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Fuzzy Rough Sets: the Forgotten Step Martine De Cock, Chris Cornelis, and Etienne E

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Fuzzy Rough Sets: the Forgotten Step Martine De Cock, Chris Cornelis, and Etienne E IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 1, FEBRUARY 2007 121 Fuzzy Rough Sets: The Forgotten Step Martine De Cock, Chris Cornelis, and Etienne E. Kerre Abstract—Traditional rough set theory uses equivalence rela- After a public debate reflecting rivalry between rough set tions to compute lower and upper approximations of sets. The theory and the slightly older fuzzy set theory, many researchers corresponding equivalence classes either coincide or are disjoint. started working towards a hybrid theory (e.g., [11], [18], [20], This behaviour is lost when moving on to a fuzzy T-equivalence relation. However, none of the existing studies on fuzzy rough set [25], [26], [28], and [30]). In doing so, the central focus moved theory tries to exploit the fact that an element can belong to some from elements’ indistinguishability (objects are indistinguish- degree to several “soft similarity classes” at the same time. In able or not) to their similarity (objects are similar to a certain de- this paper we show that taking this truly fuzzy characteristic into gree), represented by a fuzzy relation . As a result, objects are account may lead to new and interesting definitions of lower and categorized into classes with “soft” boundaries based on their upper approximations. We explore two of them in detail and we investigate under which conditions they differ from the commonly similarity to one another; abrupt transitions between classes are used definitions. Finally we show the possible practical relevance replaced by gradual ones, allowing that an element can belong of the newly introduced approximations for query refinement. (to varying degrees) to more than one class. Index Terms—Fuzzy rough set, lower and upper approximation, Soon, researchers started exploring possible applications of query refinement, transitivity. the new paradigm of fuzzy-rough hybridization. For a compre- hensive literature review up to 1999, we refer to [15]. Among the more recent work is that of Drwal [10] and Jensen and Shen I. INTRODUCTION [13], [14], who studied extensions of the well-known rough set INCE its introduction in the 1960s, fuzzy set theory has approaches to data reduction and classification. Shad a significant impact on the way we represent and com- In Section II, we recall the necessary background leading to pute with vague information. More recently it has become part the definition of a fuzzy rough set as presented in [26]. This of the larger paradigm of soft computing, a collection of tech- definition is an elegant fuzzification of the concept of a rough niques that are tolerant of typical characteristics of imperfect set and at the same time absorbs earlier suggestions in the same data and knowledge—such as vagueness, imprecision, uncer- direction. The most striking aspect of all the studies on fuzzy tainty, and partial truth—and hence adhere closer to the human rough set theory mentioned above however is that none of them mind than conventional hard computing techniques. During the tries to exploit the fact that an element of can belong to last decades new approaches have been developed that gener- some degree to several “soft similarity classes” at the same time. alize the original fuzzy set theory (which is also called type-1 This property does not only lie at the heart of fuzzy set theory but fuzzy set theory in this context). Type-2 fuzzy sets, intuition- is also crucial in the decision on how to define lower and upper istic fuzzy sets, interval-valued fuzzy sets and fuzzy rough sets approximations. For instance, in traditional rough set theory, have in common that they can all be formally characterized by belongs to the lower approximation of if the equivalence class membership functions taking values in a partially ordered set to which belongs is included in . But what happens if , which is no longer the same (but an extension of) the set of belongs to several “soft similarity classes” at the same time? membership degrees used in fuzzy set theory. The intro- Do we then require that all of them are included in ? Most of duction of such new, generalizing theories is often accompanied them? Or just one? And then, which one? In Sections III and by lengthy discussions on issues such as the choice of termi- IV, we continue this discussion touched upon for the first time nology and the added value of the generalization. in [5]. In this paper we focus on fuzzy rough set theory. Pawlak [23] As such it becomes clear that there is still significant room launched rough set theory as a framework for the construction for improvement and generalization of the definition of a fuzzy of approximations of concepts when only incomplete