Labeled Fuzzy Rough Sets Versus Fuzzy Flow Graphs

Alicja Mieszkowicz-Rolka and Leszek Rolka

Rzeszów University of Technology, Al. Powsta´nców Warszawy 8, 35-959 Rzeszów, Poland

Keywords:

Information Systems, Fuzzy Sets, Rough Sets, Fuzzy Rough Sets, Fuzzy Flow Graphs.

Abstract:

This paper presents the idea of labeled fuzzy rough sets which constitutes a novel approach to rough approxi-

mation of fuzzy information systems. The labeled fuzzy rough sets approach is compared with the fuzzy ﬂow

graph approach. The standard deﬁnition of fuzzy rough sets is based on comparing the elements of a universe

by using a fuzzy similarity relation. This is a complex task, especially in the case of large universes. The idea

of labeled fuzzy rough sets consists in comparison of elements of the universe to some ideals represented by

linguistic values of attributes. Every element of the universe can be bound up with a linguistic label. Fuzzy

rough approximations of any fuzzy set are obtained by describing its elements with the help of characteristic

elements of linguistic labels. In this paper, new parameterized notions of the positive, boundary, and negative

linguistic values are introduced.

1 INTRODUCTION

The fuzzy set theory (Zadeh, 1965), and the rough

set theory (Pawlak, 1991) are two complementary

paradigms, suitable for dealing with imperfect knowl-

edge. Both theories were combined together (Dubois

and Prade, 1992) in a fuzzy rough set approach

which was further developed by other researchers

(Radzikowska and Kerre, 2002). However, a signif-

icant problem in application of the standard fuzzy

rough set approach is the size and complexity of ob-

tained fuzzy similarity classes, in the case of large

universes with many linguistic values of fuzzy at-

tributes. In order to overcome this drawback, a novel

way of analysis of fuzzy information systems was

proposed recently (Mieszkowicz-Rolka and Rolka,

2016), which is called the labeled fuzzy rough set

approach. The crucial point of this method consists

in avoiding the determination of fuzzy similarity be-

tween particular elements of a universe. In the present

paper, we propose and discuss a new parameterized

version of that approach. We introduce new basic

notions of the positive (dominating), boundary, and

negative linguistic values. Determination of similar-

ity classes, approximation of fuzzy sets, and gener-

ating of decision rules can be done easier with the

help of linguistic labels. To show the effectiveness

of the method, we give an illustrating computational

example of analysis of a small fuzzy information sys-

tem, with the goal of obtaining a set of fuzzy deci-

sion rules. Flow graph-based representation is yet

another possibility of a formal description of (crisp)

decision tables (Pawlak, 2005a; Pawlak, 2005b). A

generalized fuzzy ﬂow graph approach can be applied

to evaluate the statistical properties of fuzzy informa-

tion systems. As it can also be used for determin-

ing and evaluating the quality of fuzzy decision rules

(Mieszkowicz-Rolka and Rolka, 2014), we performed

a parallel computation for the example data. Finally,

we compare and discuss results obtained with both

approaches.

2 ANALYSIS OF FUZZY

INFORMATION SYSTEMS

2.1 Fuzzy Rough Sets

The fundamental operation of the rough set theory is

to ﬁnd out classes of objects, being elements of a ﬁnite

universe of discourse, which are indiscernible with re-

spect to a subset of attributes. In the standard rough

set theory proposed by Pawlak, the attributes of ob-

jects have always crisp values. Relationship between

the classes of indiscernible elements of the universe

can be used for discovering dependencies between at-

tributes and for ﬁnding their reducts.

The attributes of elements of a universe U are

commonly divided into a subset of condition at-

tributes C and decision attributes D. In such a case,

the rough set theory is suitable for evaluating the

Rolka, L. and Mieszkowicz-Rolka, A.

Labeled Fuzzy Rough Sets Versus Fuzzy Flow Graphs.

DOI: 10.5220/0006083301150120

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 2: FCTA, pages 115-120

ISBN: 978-989-758-201-1

115

quality of decision systems, detecting contradictions,

reducing decision tables, and determining decision

rules.

