TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4290 ~ 4
2
9
8
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.482
2
4290
Re
cei
v
ed O
c
t
ober 1
7
, 201
3; Revi
se
d Decem
b
e
r
19, 2013; Accept
ed Ja
nua
ry 2
2
, 2014
The Non-equidistant Multivariable New Information
MGM(1,n) Based on New Information Background Value
Constructing
Zheming He*
,
Xiao
y
i
Che, Qiy
un Liu
Coll
eg
e of Mechan
ical En
gi
ne
erin
g, Huna
n U
n
iversi
t
y
of Arts and Scie
nce,
Chan
gd
e, 415
000, P.R.Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: hzming@
12
6
.
com
A
b
st
r
a
ct
Apply
i
ng
the pr
incip
l
e in
w
h
ic
h
new
i
n
for
m
at
ion
s
h
o
u
ld be used
fu
lly an
d mo
de
lin
g meth
od
of gre
y
system
for the problem
of lower prec
ision as
well as
low
e
r
adaptability in
non-
equidistant
multivar
iable
MGM(1,n)mo
d
e
l,
a non-
eq
uid
i
stant mu
ltiv
ari
abl
e n
e
w
infor
m
ati
on MGM(1
,n) mode
l w
a
s
put forw
ard w
h
ich
w
a
s taken the mth co
mpon
en
t as the
initia
li
zation. As the b
a
ckgro
und v
a
lu
e is an i
m
porta
nt factor affecti
n
g
the prec
ision
of grey system
m
o
del, ba
s
ed on index c
haracter
i
stic of
gr
ey
m
o
del, t
he characterist
ic of
integr
al an
d new
infor
m
ati
on pri
n
cip
l
e, the new
infor
m
ati
on b
a
ckgr
oun
d valu
e i
n
non-
eq
uid
i
stan
t
mu
ltivari
a
b
l
e
n
e
w
infor
m
ati
o
n
MGM(1
,n) w
a
s researc
h
e
d
a
nd th
e d
i
screte
functio
n
w
i
th n
on-h
o
m
og
en
eo
u
s
expo
ne
ntial
la
w
w
a
s used
to
fit the acc
u
mu
l
a
ted s
e
q
uenc
e
an
d th
e for
m
u
l
a of
new
i
n
for
m
ati
o
n
back
g
r
oun
d
valu
e w
a
s giv
e
n. T
h
is n
e
w
no
n-eq
uid
i
stant
mu
ltivari
a
b
l
e M
G
M(1,n) m
ode
l
can
be
use
d
i
n
eq
ui
distanc
e
&
non-
equ
id
istan
c
e
mo
del
ing
a
nd
has th
e c
haracter
i
stic
of
hi
gh
precis
io
n as
w
e
ll
as
hig
h
a
d
a
p
tabi
li
ty.
Exampl
es vali
d
a
te the practic
abil
i
ty
and re
li
a
b
ility of the pro
pose
d
mod
e
l
.
Ke
y
w
ords
:
Multivari
a
b
l
e,
backgr
oun
d
valu
e, new
infor
m
ati
on,
non-
equ
id
istance s
equ
en
ce, non-
equ
idista
nce M
G
M (1,n) mod
e
l
, least square
meth
od
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
Grey mo
del
as a
n
impo
rt
ant part i
n
grey sy
stem th
eory ha
s b
e
e
n
wid
e
ly use
d
in many
fie
l
d
s
s
i
nc
e
Pr
o
f
e
s
s
o
r
J
.
L. D
e
ng
pr
opo
s
e
d
th
e
g
r
e
y
system.
Th
ere
are m
a
n
y
grey
mode
ls,
foremo
st of
w
h
ich
is
GM
(1
,1), GM
(1,
n
)
, MG
M (1,
n) [1-3], GOM
(1,1)
[4], and GRM (1,1) [5].
There often contai
n multiple variable
s
which a
r
e i
n
trinsi
cally li
nke
d
each o
t
her in so
cia
l
,
eco
nomi
c
an
d engine
erin
g
systems. In spite of ex
tending from G
M
(1,1) mod
e
l
in the case
of n
variable
s
, M
G
M (1,n) m
o
del is not a
simple
combi
nation of th
e
GM (1,1)
m
odel
s, but al
so
different fro
m
the GM
(1,
n) mo
del e
s
t
ablishi
ng
a si
ngle first-o
r
d
e
r differential
equatio
n wit
h
n
variable
s
. T
h
i
s
m
odel
ne
e
d
to
esta
blish
n
differential
equ
ation
s
wi
th n va
riabl
e
s
to
solute, a
n
d
these
pa
ram
e
ters of
MG
M(1,n)
can
reflect the
re
l
a
tionship
s
of
mutual influ
e
n
ce
an
d
re
striction
among multip
le variable
s
[6]. Most of grey sy
stem model
s are ba
sed on e
quidi
stant se
quen
ce
,
but the origi
n
al data obtai
ned from the
actual
work
are mo
stly non-e
quidi
stan
t seque
nce. So
that establi
s
hing non
-e
q
u
idista
nt seq
uen
ce
mod
e
l
has a cert
ain pra
c
tical
and theoretical
signifi
can
c
e.
The optimi
z
in
g model
of MGM (1,n
)
was set up by ta
king the first compon
ent of the
seq
uen
ce
)
1
(
x
as the initial con
d
ition of the grey
differential e
qua
tion and mo
difying [2].
