TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 8, August 201
4, pp. 6346 ~ 6353
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.556
8
6346
Re
cei
v
ed
Jan
uary 3, 2014;
Re
vised Ma
rch 28, 2014; A
c
cepted Ap
ril 14, 2014
Rules Mining Based on Rough Set of Compatible
Relation
Weiy
an Xu*
1
, Ming Zhang
2
，
Bo Sun
1
, Meng
y
un Lin
1
, Rui Cheng
1
1
School of Mat
hematics a
nd
Ph
y
s
ics, Jian
g
s
u
Univ
ersit
y
o
f
Science
an
d T
e
chnolog
y,
Z
henj
ian
g
21
2
003, Ch
in
a
2
School of Co
mputer scie
n
ce
and Eng
i
n
eeri
ng, Ji
an
gsu U
n
iversit
y
of Scie
nce an
d T
e
chnolo
g
y
,
Z
henj
ian
g
21
2
003, Ch
in
a,
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: x
w
y_
ya
n
@
ho
tmail.com
A
b
st
r
a
ct
Rou
gh s
e
t
mo
del
bas
ed
on
tolera
nce r
e
lati
on, h
a
s b
e
e
n
w
i
dely
used
to
de
al w
i
th
inc
o
mpl
e
te
infor
m
ati
on sy
stems. How
e
v
e
r, this mod
e
l i
s
not so
p
e
rfe
c
t becaus
e n
o
t all of th
e el
e
m
e
n
ts in
a tole
rant
class
are
mutu
ally
toler
ant, b
u
t they
are
a
ll t
o
ler
ant w
i
th
th
e g
e
n
e
ratin
g
el
ement
of this
cl
ass. T
o
me
nd
thi
s
li
mitatio
n
, the compati
b
le re
lati
on is redefi
n
e
d
,
and t
hen the
conce
p
t of max
i
mal co
mp
lete
compati
b
le cl
a
s
s
in i
n
co
mplete
infor
m
ati
on sy
stem
is pr
ese
n
ted fo
r t
he
p
u
rpos
e that
a
n
y tw
o ele
m
e
n
ts in th
e sa
me
compati
b
le
mo
dul
e ar
e
mutu
ally
co
mp
atibl
e
. F
u
rther
more,
tw
o meth
ods
a
r
e p
u
t forw
ard
in th
e
interest
of
selecti
ng
opti
m
a
l
co
mpatib
l
e
class f
o
r an
obj
ect,
w
h
ich
can
be
used
in kn
ow
led
ge r
educti
on. Bes
i
des,
coveri
ngs on
univ
e
rse pro
d
u
ced by tol
e
r
ance a
nd co
mp
atib
le rel
a
ti
ons are d
eep
l
y
investig
ated
and
compar
ed. F
i
n
a
lly, a med
i
cal
decisi
on ta
ble i
s
analy
z
e
d
, so
me co
mpact ru
les are
mi
ned.
Ke
y
w
ords
:
ro
ugh s
e
t, inco
mp
lete
infor
m
ation syste
m
,
tolera
nce re
lati
on, co
mp
atibl
e
relati
on, o
p
ti
ma
l
compati
b
le cl
a
ss, Reductio
n
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Rou
gh set th
eory ha
s develope
d sin
c
e
Pawla
k
'
s
pap
er [1-2] as a
new math
em
atical tool
for analy
z
ing
vague an
d i
m
pre
c
i
s
e de
scriptio
ns
of
object
s
. It use
s
indi
scernibil
i
ty (equivalen
c
e)
relation
to re
pre
s
ent
cla
ssification. In
rece
nt
years,
rou
gh
set t
heory
ha
s b
een
su
ccessf
ully
applie
d to so
many fields
such a
s
Artifici
al Intelligen
ce, Data Minin
g
, Machi
ne L
earni
ng, Pattern
Re
cog
n
ition, Knowle
dge
A
c
qui
sition an
d
so on
[2
-11
].
Rough set prop
osed
by Pawla
k
is ba
sed
on the
a
s
su
mption of
co
mplete info
rmation
syste
m
s, i.e. the
r
e
are n
o
u
n
kn
own
value
s
i
n
the
informatio
n table. Ho
wev
e
r, in practi
cal appli
c
ation
s
, incomplete
information
system
s can
be
see
n
every
w
here fo
r a lot
of unpre
d
ict
able reason
s. Therefo
r
e, mining rule
s from incompl
e
te
informatio
n systems i
s
one
of the important
dire
ction
s
for the development of ro
ugh set.
In gene
ral in
compl
e
te inf
o
rmatio
n sy
stems, un
kno
w
n valu
es m
a
y have two
different
explanation
s
:
in the first case, all
un
kn
own val
u
e
s
a
r
e “do n
o
t ca
re”
co
ndition;
in the
se
con
d
ca
se, all
un
kno
w
n val
u
e
s
a
r
e l
o
st. In Refe
re
nce
[10], Grzym
a
la-Bu
s
se firstly studi
ed
the
unkno
wn val
ue ( “d
o n
o
t
care”) fro
m
the vi
e
w
p
o
int of roug
h set theo
ry, con
s
e
quent
ly,
Kryszkiewi
cz [12] transformed t
he indiscernibility relation to tolerance rel
a
tion (reflecti
ve,
symmetri
c
).
On the othe
r hand, in
comp
lete inform
ati
on syste
m
s in
which
all un
known value
s
are
lost, from th
e
viewpoi
nt of rou
gh
set th
eory, we
re
s
t
udied for the firs
t time in
Referenc
e [12],
whe
r
e two al
gorithm
s for
rule ind
u
ctio
n were
pre
s
ented. Base
d
on Grzym
a
la-Bu
s
se's
wo
rk,
Stefanowski [
13] advan
ced
the
non-sym
m
etric
simila
ri
ty relation (ref
l
ective, transit
ive).
