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Fuzzy Sets and Systems 275 (2015) 1–38

www.elsevier.com/locate/fss

A comprehensive study of implicator–conjunctor-based and

noise-tolerant fuzzy rough sets: Deﬁnitions, properties and

robustness analysis

Lynn D’eer

a,∗

, Nele Verbiest

, Chris Cornelis

a,b

, Lluís Godo

Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Belgium

Department of Computer Science and Artiﬁcial Intelligence, Research Center on Information and Communications Technology (CITIC-UGR),

University of Granada, Spain

Artiﬁcial Intelligence Research Institute (IIIA), Spanish National Research Council (CSIC), Bellaterra, Spain

Received 6

January 2014; received in revised form 1 July 2014; accepted 23 November 2014

Available

online 2 December 2014

Abstract

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.

Keywords: Fuzzy sets; Rough sets; Hybridization; Lower and upper approximation; Noise-tolerance

1. Introduction

Rough set theory was originally proposed by Pawlak [47] in 1982 to deal with uncertainty due to incompleteness

and indiscernibility. The basic idea of rough set theory is that it provides a lower and upper approximation of a concept

with respect to a binary indiscernibility relation. The lower approximation contains all the elements of the uni

verse

certainly belonging to the concept, while the upper approximation contains the elements possibly belonging to the

Corresponding author.

E-mail

addresses: Lynn.Deer@UGent.be (L. D’eer), Nele.Verbiest@UGent.be (N. Verbiest), Chris.Cornelis@decsai.ugr.es (C. Cornelis),

godo@iiia.csic.es (L. Godo).

http://dx.doi.org/10.1016/j.fss.2014.11.018

2 L. D’eer et al. / Fuzzy Sets and Systems 275 (2015) 1–38

concept. In the original model of Pawlak, an equivalence relation is used to model indiscernibility. Yet, many authors

have generalized Pawlak’s model by using binary non-equivalence relations (see e.g. [48,49] for a survey).

Applications of rough set theory are widespread and are especially prominent in data analysis [33,35] and more

speciﬁc in feature selection and classiﬁcation [53]. Ho

wever, since the traditional rough set is designed to process

qualitative (discrete) data, it faces important limitations when dealing with real-valued data sets [31]. Fuzzy set theory

proposed in 1965 by Zadeh [68] is ve

ry useful to overcome these limitations, as it can deal effectively with vague

concepts and graded indiscernibility.

It wa

s recognized early that both theories are complementary, rather than competitive. To that end, rough set theory

has been extended in two ways [14]. Rough fuzzy set theory discusses the approximation of a fuzzy set by a crisp

relation. If moreover the indiscernibility relation to distinguish different objects is fuzzy as well, fuzzy rough set

theory is considered. Since ev

ery crisp relation can be seen as a special case of a fuzzy relation, all results obtained in

fuzzy rough set theory also hold for rough fuzzy set theory.

The v

estiges of fuzzy rough set theory date back to the late 1980s, and originate from work by Fariñas del Cerro

and Prade [12], Dubois and Prade [13], Nakamura [45] and Wygralak [62]. From 1990 onwards, research on the

hybridization between rough sets and fuzzy sets ﬂourished. The inspiration to combine rough and fuzzy set theory wa

found in different mathematical ﬁelds. For instance, Lin [34] studied fuzzy rough sets using generalized topological

spaces (Frechet spaces) and Nanda and Majumdar [46] discussed fuzzy rough sets based on an algebraic approach.

Moreover, Thiele [54] examined the relationship with fuzzy modal logic. Later on, Yao [66] and Liu [39] used le

vel

sets to combine fuzzy and rough set theory.

This wo

rk focuses on fuzzy rough set models using fuzzy relations and fuzzy logical connectives. The semi-

nal papers of Dubois and Prade [14,15] are probably the most important in the evolution of these fuzzy rough set

models, since they inﬂuenced numerous authors who used different fuzzy logical connectives and fuzzy relations.

Essential wo

rk was done by Morsi and Yakout [44] who studied both constructive and axiomatic approaches and by

Radzikowska and Kerre [51] who deﬁned fuzzy rough sets based on three general classes of fuzzy implicators: S-, R-

and QL-implicators. However, despite generalizing the fuzzy connectives, they still used fuzzy similarity relations.

A ﬁrst attempt to use reﬂe

xive fuzzy relations instead of fuzzy similarity relations was done by Greco et al. [22,23].