Acco
rdi
ng to
new informati
on p
r
io
rity pri
n
cipl
e in
the grey system,
multiv
ariable
new
informati
o
n
MGM(1,n)
m
odel ta
kin
g
t
he nth
comp
onent
of
)
1
(
x
as initial
condition
was
es
tablis
hed [7].
Takin
g
the
nth co
mpon
ent
of
)
1
(
x
as i
n
itial
condition and optimi
z
ing
the modified
initial value
and the
co
efficient of b
a
ckgrou
nd valu
e
q
w
h
er
e
th
e fo
r
m
is
)
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
k
x
q
k
qx
z
i
i
i
,
])
1
,
0
[
(
q
the multivari
able ne
w inf
o
rmatio
n MG
M(1,n)
m
ode
l was
esta
bli
s
he
d [8]. These M
G
M(1,n)
model
s a
r
e
equidi
stan
ce, the no
n
-
equi
dista
n
ce
multivariabl
e MGM
(
1,n
)
model
wit
h
homog
ene
ou
s expo
nent functio
n
fitting backg
rou
nd
value wa
s e
s
tablish
ed [9]. Ho
wever, it
is
more
wide
sp
read of non
-h
omoge
neo
us
expone
nt func
tion, so the
r
e are inh
e
re
n
t
defects in the
modelin
g me
cha
n
ism
of t
h
is m
odel. T
he no
n-equi
d
i
stan
ce m
u
ltivariable
MG
M(1,n) mod
e
l
wa
s
establi
s
h
ed [10], whe
r
e its backg
roun
d
value is ge
ne
rated by mea
n
value so a
s
to bring ab
o
u
t
lowe
r a
c
cu
racy. Th
e n
on-e
quidi
stan
ce
multivari
able
GM(1,n) m
odel
b
a
se
d o
n
n
on-
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The No
n-equi
distant Multivariabl
e Ne
w Inform
ation MGM(1,n
)
Based on New… (Zhem
ing He)
4291
homog
ene
ou
s exp
one
nt functio
n
fitting
ba
ckground
value
wa
s e
s
tablish
ed [1
1], that imp
r
oves
the accu
ra
cy of the mod
e
l. The b
u
ild
ing meth
o
d
f
o
r ba
ckg
r
ou
n
d
value in
MGM(1,n)
was
analyzed a
n
d
a metho
d
of
re
con
s
tru
c
tin
g
ba
ck
groun
d
value wa
s put
forward whi
c
h wa
s
b
a
se
d
on vecto
r
con
t
inued fra
c
tio
n
s the
o
ry by
usin
g ratio
nal
interpol
ation,
trape
zoid
al rule in n
u
me
ri
cal
integratio
n a
nd extrapol
ation formula [
12]. This
mo
del can effe
ctively improve simulatio
n
and
predi
ction, b
u
t is a equi
di
stan
ce multiv
ariabl
e MGM
(
1,n) m
odel.
The con
s
tru
c
ting method f
o
r
backg
rou
nd v
a
lue is
a key
factor affe
cting the
predi
ction accu
ra
cy and t
he ad
a
p
tability, so the
optimizatio
n f
o
r
ba
ckgro
u
n
d
value
s
i
s
a
n
imp
o
rtant
mean
s
of im
proving
the
model. In
o
r
d
e
r to
improve
the accuracy
of GM(1,1
), so
me co
n
s
tru
c
t
i
ng m
e
thod
s for
ba
ckgro
und val
ue
were
prop
osed
an
d some
no
n-equidi
stan
ce
GM(1,1
)
mod
e
l were e
s
ta
blish
ed [1
3-1
7
]. In this pa
per,
the modelin
g
method in [17] was a
b
sorbe
d
. Base
d
on index ch
ara
c
teri
stic of
grey model,
th
e
cha
r
a
c
teri
stic of integ
r
al
an
d ne
w i
n
form
ation p
r
in
cipl
e, the n
e
w inf
o
rmatio
n b
a
ckgroun
d valu
e in
non-equi
dista
n
t multivaria
ble ne
w info
rmati
on MG
M(1,n)
was resea
r
ched and
the discrete
function
with
non-hom
oge
neou
s exp
o
n
ential la
w wa
s u
s
ed to
fit the a
c
cumulat
ed sequ
en
ce
and
the formul
a of new info
rmation ba
ckgrou
nd valu
e wa
s given
.
The ne
w
non-equi
dista
n
t
multivariable
MGM(1,n) m
odel ca
n
b
e
use
d
in
eq
ui
distan
ce
& n
on-e
quidi
stan
ce
mod
e
ling
and
extend the ap
plicatio
n ran
g
e
of the grey model. Th
e
r
e
is highe
r pre
c
isi
on, better
theoreti
c
al an
d
practical value in this model.
2. Non-equid
i
stan
t Multiv
ariable Ne
w
I
n
forma
t
ion Gre
y
Model
MGM (1,n
)
Definition 1: Suppo
sed th
e sequ
en
ce
,
),
(
),
(
[
2
)
0
(
1
)
0
(
)
0
(
t
x
t
x
i
i
i
X
)]
(
,
),
(
)
0
(
)
0
(
m
i
j
i
t
x
t
x
, if
const
t
t
t
j
j
j
1
, where
m
j
n
i
,
,
2
,
,
,
2
,
1
,
n
is t
he nu
mbe
r
of
variabl
es an
d
m
is
the seq
uen
ce
numbe
r of each vari
able,
)
0
(
i
X
is calle
d as n
on-e
quidi
stan
t seque
nce.