In this pap
er, all unkn
o
wn values
are
looke
d
a
s
“do not care
”,
that is to say, each
unkno
wn value co
uld be repla
c
ed by al
l values from
the domain of
the attri
bute, therefore, what
we
have
don
e are all
ba
se
d on
the fu
rth
e
r inve
stigat
i
on of tol
e
ra
nce rel
a
tion. In t
he
cla
ssifi
cati
on
prod
uced by toleran
c
e
rel
a
tion, not all of the el
eme
n
ts in the sa
me tolera
nt class are m
u
tual
tolerant, but they are all tol
e
rant with th
e
generating el
ement of this
tolerant cl
ass.
Owin
g to
su
ch limitation
of tolera
nce
rel
a
tion,
a
bina
ry relation
call
ed
comp
atibl
e
rel
a
tion
is re
-defin
ed.
Acco
rdin
g to com
patible
relation, a
complete
cove
ring o
n
unive
rse
ca
n be g
o
t.
That is to sa
y, any
two elements in th
e same
com
patible class are mutually
compatible. It is
clea
r that thi
s
kind
of cl
assi
fication of
uni
verse
in in
co
mplete info
rm
ation sy
stem
s meets with t
he
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rule
s Minin
g
Based o
n
Ro
ugh Set of Co
m
patible Rela
tion (Weiya
n Xu)
6347
pra
c
tical
ap
pli
c
ation
s
m
o
re
than that i
s
b
a
se
d on
tole
rance relation.
Furth
e
rm
ore,
sin
c
e
any o
n
e
obje
c
t in the universe i
s
likely to be incl
uded into t
w
o
or mo
re diffe
rent compatib
le cla
s
ses, two
different meth
ods a
r
e p
r
e
s
e
n
ted for choo
sing o
p
timal compatible
cla
ss.
2. Basic Con
cepts
An
in
co
mp
le
te
in
fo
rma
t
io
n s
y
s
t
em is
a q
u
a
d
r
up
le
S=
<U
,
AT
,
V
,
f>
, where
U
is a
nonem
pty finite set of objects called uni
verse a
nd
AT
is a nonempt
y finite set of
attributes, su
ch
that
a
AT
a
a
V
V
U
AT
a
:
whe
r
e
V
a
is calle
d the value set o
f
a
; any attribute dom
ain
V
a
may co
ntain
spe
c
ial
sym
bol “
” to in
dicate th
at the value
of
an attrib
ute i
s
un
kn
own;
V
is
rega
rd
ed a
s
the valu
e
set of all
attributes an
d t
hen
a
AT
a
V
V
; let us define
f
as a
n
informatio
n function
su
ch that
f
(
x,
a
)
V
a
for any
a
AT
and
x
U
.
Defini
tion 1
.
Let
S=
<
U
,
AT
,
V
,
f>
b
e
a
n
in
compl
e
te
inform
ation
system
,
AT
A
,
a
binary relatio
n
SIM
(
A
) ca
n be define
d
as [12]:
*}
)
,
(
*
)
,
(
)
,
(
)
,
(
,
:
)
,
{(
)
(
a
y
f
a
x
f
a
y
f
a
x
f
A
a
y
x
A
SIM
(1)
The
SIM
(
A
) i
s
a
tole
ran
c
e
relatio
n
sin
c
e it is reflexive an
d
symme
tric. Fu
rthe
rm
ore, let
us den
ote
by
S
A
(
x
) the
se
t of obj
ect
s
f
o
r
whi
c
h
SIM
(
A
)
ho
ld
s
,
In
o
t
he
r
w
o
rd
s
,
S
A
(
x
) is t
he
maximal set of
obje
c
ts which
a
r
e po
ssibly
in
disce
r
nible by
A
wi
th
x
an
d a
n
y one
el
ement
in
S
A
(
x
)
ha
s a t
o
lera
nce rela
tion with
x
, it is
called the tolerant class of
x
. It is
c
l
ear that
S
A
(
x
) is
actually a ki
n
d
of neighb
orhood of
x
.
Let
U/SIM
(
A
) denote
s
cl
assificatio
n
for
A
AT
, which is
the family s
e
t {
S
A
(
x
):
x
U
}.
What
sho
u
ld
be noti
c
ed i
s
that all tolerant cla
s
se
s i
n
U/SIM
(
A
)
d
o
not con
s
titute a pa
rtition in
gene
ral, but a
coverin
g
on
universe
U
, namely,
U
x
S
A
U
x
)
(
and
)
(
x
S
A
(
x
U
).
Table 1. Inco
mplete Inform
ation System
Car
Price
M
ileage
Si
z
e
M
a
x-spee
d
1
High High
Full
Low
2
Low
F
u
ll Low
3
Compact
High
4
High
F
u
ll High
5
F
u
ll High
6
Low
High
Full
In Table
1,
AT
={
Pr
i
c
e
,
Mileage
,
Siz
e
,
Max-speed
},
then we
h
a
v
e
U/SIM
(
A
)={
S
AT
(1),
S
AT
(2),
S
AT
(3),
S
AT
(4),
S
AT
(5),
S
AT
(6)}={
{1},{2,6
}, {3},{
4
,5},{4,5,6},{
2
,5,6}}.
Tolera
nt classe
s are
the ba
sis of d
e
fining lo
we
r
and u
ppe
r a
p
p
roximatio
n
s
of a set
X
U
.
The
A-
l
o
wer approximatio
n
and the
A-
u
p
per ap
proximation of
X
are
:
}
)
(
:
{
)
(
X
x
S
U
x
X
A
A
and
}
)
(
:
{
)
(
X
x
S
U
x
X
A
A
(2)
Even though
toleran
c
e relation ha
s
been
widely
used in
de
aling with in
compl
e
te
informatio
n system, it has the following
dra
w
ba
cks.
In the first place, we can see that different
two tolerant
cla
s
ses m
a
y have incl
usi
o
n relatio
n
. Fo
r insta
n
ce, in
Table 1, S
AT
(2)
S
AT
(6) a
nd
S
AT
(4)
S
AT
(5
) hold, this ki
nd of situatio
n sometim
e
s
is unrea
son
a
b
le whe
n
defi
n
ing app
roxi
mate
sets.