Thereafter, Wu et al. [60,61] were the ﬁrst to consider general fuzzy relations. Besides generalizing the fuzzy relation,

Mi et al. [40,41] considered conjunctors instead of t-norms. Furthermore, Yeung et al. [67] discussed two pairs of dual

approximation operators from both a constructi

ve and an axiomatic point of view. Hu et al. [26,28] for their part stud-

ied fuzzy relations based on kernel functions. In this work, we consider all these different proposals within a general

Implicator–Conjunctor (IC) based fuzzy rough set model that encapsulates all of them, as discussed in Section 3.1.

However, the aforementioned models only consider the wo

rst and best performing objects to determine the fuzzy

rough lower and upper approximations respectively. Consequently, these approximations are sensitive to noisy and/or

outlying samples. This, in turn, impacts the robustness of data analysis applications based on them, such as attribute

selection and classiﬁcation. To mitig

ate this problem in the crisp case, Ziarko [70] proposed the Variable Precision

Rough Set (VPRS) model in 1993. This model also served as a starting point for the design of several noise-tolerant

fuzzy rough set approaches, such as [5,7,17,18,24,25,42,43,65,69], which will be discussed in detail in Section 4.

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. A structured and critical analysis of

the current research of constructive methods for fuzzy rough set models is presented. Note that we do not consider

axiomatic approaches (see e.g. [36,37,41,44,50,59–61,67]). We re

view the deﬁnitions of noise-tolerant models, gen-

eralizing them where appropriate and in some cases applying modiﬁcations to correct errors in the original proposal.

Where applicable, we also establish relationships between these models and the corresponding IC based deﬁnitions,

as well as Paw

lak’s and Ziarko’s crisp approaches. Furthermore, we examine which theoretical properties of tradi-

tional rough sets and IC fuzzy rough sets can still be maintained for the noise-tolerant models; indeed, similarly as for

Ziarko’s VPRS model, providing mechanisms for making the approximations less strict usually involves sacriﬁcing

some desirable properties. Finally

, we evaluate whether the considered approaches really live up to the claim of being

more “robust” approximations, by performing a stability analysis on four real datasets, and comparing them to the IC

model. This will allow us to obtain a comprehensive overview of the beneﬁts and the drawbacks of the rob

ust fuzzy

rough set models, in order to acquire the expertise for future research opportunities.

The remainder of this article is structured as follo

ws: in Section 2, we summarize preliminary deﬁnitions concerning

fuzzy logical connectives, fuzzy sets and relations, and rough set theory. In Section 3, we introduce the general IC

L. D’eer et al. / Fuzzy Sets and Systems 275 (2015) 1–38 3

based fuzzy rough set model, and review speciﬁc cases of it studied in literature. We also establish sufﬁcient conditions

under which its approximations satisfy the same theoretical properties as Pawlak’s model. In Section 4, we review the

seven most important noise-tolerant fuzzy rough set models introduced in the literature: Mieszkowicz-Rolka and

Rolka’

s Variable Precision Fuzzy Rough Set model (VPFRS [42,43]), Cornelis et al.’s Vaguely Quantiﬁed Fuzzy

Rough Set model (VQFRS, [5]), Hu et al.’s Soft Fuzzy Rough Set model (SFRS, [24,25]), Ya o et al.’s Variable

Precision Fuzzy Rough Set model based on Fuzzy Granules (FG, [65]), Zhao et al.’s Fuzzy V

ariable Precision Rough

Set model (FVPRS, [69]), Fernández-Salido and Murakami’s β-Precision Fuzzy Rough Set model (β-PREC, [17,18])

and Cornelis et al.’s Ordered Weighted Average based Fuzzy Rough Set model (OWA, [7]). In Section 5, we test

the robustness of these proposals and the IC model in the presence of class and attrib

ute noise, and in Section 6, we

conclude and state future work. In Appendix A, we give counterexamples for the properties that are not maintained in

the different noise-tolerant models. Finally, we mention that a small part of the results in this paper (pertaining only

to the IC model) has been published in a conference paper [10].

2. Preliminaries

In this section, we present basic notions related to fuzzy logical connectives, fuzzy sets and relations, and the lower

and upper approximation operators in rough set theory.

2.1. Fuzzy logical connectives

We start by recalling some important fuzzy logical connectives. First, we discuss conjunctors, disjunctors and

negators that are fuzzy extensions of the Boolean conjunctor, disjunctor and negation.

• A conjunctor is a mapping C : [0, 1]

→[0, 1] that is increasing in both arguments and that satisﬁes the boundary

conditions C (0, 0) = C (0, 1) = C (1, 0) = 0 and C (1, 1) = 1. It is called a border conjunctor if for all x ∈[0, 1] it

holds that C (1, x) = x. A commutati

ve and associative border conjunctor is called a t-norm and is denoted by T .

