Int
ern
at
i
onal
Journ
al of Ele
ctrical
an
d
Co
mput
er
En
gin
eeri
ng
(IJ
E
C
E)
Vo
l.
8
, No
.
6
,
Decem
ber
201
8
, p
p.
4568
~
4576
IS
S
N:
20
88
-
8708
,
DOI: 10
.11
591/
ijece
.
v
8
i
6
.
pp
4568
-
45
76
4568
Journ
al h
om
e
page
:
http:
//
ia
es
core
.c
om/
journa
ls
/i
ndex.
ph
p/IJECE
Gener
atin
g Non
-
r
edun
da
nt Mul
tilev
el Ass
oc
i
ation
Rules Us
ing
Min
-
m
ax E
xact
Rules
R.
Vij
aya Pra
ka
sh
1
, SSV
N.
Sa
rm
a
2
, M.
S
heshikal
a
3
1
Depa
rt
m
ent
of C
om
pute
r
Scie
n
ce
and Engi
ne
ering,
S R
Engi
n
eering
Coll
ege,
Ind
ia
2
Depa
rt
m
ent
of C
om
pute
r
Scie
n
ce
and Engi
ne
ering,
Va
agde
vi
Co
ll
eg
e
of
Engi
n
eering,
Ind
ia
3
Depa
rtm
nt
of
C
om
pute
r
Scie
n
ce a
nd
Engi
n
ee
rin
g,
S R
Engi
n
ee
ri
ng
Coll
ege
,
Indi
a
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Sep
22
,
201
7
Re
vised
Ju
l
13
,
201
8
Accepte
d
J
ul
29
, 2
01
8
As
socia
ti
on
Rul
e
m
ini
ng
play
s
an
important
role
in
the
dis
cove
r
y
of
knowledge
and
i
nform
at
ion.
As
sociation
Rul
e
m
i
ning
discove
rs
h
uge
num
ber
of
rule
s
for
an
y
dat
ase
t
for
diff
ere
nt
support
an
d
conf
ide
n
ce
v
a
lue
s,
among
thi
s
m
an
y
of
t
hem
are
red
un
dant
,
espe
cially
in
the
ca
se
of
m
ult
i
-
le
ve
l
dat
ase
ts.
Mining
non
-
red
undant
As
socia
ti
on
Rul
es
in
m
ult
i
-
le
vel
dat
ase
t
is
a
big
conc
e
rn
in
fi
el
d
of
Data
m
ining.
In
thi
s
pape
r
,
we
pre
sent
a
de
fini
ti
on
fo
r
red
undancy
and
a
concise
r
epr
ese
ntation
called
Reliable
Exact
b
asis
for
rep
rese
nt
ing
non
-
red
undant
As
sociation
Rule
s
fro
m
m
ult
i
-
le
vel
da
ta
sets.
Th
e
give
n
non
-
red
un
dant
As
socia
ti
o
n
Rule
s
are
loss
le
ss
rep
rese
nt
ation
for
a
n
y
dat
ase
ts.
Ke
yw
or
d:
Associ
at
ion
R
ul
e
Fr
e
qu
e
nt I
te
m
set
s
Non
-
Re
dunda
nt
Rules
Re
li
able Rules
Copyright
©
201
8
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
R. V
i
j
ay
a Pr
a
ka
sh
,
Dep
a
rt
m
ent o
f C
om
pu
te
r
Scie
nce a
nd E
ng
i
ne
erin
g,
S R E
ng
i
neer
i
ng C
ollege,
An
at
hasa
gar
,
Hasa
np
a
rthy,
War
a
ngal
, Tel
ang
a
na, I
ndia
.
Em
a
il
:
vij
pr
a
k@h
otm
ail.co
m
1.
INTROD
U
CTION
The
hu
ge
am
ount
of
the
e
xtracted
ru
le
s
is
a
big
pro
blem
fo
r
Associ
at
ion
R
ule
m
ining
[
21
]
.
Especial
ly
,
m
a
ny
of
t
he
e
xtra
ct
ed
r
ules
a
re
c
on
si
der
e
d
re
dund
a
nt
si
nce
t
he
y
pro
du
ce
the
sam
e
m
eaning
to
the
us
er
or
e
xtract
ed
ru
le
s
can
be
re
placed
by
oth
e
r
ru
le
s.
M
any
ef
f
or
ts
ha
ve
bee
n
m
ade
on
reducin
g
the
siz
e
of
the
ext
racted
r
ule
set
.
T
he
re
are
nu
m
ber
of
represe
ntati
on
s
of
f
reque
nt
pa
tt
ern
s
hav
e
be
en
pro
po
se
d,
one
of
them
,
is
the
cl
os
e
d
it
em
se
ts,
is
of
pa
rtic
ular
interest
as
t
hey
can
be
a
ppli
ed
f
or
ge
ner
at
in
g
non
-
re
dunda
nt
r
ules
[10
]
,
[
12
]
,
[
18
]
,
[
23]
.
The
use
of
f
re
qu
e
nt
c
losed
it
em
set
s
pr
ese
nts
a
cl
ear
prom
ise
to
red
uce
t
he
num
ber
of
extracte
d
ru
le
s
[13
]
,
[
17
]
,
[
19]
.
Mult
i
-
le
vel
dataset
s
in
wh
i
ch
the
it
e
m
s
are
no
t
al
l
at
the
sa
m
e
con
cept
le
vel
con
ta
in
in
form
at
ion
at
dif
fer
e
nt abst
ract l
eve
ls.
The
a
pproache
s
us
e
d
to
fin
d
f
reque
nt
it
e
m
se
ts
in
sin
gle
le
ve
l
dataset
s
m
is
s
inf
or
m
at
ion
,
as
they
only
look at
on
e le
ve
l i
n
the
datase
t. Thus tec
hn
i
ques that c
onsid
er al
l t
he
le
vels
are neede
d [
6
]
-
[
9
]
,
[
22]
.
H
ow
ever,
ru
le
s
de
rive
d
f
r
om
m
ult
i
-
le
vel
dataset
s
ca
n
ha
ve
the
sam
e
issues
with
re
dund
a
ncy
as
th
ose
from
a
sing
le
le
vel
dataset
.
Wh
i
le
appr
oach
es
use
d
to
rem
ov
e
re
dundancy
i
n
si
ng
le
le
vel
data
set
s
[13
]
,
[
17
]
,
[
19]
ca
n
be
a
da
pted
for
us
e
i
n
on
e
r
ule
at
a
giv
e
n
le
ve
l
gi
ves
t
he
sam
e
inf
orm
at
ion
as
a
no
t
her
r
ule
at
a
di
ff
ere
nt
le
vel.
In
thi
s
pap
e
r,
w
e
pr
es
ent
a
Re
li
able
basis
re
pr
ese
nt
at
ion
of
non
-
re
dundant
As
so
c
ia
ti
on
Rules.
We
then
lo
ok
into
thi
s
hierar
c
hical
re
dundancy
an
d
pro
po
se
a
n
ap
proac
h
from
w
hich
m
or
e
non
-
re
dundant
r
ul
es
can
be
der
i
ve
d.
W
e
us
e
the
sam
e
def
i
niti
on
of
non
-
re
dundant
ru
le
s
in
si
ng
l
e
le
vel
dataset
s,
but
to
this
def
i
niti
on
we
add
a
requirem
ent that con
side
rs
th
e d
iffe
ren
t l
eve
ls of
the item
(s)
in
determ
ining
the r
e
dundan
cy
r
ule. By do
ing
s
o,
m
or
e
redu
nd
a
nt
A
sso
ci
at
io
n
Rules
ca
n
be
el
i
m
inate
d.