Definition 2:
Supposed the se
que
nce
,
),
(
),
(
{
2
)
1
(
1
)
1
(
)
1
(
t
x
t
x
i
i
i
X
)}
(
,
),
(
)
1
(
)
1
(
m
t
j
i
t
x
t
x
j
, if
)
(
)
(
1
)
0
(
1
)
1
(
t
x
t
x
i
i
and
j
j
i
j
i
j
i
t
t
x
t
x
t
x
)
(
)
(
)
(
)
0
(
1
)
1
(
)
1
(
whe
r
e
,
,
,
2
m
j
,
,
,
2
,
1
n
i
an
d
1
j
j
j
t
t
t
,
)
1
(
i
X
is one-time
accumul
a
ted
gene
ration of
non-e
quidi
stant sequ
en
ce
)
0
(
i
X
, and
it is denoted
by 1-AG0.
Suppo
sed th
e origin
al dat
a matrix:
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
}
,
,
,
{
)
0
(
2
)
0
(
1
)
0
(
)
0
(
2
2
)
0
(
2
1
)
0
(
2
)
0
(
1
2
)
0
(
1
1
)
0
(
1
)
0
(
)
0
(
2
)
0
(
1
)
0
(
m
n
n
n
m
m
n
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
T
X
X
X
X
(1)
Whe
r
e,
)
,
2
,
1
(
)]
(
,
),
(
),
(
[
)
(
)
0
(
)
0
(
2
)
0
(
1
)
0
(
m
j
t
x
t
x
t
x
t
j
n
j
j
j
X
is th
e ob
se
rvation
value
of ea
ch vari
able
at
j
t
, and the
seq
uen
ce
)]
(
,
),
(
,
),
(
),
(
[
)
0
(
)
0
(
2
)
0
(
1
)
0
(
m
i
j
i
i
i
t
x
t
x
t
x
t
x
)
,
,
2
,
1
,
,
,
2
,
1
(
m
j
n
i
is no
n-
equidi
stant, that is, the distance
1
j
j
t
t
is not con
s
tant.
In orde
r to e
s
tabli
s
h the
model, firstly
the origin
al data is a
c
cu
mulated on
e
time to
gene
rate a n
e
w matrix a
s
:
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
}
,
,
,
{
)
1
(
2
)
1
(
1
)
1
(
)
1
(
2
2
)
1
(
2
1
)
1
(
2
)
1
(
1
2
)
1
(
1
1
)
1
(
1
)
1
(
)
1
(
2
)
1
(
1
)
1
(
m
n
n
n
m
m
n
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
t
x
T
X
X
X
X
(2)
Whe
r
e,
)
,
,
2
,
1
)(
(
)
1
(
m
j
t
x
j
meets the co
nditio
n
s in t
he defi
n
ition 2, that is,
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4290 – 4
298
4292
)
1
(
)
(
)
,
,
2
)(
)(
(
)
(
1
)
0
(
1
1
)
0
(
)
1
(
k
t
x
m
k
t
t
t
x
t
x
i
k
j
j
j
j
i
j
i
(3)
Non
-
eq
uidi
stant multivari
able MGM
(1, n)
mod
e
l can b
e
expressed a
s
first-ord
e
r
differential eq
uation
s
with n
variable
s
:
n
n
nn
n
n
n
n
n
n
n
b
x
a
x
a
x
a
dt
dx
b
x
a
x
a
x
a
dt
dx
b
x
a
x
a
x
a
dt
dx
)
1
(
)
1
(
2
2
)
1
(
1
1
)
1
(
2
)
1
(
2
)
1
(
2
22
)
1
(
1
21
)
1
(
2
1
)
1
(
1
)
1
(
2
12
)
1
(
1
11
)
1
(
1
(4)
A
ssu
med
nn
n
n
n
n
a
a
a
a
a
a
a
a
a
2
1
2
22
21
1
12
11
A
,
n
b
b
b
2
1
B
, Equation (4)
ca
n be expre
ssed as:
B
AX
X
)
(
)
(
)
1
(
)
1
(
t
dt
t
d
(5)
Acco
rdi
ng to
new info
rmati
on pri
o
rity pri
n
cipl
e in the
grey sy
stem, it is inadeq
u
a
te for
utilizing n
e
w information
whe
n
the first compo
nent
of the seq
u
ence
)
,
,
2
,
1
)(
(
)
1
(
m
j
t
j
i
x
is
taken
as initi
a
l co
ndition
of grey
differential eq
uatio
n. Reg
a
rded
the
m
th comp
o
nent a
s
the
initial con
d
itio
ns of the g
r
e
y
differential equatio
n,
the continu
o
u
s
time re
spo
n
se
of Equation
(5)
is as:
B
I
A
X
X
A
)
(
)
(
)
(
1
)
1
(
)
1
(
t
m
At
e
t
e
t
(
6
)
Whe
r
e,
k
k
k
t
t
k
e
1
!
A
I
A
,
I
is
a unit matrix.
