TELKOM
NIKA
, Vol.14, No
.2, June 20
16
, pp. 772~7
7
7
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v14i1.3096
772
Re
cei
v
ed
No
vem
ber 1
8
, 2015; Re
vi
sed
April 5, 2016;
Accept
ed Ap
ril 20, 2016
H-WEMA: A Ne
w Approach of Double Exponential
Smoothing Method
Se
n
g
H
a
ns
un
*
1
, Subanar
2
1
Universitas M
u
ltime
d
ia N
u
sa
ntara, Jl. Boul
e
v
ard Gadi
ng S
e
rpo
ng, Scie
nti
a
Garden, T
angera
n
g
2
Universitas G
adj
ah Mad
a
, Jurusa
n Matem
a
tika
F
M
IPA UGM, Sekip Utara, Yog
y
akarta
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: hansu
n
@um
n
.ac.id
1
, suba
n
a
r@
yah
oo.com
2
A
b
st
r
a
ct
A po
pul
ar s
m
oothi
ng
techn
i
que
co
mmo
n
ly
use
d
i
n
ti
me
series
an
alysis
is
dou
bl
e ex
p
one
ntia
l
smo
o
thi
ng. Ba
sically, it
’
s
an
improve
m
ent o
f
simp
le ex
pon
entia
l smooth
i
ng w
h
ic
h d
oes
the expo
ne
nti
a
l
filter proc
ess t
w
ice. Many re
sear
che
r
s ha
d d
e
v
e
l
op
ed
th
e te
ch
ni
q
u
e
,
hen
ce
Bro
w
n’
s do
u
b
l
e
e
x
po
n
enti
a
l
smo
o
thi
n
g
an
d H
o
lt
’
s
do
ubl
e ex
po
nenti
a
l
smo
o
thi
ng. H
e
re, w
e
intro
d
u
c
e a
n
e
w
ap
p
r
oach
of
dou
b
l
e
expo
ne
ntial
s
m
o
o
thi
ng, ca
ll
ed H-W
E
MA,
w
h
ich co
mbin
e
s
the c
a
lcu
l
ati
on
of w
e
ight
i
n
g factor i
n
w
e
i
ghted
mov
i
n
g
avera
g
e
w
i
th Holt
’
s
d
oub
le exp
o
n
e
n
t
ial
smooth
i
ng
meth
od. The prop
osed
met
hod w
ill the
n
b
e
tested on J
a
ka
rta Stock Exchang
e (JKSE)
c
o
mpos
ite in
dex
data. The acc
u
racy a
nd ro
bu
stness lev
e
l of
the
prop
osed
meth
od w
ill the
n
b
e
exa
m
in
ed
by usin
g me
a
n
sq
uare error
a
nd me
an abso
l
ute
perce
ntage er
ror
criteria, an
d be
comp
are
d
to o
t
her conve
n
tio
nal
meth
ods.
Ke
y
w
ords
:
Holt
’
s
d
o
u
b
le
expo
nenti
a
l
smo
o
thi
ng, H-
W
E
MA, time
series a
n
a
l
ysi
s, w
e
ighted
mov
i
n
g
avera
g
e
Copy
right
©
2016 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Based o
n
the definition
given by
Orga
nisation
for Econo
mic Co
-op
e
ration and
Develo
pment
(OE
C
D)
Glo
s
sary
of Statistical
Te
rms,
a time
serie
s
i
s
a
set of
re
gula
r
time
-
orde
re
d o
b
se
rvations of
a
quantit
ative chara
c
te
risti
c
of
an
i
ndi
vidu
al or
colle
ctive ph
enom
en
on
taken
at su
ccessive, in mo
st ca
se
s e
qui
distant,
pe
rio
d
s/ poi
nts of
time [1]. To comprehe
nd t
h
e
cha
r
a
c
teri
stics of
a tim
e
serie
s
data, m
any re
search
ers h
a
ve d
e
velope
d time
seri
es a
nalysis
method
s
with
the final
aim
to find a
pattern th
at
can
be u
s
ed
to fo
recast futu
re
event or data
[2-
4]. Some researche
r
s eve
n
use
d
the soft comput
ing
methods,
su
ch a
s
fuzzy, neural networks,
or hybri
d
method
s to achi
e
v
e the same
goal [5-9].