W
e
al
so
sho
w
th
at
it
is
possibl
e
to
de
rive
al
l
of
th
e
Associ
at
ion
R
ul
es, w
it
ho
ut
los
e of in
form
at
io
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N:
20
88
-
8708
G
ene
ra
ti
ng N
on
-
r
e
dund
an
t
M
ulti
le
vel
Asso
ci
ation R
ules Us
ing
…
(
R. Vija
ya
Pr
ak
ash
)
4569
The
pap
e
r
is
orga
nized
a
s
f
ol
lows
.
Sect
ion
2
bri
efly
disc
us
ses
s
om
e
relat
ed
w
ork,
Se
ct
ion
3,
we
discuss
the
re
dunda
ncy
in
As
so
ci
at
ion
Rule
s
an
d
present
a
de
finiti
on
to
re
dundant
r
ules.
E
xperim
ent
s
a
nd
resu
lt
s a
re
pr
es
ented
i
n
Sect
i
on
4.
Finall
y, Se
ct
ion
5
c
on
cl
udes t
he pape
r.
2.
RELATE
D
W
ORK
The
a
ppro
ac
he
s
pro
po
se
d
in
[13
]
,
[
18
]
m
a
ke
us
e
of
the
cl
os
ure
of
the
Galois
co
nne
ct
ion
[
4]
to
extract
non
-
re
dundant
r
ules
f
ro
m
fr
eq
uen
t
c
losed
it
em
set
s
instea
d
of
f
rom
fr
eq
uen
t
it
em
se
ts.
On
e
di
f
fer
e
nce
betwee
n
the
tw
o
ap
proac
hes
is
the
def
init
io
n
of
re
dundancy
.
The
ap
proac
h
pr
op
os
e
d
in
[18]
extracts
the
ru
le
s
with
shorte
r
an
te
ceden
t
an
d
s
horter
co
ns
e
qu
ent
as
well
a
m
ong
r
ules
w
hic
h
ha
ve
the
sa
m
e
con
fide
nce
,
wh
il
e
the
m
et
ho
d
propose
d
i
n
[
13]
def
i
nes
t
h
at
th
e
non
-
re
dunda
nt
r
ules
a
re
th
ose
w
hich
ha
ve
m
ini
m
al
antec
eden
t
s
and m
axi
m
al
c
on
s
eq
ue
nts.
The
def
i
niti
on
pro
po
se
d
i
n
[
17
]
is
li
ke
tha
t
of
[
13]
.
H
oweve
r,
t
he
re
quirem
ent
to
re
dundancy
is
relaxe
d,
a
nd
t
he
le
sser
re
qu
ir
e
m
ent
m
akes
m
or
e
ru
le
s
t
o
be
c
onside
red
redu
nd
a
nt
a
nd
thu
s
el
i
m
inate
d.
M
os
t
i
m
po
rtantl
y,
[
17]
prov
e
d
that
the
el
i
m
inati
on
of
su
c
h
re
du
nd
a
nt
r
ules
in
creases
the
belie
f
to
the
extr
act
ed
ru
le
s
a
nd
t
he
capaci
ty
of
the
ext
racted
non
-
re
dunda
nt
ru
le
s
f
or
so
l
vin
g
pro
blem
s.
Howe
ver,
the
wor
k
m
entione
d
a
bo
ve
has
only
focuse
d
on
datas
et
s
wh
e
re
al
l
ite
m
s
are
at
the
sam
e
con
ce
pt
le
vel.
T
hu
s
,
the
y
do
no
t
nee
d
to
c
onside
r
re
dunda
ncy
that
can
oc
cur
w
hen
t
he
re
is
a
hie
rar
c
hy
am
on
g
the
it
e
m
s.
A
m
ulti
-
le
vel
dataset
is
the
one
wh
ic
h
ha
s
a
n
im
plicit
ta
xono
m
y
or
c
on
ce
pt
tree,
li
ke
show
n
i
n
Fig
ure
1.
T
he
it
em
s
i
n
the
dataset
ex
ist
at
the lo
west c
oncept le
vel
but a
re
par
t
of ahie
r
arch
ic
al
str
uctu
re a
nd org
a
niz
at
ion
.
F
o
o
d
B
a
n
a
n
a
A
p
p
l
e
W
h
e
a
t
W
h
i
t
e
S
k
i
m
m
e
d
2
p
e
r
ce
n
t
F
r
u
i
t
B
r
e
a
d
M
i
l
k
Figure
1
.
A
Si
m
ple ex
a
m
ple o
f
prod
uct tax
onom
y
In
Fi
gure
1
f
or
an
exam
ple,
the
fr
e
qu
e
nt
it
e
m
set
{
'
Dairylan
d
-
2%
-
m
il
k'
,
'
wh
it
e
-
br
ea
d'
}
is
a
cro
ss
-
le
vel
it
e
m
set
as
the
first
it
em
is
from
the
lo
w
est
le
vel,
wh
il
e
the
sec
ond
it
em
is
from
a
dif
fer
e
nt
c
on
ce
pt
le
vel.
In
fact
the
cr
oss
-
le
vel
idea
wa
s
an
add
it
io
n
to
the
wor
k
bei
ng
propose
d.
F
ur
t
her
w
ork
pr
opos
e
d
an
ap
proac
h
wh
ic
h
inclu
de
d
fin
ding
cr
os
s
-
le
vel
fr
e
quent
it
e
m
se
ts
[1
5].
This
la
te
r
wor
k
al
so
pe
rfor
m
s
m
or
e
pru
ning
of
the
dataset
to
m
ake
fin
ding
t
he
frequ
e
nt
it
em
sets
m
or
e
eff
ic
ie
nt
.
H
ow
e
ve
r,
e
ve
n
with
al
l
this
w
ork
t
he
f
oc
u
s
has
been
on
fin
ding
the
f
re
qu
e
nt
it
e
m
set
s
as
eff
ic
ie
ntly
as
po
ss
ible
and
t
he
iss
ue
of
qual
it
y
and
/
or
redu
nd
a
nc
y
in
sing
le
le
vel
dat
aset
s.
S
om
e
br
ie
f
wor
k
pr
e
sen
te
d
by
[5
]
-
[
6]
discusse
s
rem
ov
in
g
ru
le
s
wh
i
ch
are
hiera
rc
hi
cal
l
y
redu
nd
a
nt,
but
it
reli
es
on
t
he
us
er
gi
ving
a
n
exp
ect
e
d
c
onfi
den
ce
va
riat
io
n
m
arg
in
t
o
de
te
rm
ine
re
dund
ancy.
Ther
e
a
pp
ea
rs
to
be
a
void
in
deali
ng
with
hi
erarch
ic
al
re
dunda
ncy
in
As
so
ci
at
ion
Rule
s
de
rive
d
from
m
ulti
-
le
vel
dataset
s.
This w
ork
at
te
m
pts
to
fill
that
vo
i
d
an
d
s
ho
w
a
n
ap
pr
oac
h
to
deal with h
ie
rar
c
hical
re
dund
a
ncy
without l
osi
ng
any in
form
at
io
n.
Fr
om
the
beg
i
nn
i
ng
of
Assoc
ia
ti
on
Rule
m
i
ning
in
[
1
]
,
[
3
]
,
[
21
]
,
[
23
]
,
[
25]
,
the
first
ste
p
has
al
w
ay
s
been
t
o
fi
nd
t
he
fr
e
qu
e
nt
patte
rn
s
or
it
e
m
sets.