Another important and popular approach, devel-

oped in the last decades, is the fuzzy logic connected

with the fuzzy set theory founded by Zadeh. There

are many applications of fuzzy sets, e.g., in the form

of fuzzy inference systems used in automatic control

of technical plants.

The rough set theory and the fuzzy set theory ad-

dress different aspects of uncertainty. It is possible

to combine them together in one framework, but sev-

eral generalizations and assumptions need to be done.

First of all, a general form of a fuzzy information sys-

tem FS (Mieszkowicz-Rolka and Rolka, 2016) is nec-

essary that is deﬁned as the 4-tuple

FS “ xU, Q, V, f y (1)

where:

U – denotes a nonempty set, called the universe,

Q – is a ﬁnite set of fuzzy attributes,

V – is a set of linguistic values of fuzzy attributes,

V “

qPQ

f – is an information function, f : U ˆV Ñ r0, 1s,

f px, V q P r0, 1s, @V P V and @ x P U .

In a fuzzy decision system, every element x

of the universe U is described by fuzzy condi-

tion attributes C “ tc

, c

, . . . , c

u, and fuzzy de-

cision attributes D “ td

, d

, . . . , d

u. We denote

by C

“ tC

, C

, . . . , C

u the family of linguis-

tic values of the i-th condition attribute c

, and by

“ tD

, D

, . . . , D

u the family of linguistic

values of the j-th decision attribute d

, where i “

1, 2, . . . , n, j “ 1, 2, . . . m, respectively. Any element

x P U possesses a membership degree in every lin-

guistic value of all fuzzy attributes. The membership

degree has a value in the interval r0, 1s.

Furthermore, we require that the sum of member-

ship degrees in all linguistic values of each particular

fuzzy condition and decision attribute is equal to 1 for

every element x of the universe U. This is a general-

ization of the property of crisp decision systems, in

which every element x P U has a unique value of each

attribute.

powerpC

pxqq “

k“1

pxq “ 1 , (2)

powerpD

pxqq “

k“1

pxq “ 1 . (3)

Fulﬁlment of the requirements (2) and (3), that we

consider a fundamental property of a well-deﬁned

fuzzy inference system, is assumed in both the labeled

fuzzy rough set approach and the fuzzy ﬂow graph

method.

The standard rough set theory is based on com-

parison of elements of the universe of a crisp in-

formation system that is performed with the help of

an indiscernibility relation. This is a binary equiva-

lence relation which is reﬂexive, symmetric, and tran-

sitive. In a fuzzy information system, a generalized

counterpart in the form of a fuzzy similarity relation

is applied. Determination of similarity between any

two elements of a universe is not so straightforward

and unambiguous as in the crisp case. The degree

of similarity can be any value in the interval r0, 1s.

Furthermore, the obtained fuzzy similarity classes

with respect to condition attributes need to be ap-

propriately used in approximation of fuzzy similarity

classes generated with respect to decision attributes.

However, various fuzzy T-norm and implication op-

erators can be used in computation (Mieszkowicz-

Rolka and Rolka, 2004). Since there is no unique

way to determine the fuzzy rough approximations, it

seems that many degrees of freedom, which is a char-

acteristic feature of the fuzzy set theory, is not always

a preferable phenomenon. This is a motivation for the

labeled fuzzy rough set approach. We want to sim-

plify the method of comparing the elements of a fuzzy

information system. Instead of using a fuzzy similar-

ity relation, we want to act like a human expert, who

does not perform a detailed comparison of particular

objects. A human expert tries to assess how similar a

new object is to a labeled prototype, which is a pat-

tern that can be expressed with the help of linguistic

values of attributes.

