]).
Let U be an arbitrary universal set, and be the fuzzy power set of U. We call with a fuzzy covering of U, if for each x ∈ U.
In fact, there exist some limits of this definition in the practical applications. For example, let be the set of three students and be the set of four course grades. For we have where can be listed as follows. If the teacher chooses only one student of U to participate in the competition, then there naturally exists a problem, which is the critical value be given by the teacher. This example is a typical evaluation question. It is easy to find that critical “1” is a meaningless existence for this example. In other words, the teacher could not choose a suitable student to participate in the competition by using fuzzy evaluation methods. To overcome these limits, Ma [29] generalized the fuzzy covering to fuzzy β-covering by replacing 1 with a parameter β (0 < β ≤ 1).
In [29], Ma defined two new types of fuzzy covering-based rough set models by introducing the new concepts of fuzzy β-covering and fuzzy β-neighborhood and generalized these two models to L-fuzzy covering-based rough set models which defined over the fuzzy lattices. However, the concept of fuzzy β-neighborhood is a holistic definition and it is easy to lose some useful information in the practical examples in information systems. The notion of minimal description plays a key role in the studies of generalized rough sets generated by the covering. Correspondingly, this paper presents the concept of fuzzy β-minimal description in fuzzy covering-based rough sets and gives some new characterizations of fuzzy covering-based rough sets in terms of the notion of fuzzy β-minimal description. We know that attribute reduction is an important rough set based tool in the information systems and machine learning applications, especially with regard to feature subset selection. In this paper, the reduction issue of a type of fuzzy covering-based rough set is explored by means of the concept of fuzzy β-minimal description, which also shows that this notion is an essential characteristic of fuzzy covering-based rough sets.
The remainder of this paper is organized as follows. In Section 2, some preliminary definitions in fuzzy set theory and fuzzy covering-based rough set used in this paper are introduced. In Section 3, we define a new type of fuzzy covering-based rough set model by introducing the notion of fuzzy β-minimal description and address some problems for this model. In Section 4, the generalization of the fuzzy covering-based rough set model over fuzzy lattice is proposed and Section 5 concludes this paper.
Section snippets
Preliminaries
In this section, we review some notions in fuzzy set theory and fuzzy covering-based rough set used in this paper.
Definition 2.1 ([20]) Let U be a universe of discourse. A fuzzy set A, or rather a fuzzy subset A of U, is defined by a function assigning to each element x of U a value A(x) ∈ [0, 1]. We denote by the family of all fuzzy subsets of U, i.e., the set of all functions from U to [0, 1], and call it the fuzzy power set of U. For any we say that A is contained in B, denoted by A ⊆ B, if A
A novel type of fuzzy covering-based rough set for fuzzy subsets
In this section, by introducing the notion of fuzzy β-minimal description, we define a novel type of fuzzy covering-based rough set model. We will address the following issues in this section. First, we present the definition of fuzzy β-minimal description and study its properties. Then, we define a novel type of fuzzy covering-based rough set model and investigate the properties of this model. Furthermore, the axiomizations and matrix representations of the fuzzy covering lower and the same
Generalization of the fuzzy covering-based rough set model over fuzzy lattice
In [29], the authors generalized the models to L-fuzzy covering-based rough sets which are defined over fuzzy lattices. In this section, as a generalization work, we further generalize the fuzzy covering-based rough set model defined in Definition 3.3 to L-fuzzy covering-based rough set which is defined over fuzzy lattice.
Definition 4.1 ([29]) Let be a fuzzy lattice. For some β > 0 (β ∈ L), we call an L-fuzzy β-covering of U, if for each x ∈ U, where
Conclusions
Fuzzy β-covering is a new notion defined by Ma in [29] which can build a bridge between covering-based rough set theory and fuzzy set theory. In this paper, by introducing the notion of fuzzy β-minimal description, we defined a novel type of fuzzy covering-based rough set model and generalized this model over the fuzzy lattice. Main conclusions in this paper and continuous work to do are listed as follows.
- (1)
We proposed the concept of fuzzy β-minimal description, the purpose of which can be
Acknowledgments
The authors are greatly thankful to anonymous reviewers and the Editor-in-Chief, Professor Witold Pedrycz for sharing their valuable comments that significantly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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