Moving avera
ge is a popul
ar co
nvention
a
l ti
me serie
s
analysi
s
m
e
thod that ha
s bee
n
use
d
wid
e
ly b
y
people d
ue
to its easi
n
e
s
s, obje
c
tivene
ss, robu
stne
ss, and
u
s
eful
ness [10, 11].
It
is wid
e
ly em
ployed withi
n
the realm of
financi
a
l
anal
ysis, su
ch a
s
stock ma
rket. Clif Dro
k
e [12]
define
s
a
mo
ving ave
r
age
as a
n
in
dica
tor that
sh
ows the
ave
r
ag
e value
of
a
se
curity’s p
r
i
c
e
over a peri
o
d
of time. There are vario
u
s kinds
of mov
i
ng avera
ge method
s, but their unde
rlying
purp
o
se rem
a
in the sam
e
, that is to tra
ck the tr
e
nd determi
nation
of the given
time serie
s
d
a
t
a
[10, 13]. The
simple
st on
e is
simple
moving ave
r
a
ge where ea
ch p
o
int in time se
rie
s
da
ta i
s
weig
hted th
e
sa
me
reg
a
rd
less of
wh
ere
or wh
en it
o
c
curs in
the
seq
uen
ce.
Weighted
movi
ng
averag
e is a
nother type
of moving av
erag
e whic
h
gives a
different wei
ghting
factor fo
r e
a
c
h
point in tim
e
seri
es data.
A
nother type
of moving
averag
e i
s
exp
o
nential m
o
vin
g
average
wh
ich
is a va
riatio
n
of wei
ghted
moving ave
r
a
ge that u
s
e
d
expone
ntial n
u
mbe
r
a
s
the
ba
sis i
n
fo
rm
ing
weig
hting factors in time
seri
es a
nalysis. So
me oth
e
r re
se
arche
r
s even trie
d to combin
e the
moving ave
r
age meth
od
with othe
r m
e
thod
s, su
ch
as a
u
toregressive an
d n
eural
networks t
o
rep
r
e
s
ent several types of t
i
me se
rie
s
da
ta [14-16].
A new
app
ro
ach
of movi
ng average
method
whi
c
h co
mbine
s
the wei
ghted
moving
averag
e and
exponential
moving ave
r
age m
e
thod
s to fore
ca
st the future
data had b
e
e
n
introdu
ce
d in
201
3 [17,
1
8
]. Ho
wever
the expo
nent
ial moving
av
erag
e m
e
tho
d
u
s
ed
in th
ose
researches
, als
o
known as
the s
i
ngle
s
m
oothing method, doesn’t exc
e
l in time s
e
ries
data when
there i
s
a tre
nd [19]. The
r
efore, the
r
e i
s
a nee
d
to de
velop the hyb
r
id movin
g
averag
e meth
o
d
to
overcome the
limitation of fore
ca
sting time se
rie
s
dat
a whe
n
there
is a trend in it
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
H-WEMA: A Ne
w Approa
ch of Doubl
e Ex
po
nential S
m
oothing Method (Se
ng Hansun)
773
In this pape
r, we will fu
rther develop the
hy
brid meth
o
d
, by modifying and
com
b
ining the
weig
hted mo
ving averag
e
method wit
h
Holt’s d
o
u
b
le expon
ent
ial smoothi
ng
method. Ho
lt’s
double exponential
s
m
oothing met
hod is a varian of exponential smoothing met
hod
whic
h been
widely u
s
ed t
o
pre
d
ict the
pattern of a ti
me se
rie
s
dat
a with a tre
n
d
in it. The propo
sed m
e
th
od
will then be tested o
n
Ja
karta Stock Excha
nge
(JKS
E) comp
osite
index data and be compa
r
e
d
with oth
e
r m
o
ving ave
r
ag
e meth
od
s,
su
ch
as
wei
ghted movin
g
ave
r
ag
e
m
e
thod and
Holt’s
doubl
e expon
ential smo
o
th
ing method. T
he re
sults
the
n
will be com
pare
d
by usin
g mean squa
re
error a
nd m
e
an ab
sol
u
te
percenta
ge e
rro
r criteri
a
to get the a
ccura
cy and
ro
bustn
ess lev
e
l of
the prop
osed
method comp
ared to the ot
her movin
g
a
v
erage m
e
tho
d
s.