Furth
e
rmore, fo
r all
obje
c
ts i
n
S
AT
(
x
),
they ma
y have n
o
common
attrib
ute value
s
. F
o
r
example, in T
able 1, S
AT
(5
)=
{4,5,6},
f
(4
,
Pric
e
)=
{
high
} while
f
(6,
Pr
i
c
e
)=
{
Lo
w
}. From thi
s
poi
nt
of view, it is clear that obje
c
ts 4 an
d 6 a
r
e
disce
r
n
abl
e. From wh
at have been di
scusse
d abo
ve,
we shoul
d make a mo
re reasona
ble cl
assifica
tio
n
in
incompl
e
te informatio
n sy
stem.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 634
6 –
6353
6348
3. Rough Set Based o
n
O
p
timal Comp
atible Clas
s
3.1. Compati
b
le Relation
Defini
tion 3.
Let
S
be a
n
incom
p
lete
information
syst
em, ea
ch
sub
s
et of attributes
A
AT
determ
i
nes a
comp
a
t
ible relation
COM
(
A
).
*}
)
,
(
*
)
,
(
)
,
(
)
,
(
,
:
)
,
{(
)
(
a
y
f
a
x
f
a
y
f
a
x
f
A
a
y
x
A
COM
(3)
Defini
tion 4.
Let
S
be an incompl
e
te informatio
n sy
stem an
d
A
AT
, then
U
/
COM
(
A
)
(repres
ents
c
l
ass
i
fic
a
tion) is
defined as
f
o
llows
:
))}
(
})
{
(
(
),
(
:
{
)
(
/
A
COM
x
B
B
x
U
x
A
COM
B
B
U
B
A
COM
U
2
(4)
In Definition
3, com
patible
relation
is
rea
lly sam
e
t
o
the tole
ran
c
e
relation.
Ho
wever,
U/CO
M
(
A
) i
n
Definition
2 i
s
quitely
different
with
U/S
I
M
(
A
) be
ca
use any two ele
m
ents i
n
B
are
mutually com
patible. Fu
rth
e
rmo
r
e, if a
n
y
other
elem
ent in
U
i
s
ad
ded into
the
compatible
cl
a
ss,
compatible
relation in this
comp
atible
cl
ass
will be destroyed an
yway. Thi
s
kind of
com
pati
b
le
cla
ss
meet
s
with the a
c
tu
al nee
ds m
o
re than
to
lerant cla
s
s an
d as
a result, it is calle
d
the
maximal complete com
p
atible clas
s
.
For example,
Let
u
s
co
nsi
der Tabl
e
1,
U/CO
M
(
AT
)={{1},
{2,6},
{3}
,
{4,5}, {
5
,6}}.
For an
y
B
U/CO
M
(
AT
),
B
is t
he max
i
mal com
p
l
e
t
e
comp
at
ibl
e
cla
ss.
Property
1.
If
COM
(
A
) i
s
a
compatibl
e
relation, then:
A
a
a
COM
A
COM
)}
{(
)
(
(5)
Theorem 1
. Let
S
be
an in
compl
e
te inform
atio
n syste
m
an
d
A
C
AT
,
for any
M
U/C
O
M
(
C
), there mu
st be
N
U/C
O
M
(
A
)
such that
M
N.
Proof
. By
A
C
AT
and
property 1, th
ere mu
st be
COM
(
C
)=
(
CO
M
(
A
)
∩
(
∩
c
C-A
COM
({
c}
),
then
CO
M
(
C
)
COM
(
A
). If
M
U/C
O
M
(
C
), then we
h
a
ve
M
2
CO
M
(
C
)
COM
(
A
). It means
that
any two elem
ents in
M
a
r
e
mutually co
mpatible on
set of attributes
A
. If
M
U/COM
(
A
), then the
theore
m
is ob
vious. If
M
U/C
O
M
(
A
), the
n
, according t
o
the kno
w
le
dge of discret
e mathemati
c
s,
there mu
st be
one cla
s
s su
ch that
M
N
and
M
U/CO
M
(
A
).
As
far as
U/S
I
M
(
A
) is con
c
erne
d, we ca
n say about tolera
nt cla
ss
for any
x
U
, and for
U/CO
M
(
A
)
we ca
n only
say ab
out n
on-ex
clu
s
ive
coverage
of
U
by all maximal com
p
lete
comp
atible
cl
asse
s. Howe
ver, the follo
wing
theo
rem
tells us
that maximal com
p
lete compatible
cla
s
ses a
r
e so tightly related with tolera
nt classe
s.
Theorem 2
. Let
S
be an in incom
p
lete informatio
n system and
A
AT
, then for any
x
U
,
)
(
}
,
/
:
{
X
S
B
x
COM
U
B
B
A
hold
s
.
Proof
.
y
∪
{
B:B
U/C
O
M
(
A
),
x
B
}, we have
(
x,
y
)
COM
(
A
)=
SIM
(
A
). By se
ction 2,
S
A
(
x
)=
{
y
U
:(
x,
y
)
SIM
(
A
)}, then th
ere
must
be
y
S
A
(
x
)
.
Sinc
e
y
i
s
a
r
bitrary
,
then
we
m
u
st
have
∪
{
B:B
U/CO
M
(
A
),
x
B
}
S
A
(
x
).
For any
y
S
A
(
x
), (
x,
y
)
SIM
(
A
)=
COM
(
A
)
holds
. If
{
x,
y
}
is the maximal complete
comp
at
ible cl
as
s
B
, then t
he theo
rem i
s
tru
e
. If {
x,
y
}
is not th
e m
a
ximal one, t
here
mu
st be
B
su
ch t
hat
{
x,
y
}
B
and
B
U/C
O
M
(
A
), then
y
∪
{
B:B
U/COM
(
A
),
x
B
}ho
l
ds. Since
y
is
arbitrary, then
S
A
(
x
)
∪
{
B:B
U/COM
(
A
),
x
B
}.
From the a
b
o
v
e discusse
d,
the theorem
2 is prove
d
.
3.2. Optimal Compa
t
ible Class
es
Let
S
be
an i
n
com
p
lete i
n
formatio
n sy
stem,
U/C
O
M
(
A
)={
B
1
,
B
2
,...,
B
n
} wh
ere
A
AT
, for
any
x
U
, we
call
B
1
t
he max
i
mal com
p
l
e
t
e
com
pat
ibl
e
cla
ss
of
x
if and only if 1
i
n
an
d
x
B
i
.