Example 1. Fo

r x, y ∈[0, 1], the minimum t-norm or minimum operator T

is deﬁned by T

(x, y) = min(x, y),

the product t-norm T

is deﬁned by T

(x, y) = x · y and the Łukasiewicz t-norm T

is deﬁned by T

(x, y) =

max(0, x + y − 1).

• A disjunctor is a mapping D: [0, 1]

→[0, 1] that is increasing in both arguments and that satisﬁes the boundary

conditions D (1, 1) = D (0, 1) = D (1, 0) = 1 and D (0, 0) = 0. It is called a border disjunctor if for all x ∈[0, 1]

it holds that D (0, x) = x. A commutati

ve and associative border disjunctor is called a t-conorm and is denoted

by S .

Example 2. Fo

r x, y ∈[0, 1], the maximum t-conorm or maximum operator S

is deﬁned by S

(x, y) = max(x, y),

the probabilistic sum S

is deﬁned by S

(x, y) = x + y − x · y and the Łukasiewicz t-conorm S

is deﬁned by

(x, y) = min(1, x + y).

• A ne

gator is a decreasing mapping N : [0, 1] →[0, 1] that satisﬁes N (0) = 1 and N (1) = 0. A negator is called

involutive if for all x ∈[0, 1] it holds that N (N (x)) = x.

Example 3. Fo

r x ∈[0, 1], the standard negator N

is deﬁned by N

(x) = 1 − x.

When we use an in

volutive negator, we can deﬁne a dual couple of a conjunctor and a disjunctor with respect to the

chosen negator. Given an involutive negator N , a conjunctor C and a disjunctor D , the N -dual of C is a disjunctor

C ,N

deﬁned by

∀x,y ∈[0, 1]: D

C ,N

(x, y) = N



N (x), N (y)



and the N -dual of D is a conjunctor C

D,N

deﬁned by

4 L. D’eer et al. / Fuzzy Sets and Systems 275 (2015) 1–38

∀x,y ∈[0, 1]: C

D,N

(x, y) = N



N (x), N (y)



It can be veriﬁed that the N -dual of a t-norm is a t-conorm and vice versa.

Besides conjunctors, disjunctors and ne

gators, we recall the notion of implicators. They extend the Boolean impli-

cation to the fuzzy setting.

An implicator is a mapping I : [0, 1]

→[0, 1] that is decreasing in the ﬁrst and increasing in the second argument

and that satisﬁes the boundary conditions I (0, 0) = I (0, 1) = I (1, 1) = 1 and I (1, 0) = 0. It is called a border

implicator if for all x ∈[0, 1] it holds that I (1, x) = x. It satisﬁes the weak conﬁnement principle if ∀x, y ∈[0, 1] :

x ≤ y ⇒ I (x

, y) = 1. An implicator that satisﬁes the weak conﬁnement principle is called a WCP-implicator in this

paper.

Let I be an implicator

. The induced negator of I is the negator N

deﬁned by

∀x ∈[0, 1]: N

(x) = I (x, 0).

There are two important classes of implicators: S-implicators based on disjunctors and negators and R-implicators

based on border conjunctors.

• Let D be a disjunctor and N a ne

gator. The S-implicator I

D,N

based on D and N is deﬁned by

∀x,y ∈[0, 1]: I

D,N

(x, y) = D



N (x), y



• Let C be a border conjunctor. The R-implicator I

based on C is deﬁned by

∀x,y ∈[0, 1]: I

(x, y) = sup



z ∈[0, 1]|C (x, z) ≤ y



Both S- and R-implicators are border implicators. R-implicators are also WCP-implicators. A special class of

R-implicators are IMTL-implicators, where IMTL stands for ‘Involutive Monoidal T-norm based Logic’ [16,21]:

these R-implicators are based on a left-continuous t-norm and have an involutive induced negator. A left-continuous

t-norm of which the R-implicator is an IMTL-implicator, is called an IMTL-t-norm.

Example 4. The Kleene–Dienes (KD) implicator I

is an S-implicator deﬁned by, for x, y ∈[0, 1], I

(x, y) =

max(1 − x, y). The Łukasiewicz implicator I

is both an S-implicator and R-implicator deﬁned by, for x, y ∈[0, 1],

(x, y) = min(1, 1 − x + y).

Finally, we recall that we can construct conjunctors based on implicators and in

volutive negators. Given an in-

volutive negator N and an implicator I , the induced conjunctor of I and N is the conjunctor C

I ,N

deﬁned

∀x,y ∈[0, 1]: C

I ,N

(x, y) = N



x,N (y)



which is not necessarily a t-norm.