2. The Propo
sed Me
thod
Basically the prop
osed me
thod will com
b
ine the weig
hted moving
averag
e met
hod with
Holt’s
dou
ble
expone
ntial
smo
o
thing
method. Th
e
r
efore we
wil
l
begin
this
cha
p
ter
with
the
discu
ssi
on of weig
hted mo
ving averag
e method.
2.1. Weighte
d
Mov
i
ng A
v
erage
Weig
hted mo
ving averag
e
(WMA) i
s
an
improveme
n
t
form of simple moving a
v
erage,
whi
c
h
gives a
greater weig
ht to mo
re
re
cent
data
th
a
n
the
olde
r
on
es [2
0]. The
weig
hting fa
ct
ors
are
cal
c
ulate
d
from the su
m of days u
s
ed in time
se
ries
data, also kno
w
n a
s
t
he ‘su
m
of digits’
[11]. The formula of WMA
can be d
e
scribed a
s
:
⋯
⋯
(
1
)
Whe
r
e
refers to the period
or spa
n
num
ber of fore
ca
sting formula a
nd
refers to the value of
time seri
es d
a
ta at point
[
20].
2.2. Holt’s Double Expon
ential Smoothing
Holt’s do
ubl
e exponenti
a
l smoothi
n
g
, also
kn
o
w
n a
s
Holt’
s
linea
r expone
ntial
smoothi
ng, i
s
a type of
dou
ble
expo
nenti
a
l sm
oothin
g
widely u
s
e
d
b
y
people.
Thi
s
te
chni
que
n
o
t
only smooth
the tre
nd
an
d the
slo
p
e
dire
ctly by
u
s
ing
different
sm
oothin
g
consta
nt, but
also
provide
s
mo
re flexibility in
sele
cting the
ra
tes at
which trend an
d sl
ope
s are tracked [21].
There are th
ree equ
ation
s
inco
rpo
r
ate
d
in this tech
niq
ue [22, 23]:
1
(
2
)
1
(
3
)
(
4
)
Whe
r
e:
refers
to the ac
tual value in time
refers
to the proc
e
s
s
s
m
oothing
cons
tant,
0
1
refers to the trend smoothi
ng co
nsta
nt,
0
1
refers to the smooth
ed co
nstant p
r
ocess value for pe
riod
refers to the smooth
ed tre
nd value for p
e
riod
refers
to the forecas
t
value for period
, where
0
is the cu
rre
nt time period
As su
gge
ste
d
by NIST [19], to set the initial val
ues fo
r
and
we will use the
followin
g
equ
ations:
(
5
)
(
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 2, June 20
16 : 772 – 77
7
774
2.3. H-WEMA: Holt’s Weighted
Expo
nential Mov
i
ng Av
erage
In
this re
sea
r
ch, we
u
s
e
weig
hted
m
o
ving
average’
s
weightin
g f
a
ctor calculat
ion an
d
combi
ne it
wi
th the Holt’s
doubl
e expo
n
ential
smooth
i
ng meth
od.
The p
r
op
ose
d
metho
d
will
be
calle
d Holt’
s
weig
hted exp
onential m
o
ving aver
age
(H-WEMA). T
he procedu
re
s of the p
r
op
ose
d
method can b
e
descri
bed a
s
followi
ng st
eps.
(1)
Cal
c
ul
ate
the ba
se val
ue,
, usi
ng e
quation
(1
) f
o
r a
given ti
me serie
s
da
ta and
perio
ds.
(2)
Usin
g the
base value
obtaine
d, cal
c
ulate the foreca
sting valu
e using fo
rm
ula (2) –
(4), wherea
s:
(
7
)
(
8
)
Will be used t
o
substitute the initial values for
and
as stated in th
e Equation (5
) and (6).
(3)
Return to step (1
) until
each data poi
nt in
the time seri
es d
a
ta gi
ven have end
ed.
In orde
r to know the a
ccura
cy and ro
bustn
ess lev
e
l of the pro
posed metho
d
again
s
t
other movin
g
average m
e
thods, we use two mo
st
comm
on crit
eria, i.e. mean sq
uare error
(MSE) and m
ean absol
u
te
percentage e
rror (MAPE).