Neverth
e
le
ss,
it is not difficult to find that there
may be two
or mo
re m
a
ximal com
p
l
e
te
comp
at
ible cl
as
se
s f
o
r
x
U
.
For exam
ple,
the obje
c
t 5
in Table 1,
{4,5
} an
d {5,
6
} are all it's maximal co
mplete
comp
at
ible cl
as
se
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rule
s Minin
g
Based o
n
Ro
ugh Set of Co
m
patible Rela
tion (Weiya
n Xu)
6349
It is natu
r
al
to con
s
ide
r
h
o
w
to
ch
oo
se
a cl
ass
(calle
d
optimal compatible class
) fr
om
several maxi
mal complet
e
compatible classes
of
object
x
fo
r
comp
uting
re
ductio
n
. In the
followin
g
, two
different method
s are p
r
e
s
ented.
Metho
d
1.
The first meth
od of cho
o
si
n
g
optimal co
mpatible cl
ass ro
ots from t
he ba
sic
idea of value
d
toleran
c
e
relation [13]. Simply, for
x
U
,
we sh
o
u
ld ch
oo
se a
max
i
mal com
p
let
e
comp
atible
cl
ass in
whi
c
h
element
s a
r
e most
po
ssi
bly having
sa
me value
s
of
attribute
s
a
s
x
has. Assumi
n
g
that the set
of possible v
a
lue
s
on
e
a
ch
attribute
is discrete, we make hypoth
e
si
s
that there ex
its a
uniform
pro
bability
di
strib
u
tion a
m
ong su
ch
values. Co
nsider
a
AT
in
incom
p
lete i
n
formation
system
S
and
a
s
so
ciate it to t
he
set {
a
1
,
a
2
,
...,
a
m
} of all possibl
e valu
es,
given an obj
e
c
t
x
U
, if
f
(
x,
a
)=
, then we assume th
a
t
the proba
bility
f
(
x,
a
)=
a
j
(
j
=
1
,2,…,
m
) is
equal to 1/
Car
d
({
a
1
,
a
2
,...,
a
m
}).
Defini
tion
5
.
The probabilit
y of two objects
x
,
y
h
a
ving
sam
e
valu
e
on a
set
of attribute
s
A
is
pr
A
(
x
,
y
)=
)
,
(
y
x
pr
a
A
a
wher
e
pr
a
(
x
,
y
) i
s
the probability of two
obj
ect
s
having
same val
ue
on sin
g
le attri
bute
a
.
For any
x
,
y
U
and
a
AT
,
pr
a
(
x
,
y
)
coul
d
be comp
uted
as follows:
)
,
(
)
,
(
:
)
,
(
)
,
(
:
*
)
,
(
*
)
,
(
:
))
(
/(
*)
)
,
(
*
)
,
(
(
:
)
(
/
)
,
(
a
y
f
a
x
f
a
y
f
a
x
f
a
y
f
a
x
f
V
card
a
y
f
a
x
f
V
card
y
x
Pr
a
a
a
0
1
1
1
2
(6)
Defini
tion 6
.
Let
S
be
an
incom
p
lete in
formation
system in which
A
AT
, for any
x
U
,
the optim
al
co
mpatible
cla
s
s of
x
is
S
A
OPT
(
x
)=
B
i
if
and
only if
the valu
e
of
}
{
})
{
(
/
)
,
(
x
B
y
i
A
i
x
B
card
y
x
Pr
is maximal for any
B
i
U/C
O
M
(
A
)
x
B
i
.
Not
e
1
. For
x
U
, if the
r
e a
r
e t
w
o
or
more
compati
b
le
cla
s
ses h
a
ve the
sa
me
maximal
value of
{}
(,
)
/
(
{
}
)
i
yB
x
A
i
Pr
x
y
C
a
rd
B
x
, th
en their inte
rse
c
tion is a
c
cepta
b
le a
s
the optimal
comp
at
ible cl
as
s of
x
.
For exa
m
ple,
the obj
ect
5 in Ta
ble 1
,
there a
r
e t
w
o m
a
ximal
compl
e
te
co
mpatible
cla
s
ses, {4, 5
}
and {5, 6}. For maximal
compl
e
te co
mpatible cl
ass {4, 5},
11
1
24
8
5,
4
AT
pr
, for
maximal complete compatible
class {5,6},
11
1
1
22
2
8
5,
6
AT
pr
. The
r
efore,
5{
4
,
5
}
{
5
,
6
}
{
6
}
OPT
AT
S
.
Metho
d
2
. In
method
2,
we only
co
nsi
d
er tho
s
e
valu
es
are
all
kn
own. In
all
m
a
ximal
compl
e
t
e
co
mpat
ible cla
s
se
s
of
x
U
,
there
is at le
ast o
ne i
n
whi
c
h
eleme
n
ts
have the
max
i
mal
numbe
rs of attributes
who
s
e certai
n valu
es ar
e equ
al to the attribute
s
' certain valu
es of
x
.
In incom
p
lete information
system
S
, let
x
,
y
U
and
a
A
AT
, su
ppo
se that
t
=1 if and
only if
f
(
x
,
a
)=
f
(
y
,
a
) whil
e
f
(
x
,
a
) a
nd
f
(
y
,
a
) are all kno
w
n, othe
rwi
s
e
t
=0. Therefore, for any
x
U
,
A
AT
, we define
{}
()
ii
B
yB
x
a
A
x
t
whe
r
e
x
B
i
.
Defini
tion 7
.
In incom
p
let
e
information system
S
, in which
A
AT
, for any
x
U
, the
optimal com
p
atible cla
ss o
f
x
is
S
A
OPT
(
x
)=
B
if and o
n
ly if
the value of
()
/
(
{
}
)
i
Bi
x
Ca
r
d
B
x
is
maximal for any
B
i
U/CO
M
(
A
)
x
B
i
.