2.2. Fuzzy sets and relations

In this section, we recall basic notions about fuzzy sets and fuzzy relations [68].

• A fuzzy set A in a non-empty uni

verse U is a mapping A: U →[0, 1]. The collection of all fuzzy sets in U is

denoted by F (U ). If U is ﬁnite, the cardinality of A is deﬁned by

|A|=



x∈U

A(x).

The support of A is the crisp set supp(A) ={x ∈ U | A(x) > 0}. Given α ∈[0, 1], the α-level set A

of A in U is

a crisp set in U such that x ∈ A

if and only if A(x) ≥ α.

• Gi

ven α ∈[0, 1], the constant (fuzzy) set ˆα in U is deﬁned by

∀x ∈ U: ˆα(x) = α.

In the crisp case, the only constant sets in U are

0 =∅and

1 = U .

L. D’eer et al. / Fuzzy Sets and Systems 275 (2015) 1–38 5

• Let A, B ∈ F (U ). Given a negator N , a conjunctor C and a disjunctor D , the N -complement A

of A is given

∀x ∈ U: A

(x) = N



A(x)



The C -intersection of A and B is the fuzzy set A ∩

B deﬁned by

∀x ∈ U: (A ∩

B)(x) = C



A(x), B(x)



The D-union of A and B is the fuzzy set A ∪

B given by

∀x ∈ U: (A ∪

B)(x) = D



A(x), B(x)



For C = T

and D = S

, we shortly denote A ∩ B and A ∪ B.

• A binary fuzzy relation R in U is a fuzzy set in U × U . We deﬁne its in

verse fuzzy relation R



by R



(x, y) =

R(y, x) for x, y ∈ U . The relation R is called reﬂexive if for all x in U it holds that R(x, x) = 1 and it is called

serial if for all x ∈ U : sup

y∈U

R(x, y) = 1. We call the relation symmetric if for all x, y ∈ U , R(x, y) = R(y, x).

It is obvious that for a symmetric binary fuzzy relation R it holds that R = R



. Given a t-norm T , the relation R

is called T -transitive if for all x, y and z in U it holds that



R(x,y),R(y,z)



≤ R(x,z).

Furthermore, the relation R is called T -Euclidean if for all x, y and z in U it holds that



R(y,x),R(y,z)



≤ R(x,z).

Every symmetric and T -transitive relation is T -Euclidean. If a relation R is reﬂexive, symmetric and

T -transitive, it is called a (fuzzy) T -similarity relation. When T = T

, we also call it a similarity relation.

Since the minimum operator is the largest t-norm, a similarity relation is T -transitive for every t-norm T .

• Let R be a binary fuzzy relation and x ∈ U , then the R-for

eset of x is the fuzzy set Rx such that for every y ∈ U ,

Rx(y) = R(y, x). When R is a crisp binary relation, the R-foreset of x is the crisp set {y ∈ U | (y, x) ∈ R}.

2.3. Rough sets

In this section, we recall the deﬁnition of the lower and upper approximation in a Pawlak approximation space

[47].

A classic or Pawlak approximation space is a couple (U, R) consisting of a non-empty universe U and an equiv-

alence relation R in U . The rough approximation of a crisp set A in U by R is the pair of sets (apr

(A), apr

(A))

deﬁned by, for x ∈ U ,

x ∈ apr

(A) ⇔ (∀y ∈ U)



(y, x) ∈ R ⇒ y ∈ A



, (1)

x ∈

apr

(A) ⇔ (∃y ∈ U)



(y, x) ∈ R ∧ y ∈ A



. (2)

A pair (A

, A

) of sets in U is called a rough set in (U, R) if there is a set A in U such that A

= apr

(A) and

= apr

(A). A

is called the lower approximation of A and A

is its upper approximation. A set A is deﬁnable

when apr

(A) = A = apr

(A) holds.

We list the most important properties of the rough approximation in a Pawlak approximation space in Table 1. The

properties (D), (INC), (SM), (IU), (ID), (LU) and (UE) were already stated in [47].

The idea of property (A) comes from the concept of a Galois connection [30,56]: let U

and U

be two universes

and f : U

→ U

and g: U

→ U

two mappings between these universes. We call (f, g) a Galois connection in

, U

) if for all x ∈ U

and y ∈ U

it holds that

f(x)≤

y ⇔ x ≤

g(y),

where ≤

is an order relation in U

. It is clear that if f and g are the upper and lower approximation operator

respectively, then they form a Galois connection in (F (U ), F (U )) with ≤

=≤

=⊆. In the special case that B = A

in Table 1, the properties (A) and (INC) imply that

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