The
sim
ples
t
way
to
do
thi
s
is
thr
ough
th
e
us
e
of
th
e
A
pr
i
or
i
al
gorithm
[2
]
.
Howe
ver,
A
pri
or
i
is
not
de
sig
ned
to
w
ork
on
e
xtracti
ng
frequ
e
nt
it
em
sets
at
m
ulti
ple
lev
el
s
in
a
m
u
lti
-
le
vel
dataset
.
It
is
design
e
d
f
or
use
on
si
ng
le
le
vel
dataset
s.
But,
it
has
been
ad
apted
f
or
m
ulti
-
le
vel
dataset
s.
On
e
ada
ptati
on
of
Aprio
ri
to
m
ul
ti
-
le
vel
dat
ase
ts
is
the
ML_T
2L
1
al
go
rithm
[5
]
-
[
6].
T
he
ML
_T
2L
1
al
gorithm
us
es
a
transacti
on
ta
ble
that
h
as
t
he
hierar
c
hy
in
f
or
m
at
ion
enc
oded
i
nto
it
.
Eac
h
le
vel
in
the
da
ta
set
is
processe
d
in
div
id
ually
.
Firs
tl
y,
le
vel
1
ana
ly
zed
for
la
rge
1
-
it
em
set
s
us
i
ng
A
pr
io
ri.
T
he
li
st
of
le
vel
1
la
rg
e
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
201
8
:
4568
-
4576
4570
1
-
it
em
set
s
is
then
us
e
d
to f
il
te
r
an
d
pru
ne
the
transacti
on
dat
aset
o
f
a
ny
it
e
m
that
do
es
no
t
hav
e
a
n
ances
tor
i
n
the
le
vel
1
la
r
ge
1
-
it
em
se
t
list
and
rem
ov
e
any
transacti
on
w
hich
has
no
f
reque
nt
it
em
s.
Fr
om
the
le
vel
1
la
rg
e
1
-
it
em
set
li
st,
le
vel
1
lar
ge
2
-
it
em
set
s
are
de
rive
d.
T
hen
le
vel
1
la
r
ge
3
-
it
e
m
set
s
a
re
der
i
ve
d
an
d
so
on,
un
ti
l
there
are
no
m
or
e
fr
e
qu
ent
it
e
m
set
s
to
disco
ver
at
le
vel
1.
Si
nce
ML_T
2L
1
de
fin
es
that
on
ly
the
it
e
m
s
that
are
desce
ndant
f
r
om
fr
eq
uen
t
it
em
s
at
lev
el
1
ca
n
be
frequ
e
nt
them
selv
es,
the
le
vel
2
it
e
m
set
s
are
der
ive
d
from
the
filt
ered
transacti
o
n
ta
ble.
For
le
ve
l
2,
the
la
rg
e
1
-
it
e
m
set
s
are
discov
e
red,
fro
m
wh
ic
h
the
la
rg
e
2
-
it
e
m
set
s
are
der
ive
d
an
d
t
hen large
3
-
it
em
sets
et
c.
A
fter
al
l t
he
f
reque
nt
it
em
se
ts
are
disc
overe
d
at
le
vel 2
,
t
he
le
vel
3
la
r
ge
1
-
it
e
m
set
s
are
disco
ver
e
d
a
nd
s
o
on.
ML_
T2
L
1
re
peats
unti
l
ei
ther
al
l
le
vel
s
are
sea
rc
hed
us
in
g
Aprio
ri or
no l
arg
e
1
-
it
em
se
ts are a
foun
d
at
a level
.
As
the
ori
gin
al
wor
k
s
hows
[
5
]
-
[
6],
ML
_T2L1
do
e
s
not
fi
nd
cr
os
s
-
le
vel
fr
e
qu
e
nt
it
em
s
et
s.
W
e
ha
ve
add
e
d
the ab
il
it
y fo
r
it
to
do
this. A
t eac
h
le
vel b
e
lo
w
1,
w
hen
lar
ge
2
-
it
em
se
ts or
lat
er a
re d
eri
ved
the
Aprio
ri
al
gorithm
is
no
t
restrict
ed
to
just
us
in
g
the
la
rge
n
-
1
-
it
e
m
set
s
at
th
e
curre
nt
le
ve
l,
but
can
ge
ner
at
e
com
bin
at
ion
s
us
in
g
the
la
r
ge
it
e
m
set
s
fr
om
higher
le
vels.
The
only
rest
rict
ion
s
on
thi
s
ar
e
that
t
he
der
i
ved
fr
e
qu
e
nt
it
em
s
et
(s)
ca
n
not
c
on
ta
in
a
n
it
em
that
has
a
n
a
ncesto
r
-
desce
ndant
relat
ionsh
ip
with
a
no
t
he
r
it
e
m
within
the
sa
m
e
i
tem
set
and
that
the
m
in
i
m
u
m
su
pp
ort
threshold
use
d
is
that
of
the
cur
re
nt
le
vel
being
processe
d.
3.
GENER
ATIO
N
O
F
NON
-
R
EDU
NDA
NT
MU
LT
I
-
LE
V
EL
A
SS
OCIA
TION
R
ULES
The
us
e
of
f
re
qu
e
nt
it
e
m
set
s
as
the
basis
for
Asso
ci
at
io
n
Rule
m
ining
of
t
en
res
ults
in
the
gen
erati
on
of
m
any
ru
le
s.
More
rece
nt
w
ork
has
dem
on
strat
ed
that
the
us
e
of
cl
ose
d
it
e
m
set
s
and
ge
ner
at
or
s
ca
n
r
edu
c
e
the
nu
m
be
r
of
ru
le
s
ge
ner
at
ed
[14
]
,
[
16
]
-
[
18
]
,
[
20
]
-
[
22]
.
Desp
it
e
this,
r
edun
dan
cy
sti
ll
exists
in
the
ru
le
s
gen
e
rated
fro
m
m
ult
i
-
le
vel
dataset
s
eve
n
wh
e
n
us
in
g
s
om
e
of
the
m
eth
ods
desig
ne
d
to
rem
ov
e
re
dunda
ncy.
This
redu
nd
a
nc
y
we
cal
l
hi
erarc
hical
re
dunda
ncy.
Her
e
in
t
his
sect
i
on
we
fi
rst
i
ntr
oduce
hiera
rch
ic
al
redu
nd
a
ncy
in
m
ulti
-
le
vel
dataset
s
and
then
we
detai
l
ou
r
wo
r
k
to
rem
ov
e
this
redu
ndancy
without
losin
g
inf
or
m
at
ion
.
3.1.
Hier
archic
al
Redund
an
c
y
Wh
et
her
a
r
ule
is
interest
ing
and
/
or
us
ef
ul
is
us
ually
deter
m
ined
thr
ough
the
su
pp
or
t
a
nd
co
nf
i
den
ce
values
that
it
ha
s.
H
ow
e
ve
r,
this
does
not
guara
ntee
that
a
ll
of
the
r
ules
that
ha
ve
a
high
en
ough
s
upport
a
nd
confide
nce
act
ually
co
nv
ey
new
in
form
at
i
on.
F
ollo
wing
is
an
e
xam
pl
e
tran
sact
ion
t
able
f
or
a
m
ulti
-
le
vel
d
at
aset
Ta
ble 1
.
Table
1.