In orde
r to id
entify
A
and
B
, Equation
(4) i
s
made the int
egratio
n in
]
,
[
1
j
j
t
t
and
we can obtai
n:
)
,
,
3
,
2
;
,
,
2
,
1
(
)
(
)
(
1
)
1
(
1
)
0
(
m
j
n
i
t
b
dt
t
x
a
t
t
x
j
i
t
t
j
i
n
l
il
j
j
i
j
j
(7)
Noting
dt
t
x
t
z
j
j
t
t
j
i
j
i
)
(
)
(
1
)
1
(
)
1
(
, the
comm
on fo
rmula fo
r ba
ckgro
und
val
ue that i
s
actually
ba
se
d on
the t
r
ap
ezoi
dal
are
a
j
j
i
t
t
z
)
(
)
1
(
is a
pprop
riate
wh
en th
e tim
e
interval
is
small,
that is, the
ch
ange
of
sequ
ence d
a
ta is
slo
w
. Bu
t wh
en thi
s
chan
g
e
is sudde
n, the b
a
ckg
r
ou
n
d
value u
s
in
g t
he
comm
on
formul
a often
bring
s
out th
e la
rge
r
error, and
the
mo
del p
a
ramete
rs
obtaine
d by the com
m
on
formula for b
a
ckgroun
d
value in the n
e
w informatio
n model is the
same
a
s
the
one in th
e
ordin
a
ry mo
d
e
l. It is
also
unreasona
bl
e, so it i
s
m
o
re
suitabl
e
for
Equation
(4
)
that pa
ramet
e
r m
a
trix
A
ˆ
an
d
B
ˆ
estimate
d
by the
ba
ckg
r
oun
d valu
e i
n
]
,
[
1
j
j
t
t
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TELKOM
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ISSN:
2302-4
046
The No
n-equi
distant Multivariabl
e Ne
w Inform
ation MGM(1,n
)
Based on New… (Zhem
ing He)
4293
are obtaine
d by
dt
t
x
t
z
j
j
t
t
j
i
j
i
)
(
)
(
1
)
1
(
)
1
(
s
u
bs
tituting for
)
(
)
1
(
j
i
t
x
. Base
d on q
u
a
s
i-expone
ntially
law of the
grey mod
e
l and the mo
deling p
r
in
ci
ples a
nd m
e
thod
s in [7, 9], we set that
i
t
t
a
i
i
C
e
G
t
x
i
)
(
)
1
(
1
)
(
, where
i
i
i
C
G
a
,
,
are th
e
undete
r
mi
ned
coefficients a
nd
i
t
t
a
i
j
i
C
e
G
t
x
i
)
(
)
1
(
1
)
(
is meet. It
ca
n be
det
ermi
ned
by the
grey mod
e
ling
whe
n
the
dat
a a
r
e
kno
w
n.
)
(
)
1
(
j
i
t
x
is regre
ssively ge
nerate
d
to ob
tain:
)
(
)
(
1
)
1
(
)
1
(
)
0
(
)
1
(
)
(
)
(
)
(
m
j
i
m
j
i
j
i
t
t
a
i
t
t
a
j
t
a
i
j
j
i
j
i
j
i
e
g
e
t
e
G
t
t
x
t
x
t
x
(8)
Whe
r
e,
j
j
i
j
i
i
j
t
a
i
i
t
t
a
t
a
G
t
e
G
g
j
)
!
2
)
(
)
(
1
(
1
(
)
1
(
2
Whe
n
i
a
and
j
t
are
small, th
e first two ite
m
s a
r
e taken
after expand
ing
j
i
t
a
e
, they
can b
e
obtain
ed as follo
ws:
i
i
j
j
i
i
j
t
a
i
i
a
G
t
t
a
G
t
e
G
g
j
i
)
(
)
1
(
j
i
m
j
i
m
j
i
t
a
t
t
a
t
t
a
j
i
j
i
e
e
e
t
x
t
x
)
(
)
(
1
)
0
(
)
0
(
1
)
(
)
(
Then,
)
,
,
3
,
2
(
)
(
ln
)
(
ln
1
)
0
(
)
0
(
m
j
t
t
x
t
x
a
j
j
i
j
i
i
(9)
That Equatio
n (9)
sub
s
tituting into Equat
ion (8
), we ca
n obtain:
)
(
)
(
1
)]
(
/
)
(
[
)
(
)]
(
/
)
(
[
)
(
)
(
)
0
(
1
)
0
(
1
)
0
(
1
)
0
(
)
0
(
1
)
0
(
)
0
(
)
0
(
)
0
(
)
(
)
0
(
j
i
j
i
t
t
t
j
i
j
i
j
j
i
i
t
t
t
j
i
j
i
j
i
t
t
a
j
i
i
t
x
t
x
t
x
t
x
t
t
x
G
t
x
t
x
t
x
e
t
x
g
j
j
m
j
j
m
m
j
i
(10
)
Accounting to the initial conditions as
i
i
i
t
t
a
i
i
C
G
C
e
G
t
x
i
)
(
1
)
1
(
1
1
)
(
, it
can b
e
obtain
ed:
)
(
)
(
1
)]
(
/
)
(
[
)
(
)
(
)
(
1
)
0
(
1
)
0
(
1
)
0
(
)
0
(
)
0
(
)
0
(
)
0
(
j
i
j
i
t
t
t
j
i
j
i
j
j
i
m
i
i
m
i
i
t
x
t
x
t
x
t
x
t
t
x
t
x
G
t
x
C
j
j
m
(11)
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TELKOM
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KA
Vol. 12, No. 6, June 20
14: 4290 – 4
298
4294
That Equatio
n (9) a
nd Eq
uation (1
1)
substi
tuting fo
r the formula
for ba
ckgro
u
nd value
j
j
t
t
j
i
dt
t
x
1
)
(
)
1
(
can be o
b
tai
ned:
)
(
)
(
1
)]
(
/
)
(
[
)
)(
(
)
(
)
(
ln
)
(
ln
)