Hence, our task consists in discovering the labeled

prototypes. We need to ﬁnd out "active" linguistic

values of attributes which are dominating in the infer-

ence process. To formalize this goal, we use a level β

which satisﬁes the following inequality

0.5 ă β ď 1 . (4)

A selected value of the parameter β will be used

in criteria for classifying particular linguistic values

of attributes. Given a fuzzy information system FS,

we deﬁne for any element x of the universe U and any

fuzzy attribute q P Q:

the set

pxq Ď V

of positive (or dominating) lin-

guistic values

pxq “ tV P V

: f px, V q ě βu, (5)

the set

pxq Ď V

of boundary linguistic values

pxq “ tV P V

: 0.5 ď f px, V q ă βu, (6)

and the set

pxq Ď V

of negative linguistic values

pxq “ tV P V

: 0 ď f px, V q ă 0.5u. (7)

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

116

Taking into account the constraints (2) and (3), we

get the properties of the sets (5), (6), and (7):

1. the set

pxq of dominating linguistic values can

contain at most one element,

2. the set

pxq of boundary linguistic values can

contain at most two elements,

3. the set

pxq of negative linguistic values can con-

tain at most |V

| elements.

Every element x of the universe U is represented

by a row in a decision table. Since we want to com-

pletely describe any element x P U in terms of linguis-

tic values, we should determine a combination of the

linguistic values that are dominating for the selected

element x. In this way, we get a linguistic label of

the element x. We deﬁne the set of linguistic labels

pxq of any element x P U, for a subset of fuzzy at-

tributes P Ď Q, as the cartesian product of the sets of

dominating linguistic values

, for p P P

pxq “

pPP

pxq. (8)

From deﬁnition (5) and property 1, we conclude

that every element x P U can possess at most one lin-

guistic label. We want to generate the family

linguistic labels for the entire universe U. When the

value of the parameter β is increased, the number of

linguistic labels belonging to the family

can be

only decreasing.

Each dominating linguistic label is bound up with

a set, denoted by X

, that represents a linguistic label

, for a subset of fuzzy attributes P Ď Q. This

is a subset of those elements x P U which have the

linguistic label E

P E

“ tx P U : E

pxqu. (9)

We call X

the set of characteristic elements of the

linguistic label E

P E

Linguistic label E

P E

has the form of an or-

dered tuple of dominating linguistic values for all at-

tributes p P P

“ p

, . . . ,

q, (10)

and the resulting membership degree of x P U in the

linguistic label E

P E

can be determined as follows

pxq “ minpµ

pxq, µ

pxq, . . . , µ

|P|

pxqq. (11)

By ﬁnding the membership degree for all elements

of a universe U in a linguistic label E

P E

, we get a

fuzzy similarity class denoted by

“ tµ

q{x

, µ

q{x

, . . . , µ

q{x

(12)

The lower and upper approximations of a set con-

stitute basic notions of the rough set theory. There are

many possibilities to deﬁne fuzzy rough approxima-

tions of a fuzzy set (Radzikowska and Kerre, 2002;

Mieszkowicz-Rolka and Rolka, 2004). As we prefer

to propose a simple method, which avoids problems

with selecting appropriate fuzzy operators, we want

to deﬁne the approximations by using characteristic

elements of a fuzzy set to be approximated.

Given a fuzzy set A, we deﬁne the set X

of char-

acteristic elements of the set A as follows

“ tx P U : µ

pxq ě 0.5u (13)

Now, we use the sets of the characteristic elements

of linguistic labels to approximate the set of charac-

teristic elements of a fuzzy set A. We deﬁne lower

approximation E

pAq of a fuzzy A by the set of lin-

guistic labels E

, which are obtained with respect to

a subset of fuzzy attributes P Ď Q, as follows

pAq “

: X

Ď X

(14)

Upper approximation E

pAq of a fuzzy set A by

the set of linguistic labels E

, which are obtained with

respect to a subset of fuzzy attributes P Ď Q, is deﬁned

pAq “

: X

X X

‰ H (15)

In analysis of an information system given in the

form of a decision table, two families of linguistic la-

bels will be generated:

for the condition attributes

C, and

, for the decision attributes D, respectively.