Sim
pl
e
M
ulti
-
le
vel
T
ran
sact
io
n Dat
aset
.
Tr
an
sactio
n
I
D
Ite
m
s
1
[1
-
1
-
1
,
1
-
2
-
1
,
2
-
1
-
1
,
2
-
2
-
1]
2
[1
-
1
-
1
,
2
-
1
-
1
,
2
-
2
-
2
,
3
-
2
-
3]
3
[1
-
1
-
2
,
1
-
2
-
2
,
2
-
2
-
1
,
4
-
1
-
1]
4
[1
-
1
-
1
,
1
-
2
-
1]
5
[1
-
1
-
1
,
1
-
2
-
2
,
2
-
1
-
1
,
2
-
2
-
1
,
4
-
1
-
3]
6
[1
-
1
-
3
,
3
-
2
-
3
,
5
-
2
-
4]
7
[1
-
3
-
1
,
2
-
3
-
1]
8
[3
-
2
-
3
,
4
-
1
-
1
,
5
-
2
-
4
,
7
-
1
-
3]
This
sim
ple
m
ulti
-
le
vel
trans
act
ion
al
datase
t
has
3
le
vels
with
each
it
e
m
belonging
to
the
lo
wes
t
le
vel.
The
it
em
ID
in
the
ta
ble
store/h
ol
ds
the
hierar
c
hy
info
rm
at
ion
for
each
it
em
.
Thu
s
,
the
it
em
1
-
2
-
1
belo
ngs
to
the
first
cat
egory
at
le
vel
1
and
f
or
le
vel
2
it
belo
ng
s
t
o
the
seco
nd
s
ub
-
cat
e
gor
y
of
the
first
le
vel
1
cat
egory.
Final
ly
,
at
le
vel
3
i
t
belongs
to
th
e
first
subcat
egory
of
t
he
pa
ren
t
cat
eg
or
y
at
le
vel
2.
Fro
m
this
transacti
on
se
t
we
us
e
the
ML
_T
2L1
al
gorith
m
with
the
cross
-
le
vel
ad
d
-
on
and
a
m
ini
m
u
m
su
ppor
t
valu
e
of
4
f
or
le
vel
1
a
nd
3
f
or
le
vels
2
an
d
3.
Fro
m
these
fr
eq
ue
nt
it
e
m
set
s,
the
cl
os
ed
it
em
s
et
s
and
gen
e
ra
tors
are
der
i
ved Table
2
. T
he
it
e
m
set
s
, close
d
it
em
se
ts an
d gen
e
ra
to
rs
c
om
e fr
om
all
thr
ee le
vels
.
Finall
y,
from
the
cl
ose
d
it
e
m
set
s
an
d
gen
e
r
at
or
s
t
he
Asso
ci
at
ion
Rules
can
be
gen
e
rat
ed.
I
n
this
exam
ple,
we
us
e
the
Re
li
ableExact
Rule
ap
proac
h
prese
nted
in
[17
]
-
[
18]
to
ge
ner
at
e
the ex
act
basis
r
ules.
The
disco
ver
e
d
ru
l
es
are
f
r
om
mu
lt
iple
le
vels
a
nd
i
nclu
de
cr
oss
-
le
vel
r
ules.
The
Re
li
ableE
xactR
ule
ap
pro
ach
ca
n
rem
ov
e
re
dund
ant
r
ules,
bu
t
as
we
will
sh
ow,
it
does
not
rem
ov
e
hiera
rc
hy
re
dundancy
.
The
r
ules
gi
ve
n
in
Table
3
are
de
rive
d
from
the
cl
os
ed
it
em
s
et
s
and
gen
e
ra
tors
in
Ta
ble
3
w
he
n
the
m
ini
m
u
m
con
fi
de
nce
thres
ho
l
d
is set
to 0.
5 or 5
0%
as sho
wn in
Ta
ble 3
.
The
Re
li
ableE
xactR
ule
a
lg
or
it
h
m
li
sts
al
l
t
he
ru
le
s
i
n
Ta
ble
3
as
im
portant
a
nd
no
n
-
redu
nd
a
nt.
Howe
ver,
we
argue
that
the
re
are
sti
ll
red
un
dan
t
ru
le
s.
This
ty
pe
of
r
ed
unda
ncy
is
beyo
nd
w
ha
t
the
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N:
20
88
-
8708
G
ene
ra
ti
ng N
on
-
r
e
dund
an
t
M
ulti
le
vel
Asso
ci
ation R
ules Us
ing
…
(
R. Vija
ya
Pr
ak
ash
)
4571
Re
li
ableExactR
ule
al
go
rith
m
was
designed
for.
L
ooki
ng
at
the
ru
le
s
in
Table
3
we
cl
aim
that
ru
le
4
is
redu
nd
a
nt
to
r
ule
1,
r
ule
7
is
red
un
dan
t
to
ru
le
5,
ru
le
8
is
redunda
nt
to
ru
le
6
an
d
ru
l
e
12
is
redu
ndant
to
ru
le
10.
For
e
xam
ple,
the
it
e
m
2
-
2
-
1
(fro
m
ru
le
4)
is
a
child
of
the
m
or
e
ge
ner
al
/a
bst
ract
it
e
m
2
-
2
-
*
(f
r
om
ru
le
1).
Th
us
r
ule
4
is
in
fact
a
m
or
e
sp
eci
fic
ver
sio
n
of
r
ule
1.
Be
cause
we
know
that
ru
le
1
say
s
2
-
2
-
*
is
enou
gh
to
fire
the
r
ule
with
c
on
s
eq
ue
nt
C,
wh
e
reas
ru
le
4
requires
2
-
2
-
1
to
fire
with
co
ns
e
qu
e
nt
C,
a
ny
it
e
m
that
is
a
desce
nd
a
nt
of
2
-
2
-
*
will
cause
a
r
ule
to
fire
with
con
se
quent
C.
It
do
e
s
not
ha
ve
to
be
2
-
2
-
1.
Thus
ru
le
4
is
m
or
e
restrict
ive
.
B
ecause
2
-
2
-
1
i
s
pa
rt
of
2
-
2
-
*
ha
ving
ru
le
4
does
not
act
ually
br
i
ng
a
ny
ne
w
inf
or
m
at
ion
to
the
use
r,
as
t
he
inf
or
m
at
ion
c
on
ta
ine
d
in
it
i
s
act
ually
pa
rt
of
the
i
nfor
m
at
ion
c
onta
ine
d
i
n
ru
le
1.
T
hus
ru
le
4
is
re
dunda
nt
.
W
e
de
fine
hi
erarch
ic
al
redunda
ncy
in
e
xa
ct
Associ
at
ion
Rules
t
hro
ugh
the
fo
ll
owin
g defi
niti
on
Ta
ble
2.