(
)
(
)
(
)
(
)
0
(
1
)
0
(
1
)
0
(
)
0
(
2
)
0
(
)
0
(
)
0
(
)
0
(
)
0
(
2
)
0
(
)
1
(
)
1
(
1
j
i
j
i
t
t
t
j
i
j
i
j
m
i
j
m
i
j
i
j
i
j
i
j
j
i
i
j
i
j
t
t
i
j
i
t
x
t
x
t
x
t
x
t
t
x
t
t
x
t
x
t
x
t
x
t
t
C
a
t
x
t
dt
x
t
z
j
j
m
j
j
(12)
Noting
)
,
,
2
,
1
(
)
,
,
,
,
(
2
1
n
i
b
a
a
a
i
in
i
i
i
T
a
, the identified value
i
a
ˆ
of
i
a
can be
obtaine
d by using the le
ast
squa
re meth
od:
n
i
b
a
a
a
i
T
T
i
in
i
i
i
,
,
2
,
1
,
)
(
]
ˆ
,
ˆ
,
,
ˆ
,
ˆ
[
ˆ
1
2
1
Y
L
L
L
a
T
(13
)
Whe
r
e,
m
m
n
m
m
n
n
t
t
z
t
z
t
z
t
t
z
t
z
t
z
t
t
z
t
z
t
z
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1
(
)
1
(
2
)
1
(
1
3
3
)
1
(
3
)
1
(
2
3
)
1
(
1
2
2
)
1
(
2
)
1
(
2
2
)
1
(
1
L
(14)
T
Y
]
)
(
,
,
)
(
,
)
(
[
)
0
(
3
3
)
0
(
2
2
)
0
(
m
m
i
i
i
i
t
t
x
t
t
x
t
t
x
(15
)
Then the ide
n
t
ified values o
f
A and B can be obtaine
d:
nn
n
n
n
n
a
a
a
a
a
a
a
a
a
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
2
1
2
22
21
1
12
11
A
，
n
b
b
b
ˆ
ˆ
ˆ
ˆ
2
1
B
(16
)
The cal
c
ul
ate
d
value in ne
w inform
ation
MGM(1,n
)
m
odel is:
m
j
e
t
e
t
m
j
m
j
t
t
m
t
t
j
i
,
,
2
,
1
,
ˆ
)
(
ˆ
)
(
)
(
ˆ
)
(
ˆ
1
)
1
(
)
(
ˆ
)
1
(
B
I
A
X
X
A
A
(17
)
After resto
r
in
g the fitting value of the ori
g
inal data i
s
:
m
j
t
t
t
t
j
t
t
t
t
t
t
j
j
j
i
j
i
i
i
j
i
,
,
3
,
2
),
/(
))
(
ˆ
)
(
ˆ
(
1
,
)
(
)
(
0
lim
)
(
ˆ
1
1
)
1
(
)
1
(
)
1
(
1
)
1
(
)
0
(
X
X
X
X
X
(18
)
The ab
solute
error of the
i
th variable:
)
(
)
(
ˆ
)
0
(
)
0
(
j
i
j
i
t
x
t
x
.
The rel
a
tive erro
r of the
i
th variable:
100
)
(
)
(
)
(
ˆ
)
(
)
0
(
)
0
(
)
0
(
j
i
j
i
j
i
j
i
t
x
t
x
t
x
t
e
.
The mea
n
of the relative error of the
i
th variabl
e:
m
j
j
i
t
e
m
1
)
(
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The No
n-equi
distant Multivariabl
e Ne
w Inform
ation MGM(1,n
)
Based on New… (Zhem
ing He)
4295
The average
error of the whole data:
)
)
(
(
1
1
1
m
j
j
i
n
i
t
e
nm
f
It can be
se
e
n
that non
-eq
u
idista
nt ne
w informatio
n
MGM(1,n) m
odel i
s
de
gra
ded into
GM(1,n
) whe
n
1
n
, and thi
s
MGM(1,n) m
odel i
s
a
co
mbination
of
n GM(1,n)
m
odel
s when
0
B
. This MG
M(1,n)
ca
n b
e
u
s
ed
not
only f
o
r m
odeli
ng
and
predi
cting but
al
so fo
r data
fitting
and p
r
ocessi
ng. The valu
e of n accou
n
ting to t
he spe
c
ific
circu
m
stan
ce
s ca
n get the ne
eded
model a
s
MG
M (1,2), MG
M (1,3) a
nd
MGM (1,4
).
3.
Precision Inspec
ting for the Mod
e
l
The inspe
c
tin
g
mean
s co
n
t
ain resi
dual
analysi
s
, co
rrelation deg
re
e analysi
s
, and post
-
error a
nalysi
s
[1], [19-22]
. The displa
cement relative deg
ree, th
e sp
eed
rela
ted deg
ree, t
he
accele
ration
degree, a
nd
the total relat
ed de
gr
ee a
r
e calcul
ated
simultan
eity. These
kind
s
of
related d
e
g
r
e
e
s are call
ed
related de
grees of C-type
[22], which
can be u
s
e
d
to both of the
whol
e analy
s
i
s
an
d the dy
namic
analy
s
is. The follo
wing rel
a
ted d
egre
e
in
spe
c
t
i
on of C-type
is
employed in t
h
is pa
per.
1) To calculat
e the three
-
la
yer relate
d de
gree
s
Displa
ceme
nt related de
gree
)
(
)
0
(
k
t
d
m
k
x
x
t
d
k
k
t
t
k
,
,
2
,
1
,
ˆ
/
)
(
)
0
(
)
0
(
)
0
(
(19
)
Speed relate
d degree
)
(
)
1
(
k
t
d
.
1
,
,
2
,
1
,
ˆ
ˆ
)
(
)
0
(
)
0
(
)
0
(
)
0
(
)
1
(
1
1
m
k
x
x
x
x
t
d
k
k
k
k
t
t
t
t
k
(20
)
Accel
e
ration related de
gre
e
)
(
)
2
(
k
t
d
).