Not
e
2
. Fo
r
any
x
U
, if there
are two
or mo
re
co
mpatible
cla
s
se
s have th
e
sam
e
maximal value of
()
/
(
{
}
)
i
Bi
x
Ca
r
d
B
x
, then their inte
rse
c
tion is
a
c
ceptabl
e as optimal
comp
at
ible cl
as
s of
x
.
For exampl
e, in Table 1, there are t
w
o ma
x
i
mal
comp
at
ible cl
as
se
s f
o
r ob
ject
5,
N
1
={
4,5},
N
2
={5,6}, respec
tively. Ac
c
o
rdi
ng to d
e
finition 7, it is ea
sy to work o
u
t
1
(5
)
N
=
2
a
nd
2
(5
)
N
=1,
so
Ma
x
(
1
(5
)
N
,
2
(5
)
N
)=
2,
5
OPT
AT
S
={
4,5}.
Defini
tion
8
.
Giv
e
n
an
in
compl
e
t
e
i
n
f
o
rmat
ion
sy
st
e
m
S
a
nd a n
on-e
m
pty
su
bset of
attributes
A
AT
, with each
sub
s
et of obj
ects
X
U
we
asso
ciate two
sets:
()
{
:
(
)
}
OPT
OPT
A
A
Xx
U
S
x
X
and
()
{
:
(
)
}
OP
T
OP
T
A
AX
x
U
S
x
X
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 634
6 –
6353
6350
Not
e
3
. Acco
rding to the compatible rel
a
tion of roug
h set theory,
A
-lower an
d
A
-upp
er
of
X
have two instan
ce
s i
n
respe
c
t that two diffe
ren
t
methods of
cho
o
si
ng opti
m
al com
patib
le
cla
ss.
3.3. Compari
s
on bet
w
e
e
n
Tolerance a
nd Compa
t
ib
le Relation
s
From the vie
w
poi
nt of gra
nular
comp
ut
ing
[14], classe
s are the
basi
c
buildi
n
g blocks
and
call
ed
el
ementa
r
y g
r
a
nule
s
[15]. T
hey a
r
e th
e
smalle
st
non
empty sub
s
e
t
s that
ca
n b
e
defined, ob
served o
r
me
asu
r
ed. Of course, diffe
re
nt binary rel
a
tions m
a
y produ
ce diffe
rent
elementa
r
y g
r
anul
es.
Fro
m
what h
a
ve
bee
n di
scu
s
sed
ab
ove,
classe
s p
r
o
d
u
c
ed
by tole
ra
nce
and compatib
le relation
s re
spe
c
tively, all form cove
rin
g
s on the u
n
i
v
erse.
Each
cove
rin
g
rep
r
e
s
e
n
ts
one g
r
an
ulat
ed view of th
e universe.
Due to the diff
eren
ce
of
method
s in
cl
assifying, tole
ran
c
e
and
co
mpatib
le relat
i
ons
produ
ce different cove
ring
s calle
d
1
and
2
, re
sp
ectively. Accordin
g to T
h
eore
m
2, it i
s
clea
r that
coveri
ng
2
i
s
a
refin
e
me
nt of
coveri
ng
1
o
r
equival
ently
1
is a
coa
r
sening
of
2
, denote
d
by
1
2
or
2
1
, for the rea
s
on
that every block of
2
is contai
ned in
some bl
ock
of
1
. Given two cove
ring
s
1
and
2
, their
meet
1
2
is the large
s
t covering
whi
c
h
is a refineme
n
t of both
1
and
2
, and their join
1
2
is
the small
e
st
coveri
ng
whi
c
h is a
coa
r
se
ning of both
1
and
2
. Fro
m
what h
a
ve
been
discu
s
sed,
coveri
ng
2
h
a
s a smalle
r level of granul
ation for probl
em solving th
an cove
ring
1
.
Let
U
be
a
n
on-e
m
pty finite set of
obje
c
ts, a
n
d
let
RU
U
denote
an
bi
nary
relatio
n
on
U
. The pai
r
apr
=<
U
,
R
> is called a
n
approximatio
n spa
c
e [1]. The cove
ring o
f
the universe
is
calle
d the qu
otient set in
d
u
ce
d by
R
a
nd is
denote
d
by
U/R
. E
v
en thoug
h rough
set d
a
ta
analysi
s
i
s
a
symboli
c
met
hod
of analy
s
is, it u
s
es
co
unting info
rm
ation p
r
ovide
d
by the
cla
s
se
s
of the bina
ry
relation
s u
n
d
e
r
con
s
ide
r
ati
on. T
he i
nhe
rent statisti
c o
f
an app
roxim
a
tion spa
c
e
<
U
,
R
> is the
accura
cy me
asure of
rou
gh
set
app
roximatio
n
[1]
()
(
(
)
)
/
(
(
)
)
A
Ca
r
d
A
X
Ca
r
d
A
X
. It may
be interpreted as the probability t
hat an element bel
ongs to the lo
wer approximation, given that
the element
belon
gs to the uppe
r app
roximation an
d expre
s
ses t
he deg
ree of
compl
e
tene
ss of
o
u
r
kn
ow
le
dge
o
f
X
.
Toleran
c
e an
d comp
atible
relation
s pro
duce differen
t
accu
ra
cy measure
s
of ro
ugh set
approximatio
n named a
s
1
()
A
and
2
()
A
, respe
c
ti
vely. Accordi
ng to theore
m
2, it is easy to
validate that
()
()
()
()
OP
T
OP
T
A
XA
XX
A
X
A
X
, therefo
r
e,
(
(
))
(
(
))
OPT
Card
A
X
C
a
rd
A
X
an
d
(
(
))
(
(
))
OP
T
Card
A
X
Card
A
X
, so
we
hav
e
12
0(
)
(
)
1
AA
. That is to say, with
c
o
mpatible
relation, we can get more knowl
edge of
X
than tolera
nce relation.
4. Kno
w
l
e
d
g
e Redu
ction
An incompl
e
te de
ci
sion t
a
ble [4] i
s
a
n
i
n
com
p
lete i
n
formatio
n
syst
em
DT
=<
U
,
AT
∪
{
d
},
V
,
f
> where
d
is call
ed de
cision attri
bute
and
V
d
is the
value domai
n of the deci
s
ion attribute
d
,
corre
s
p
ondin
g
ly, element
s in
AT
a
r
e
called
con
d
itio
n attribute
s
.