Fr
e
quent
Cl
os
e
d Item
se
ts
and
Ge
ne
rators
Der
ive
d
from
t
he
Fr
e
qu
e
nt
Item
set
s
in Ta
ble 1
Clo
sed
I
te
m
sets
Gen
erators
[1
-
*
-
*]
[1
-
*
-
*]
[1
-
1
-
*]
[1
-
1
-
*]
[1
-
1
-
1]
[1
-
1
-
1]
[1
-
*
-
*
,
2
-
2
-
*]
[2
-
2
-
*]
[2
-
*
-
*
,
1
-
1
-
*]
[2
-
*
-
*
,
1
-
1
-
*]
[1
-
1
-
*
,
1
-
2
-
*]
[1
-
2
-
*]
[1
-
1
-
*
,
2
-
2
-
*]
[2
-
2
-
*]
[1
-
*
-
*
,
2
-
2
-
1]
[2
-
2
-
1]
[2
-
*
-
*
,
1
-
1
-
1]
[2
-
*
-
*
,
1
-
1
-
1]
[1
-
2
-
*
,
1
-
1
-
1]
[1
-
2
-
*
,
1
-
1
-
1]
[1
-
*
-
*
,
2
-
1
-
*
,
2
-
2
-
*]
[2
-
1
-
*]
[2
-
*
,
1
-
1
-
*
,
1
-
2
-
*]
[2
-
*
-
*
,
1
-
2
-
*]
[1
-
1
-
*
,
1
-
2
-
*
,
2
-
2
-
*]
[1
-
2
-
*
,
2
-
2
-
*]
[1
-
1
-
*
,
2
-
1
-
*
,
2
-
2
-
*]
[2
-
1
-
*]
[1
-
*
-
*
,
2
-
1
-
1
,
2
-
2
-
*]
[2
-
1
-
1]
[1
-
1
-
*
,
2
-
1
-
1
,
2
-
2
-
*]
[2
-
1
-
1]
[1
-
1
-
*
,
2
-
2
-
1
,
1
-
2
-
*]
[2
-
2
-
1]
[2
-
1
-
*
,
1
-
1
-
1
,
2
-
2
-
*]
[2
-
1
-
*
]
[
2
-
2
-
*
,
1
-
1
-
1]
[2
-
2
-
*
,
1
-
1
-
1
,
2
-
1
-
1]
[2
-
1
-
1
]
[
2
-
2
-
*
,
1
-
1
-
1]
Table
3.
E
xact
basis
Associ
at
ion R
ules
De
riv
ed fr
om
Cl
os
ed Ite
m
se
ts
and
G
ene
rato
rs
i
n Table
2
No
.
Ru
le
Su
p
p
1
[2
-
2
-
*
]
==> [
1
-
*
-
*]
0
.57
1
2
[1
-
2
-
*
]
==> [
1
-
1
-
*]
0
.57
1
3
[2
-
2
-
*
]
==> [
1
-
1
-
*]
0
.57
1
4
[2
-
2
-
1
]
==> [
1
-
*
-
*]
0
.42
8
5
[2
-
1
-
*
]
==> [
1
-
*
-
*
,
2
-
2
-
*]
0
.42
8
6
[2
-
1
-
*
]
==> [
1
-
1
-
*
,
2
-
2
-
*]
0
.42
8
7
[2
-
1
-
1
]
==> [
1
-
*
-
*
,
2
-
2
-
*]
0
.42
8
8
[2
-
1
-
1
]
==> [
1
-
1
-
*
,
2
-
2
-
*]
0
.42
8
9
[2
-
2
-
1
]
==> [
1
-
1
-
*
,
1
-
2
-
*]
0
.42
8
10
[2
-
1
-
*
]
==> [
1
-
1
-
1
,
2
-
2
-
*]
0
.42
8
11
[2
-
2
-
*
,
1
-
1
-
1
]
==>
[
2
-
1
-
*]
0
.42
8
12
[2
-
1
-
1
]
==> [
2
-
2
-
*
,
1
-
1
-
1]
0
.42
8
13
[2
-
2
-
*
,
1
-
1
-
1
]
==>
[
2
-
1
-
1]
0
.42
8
Def
i
niti
on
1
:
L
et
R1
= X1
=
> Y
a
nd
R2
=
X
2
=> Y
be
t
wo ex
act
A
sso
ci
at
i
on
Rules,
with
ex
act
ly
the
sam
e
it
e
m
se
t
Y
as
t
he
co
ns
e
qu
e
nt.
R
ule
R1
is
redu
nd
a
nt
t
o
r
ule
R2
if
(1)
the
it
em
set
X1
is
m
ade
up
of
it
e
m
s
wh
e
re
at
le
ast
on
e
it
em
in
X1
is
desce
nd
a
nt
from
the
it
e
m
s
in
X
2
a
nd
(
2)
the
i
tem
set
X2
is
entirel
y
m
a
de
up
of
it
em
s
wh
ere
at
le
ast
on
e
it
e
m
in
X2
is
an
ancesto
r
of
th
e
it
e
m
s
in
X1
and
(
3)
the
othe
r
no
n
-
a
ncesto
r
it
e
m
s
in X2 a
re all
present in
it
e
m
set
X
1.
Fr
om
this
def
i
niti
on
,
if
for
a
n
exact
A
sso
ci
at
ion
Rule
X
1
=>
Y1
the
re
does
not
exist
a
ny
oth
e
r
r
ule
X2
=>
Y2
su
c
h
that at
least
o
ne
it
e
m
in
X1
s
har
es a
n
a
ncestor
-
de
scen
da
nt r
el
at
ion
s
hip
with X2 contai
ni
ng
t
he
ancesto
r(
s
)
an
d
al
l
oth
er
it
em
s
X2
are
pre
sent
in
X
1,
th
en
X
1
=>
Y1
is
a
no
n
-
re
dundant
r
ule.
To
t
est
fo
r
redu
nd
a
ncy,
w
e
ta
ke
t
his
de
finiti
on
a
nd
a
dd
a
nothe
r
co
nd
it
io
n
f
or
a
r
ule
to
be
c
ons
idere
d
valid
.
A
r
ule
X
=>
Y
is
valid
if
it
ha
s
no
a
ncesto
r
-
desce
ndant
relat
io
ns
hi
p
betwee
n
a
ny
it
e
m
s
in
it
em
se
ts
X
a
nd
Y
.
T
hu
s
,
for
exam
ple
1
-
2
-
1
=>
1
-
2
-
*
i
s
not
a
valid
r
ul
e,
but
1
-
2
-
1
=
>
1
-
1
-
3
is
a
va
li
d
ru
le
.
If
this
conditi
on
is
not
m
e
t
by
any
r
ule
X
2
=>
Y2
w
hen
t
est
ing
to
s
ee
if
X1
=>
Y
1
is
r
edun
dan
t
to
X
2
=>
Y
2,
t
hen
X1
=>
Y
1
is
a
non
-
redu
nd
a
nt rule
as X2 =>
Y2 is
not a
valid
ru
l
e. Subm
it
you
r
m
anu
script ele
ct
ronical
ly
f
or
rev
ie
w.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
201
8
:
4568
-
4576
4572
3.2.
Genera
t
in
g
E
xa
c
t
B
as
is
Ru
l
es
As
p
re
vious
work
has
s
ho
wn
[14,
17
-
18]
us
in
g
f
requ
ent
cl
os
ed
it
e
m
se
ts
in
the
gen
e
rati
on
of
Associ
at
ion
R
ules
can
re
duc
e
the
qua
ntit
y
of
disc
ov
e
red
ru
le
s.
Be
cause
we
wish
t
o
re
m
ov
e
redunda
ncy
on
top
of
the
re
dunda
ncy
al
read
y
bein
g
rem
o
ve
d,
our
ap
proac
h
us
e
s
the
cl
os
ed
it
e
m
set
s
and
ge
ner
at
or
s
to
disco
ver
th
e
non
-
re
dunda
nt
r
ules.
[14,
17
-
18]
ha
ve
both
pro
posed
co
nde
ns
e
d/m
or
e
conci
se
bases
to
re
pr
ese
nt
non
-
re
dundant
exact
r
ules.