1
,
,
2
,
1
(
ˆ
ˆ
2
ˆ
2
)
(
)
0
(
)
0
(
)
0
(
)
0
(
)
0
(
)
0
(
)
2
(
1
1
1
1
m
k
x
x
x
x
x
x
t
d
k
k
k
k
k
k
t
t
t
t
t
t
k
(21
)
2) To calculat
e the three
-
la
yer relate
d co
mpre
hen
sive
degree at
k
t
)
(
)
(
,
2
)
(
)
(
)
(
)
0
(
1
)
0
(
1
)
1
(
1
m
m
t
d
t
D
t
d
t
d
t
D
(22
)
)
1
,
,
3
,
2
(
3
)
(
)
(
)
(
)
(
)
2
(
)
1
(
)
0
(
m
k
t
d
t
d
t
d
t
D
k
k
k
k
(
2
3
)
3) To calculat
e the total related deg
ree
of the model
m
k
t
x
k
,
,
2
,
1
),
(
ˆ
)
0
(
m
k
k
t
D
n
D
1
)
(
1
(24)
Whe
n
1
60
.
0
D
or
1
D
and
60
.
0
1
D
, the preci
s
i
on of the m
o
del is "
G
oo
d". When
60
.
0
3
.
0
D
, the preci
s
i
on of the model is "b
etter". Whe
n
30
.
0
D
or
30
.
0
1
D
, the
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4290 – 4
298
4296
pre
c
isi
on of the model i
s
"bad" [22].
4. Examples
Example 1
: The affectin
g
data of wat
e
r ab
so
rption
to mech
ani
cal pro
pertie
s
of pure
PA66 are se
en in [10]. After PA66 sa
mples
with the different water ab
sorption we
re test
ed in
mech
ani
cal p
r
ope
rty, the followin
g
exp
e
rime
ntal
dat
a of PA66 can be obtain
ed, as shown in
Table 1,
whe
r
e
)
0
(
1
X
is b
endi
ng
stren
g
th
(Mp
a
),
)
0
(
2
X
is b
endi
n
g
ela
s
tic mod
u
lus
(Gpa
)
an
d
)
0
(
3
X
is tensil
e stre
ngth (Mp
a
).
Table 1. The
Affecting Dat
a
of Water Ab
sorption to M
e
ch
ani
cal Pro
pertie
s
of Pure PA66
No.
1 2 3
4 5 6 7
8 9
w
a
t
e
r abso
r
ption
%
/
j
t
0
0.0607
0.1071
0.1662
0.2069
0.4344
0.5243
0.8524
0.9756
)
0
(
1
X
83.4
84.9 84.5
84.2 84.4 78.4 75.4
59.5 54.1
)
0
(
2
X
2.63
2.64 2.61
2.65 2.66 2.52 2.32
1.90 1.72
)
0
(
3
X
84.2
84.4 86.3
84.3 81.3 74.9 75.7
73.2 66.9
The no
n-e
qui
distant n
e
w i
n
formatio
n M
G
M(1,
3
)
mo
d
e
l wa
s e
s
tab
lishe
d by usi
ng the
prop
osed met
hod in this p
a
per. The p
a
ra
me
ters of this model are as follows:
0.2612
-
0.7070
-
0.0954
0.0011
-
0.6888
-
0.0069
0.0620
-
15.6730
-
0.0468
A
,
81.7480
2.5492
80.5759
B
The fitting value of
)
0
(
3
X
:
)
0
(
3
ˆ
X
=[82.57
75,82
.1106,81.2
9
6
5
,80.497,7
9
.7
43,
77.7316
,75.3726,7
2
.3
237,69.0
904]
The ab
solute
error of
)
0
(
3
X
:
q
[-1.6225,
-2.2
894,-5.0
035,
-3.803,-1.5
57,
0.8316, -0.3
2741,-0.876
2
6
,2.1904]
The rel
a
tiv
e
erro
r of
)
0
(
3
X
(%
):
e
[-1.9269,
-2.7
125,-5.7
978,
-4.5113,-1.91
52, 3.7804,-0
.43251,-1.19
71,3.274
2]
The me
an of
the rel
a
tive error
of
)
0
(
3
X
is 2.
8387%, so th
is mo
del ha
s highe
r p
r
e
c
i
s
ion
and the preci
s
ion of
)
0
(
3
X
is "Good". The
maximum rel
a
tive erro
r of
)
0
(
3
X
-5.7978% i
s
sm
aller
than the one
-6.104
8% in [10].
Example 2:
I
n
the co
nditio
n
s of the lo
ad
600
N and th
e relative
slidi
ng speed
0.3
14m/s,
0.417m/s,
0.6
28m/s, 0.9
4
2
m
/s an
d 1.04
6m/s respe
c
ti
vely, the test data of the th
in film with Ti
N
coat are sh
o
w
n a
s
in Tabl
e 2 [18].
Table 2. Tes
t
Data of the thin Film with TiN Coat
No.