In addition,
AT
∩
{
d
}=
a
nd
V
d
.
Any deci
s
ion
table may be
rega
rd
ed a
s
a set of d
e
ci
sion rul
e
s
of the form:
(,
)
(
,
)
av
d
w
, where
a
AT
,
v
V
c
,
w
V
d
. Owing to difference o
f
classificatio
n
betwee
n
compatible a
n
d
toleran
c
e
rel
a
tions, the computation o
f
genera
lize
d
deci
s
ion fu
nction [12] that is used
in
kno
w
le
dge re
ductio
n
sh
oul
d be modified
.
Defini
tion
9
. Let
DT
=<
U
,
AT
∪
{
d
},
V
,
f
>
be an in
comp
lete deci
s
io
n system a
nd then th
e
gene
rali
zed d
e
ci
sion fun
c
ti
on is defin
ed
as follo
ws:
:
(
),
,
{
:
(
),
(
)
}
OP
T
Ad
A
A
UP
V
A
A
T
i
i
d
y
y
S
x
(8)
Defini
tion
10
. Let
DT
=<
U
,
AT
∪
{
d
},
V
,
f
> be a
n
in
co
mplet
e
de
ci
si
on sy
st
em,
A
AT
is a
redu
ction of
DT
(relative re
ductio
n
) iff
AA
T
and for any
C
A
,
CA
.
In the process of investi
gation, it i
s
conv
enient t
o
use
discernibility functi
on [12] to
comp
ute re
du
ction in in
com
p
lete inform
ation and d
e
ci
si
on syste
m
s.
In inco
mplete
deci
s
io
n sy
stem, for any
A
AT
, let
(,
)
A
x
y
be a
set of att
r
ibute
s
a
A
su
ch t
h
at
(
x
,
y
)
COM
({
a
}
)
a
nd a
s
a result if (
x
,
y
)
COM
({
a
}
)
th
en
(,
)
A
xy
. Let
(,
)
A
x
y
be
a
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rule
s Minin
g
Based o
n
Ro
ugh Set of Co
m
patible Rela
tion (Weiya
n Xu)
6351
Boolean
expression th
at is equal
to 1 if
(,
)
A
xy
, otherwise,
(,
)
A
x
y
be a
disj
un
ction of
variable
co
rre
spo
ndin
g
to attributes
cont
ained in
(,
)
A
x
y
.
Defini
tion
11
.
is a di
scerni
bility function for
incomplet
e
deci
s
ion sy
stem if:
(,
)
{
:
(
)
(
)
}
(,
)
A
A
xy
U
z
U
d
z
x
x
y
(9)
Defini
tion
12
.
()
x
is a discernibility function for object
x
in incomplete decisi
on
sy
st
em if
:
()
{
:
(
)
(
)
}
()
(
,
)
A
A
yz
U
d
z
x
x
xy
(10)
5. Illustrativ
e
Example
In this se
ctio
n, a medical treatment de
ci
si
on table
will be analyze
d by roug
h set base
d
on
comp
atibl
e
relation.
Of co
urse, t
w
o
method
s of
choo
sing
opti
m
al
comp
atib
le cl
asse
s
are all
work
ed out.
Table 2 dep
icts an in
co
mplete de
cision table ab
out medical
treatment deci
s
ion.
Age
,
Sarcou
s pain
,
Fe
ve
re
d
, and
H
e
ada
c
h
e
are
the con
d
itional attributes,
Rem
edial schem
e
is
the deci
s
io
n attribute (in t
he se
quel,
a
1
,
a
2
,
a
3
,
a
4
, and
d
will
stand for
Age
,
Sarco
u
s pai
n
,
Feve
re
d
,
He
ada
ch
a
nd
Rem
edial schem
, resp
ect
i
v
e
ly
).
V
a
1
={
a
dult
,
enfant
,
infant
}={1,2,
3
},
V
a
2
={
no pain
,
pain
}={1,2},
V
a
3
={
no
rm
al
,
hyperpyre
xi
a
}={1,2},
V
a
4
={
n
o
he
ada
che
,
hea
da
ch
e
}
={1,2
}
, an
d
V
d
={
m
edical
therap
y
,
ph
ys
ica
l
th
er
a
p
y
,
com
b
inatio
n
of m
edicati
on a
nd
physics
}
={1,2,3
}
.
Table 2. Medi
cal Deci
sio
n
Table
U
a
1
a
2
a
3
a
4
d
1 1
2 1
2 1
2
2 1
3 1
2 2
1
4 1
1
1
2
2
5 1
1
2
6 2
1
1
2
2
7
2 1
2
8
2 2
1
3
9 2
1
2
1
3
10 3
1 3
The followi
ng
are ten de
cision rule
s for T
able 2:
r
1
: (
a
1
, 1)
(
a
2
,
)
(
a
3
,
)
(
a
4
, 2)
(
d
, 1)
r
2
: (
a
1
, 1)
(
a
2
, 2)
(
a
3
,
)
(
a
4
, 2)
(
d
, 1)
r
3
: (
a
1
, 1)
(
a
2
,
)
(
a
3
, 2)
(
a
4
, 2)
(
d
, 1)
r
4
: (
a
1
, 1)
(
a
2
, 1)
(
a
3
, 1)
(
a
4
, 2)
(
d
, 2)
r
5
: (
a
1
, 1)
(
a
2
, 1)
(
a
3
, 1)
(
a
4
,
)
(
d
, 2)
r
6
: (
a
1
, 2)
(
a
2
,
)
(
a
3
, 1)
(
a
4
, 2)
(
d
, 2)
r
7
: (
a
1
,
)
(
a
2
, 1)
(
a
3
, 1)
(
a
4
,
)
(
d
, 2)
r
8
: (
a
1
,
)
(
a
2
, 2)
(
a
3
, 2)
(
a
4
, 1)
(
d
, 3)
r
9
: (
a
1
, 2)
(
a
2
, 1)
(
a
3
, 2)
(
a
4
, 1)
(
d
, 3)
r
10
: (
a
1
, 3)
(
a
2
,
)
(
a
3
,
)
(
a
4
, 1)
(
d
, 3)
From
Table
2. we
ha
ve
U/C
O
M
(
AT
)={{1,2,3
},{
1
,2,5,7},{1,4,
5
},{6},{
7
,10},{
8,10},{9
}}.