E
xact
r
ules
ref
e
r
to
r
ules
w
hose
co
nfi
de
nce
is
1.
T
he
pro
pose
d
a
ppr
oach
will
be
exten
ded to
ot
her r
ules (
i.e
., so
cal
le
d ap
pro
xim
a
te
r
ules)
. T
he follo
wing
def
i
niti
on
s
outl
ine these
tw
o b
ases:
Def
i
niti
on
2:
Fo
r
the
Mi
n
-
Ma
xEx
act
(MM
E)
basis
,
C
is
the
set
of
th
e
disco
ver
e
d
f
reque
nt
cl
os
e
d
it
e
m
set
s.
Fo
r
e
ach
cl
os
e
d
it
e
m
se
t
c
in
C,
G
c
is
the
set
of
gen
e
rato
rs
for
c.
From
this
the
exact
basis
f
or
m
in
-
m
ax
is:
MMEHR
=
{
g
→
c
|
c
∈
C
,
g
∈
G
C
,
g
≠
c
and
t
her
e
e
xists
no
ru
le
s
g
′
→
c
′
wh
e
re
c
′
ϵ
C,
g
′
ϵ
Gc
,
c
≠
c
′,
g′
≠
c′
a
nd g is
descend
ant set
of
g
′
, g
′
has n
o
a
ncesto
r or
desce
ndant
of c
′
or
g
′
Def
i
niti
on
8:
(
Re
li
able
Exact
Ba
sis
with
out
Hiera
rch
y
Re
dundancy
)
Let
C
be
t
he
set
of
freq
ue
nt
cl
os
ed
it
e
m
set
s
.
F
or
eac
h
fr
e
quent
cl
os
e
d
it
em
se
t
c,
le
t
Gc
be
the
set
of
m
ini
m
al
gen
erat
or
s
of
c.
T
he
R
el
ia
ble
exact ba
sis i
s:
REH
R
=
{
g
→
c
|
c
∈
C
,
g
∈
G
C
,
REH
R
=
{
g
−
→
c
|
c
∈
C
,
g
∈
G
C
,
¬
(c
or
c
′
∪
g
′
),
wh
e
re
c
′
ϵ
C
,
c
′
⸦
c, g
′
ϵ
Gc
, and t
her
e e
xists n
o ru
le
s
g
′
→
c
′
w
her
e c
≠
c
′
, g
′
≠
c
′
a
nd
g
is de
scen
dan
t set
of g
′
, g
′
has no a
ncesto
r
or
descenda
nt
of
c
′
or
g
′
.
Th
us
the
al
go
rithm
s
to
extract
non
-
redu
ndant
m
ulti
-
le
vel
ru
le
s
us
in
g
ei
ther
Mi
nm
axEx
act
HR or Rel
ia
ble
ExactHR a
re
giv
en
as
fo
ll
ows:
Algori
th
m
1:
Minm
ax
E
xa
c
t
HR()
Inp
ut:
C:
a
set
of
fr
e
qu
e
nt
cl
ose
d
it
e
m
set
s
G:
a
set
of
m
inim
al
gen
erato
rs
.
For
g
ϵ
G,
g.cl
os
ure
is
the
cl
os
ed
it
e
m
set
o
f
g.
Ou
t
pu
t:
A
set
of
non
-
re
dundan
t
m
ulti
le
vel r
ules.
1.
Mi
nMaxE
xa
ct
: =
ф
2.
f
or
eac
h k=
1 t
o v
3.
f
or
eac
h k
-
ge
ner
at
or
g
ϵ
Gk
4.
nonRe
dunda
nt = tr
ue
5.
i
f g
≠
g
cl
osu
re
6.
f
or
all
g′
ϵ
G
7.
i
f
(
g ′
≠ g)
8.
i
f
(
g′ is a
nc
est
or
set
of
g )
and (
(c′ =c)
or
( g=
g
′
)) an
d(g
′
is
no
t a
ncest
or set
of c′
)
9.
t
hen no
nRed
unda
nt =
false
10. brea
k
14. if
no
nRed
unda
nt = t
ru
e
15. inse
rt {
(g
→
c)
,
g.
s
upp
}
in
Mi
nMa
xE
xa
ct
20. ret
urn
Mi
nM
axEx
act
Algori
th
m
2:
Reli
ab
le
Ex
actHR
()
Inp
ut:
C:
a
set
of
fr
e
qu
e
nt
cl
ose
d
it
e
m
set
s
G:
a
set
of
m
inim
al
gen
erato
rs
.
For
g
ϵ
G,
g.cl
os
ure
is
the
cl
os
ed
it
e
m
set
o
f
g.
Ou
t
pu
t:
A
set
of
non
-
re
dundan
t
m
ulti
le
vel r
ules.
1.
Rel
ia
bleE
xa
ct
: =
ф
2.
f
or
all
c
ϵ
C
3.
f
or
all
g
ϵ
Gc
4.
nonRe
dunda
nt =
false
5
. i
f
c
C s
uch that c
′
⸦
c
and c
g
′
ϵ
Gc,
we have
(
c
or c
′
)
∪
g
′
)
⊆
g)
6.
t
hen no
nRed
unda
nt = tr
ue
7.
else
8.
nonRe
dunda
nt =
false
9.
break
11. fo
r
al
l g
′
∈
G
12. if
g ′ ≠
g
13.
if
(g′
is
an
cest
or
set
of
g)
and
(c
′
=c
or
g
′
=
g)
an
d
(g′
is
no
t
ance
stor
set
of
(c
′
or
g
′
)
a
nd
(g′
is
no
t
desce
nd
a
nt set
of (
c
′
or
g′)
14. th
e
n n
on
Re
dundant =
tr
ue
;
break
19. if
no
nRed
unda
nt = t
ru
e
20. inse
rt {
(g
→
c
or g,
g.
s
upp
}
in
Re
li
able
Exact
24. ret
urn
Re
li
ableExact
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N:
20
88
-
8708
G
ene
ra
ti
ng N
on
-
r
e
dund
an
t
M
ulti
le
vel
Asso
ci
ation R
ules Us
ing
…
(
R. Vija
ya
Pr
ak
ash
)
4573
3.3.
Derivin
g
E
xa
c
t
R
ules f
rom
the Ex
act Basi
s
Ru
le
s
The
Mi
n
-
Ma
xE
xact
ap
proac
h
a
nd
Re
li
able
Exact
ap
proac
h
ha
ve
pro
ve
n
that
they
can
deduce
al
l
of
the
exact
ru
le
s
from
their
ba
sis
set
[17].
C
om
par
ing
with
the
Mi
n
-
Ma
xE
xact
ap
proac
h
an
d
Re
li
able
Exact
appr
oach,
our
work
re
su
lt
s
i
n
a
sm
aller
ex
act
basis
set
by
no
t
only
re
m
ov
ing
t
he
re
dundant
r
ules
that
ar
e
rem
ov
ed
by
th
e
Mi
n
-
Ma
xE
xa
ct
appro
ac
h
a
nd
Re
li
ableE
xa
ct
appro
ac
h,
bu
t
al
so
rem
oving
the
hiera
rc
hical
ly
redu
nd
a
nt r
ule
s.
If
w
e ca
n
rec
ov
e
r
al
l t
he
hierar
c
hical
ly
r
edu
nda
nt r
ules, t
hen
we
can d
e
r
ive all
the e
xact ru
le
s
by
us
in
g
the
Mi
n
-
Ma
xE
xact
or
Re
li
ableExac
t
reco
ve
ry
al
gorithm
.