We need a convenient measure for evaluating the con-

sistency of a fuzzy information system. Approxima-

tion quality γ

q of the fuzzy similarity classes

by the fuzzy similarity classes

is deﬁned as

q “

powerpPos

cardpUq

(16)

Pos

q “

q (17)

The value of approximation quality γ

q be-

longs to the interval r0, 1s. It is decreasing in the

case of inconsistency, i.e., when different decisions

are taken for the same condition.

2.2 Fuzzy Flow Graphs

Another method for describing and analyzing of in-

formation systems utilizes the idea of ﬂow graph

(Pawlak, 2005a; Pawlak, 2005b). A decision ﬂow

graph has the form of layers of nodes, which repre-

sent particular values of condition and decision at-

tributes, connected by branches. Every element of a

Labeled Fuzzy Rough Sets Versus Fuzzy Flow Graphs

117

universe, corresponding to a row of the decision ta-

ble, ﬂows through the graph by taking a unique path.

The original ﬂow graph concept of Pawlak was pro-

posed for dealing with decision tables with crisp at-

tributes. A generalized fuzzy ﬂow graph approach

(Mieszkowicz-Rolka and Rolka, 2006) can help to

evaluate quality and statistical properties of fuzzy in-

ference systems. It should be emphasized that this

generalization is valid for the product T-norm opera-

tor only. Furthermore, the membership functions of

the linguistic values for all attributes must satisfy the

inequalities (2) and (3). In the case of information

systems with fuzzy attributes, each element of the uni-

verse U can ﬂow through more than one path in the

ﬂow graph.

A selected path in a fuzzy ﬂow graph represents a

single fuzzy decision rule. We denote by R

the k-th

decision rule from the set of r possible decision rules

: IF c

is C

AND c

is C

. . . AND

is C

THEN d

is D

AND d

is D

. . . AND

is D

(18)

where k “ 1, 2, . . . , r ,

P C

, i “ 1, 2, . . . , n ,

P D

, j “ 1, 2, . . . , m.

For every element x of the universe U, we can de-

termine the degree of conﬁrmation cdpx, kq of the de-

cision rule R

, by using a T-norm operator as follows

cdpx, kq “ Tpcdapx, kq, cdcpx, kqq, (19)

where cdapx, kq denotes the conﬁrmation degree of the

decision rule’s antecedent

cdapx, kq “ Tpµ

pxq, µ

pxq, . . . , µ

pxqq, (20)

and cdcpx, kq is the conﬁrmation degree of the deci-

sion rule’s consequent, respectively

cdcpx, kq “ Tpµ

pxq, µ

pxq, . . . , µ

pxqq. (21)

By computing the conﬁrmation degrees (20), (21)

and (19), for all elements of the universe U, we get

the support set of the decision rule’s antecedent

supportpcdapx, kqq “ tcdapx

, kq{x

, cdapx

, kq{x

. . . , cdapx

, kq{x

(22)

the support set of the decision rule’s consequent

supportpcdcpx, kqq “ tcdcpx

, kq{x

, cdcpx

, kq{x

. . . , cdcpx

, kq{x

(23)

and the support of the decision rule R

, respectively

supportpR

q “ tcdpx

, kq{x

, cdpx

, kq{x

. . . , cdpx

, kq{x

u .

(24)

Finally, we can evaluate the quality of a decision

rule R

by taking into account the relative throughﬂow

in the corresponding path of the ﬂow graph. This is

done by determining the fuzzy cardinality (power) of

the obtained support sets. The certainty factor cerpR

cerpR

q “

powerpsupportpR

powerpsupportpcdapx, kqqq

(25)

expresses the determinism of the decision rule R

whereas the measure strengthpR

strengthpR

q “

powerpsupportpR

cardpUq

(26)

indicates how many elements of the universe U ﬂow

through the selected path of the ﬂow graph.

3 EXAMPLE

In the following, we perform an analysis of a fuzzy in-

formation system (Table 1) including a universe U of

ten elements, which are described by three fuzzy con-

dition attributes c

, c

, and one decision attribute

. All condition and decision attributes have three

linguistic values. Observe, that for every element of

the universe U, and every attribute, the constraints (2)

and (3) are always satisﬁed.