informa- rough set, beyond the most “obvious” fuzzification established tion is available. The available information consists of a set so far. Furthermore Section V reveals that this generalization of examples (a subset of a universe being a nonempty set is not just of theoretical interest but becomes crucial in a top- of objects we want to say something about) of a concept , ical application such as query refinement for searching on the and a relation in . models “indiscernibility” or “indistin- WWW. guishability” and therefore generally is a tolerance relation (i.e., a reflexive and symmetrical relation) and in most cases even an II. FROM ROUGH SETS TO FUZZY ROUGH SETS equivalence relation (i.e., a transitive tolerance relation). Rough set analysis makes statements about the membership of some element of to the concept of which is a set of ex- Manuscript received February 15, 2005; revised December 18, 2005, March amples, based on the indistinguishability between and the el- 20, 2006, and June 16, 2006. ements of . To arrive at such statements, is approximated in The authors are with the Department of Applied Mathematics and Computer two ways. An element of belongs to the lower approxima- Science, Ghent University, B-9000 Gent, Belgium (e-mail: Martine.De- [email protected]; [email protected]; [email protected]). tion of if the equivalence class to which belongs is included Digital Object Identifier 10.1109/TFUZZ.2006.889762 in . On the other hand belongs to the upper approximation 1063-6706/$25.00 © 2007 IEEE 122 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 1, FEBRUARY 2007 TABLE I WELL-KNOWN T-NORMS; x AND y IN HY I TABLE II WELL-KNOWN S-IMPLICATORS; x AND y IN HY I Fig. 1. Lower and upper approximation of a set e. The dark shaded area is the boundary region. TABLE III WELL-KNOWN RESIDUAL IMPLICATORS; x AND y IN HY I of if its equivalence class has a non-empty intersection with . Formally, let be a universe and an equivalence relation. The lower and the upper approximation (in the sense of Pawlak [23]) of a subset of in the approximation space are the sets and such that for all in for all in . The fuzzy logical counterparts of the connectives (1) in (3) and (4) play an important role in this paper; we there- (2) fore recall some preliminaries in detail. Throughout this paper, let and denote a triangular norm and an implicator, respec- In other words tively. Recall that a triangular norm (t-norm for short) is any increasing, commutative and associative map- (3) ping satisfying , for all in . A negator is (4) a decreasing mapping satisfying and . is called involutive if for all in The underlying meaning is that is the set of elements nec- . Finally, an implicator is any -mapping essarily satisfying the concept (strong membership), while satisfying , for all in . More- is the set of elements possibly belonging to the concept (weak over we require to be decreasing in its first, and increasing in membership); for belongs to if all elements of in- its second component. If is a t-norm, the mapping defined distinguishable from belong to (hence, there is no doubt by, for all and in [0, 1], that also belongs to ), while belongs to as soon as an element of is indistinguishable from . It holds that .If belongs to the boundary region , and (5) then there is some doubt, because in this case is at the same time indistinguishable from at least one element of and at least is an implicator, usually called the residual implicator (of ). If one element of that is not in . This is illustrated graphically is a t-norm and is an involutive negator, then the mapping in Fig. 1. We call a rough set (in ) as soon as defined by, for all and in [0, 1], there is a set in such that and (see e.g., [26]). For completeness we mention that a second stream con- (6) cerning rough sets in the literature was initiated by Iwinski [12] is an implicator, usually called the S-implicator induced by who did not use an equivalence relation or tolerance relation and . In Tables I–III, we mention some well known t-norms, as an initial building block to define the rough set concept. S- and residual implicators. The S-implicators in Table II are Although his formulation provides an elegant mathematical induced by means of the standard negator which is defined model, the absence of the equivalence relation makes his model, by , for all in . as well as the fuzzy rough set theoretical models developed in Because equivalence relations are used to model equality, the same spirit (see, e.g., [21]), hard to interpret. Therefore, we fuzzy -equivalence relations are commonly considered to rep- do not deal with it in this paper. resent approximate equality or similarity. Recall that a fuzzy re- In the context of fuzzy rough set theory, is a fuzzy set in lation in is called a fuzzy -equivalence relation iff for all , i.e., an mapping, while is a fuzzy relation in and in , i.e., a fuzzy set in .
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