This
wil
l
ensu
re
that
al
l
the
exact
ru
le
s
ca
n
sti
ll
be
der
ived
and
by
ac
hiev
ing
this,
our
ap
proac
h
will
be
a
lossless
repre
sentat
ion
of
th
e
exact
Associ
at
io
n
Rules.
The
f
ollo
wing
al
go
rit
hm
is
desig
ne
d
to
re
cov
e
r
the
hier
arch
ic
al
ly
redu
nd
a
nt
r
ules
f
rom
the
exact
basis.
By
ad
din
g
it
to
the
al
gorithm
s
us
ed
by
Mi
n
-
Ma
xE
xa
ct
and
Re
li
ableExact
to
de
riv
e
the
exact
r
ules
it
is
then
able
for
t
he
e
xisti
ng
Re
li
ableExact
rec
ov
e
ry
al
go
rith
m
to
de
r
ive
al
l
the
e
xact
ru
le
s.
T
his
is
beca
us
e
our
al
gorithm
will
giv
e
t
hem
a
basis
set
that
incl
ud
e
s
the
hiera
r
chical
ly
redundan
t
ru
le
s
(
wh
i
ch
the
Re
li
ableExact
appr
oach
w
ou
l
d
not
hav
e
re
m
ov
ed
in
the
f
irst
place).
Th
e
basic
idea
is
that,
f
or
each
exact
basis
r
u
le
,
fir
s
t
from
gen
erato
r
s
to
const
ru
ct
a
ll
po
ssible
exa
ct
basis
ru
le
s
whose
a
nteced
ent
is
a
descendant
of
the
e
xa
ct
basis
ru
le
(steps
4
to
7
i
n
Algorith
m
3)
.
T
hese
rul
es
are
pote
ntial
exact
basis
rul
es
that
m
igh
t
hav
e
be
en
el
i
m
inat
e
d
du
e
to
t
he
a
nc
est
or
-
de
sce
nda
nt
relat
io
ns
hi
p.
The
n
c
heck
to
m
ake
su
re
t
he
se
pote
ntial
r
ules
are
valid
(st
eps
8
to
12
),
finall
y,
from
the
pote
ntial
exact
r
ul
es
to
fin
d
e
xa
ct
basis
ru
le
s.
These
e
xact
ba
sis
r
ules
ha
ve
bee
n
el
i
m
inate
d du
e
to
the
an
c
est
or
-
desce
ndant
rel
at
ion
s
hip
(ste
ps 1
3
t
o 18).
Algori
th
m
3:
DeriveE
xa
c
tH
R
(
)
Inpu
t:
Set o
f
e
xact b
a
sis r
ules
d
e
no
te
d
as
Ex
ac
tba
sis
,
set
of
frequ
e
nt clo
sed
it
e
m
set
s
C
andg
ener
at
or
s
G.
Out
p
ut:
Set
of
rules t
hat c
ove
rs
the
ex
act
bas
isa
nd the
hiera
rch
ic
al
ly
r
e
dundant
r
ules.
1.
Re
c
overe
d:
=
2.
r
Exact
basi
s
3.
Ca
ndidate
B
asi
s
: =
4.
f
or
all
gen
e
r
at
or
g
i
n
G
5.
i
f
a
ny of the
it
e
m
x
in the
a
ntecede
nt
X
of
ru
le
r: X
Y
i
s the a
ncesto
r of
g
.
6.
t
hen ad
d
al
l
the possi
ble s
ubset
s
of
g
into
S
8.
f
or
all
s
in
S
,
ch
ec
k
e
ver
y,
x
X
if
x
do
e
sn’t ha
ve
a
d
esc
e
nd
a
nt in
s
,
add
x
to
s
t
o
m
ake
s
a
descenda
nt
set
of
X
9.
i
f
s
has
no a
ncesto
rs
i
n
Y
a
nd
s
has n
o des
cend
a
nts i
n
Y
and f
or all
it
e
m
s
i
s
t
her
e a
re
no a
ncesto
r
-
desce
nd
a
nt r
el
at
ion
s
with it
e
m
i
s
a
nd fo
r
all
it
e
m
i
Y
there
are n
o
a
nc
e
stor
-
desce
nda
nt r
el
at
io
n wit
h i
tem
i
Y
10. th
e
n
i
ns
ert
s
Y
in
Ca
nd
i
da
te
Ba
sis
13. fo
r
al
l
B
D
Ca
nd
i
dateB
asi
s
14. if
B
U
D
= it
e
m
set
i
C
an
d
B
=
g
G
i
15. inse
rt {
B
D
,
g. s
upp
} i
n
Re
cov
e
red
19. ret
urn
Exac
tbasis
U
Re
co
ve
red
4.
E
X
PERI
MEN
TS
Ex
per
im
ents
wer
e
c
onduct
ed
to
te
st
and
e
valuate
the
ef
f
ect
iveness
of
t
he
pro
pose
d
hi
erarch
ic
al
ly
non
-
re
dundant
exact
basis
a
nd
toc
onfirm
that
it
is
al
so
a
l
os
sle
ss
basis
s
et
.
This
sect
io
n
pr
ese
nts
a
nd
detai
ls
the expe
rim
ent
s and t
heir resu
lt
s.
4.1.
Datasets
We
us
ed
6
dat
aset
s
t
o
te
st
our
a
ppr
oac
h
to
disco
ve
r
w
het
her
it
r
ed
uce
d
the
siz
e
of
the
exact
basis
ru
le
set
an
d
t
o
te
st
that
the
ba
sis
set
was
lo
ssless,
m
eaning
al
l
the
r
ules
cou
l
d
be
recov
ered.
T
hese
da
ta
set
s
wer
e
c
om
po
se
d
of
10
0,
20
0,
500,
2000
an
d
50
00
tra
ns
ac
ti
on
s
an
d
are
nam
ed
A
to
F
r
especti
vely
.
T
he
key
sta
ti
sti
cs f
or
t
he
se buil
tdata
set
s ar
e
detai
le
d
i
n
Ta
ble
4.
Table
4.
O
btained
F
reque
nt it
e
m
set
s u
sin
g
E
xact Ba
sis
Dataset
MM
E
MM
EHR
RE
REHR
A
15
10
13
9
B
106
68
80
58
C
174
134
113
89
D
577
429
383
305
E
450
405
315
287
F
725
602
91
80
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
201
8
:
4568
-
4576
4574
Wh
e
re
as
the
pro
po
se
d
al
gor
it
h
m
s
MM
E
re
du
ce
d
le
ss
nu
m
ber
of
ru
le
s
com
par
ed
to
MM
EHR.
Th
e
MM
EHR
al
gorithm
s
con
side
rs
the
cr
os
s
le
vel
hierar
c
hy
a
nd
c
on
ta
in
s
m
or
e
data,
at
this
le
vel
it
red
uc
ed
the
m
or
e
nu
m
ber
of
r
ules,
it
in
di
cat
ed
that
at
hierar
c
hy
t
he
data
is
duplica
te
d
an
d
the
propose
d
al
gorithm
s
is
reduce
d
th
os
e
r
eplic
at
ed dat a
nd pr
oduce
d
m
or
e
r
el
ia
ble a
nd acc
ur
at
e
r
ules.
Si
m
il
arly
,
fo
r
RE
an
d
RE
HR
al
gorithm
s
has
ge
ner
at
e
d
the
dif
fer
e
nt
r
ules
at
with
a
nd
w
it
ho
ut
c
r
oss
le
vels
hie
ra
rchy
.