Let us assume β equal to 0.55 in application of

the labeled fuzzy rough set approach. Only positive

linguistic values of attributes for each element of the

universe U are taken into account. After inspecting

the decision table, we get six linguistic labels with

respect to the condition attributes C:

“ pC

, C

q, E

“ pC

, C

“ pC

, C

q, E

“ pC

, C

“ pC

, C

q, E

“ pC

, C

and three linguistic labels with respect to the decision

attributes D:

“ pD

q, E

“ pD

q, E

“ pD

With each of these linguistic labels, a respective sim-

ilarity class in the form of an appropriate fuzzy set is

connected (Tables 2 and 3).

In the next step, we calculate the lower and up-

per approximations of every fuzzy similarity class to

the linguistic labels for the decision attributes D, by

the fuzzy similarity classes to the linguistic labels for

the condition attributes C, according to the formulae

(14) and (15). It turns out that for all decision sim-

ilarity classes, the lower approximation is equal to

the upper approximation. Hence, we get six certain

decision rules, presented in Table 4, and denoted by

FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

118

Table 1: Decision table with fuzzy attributes.

0.75 0.25 0.00 0.90 0.10 0.00 0.00 0.80 0.20 0.85 0.15 0.00

0.35 0.65 0.00 0.15 0.85 0.00 0.90 0.10 0.00 0.10 0.90 0.00

0.00 0.25 0.75 0.00 0.20 0.80 0.00 0.25 0.75 0.00 0.10 0.90

0.00 0.45 0.55 0.60 0.40 0.00 0.30 0.70 0.00 0.00 0.20 0.80

0.80 0.20 0.00 1.00 0.00 0.00 0.00 0.75 0.25 0.90 0.10 0.00

0.20 0.80 0.00 0.10 0.90 0.00 1.00 0.00 0.00 0.05 0.95 0.00

0.00 0.10 0.90 0.00 0.10 0.90 0.00 0.00 1.00 0.00 0.00 1.00

0.00 0.10 0.90 0.00 0.25 0.75 0.00 0.10 0.90 0.00 0.15 0.85

0.90 0.10 1.00 0.85 0.15 0.00 0.90 0.10 0.00 1.00 0.00 0.00

0.25 0.75 0.00 0.00 0.90 0.10 0.10 0.90 0.00 0.00 0.90 0.10

Table 2: Fuzzy similarity classes to linguistic labels for the

condition attributes C.

0.75 0.00 0.00 0.00 0.00 0.10

0.10 0.65 0.00 0.00 0.15 0.10

0.00 0.00 0.75 0.00 0.00 0.20

0.00 0.30 0.00 0.55 0.00 0.40

0.75 0.00 0.00 0.00 0.00 0.00

0.00 0.80 0.00 0.00 0.10 0.00

0.00 0.00 0.90 0.00 0.00 0.00

0.00 0.00 0.75 0.00 0.00 0.10

0.10 0.10 0.00 0.00 0.85 0.10

0.00 0.10 0.00 0.00 0.00 0.75

Table 3: Fuzzy similarity classes to linguistic labels for the

decision attributes D.