1 2 3
4
5
Sliding
speed
(m/s)
0.314
0.471
0.628
0.942
1.046
Friction coefficie
n
t
0.251
0.258
0.265
0.273
0.288
Wear rate
5
10
*
(mg/m)
7.5 8
8.5
9.5 11
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The No
n-equi
distant Multivariabl
e Ne
w Inform
ation MGM(1,n
)
Based on New… (Zhem
ing He)
4297
Assu
med sli
d
ing
spee
d
j
t
,
fric
tion coeffic
i
ent
)
0
(
1
x
and we
ar rate
)
0
(
2
x
, non-
equidi
stant n
e
w info
rmatio
n MGM
(1,2
)
model
wa
s
establish
ed by
usin
g the
pro
posed m
e
tho
d
in
this pap
er. Th
e para
m
eters of
this model
are as follo
ws:
3.6243
101.2603
-
0.0407
1.1618
-
A
,
0.8559
0.1817
B
The fitting value of
)
0
(
1
x
:
)
0
(
1
ˆ
x
= [0.25494,0.2
562
1
,
0.25994,0.2
7
114,0.28
825]
The ab
solute
error of
)
0
(
1
x
:
q
3
10
[3.9425,- 1.79
4,-
5.0558,- 1.8
5
65,
0.24543]
The rel
a
tiv
e
erro
r of
)
0
(
1
x
(%
):
e
[1.5707,-0.6
9
535,-1.9
078,
-0.6800
4,0.08
5217]
The m
ean
of
the
relative
erro
r of
)
0
(
1
x
is 0.987
83% a
nd the
on
e
of this
mod
e
l
is
1.4981%. So this model
ha
s high
er p
r
e
c
i
s
ion.
When e
quidi
stant MGM (1,3
) mo
del wa
s u
s
ed
in
[18], the mean of the relative erro
r of
)
0
(
1
x
is 1.6225%.
4. Conclusio
n
Aimed to no
n-eq
uidi
stant
multivariable
seq
uen
ce
with mutual infl
uen
ce an
d re
stri
ction
among
multi
p
le va
riable
s
,
ba
sed
on
in
dex cha
r
a
c
te
ristic of
grey mod
e
l,
the cha
r
a
c
teri
stic
of
integral a
nd n
e
w informatio
n prin
ciple, th
e new
info
rm
ation ba
ckgro
und value in
non-equi
dista
n
t
multivariable
new
info
rmati
on
M
G
M (1,n
) wa
s re
sea
r
che
d
a
nd th
e
discrete fu
n
c
tion
with
no
n
-
homog
ene
ou
s expo
nential
law was
use
d
to fit t
he accumulate
d seque
nce and
the formula
of
new i
n
form
ation ba
ckgrou
nd value
wa
s given. Th
e
new M
G
M
(1,n) m
odel
can
be u
s
ed
in
equidi
stan
ce
& non-equi
distance
and
it e
x
tents t
he a
p
p
licatio
n sco
p
e of grey mo
del. Ne
w m
o
del
has the
cha
r
acte
ri
stic of
high preci
s
i
on as
well
as ea
sy to use. Exampl
es validate t
he
pra
c
tica
bility and the
relia
bility of the prop
osed m
odel. The
r
e
are im
porta
n
t
practi
cal a
nd
theoreti
c
al si
gnifica
nce an
d this model
sho
u
ld be
wo
rthy of promo
t
ion.
Ackn
o
w
l
e
dg
ements
This
re
sea
r
ch is
sup
porte
d by the g
r
a
n
t of the 12t
h Five-Yea
r
Plan for the
con
s
tru
c
t
prog
ram of
the key discipline
(Mech
ani
cal
De
sign a
n
d
Theory
)
in
Huna
n province
(XJF
2011[7
6
]).
Referen
ces
[1]
SF
Liu, YG Dang, Z
G
F
ang, et al. Gre
y
S
y
s
t
em
s and Ap
pli
c
ation (Ed
i
tio
n
3). Beiji
ng: Ch
i
na Scie
nc
e
Press. 2004.
[2]
YX
Luo, JY Li.
Applic
ation
of Multi-varia
b
l
e
Optimi
z
i
n
g
Grey Model MGM (1,n,q,r) to the Load-stra
i
n
Relati
on.
T
he 200
9 IEEE Internati
ona
l Co
n
f
erence o
n
Me
chatron
i
cs an
d
Automation (I
CMA). 2009:
402
3-40
27.
[3]
YX
Luo,
X W
u
,
M Li, et al. Gr
e
y
d
y
n
a
mic m
ode
l GM
(1,N)
for the rel
a
tio
n
s
hip
of cost an
d vari
abi
lit
y
.
K
y
b
e
rn
etes. 20
09; 38(3): 4
35-
440.
[4]
Z
M
Song, JL Deng. T
he Accumulate
d Gener
ating O
per
atio
n in Opposit
e Directio
n
an
d Its Use in Gre
y
Model GOM (1,1).
Systems Engi
neer
in
g.
20
01; 19(1): 6
6
-6
9.
[5]
BH Yan
g
, Z
Q
Z
hang. T
he gre
y
mod
e
l
h
a
s be
en acc
u
mulate
d ge
ner
ating
oper
atio
n in rec
i
proc
al
numb
e
r an
d its appl
icatio
n. Mathematics i
n
Practice an
d T
heor
y. 20
03; 3
3
(10): 21-
25.
[6]
J Z
hai, JM Sheng. Gre
y
Mo
del a
nd its Ap
plicati
o
n
. Syst
ems E
ngi
ne
eri
ng-T
h
e
o
ry & Practice.
19
97
;
15(5): 10
9-1
1
3
.
[7]
ZM He, YX
Luo.
Appl
icatio
n
of New
Information M
u
lti-var
i
abl
e Grey Mod
e
l NMGM (1,n)
to the Lo
ad-
strain R
e
lati
on
. Internatio
nal
Confer
enc
e
on Intel
lig
ent
Comp
utation
T
e
chnolog
y a
nd
Autom
a
tio
n
(ICICT
A). 2009
.