Acco
rdi
ng to Method 1, we
have:
(1)
{
1
,
4
,
5}
OPT
AT
S
;
(1)
{
1
,
2
,
3
}
OPT
AT
S
;
(
3
)
{
1,
2
,
3}
OPT
AT
S
;
(4
)
{
1
,
4
,
5
}
OPT
AT
S
;
(5
)
{
1
,
4
,
5
}
OPT
AT
S
;
(6)
{
6
}
OP
T
AT
S
;
(7
)
{
1
,
2
,
5
,
7
}
OPT
AT
S
;
(8
)
{
8
,
1
0
}
OP
T
AT
S
;
(9)
{
9
}
OP
T
AT
S
;
(
10)
{
8
,
1
0
}
OPT
AT
S
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 634
6 –
6353
6352
Acco
rdi
ng to Method 2, we
have:
(1)
{
1
,
2
,
3
}
OPT
AT
S
;
(1)
{
1
,
2
,
3
}
OPT
AT
S
;
(
3
)
{
1,
2
,
3}
OPT
AT
S
;
(4
)
{
1
,
4
,
5
}
OPT
AT
S
;
(5
)
{
1
,
4
,
5
}
OPT
AT
S
;
(6)
{
6
}
OP
T
AT
S
;
(7
)
{
1
,
2
,
5
,
7
}
OPT
AT
S
;
(8
)
{
8
,
1
0
}
OP
T
AT
S
;
(9)
{
9
}
OP
T
AT
S
;
(
10)
{
8
,
1
0
}
OPT
AT
S
Owin
g to we
have two method
s of cho
o
si
ng opt
imal com
pati
b
le cla
s
se
s, then two
different gen
e
r
alized de
ci
si
on functio
n
s (
1
AT
,
2
AT
, respe
c
tively) are
worke
d
out in Table 3
.
Table 3. Gen
e
rali
zed
De
ci
sion Fu
nctio
n
s
U\AT
1
2
3 4 5
6 7 8
9
10
η
AT
1
1,2
1
1 1,2
1,2
2 1,2
3
3
3
η
AT
2
1 1
1 2
1,2
2
1,2
3
3 3
Owing to different generaliz
ed decisi
on functions,
the
computation of discernibility
function
will be different, too. Formally
, using
1
AT
we
can
com
pute
out all of the rel
a
tive
redu
ction
s
of Table 2:
4
(1
)
a
;
24
(2
)
aa
;
34
(3
)
aa
;
12
1
3
(4
)
aa
aa
;
13
(5
)
aa
;
13
1
4
(6)
aa
a
a
;
3
(7
)
a
;
34
(8
)
aa
;
13
1
2
4
(9
)
aa
a
a
a
;
1
(1
0
)
a
;
1
234
aa
a
a
.
From relative redu
ction
s
of obje
c
ts, we
can get followi
ng com
p
a
c
t rules:
r
1
’
: (
a
4
, 2)
(
d
, 1)
(
d
, 2)
r
2
’
: (
a
1
, 1)
(
a
2
, 2)
(
d
, 2)
r
3
’
: (
a
3
, 2)
(
d
,2)
r
4
’
: (
a
1
, 2)
(
a
4
, 2)
(
d
, 2)
r
5
’
: (
a
3
, 2)
(
a
4
, 1)
(
d
, 3)
r
6
’
: (
a
1
, 2)
(
a
3
, 2)
(
d
,3)
r
7
’
: (
a
1
, 2)
(
a
2
, 1)
(
a
4
, 1)
(
d
, 32)
r
8
’
: (
a
3
, 2))
(
d
, 3)
Usi
ng
2
AT
we ca
n also
comp
u
t
e out all of th
e redu
ction
s
i
n
Table 2:
12
(1)
aa
;
24
(2
)
aa
;
34
(3
)
aa
;
12
1
3
4
(4
)
aa
aa
a
;
13
(5
)
aa
;
13
1
4
(6
)
aa
a
a
;
3
(7
)
a
;
34
(8
)
aa
;
13
1
2
4
(9
)
aa
a
a
a
;
1
(
10)
a
;
1
234
aa
a
a
.
Similarly to ru
les ind
u
ci
ng
by method 1
of choo
sin
g
o
p
timal com
p
a
t
ible cla
s
ses,
it is no
t
hard to b
r
ing
out the rule
s by method 2 and the out
co
mes a
r
e sam
e
to
r
1
',…,
r
8
'.
6. Conclusio
n
Rou
gh set theory a
s
sum
e
s that kn
owledge
co
me
s from the ab
ility of classif
i
cation.
Ho
wever,
an
explicit hyp
o
thesi
s
in
ro
ugh
set is
th
at all availab
l
e obje
c
ts
ca
n be
com
p
let
e
ly
descri
bed
by
the set of attri
butes. In
ord
e
r to
m
ana
ge
obje
c
ts
wh
o
have in
com
p
l
e
te de
scriptio
ns
of attribute
s
,
so m
any
sch
o
lars h
a
ve d
o
ne ex
cellent j
obs. In
this p
aper, th
e
co
mpatible
relat
i
on
and
maximal
co
mplete
co
mpatible
cla
s
se
s a
r
e
presented. T
he
main
advanta
ge of
compat
ible
cla
ss i
s
that it
can m
a
ke su
re that all el
e
m
ents
in t
h
e
same
cla
s
s a
r
e mu
tually compatible
while
tolerant cl
ass ca
nnot. From the kno
w
led
ge re
du
ction ba
se
d on com
patibl
e
relation, som
e
comp
act rul
e
s are min
ed.