0.85 0.15 0.00

0.10 0.90 0.00

0.00 0.10 0.90

0.00 0.25 0.75

0.90 0.10 0.00

0.05 0.95 0.00

0.00 0.00 1.00

0.00 0.15 0.85

1.00 0.00 0.00

0.00 0.10 0.90

, . . . , R

(` in column LFRS). By applying the for-

mulae (16) and (17), we determine the approximation

quality, which has the value equal to 0.75. We see that

the value of approximation quality is less than 1, al-

though all decision rules are certain. This is caused by

intersection of the membership functions of the neigh-

bouring linguistic values of fuzzy attributes. Hence,

the approximation quality can reach the value of 1

only in the case of a consistent crisp information sys-

tem. The measure of approximation quality can also

be used for determining which of the condition at-

tributes could be removed from the information sys-

tem. The results of attribute reduction are presented in

Table 5. Only the attribute c

can be removed, and we

obtain the same decision rules in the reduced infor-

mation system. The approximation quality decreases,

when we remove the remaining attributes. In other

words, the number of certain decision rules becomes

smaller. For example, after removing the attribute c

we get only three certain decision rules: pC

, C

q Ñ

q, pC

, C

q Ñ pD

q, and (C

, C

q Ñ pD

Without the attribute c

, the decision rules R

and R

become uncertain, and the decision rules R

and R

merge together. The results indicate that the attributes

and c

are indispensable in the information system.

Another problem to be considered is the inﬂu-

ence of the parameter β. The presented results were

obtained for β equal to 0.55. When we increase

β to 0.6, we observe that the linguistic label E

“

, C

q disappears, and we obtain ﬁve certain

decision rules R

, . . . , R

In order to make a comparison, we generate deci-

sion rules with the help of the fuzzy ﬂow graph ap-

proach. In this method, all combinations of linguistic

values for all attributes are taken into account. Since

the decision table contains four attributes, and each at-

tribute possesses three linguistic values, there are 81

possible decision rules. Only part of these rules will

be activated, for which the ﬂow in the corresponding

path is not equal to zero. We select the most signif-

icant decision rules by applying threshold values for

the factors of certainty and strength. By setting the

limit of the certainty factor to 0.7, we get 21 deci-

sion rules. Together with a limit value of the factor of

strength equal to 1.7%, we eventually obtain eight de-

cision rules R

, . . . , R

. Table 4 contains the decision

Labeled Fuzzy Rough Sets Versus Fuzzy Flow Graphs

119

rules generated by the labeled fuzzy rough set (LFRS)

and the fuzzy ﬂow graph (FFG) approaches. Two ad-

ditional rules R

and R

, obtained with the fuzzy ﬂow

graph approach, are denoted by ´ in column LFRS.

As we can see, the rules R

and R

are composed of

such linguistic values that do not form linguistic la-

bels determined with the labeled fuzzy rough set ap-

proach. On the other hand, the decision rule R

has a

low value of strength. This rule would be discarded,

when we set a slightly higher threshold of the strength

of rules.

Table 4: Decision rules.

Decision rule LFRS FFG

strength [%] cer

: pC

, C

q Ñ pD

q ` 6.94 0.92

: pC

, C

q Ñ pD

q ` 10.76 0.88

: pC

, C

q Ñ pD

q ` 11.52 0.85

: pC

, C

q Ñ pD

q ` 17.31 0.92

: pC

, C

q Ñ pD

q ` 6.55 0.79

: pC

, C

q Ñ pD

q ` 1.73 0.75

: pC

, C

q Ñ pD

q ´ 4.14 0.70

: pC

, C

q Ñ pD

q ´ 3.63 0.89

Table 5: Approximation quality of the information system.

Removed attribute

none c

0.75 0.64 0.77 0.56

4 CONCLUSIONS

Both the labeled fuzzy rough set approach and the

fuzzy ﬂow graph method generate comparable results.

Due to a new way of determination of fuzzy similar-

ity classes, which helps to reduce the computational

complexity of the standard fuzzy rough set algorithm,

the labeled fuzzy rough set approach is a preferable

method. It is also less computationally demanding in

comparison with the fuzzy ﬂow graph approach. By

acting like a human expert, who takes into account

only the most signiﬁcant features of the observed phe-

nomenon, a system of fuzzy decision rules can be

easy obtained. The presented example of analysis

of a simple information system with fuzzy attributes

conﬁrms the effectiveness of the developed method.

Nevertheless, due to a detailed analysis of informa-

tion system with a fuzzy ﬂow graph representation,

all possible decision rules can be taken into consid-

eration, which can help to reﬁne the fuzzy inference

system. However, selecting suitable threshold values

for the factors of certainty and strength of rules can

be problematic. In future work, another variants of

parameterized fuzzy rough set approach will be in-

vestigated.

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