[8]
YX L
uo, W
Y
Xi
ao.
New Information Grey Multi-
variable Optim
i
z
i
ng Mode
l NMGM (1,n,q,r) for the
Relati
ons
hip
of
Cost an
d Vari
abil
i
ty.
Internati
ona
l Co
nfere
n
c
e on Inte
lli
ge
nt Comp
utatio
n T
e
chnolo
g
y
and Autom
a
tio
n
(ICICT
A). 2009.
[9]
F
X
W
a
ng. Mu
l
t
ivaria
ble
n
on-
equ
idista
nc
e
GM (1,m) mo
del
an
d its
ap
plic
ation.
System
s En
gi
n
e
e
r
i
n
g
and El
ectron
ics
. 2007; 9(3): 3
88-3
90.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4290 – 4
298
4298
[10]
PP Xi
on
g, YG Dan
g
, H Z
h
u
.
Researc
h
of
m
ode
lin
g of
multi-vari
abl
e
non-
equ
id
istan
t
MGM (1,m)
mode
l.
Control and
D
e
cisi
on.
201
1; 26(1): 49
-53.
[11]
PP Xiong, YG
Dang, Y Yang.
T
he Opti
mi
z
a
t
i
on
of Back
grou
nd V
a
lu
e i
n
M
u
lti-Varia
b
l
e
N
o
n-Equ
i
dista
n
t
Mode
l
. 19th Ch
ines
e Conf
eren
ce on Gre
y
S
ystems. 2010; 2
77-2
81.
[12]
LZ
Cui, SF
Li
u, Z
P
W
u
. MGM
(1,m) bas
e
d
on
vector conti
n
u
ed fractio
n
s th
eor
y
.
Syst
e
m
s Engi
neer
in
g.
200
8; 26(1
0
): 47-51.
[13]
YM W
ang, YG Dang, Z
X
W
ang. T
he Optimizati
on of Back
grou
nd Va
lue
i
n
Non-E
q
u
i
dist
ant GM(1,1)
Mode
l.
Chin
es
e Journ
a
l of Mana
ge
me
nt Sci
ence.
20
08; 16
(4): 159-1
62.
[14]
W
Z
Dai, JF
Li
. Mode
lin
g R
e
search
on
No
n-eq
uid
i
stanc
e
GM (1,1) Mo
del.
Syste
m
s
Engi
neer
in
g-
T
heory & Practice.
200
5; 25(9
)
: 89-93.
[15]
Z
X
W
ang, YG Dan
g
, SF
Liu. An optima
l
GM (1
,1) based o
n
the discrete fu
nction
w
i
t
h
e
x
p
one
ntial l
a
w
.
Systems En
gin
eeri
ng-T
h
e
o
ry & Practice
. 200
8; 28(2): 61-6
7
.
[16]
YX L
uo, Z
M
He.
T
he new
Non-e
qui
dista
n
t Optimu
m GM (1,1) of Line-
D
r
aw
ing Data P
r
ocessi
ng in
Co
mp
uter Ai
de
d Des
i
g
n
. Proc
eed
ings
of 2
0
0
9
4th I
n
ternati
ona
l C
onfere
n
c
e o
n
C
o
mput
er Scie
nce
&
Educati
on (ICC
SE200
9). 200
9
;
971-97
4.
[17]
YX L
uo. No
n-
equ
idista
nt ne
w
i
n
forma
ti
on
GM (1,1) mod
e
l an
d its ap
pli
c
ation.
Jo
urna
l
of Sheny
an
g
Univers
i
ty of Techn
o
lo
gy.
201
0; 32(5): 55
0-5
54.
[18]
YX Lu
o,
XY
C
he. T
he Gre
y
Mu
lti-vari
abl
e
Optimizin
g
Mo
del
an
d Its Ap
plicati
o
n
to A
n
al
ysis
of t
h
e
T
r
ibologic
a
l Be
havi
o
rs of the F
ilm.
Lubric
atio
n Engi
ne
erin
g.
200
8; 33(3): 58
-61.
[19]
Z
M
He, YX L
U
O. Gra
y
mo
del
an
d meth
od of c
o
mp
uter ai
de
d li
ne-
dra
w
i
ng
data
process
i
ng
(
i
n
Chin
ese).
T
r
an
sactions of the
Chin
ese Soc
i
et
y for Agricultur
al Mach
inery
. 2
002; 33(
2): 94-
96.
[20]
YX
LUO, DG
Lia
o
, QX T
ang, et al. A Ste
p
b
y
St
e
p
Opti
mum Mod
e
li
ng
metho
d
of G
M
(1,1) a
nd Its
Appl
icatio
n to F
ault Dia
gnos
i
s
in NC S
y
ste
m
of NC Machine T
ool.
Mach
ine too
l
& Hyd
r
aulics
. 2
002
;
(6): 229-2
31.
[21]
YX
LUO, LT
Z
hang, M
Li.
Gre
y
s
y
stem t
heor
y
an
d its
app
licati
on to
mecha
n
ica
l
e
n
g
in
eeri
ng (
i
n
Chin
ese). Sha
ngSh
a
: Natio
n
a
l Univ
ersit
y
of
Defense tec
h
n
o
lo
g
y
Pr
ess. 2001.
[22]
QY W
ang. Un
certaint
y math
em
atica
l
mod
e
l
of forecast
& decis
ion-m
a
king (i
n C
h
in
e
s
e). Beiji
ng:
Metall
urgic
a
l in
dustr
y
Pr
ess. 2001.
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