2.4. Mean Square Error
Mean squa
re
erro
r (MSE) is the average
of the square of er
ror su
m between th
e
forecaste
d
da
ta and the re
al (actu
a
l) dat
a. As
describ
ed by Lawren
c
e et al [24], the formula
ca
n
be written a
s
follows:
∑
(
9
)
Whe
r
e
denot
es the n
u
mbe
r
of data an
d
denote
s
the
forecastin
g error from
. Here,
is the actu
al data and
is the fore
ca
sted
data.
2.5. Mean Ab
solute Per
c
e
n
tag
e
Error
Mean Ab
sol
u
te Percentag
e Error
(MAP
E) value
give
s u
s
a
n
indi
cation ab
out h
o
w m
u
ch
the average
of ab
solute
error of the
fore
ca
sted
data
compa
r
e to
th
e a
c
tual
data,
and
den
otes
by
the formula [2
4],
∑
100
(
1
0
)
Whe
r
e
denotes the numb
e
r of data and
denotes the fore
castin
g erro
r from
. The
actual d
a
ta is denoted by
, while
denote
the forecaste
d
data.
3. Results a
nd Discu
ssi
on
The expe
rim
ent to test the
accu
ra
cy and ro
bu
stne
ss level of the
prop
osed m
e
thod will
be d
one
by i
m
pleme
n
ting
the p
r
o
p
o
s
e
d
meth
od to
forecast
Jaka
rta Stock Excha
nge
(JKS
E)
c
o
mpos
ite index data.
The number of
dat
a been
used were 100
J
KSE
data tak
en
monthly f
r
om
April 200
7 to
July 201
5 fro
m
Yahoo! Fin
ance [25
]. Th
e length o
r
span data
as
well a
s
the in
itial
data ca
n be
chosen fre
e
ly by the use
r
. Mean
squ
a
re
erro
r an
d me
an ab
solute p
e
rcentag
e error
will be
u
s
ed
to cal
c
ul
ate
and
com
p
a
r
e the a
c
cu
ra
cy and
ro
bu
stne
ss level
of the p
r
opo
sed
method
agai
nst the
othe
r two
moving
averag
e m
e
thod, i.e.
wei
ghted m
o
vin
g
average
(WMA
)
and Holt’s do
uble expo
nen
tial smoothin
g
(H-DES).
The inte
rface
of the syste
m
is
sho
w
n
by Fi
gure 1.
User
can
ch
o
o
se
any valu
e for the
initial data to
start with and
span data a
s
describ
ed be
fore. As sho
w
n in Figure 1,
the initial data
been
u
s
ed
in
the first
exp
e
rime
nt is 29
and
the
spa
n
data
be
en
use
d
i
s
5,
which
me
an
s t
he
forecast
cal
c
ulation
will be started f
r
om the 30
th
d
a
ta pe
riod
co
nsid
erin
g the
last
5 data
take
n
su
cc
es
siv
e
ly
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
H-WEMA: A Ne
w Approa
ch of Doubl
e Ex
po
nential S
m
oothing Method (Se
ng Hansun)
775
Figure 1. Interface of the
system
The
gra
p
h
o
f
fore
ca
sted
data
which
had
b
een
cal
c
ulate
d
u
s
ing
weighte
d
movin
g
averag
e (WM
A
), Holt’s do
u
b
le expon
ential sm
oothi
ng
(H-DES), an
d
Holt’s weight
ed expon
enti
a
l
moving ave
r
a
ge (H-WEMA
) a
r
e
sho
w
n
on Fig
u
re
2,
Figure 3, a
n
d
Figu
re 4
con
s
e
c
utively. The
actual
data i
s
den
oted by t
he bl
ue lin
e a
nd the
fo
re
ca
sted
data i
s
d
enoted
by the
red
line
with
a
triangle m
a
rk on ea
ch fore
casted p
o
int.
Figure 2. Wei
ghted Moving
Average fore
ca
sting re
sult
s
Figure 3. Holt
’s Do
uble Expone
ntial Smoothing forecasting results
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 2, June 20
16 : 772 – 77
7
776
Figure 4. Holt
’s Wei
ghted
Exponential
Moving Avera
ge fore
ca
stin
g results
The exp
e
rim
ents the
n
b
e
contin
ued
b
y
using
different num
ber
of initial data
for ea
ch
moving average meth
od.