In the furth
e
r re
sea
r
che
s
, we are g
o
ing to defin
e some p
r
e
c
ise
measurement
s to
weig
h
the reli
ability of thos
e ru
les mi
ned f
r
om in
compl
e
te informatio
n
sy
st
em
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rule
s Minin
g
Based o
n
Ro
ugh Set of Co
m
patible Rela
tion (Weiya
n Xu)
6353
Ackn
o
w
l
e
dg
ements
This
wo
rk i
s
sup
porte
d by
the Natu
ral
Scien
c
e F
o
u
ndation
of China (No.6
1
1
0011
6),
Natural Sci
ence Foun
d
a
tion of Jiang
su
Provi
n
ce of Ch
ina (No.BK2011
492
) a
nd
(No.BK20
130
472).
Referen
ces
[1
]
Pa
w
l
a
k
Z. R
o
ug
h
se
ts.
I
n
t
e
rn
at
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a
l J
ourn
a
l
of
Co
mput
er
a
nd I
n
f
o
r
m
at
ion
Scienc
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.
19
8
2
;
11(5):
34
1-
356.
[
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]
Pa
w
l
ak Z
,
Skow
r
o
n A.
Rou
g
h
set
s
:
some ext
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ons,
I
n
f
o
rmat
i
on Sci
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.
2007;
1
77(
1):
28-40.
[3]
Qian Y
H
, L
i
a
ng JY,
Pedr
ycz W, Dan
g
CY. Positiv
e
appr
o
x
imati
on:
an
acc
e
ler
a
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attribut
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reduct
i
on in ro
ugh set
t
h
e
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r
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201
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[
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Yang
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t
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Credi
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u
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inc
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t
e
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isio
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.
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ge-B
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[
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Xi
on
g Yus
hu.
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e
r
i
n
g
A
l
gorit
hm Bas
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d on
Ro
ug
h S
e
t
an
d Gen
e
t
i
c Algor
it
hm.
TEL
K
OMNIKA
I
ndon
esi
an Jou
r
nal of
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r
ic
al Eng
i
ne
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n
g
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1
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578
2-57
88.
[
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]
Z
hang
Y.
Ro
u
gh S
e
t
a
nd
D
EA of
St
rat
e
g
i
c Alli
anc
e St
a
b
le
Dec
i
sio
n
-m
akin
g Mo
del.
TEL
K
OMNIKA
I
ndon
esi
an Jou
r
nal of
Elect
r
ic
al Eng
i
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n
g
.
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729
5-73
01.
[7]
Qian YH,
Li
a
ng JY, P
edr
ycz W, Dang
CY. An e
ffici
e
n
t acce
lerator
for attribut
e
reducti
on from
in
co
mp
le
te
da
t
a
in
ro
ug
h
se
t fra
m
e
w
o
r
k.
Pat
t
ern Rec
ogn
it
io
n
.
2011;
4
4
(8):
165
8-16
70.
[
8
]
Xu WH,
L
i
Y,
Li
ao
XW.
Appro
a
c
hes t
o
at
t
r
ib
ut
e red
u
ct
ions
b
a
sed
on ro
ug
h
set
and m
a
t
r
i
x
comput
at
i
on
in inc
onsist
e
nt
order
ed inf
o
rmat
ion s
y
st
ems.
Know
led
ge-B
a
sed Syst
e
m
s.
201
2;
27:
78-9
1
.
[
9
]
Lia
ng JY,
Wan
g
F
,
Dang CY,
Qian YH,
An
ef
f
i
cient
ro
ugh
f
eat
ure sel
e
ct
i
on al
gor
it
hm
w
i
t
h
a mult
i-
gran
ulat
i
on vi
e
w
,
I
n
t
e
rn
at
ion
a
l
Journa
l of
App
r
oxi
m
at
e R
eas
oni
ng
.
20
12;
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3
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92
6.
[
10]
Grz
y
mal
a
-Bus
se JW.
Dat
a
w
i
t
h
missin
g
at
t
r
i
but
e va
lues:
g
ener
aliz
at
io
n o
f
indisc
erni
bil
i
t
y
re
lat
i
o
n
a
n
d
ru
le
red
u
c
tion
.
Lect
u
re N
o
t
e
s i
n
Co
mp
ut
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e
.
200
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11]
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y
mal
a
-Bus
se JW,
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Modif
i
e
d
al
gorit
hms
LEM1
and
LEM2
f
o
r
rule
ind
u
ct
io
n f
r
om dat
a
w
i
t
h
missing
at
t
r
ibu
t
e valu
es.
Pro
c
eed
ing
of
t
h
e
5t
h I
n
t
e
rn
at
io
nal Works
hop
on R
o
u
gh S
e
t
s
and
Sof
t
Co
mp
ut
ing
at
t
he 3r
d J
o
int
C
onf
ere
n
ce
on I
n
f
o
rmat
io
n Sci
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,
R
e
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2
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n
t
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I
n
f
o
rmat
i
o
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Scienc
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199
9;
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4):
27
1-29
2.
[
13]
St
ef
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w
s
k
i
J
.
I
n
compl
e
t
e
i
n
f
o
rmat
io
n t
a
b
l
es a
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o
u
g
h
class
i
f
i
cat
i
o
n
.
I
n
t
e
rnat
i
o
n
a
l
Jour
na
l of
Co
mp
ut
at
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a
l I
n
t
e
lli
genc
e
.
20
01;
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45-
566.
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14]
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I
n
f
o
rm
at
ion
gran
ul
at
i
on a
nd ro
ug
h a
ppro
x
imat
io
n.
I
n
t
e
rnat
i
o
n
a
l Jo
urna
l of
I
n
t
e
ll
ig
ent
Syst
e
m
s
,
200
1;
16(1):
87
-104.
[
15]
Xu WH,
Sun
WX,
Z
hang
XY,
Z
hang WX.
Mult
iple
gra
nul
at
ion r
oug
h
set
appro
a
ch
t
o
ordere
d
inf
o
rmat
i
on s
y
s
t
ems
.
I
n
t
e
rnat
iona
l Journ
a
l of
General Syst
e
m
s.
20
12;
41(
5
)
:
475-50
1.
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