At
cro
ss
le
ve
l
there
will
be
m
or
e
data
so
m
or
e
ru
le
s
are
gen
e
rated
by
REHR
al
gorith
m
than
the
RE
al
gorit
hm
wh
ere
as
the
ML
_T2L1
al
gorithm
has
gen
e
rated
th
sa
m
e
ru
le
s
wit
h
and
with
out
cr
os
s
le
vel
hierar
c
hy.
T
hu
s
the
pro
pose
d
work
has
ge
ne
rated
m
or
e
re
li
able
and
acc
ur
at
e
al
gorit
hm
than
the
M
L_T
2L
1
al
gorithm
.
Fig
ure
2. Mult
i Level
Associ
a
ti
on
R
ules
Fig
ure
3. Com
par
is
on of MM
E
, MMEHR
A
l
gorithm
s w
it
h M
l_T2
L
1
40%
50%
60%
70%
80%
90%
100%
MME
ML_
T2L
1
MM
EHR
ML_
T2L
1
Redu
ction
Ru
le
s
in %
Co
mpa
ri
so
n
of
MM
E,
MM
EHR
Algo
ri
th
ms
an
d
ML_T2
L1
A
B
C
D
E
F
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
&
C
om
p
En
g
IS
S
N:
20
88
-
8708
G
ene
ra
ti
ng N
on
-
r
e
dund
an
t
M
ulti
le
vel
Asso
ci
ation R
ules Us
ing
…
(
R. Vija
ya
Pr
ak
ash
)
4575
Figure
4. Com
par
is
on of RE
, R
EHR A
l
gorithm
s w
it
h
Ml
_T
2L
1
5.
CONCL
US
I
O
N
Re
dundancy
i
n
Associ
at
ion
Rules
af
fects
th
e
qual
it
y
of
th
e
inf
or
m
at
ion
pr
ese
nted
an
d
this
aff
e
ct
s
and
reduces
t
he
us
e
of
t
he
ru
le
set
.
T
he
go
al
of
redu
ndancy
el
im
inati
on
is
to
im
pr
ov
e
t
he
qual
it
y,
thu
s
al
lowing
them
to
bette
r
s
olve
pro
blem
s
bei
ng
face
d.
Our
work
ai
m
s
to
rem
ov
e
hiera
rc
hical
redu
nd
a
nc
y
in
m
ul
ti
-
le
vel
dataset
s,
thu
s
reducin
g
the
siz
e
of
the
ru
le
se
t
to
i
m
pr
ov
e
t
he
qual
it
y
and
us
ef
uln
es
s,
w
it
ho
ut
causin
g
t
he
los
s
of
a
ny
inf
orm
at
ion
.
W
e
ha
ve
propose
d
a
n
ap
proac
h
w
hi
ch
rem
ov
es
hie
rar
c
hical
re
dundanc
y
thr
ough
t
he
use
of
f
re
qu
e
nt
cl
os
ed
it
em
set
s
and
ge
ner
at
ors
.
This
al
lows
it
to
be
ad
ded
t
o
ot
her
a
ppro
a
che
s
wh
ic
h
al
s
o
re
m
ov
e
redu
nd
a
nt
r
ules,
t
her
e
by
al
lowing
a
use
r
to
rem
ov
e
as
m
uch
re
dundancy
a
s
po
ssi
ble.
T
he
nex
t
ste
p
i
n
our
work
is
to
ap
ply
this
a
ppr
oa
ch
to
th
e
ap
pro
xim
a
te
basis
r
ul
e
set
to
rem
ov
e
re
dundancy
t
here
.
We
will
al
so
r
eview
our
wor
k
to
se
e
if
the
re
are
oth
e
r
hierar
c
hical
redu
nd
a
ncies
in
th
e
basis
r
ule
se
ts
that
sh
oul
d
be
rem
ov
e
d
a
nd
will
inv
e
sti
gate
w
ha
t
sh
ould
an
d
ca
n
be
do
ne
to
further
im
pr
ov
e
t
he
q
ualit
y
of
m
ul
ti
-
le
vel A
ss
ociat
ion R
ules.
REFERE
NCE
S
[1]
Ba
y
ard
o,
R
.
J.,
A
gra
wal,
R
.
&
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D.
“
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nt
-
b
ase
d
rule
m
ini
ng
in
la
rg
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ase
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Berr
y
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“
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m
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ark
et
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e
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a
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il
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R
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eve
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at
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ab
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EE
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ansacti
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n
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ie
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fuz
z
y
m
ul
ti
pl
e
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le
ve
l
As
socia
t
io
n
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s
from
q
uant
itati
v
e
d
at
a
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Applie
d
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y
a,
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haj
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“
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g
m
ult
i
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cro
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level
fuz
z
y
we
igh
te
d
As
socia
t
ion
Rule
s”
2nd
Int
ernati
onal
I
EE
E
Confe
renc
e
on
I
nte
lligent Sy
st
ems
,
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225
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230,
2
004.
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y
szki
ewic
z
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M.,
R
y
binski
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H.
&
Ga
je
k
,
M.
“
Data
le
ss
tra
nsi
t
ions
b
et
wee
n
co
nci
se
rep
rese
nt
a
ti
ons
of
fre
qu
en
t
pat
t
ern
s”,
Journ
al
of
Intelli
g
ent I
nformation
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e
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,
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aks
hm
ana
n,
V.,
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a
n,
J.
&
Pang
,
A.
“
Expl
ora
tor
y
m
ini
ng
and
pr
uning
oti
m
izati
o
ns
of
constra
ined
As
socia
ti
on
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l
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ere
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Pasquier,
N.
,
B
asti
de
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ou
il
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.
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akhal,
L.
“
Eff
i
ci
en
t
m
ini
ng
of
associa
ti
on
rultes
using
cl
osed
i
te
m
set
la
ttic
es”
.
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l
of
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g
ent In
formation
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e
ms
,
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24
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[13]
Pasquier,
N.
,
T
aoui
l
,
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,
B
astide,
Y.
,
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m
e
,
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&
La
kh
al,
L.
“
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rating
a
conde
nsed
r
epr
ese
nt
at
ion
fo
r
As
socia
ti
on
Ru
l
es”
,
.
Journal
of
I
nte
lligent Inf
orm
ati
on
S
yste
ms
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ol.
24
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Srikant
,
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,
Vu,
Q.
&
Agrawal
,
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“
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Associ
at
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Ru
le
s
with
it
em
cons
tr
ai
nts”
,
Proceedi
ngs
of
the
KDD
Confe
renc
e
,
pp.
67
-
73,
1997
.
50%
60%
70%
80%
90%
100%
RE
ML_
T2L
1
REH
R
ML_
T2L
1
Redu
ction
Ru
le
s
in %
Comparison of
RE
,
RE
HR
Algorit
hms an
d
ML_T
2L1
A
B
C
D
E
F
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8708
In
t J
Elec
&
C
om
p
En
g,
V
ol.
8
, N
o.
6
,
Dece
m
ber
201
8
:
4568
-
4576
4576
[15]
Tha
kur,
R.
S.
,
J
ai
n,
R.
C
.
&
Pa
rda
sani,
K.P.
“
Mining
le
v
el
-
cr
oss
ing
As
socia
tion
Rule
s
from
la
rge
databa
ses”
,
Journal
of
Computer
Sc
ie
nc
e
,
pp.
76
-
81,
2006.
[16]
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