We will u
s
e
10 different
numbe
rs
of initial data and calculate
the
accuracy
an
d
ro
bu
stne
ss l
e
vel u
s
ing
m
ean
sq
ua
re
e
rro
r
(MSE) a
nd m
ean
ab
solute p
e
rce
n
tage
error (MAPE
) criteri
a
as
ca
n be se
en in
Table 1.
Table 1. MSE and MAPE comparison of each method
∑
of in
itial
data
MSE M
A
PE
WM
A
H-DES
H-WEM
A
WM
A
H-DES
H-WEM
A
9
92683.89
33536.94
32754.80
8.7223
4.5829
4.7246
14
95814.61
35005.28
32699.05
8.8300
4.8062
4.6059
19
87023.81
38260.21
29973.83
7.8731
4.9383
4.0256
24
83357.75
32254.71
31092.93
6.8425
3.7853
3.8525
29
72435.79
34562.30
30797.87
5.6890
3.6294
3.6000
34
74278.44
33061.38
32411.76
5.5308
3.2849
3.5962
39
75874.49
37851.70
32802.82
5.4092
3.5967
3.4391
44
69022.13
33952.80
31735.53
4.9599
3.0839
3.3439
49
71607.38
36439.93
31230.47
4.9678
3.2198
3.1808
54
71058.42
39563.08
29629.27
4.8121
3.4138
3.0225
A
v
e
r
age
79315.67
35448.83
31512.83
6.36367
3.83412
3.73911
Table 1 shows us the di
fferent MSE and M
APE values for each method and each
experiment. As
c
an be s
e
en
on
the
table,
the
average MSE and
MAPE values of the
propos
ed
method
are the
smalle
st a
m
ong th
e oth
e
r m
e
thod
s,
whi
c
h m
ean
s that H-WEM
A
gives
a b
e
tte
r
f
o
rec
a
st
in
g
r
e
sult
s
(bet
t
e
r
ac
cu
ra
cy
a
n
d
r
obu
stne
ss) rathe
r
tha
n
WMA
and
H-DES meth
od.
Therefore, th
e pro
p
o
s
ed
method
can b
e
use
d
a
s
a
b
e
tter fore
ca
sti
ng tool in tim
e
se
rie
s
anal
ysis
rathe
r
than th
e other two m
o
ving avera
g
e
method
s.
4. Conclusio
n
In this pape
r, we develop
a new ap
pro
a
ch of movin
g
averag
e method, whi
c
h
combi
n
e
s
the ba
sic fo
rmula of
weig
hted moving
averag
e (W
M
A
) to get a
b
a
se val
ue, a
nd u
s
e the
b
a
se
value to get the fore
ca
sted
value usin
g Holt
’s d
ouble
expone
ntial smoothing
(H-DES) form
ula
.
The experim
ental results on
100 Jakart
a
St
ock Exchange (JKSE)
co
mposite index data
sho
w
a
prom
ising
re
sult.
The a
c
curacy and robu
st
ness level
of
the propo
se
d metho
d
ex
cel
s
both the
wei
ghted movin
g
averag
e an
d
the Holt’
s
d
ouble
expone
ntial smo
o
thi
ng metho
d
s,
as
can
be con
c
l
uded from th
e small m
e
a
n
sq
uare error an
d mea
n
absolute pe
rce
n
tage
error
values.
For th
e futu
re
re
sea
r
ch,
we
ca
n try to
ta
ke
a m
o
re
co
mpre
hen
sive
study to
anal
yze the
advantag
es
and disadva
n
tage
s of
the prop
osed method com
pare to othe
r hybrid mov
i
ng
averag
e met
hod, such
a
s
the weight
ed expon
enti
a
l moving a
v
erage
(WEMA) metho
d
and
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
H-WEMA: A Ne
w Approa
ch of Doubl
e Ex
po
nential S
m
oothing Method (Se
ng Hansun)
777
autore
g
ressiv
e integrate
d
moving average (A
RIMA
) method. Another st
udy to combi
ne ot
her
moving ave
r
a
ge meth
od
s,
su
ch
as
Holt
-Winters t
r
iple
expone
ntial
smoothi
ng
ca
n al
so b
e
ta
ken
in the future.
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