ISSN: 1411-5735
V o l u m e1 0 No mo r2
J u r n a lI LMUD A S A R V o l .1 0 No.2
Hlm. 109-244
Jember
Juli 2009
ISSN
14',11-5735
lssN 1411-57:5
,
Jurnal ILMUDASAR
Volume10 Nomor2 Juli2009
PimpinanEditor
I MadeTirta
Sekretaris
KartikaSenjarini
EditorPelaksana
EvaTyas Utami
EdySupriyanto
DewanEditor
Moh.Hasan
Sujito
WuryantiHandayani
SattyaArimurti
EditorTeknik
Kusbudiono
Adminisffi:'g,,ilfffansan
Nur SyamsiyahHarpanti
JurnalllmuDasarditerbitkan
oleh:
FakultasMatematika
dan llmuPengetahuan
Alam,Universitas
Jember.
TerbitsejakJanqpri2000denganfrekuensi
penerbitan
dua kalisetahun.
Terakreditasi
berdasarkan
SK DirjenDiktiNOMOR:65a/DlKTl/Kep/2008
tanggal15 Desember
2008
AlamatEditor/Penerbit:
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Keterangan
sampul:
Atas
: Cormelexplantswithshootintiationof threegladiolus
(a) Nabila,
cultivars;
(b) Claraand (c) Kaifain 18 days"afterplantingon MS freehormonemedia.
(a)cv. Nabilaon 2 mgilBAr
Bawah : Plantletperformance
of gladiolus
cultivars:
(b)cv.Claraand (c)cv. Kaifa on 3 mg/lBA supplemented
in MS + 0.5 mg/l
NAA after60 daysculture.
UCAPANTERIMAKASIH
Diucapkan
terimakasihkepadaMITRABESTARIatas kontribusinya
pada penerbitan
Jurnal
ILMUDASARVolume10tahun2009:
1.
2.
Prof.AgusSubekti,
M.Sc.,Ph.D.
ph.D.
Prof.Prof.Dra.SusantiLinuwih,
M.Stats.,
4.
Prof.Dr.rer.nat.
KarnaWijaya,M.Eng.
Prof.Dr.WiwikTriWahyuni,
M.Sc.
5.
Triyanta,
Ph.D.
6.
Prof.Dr.Tejasari,
M.Sc.
7.
Slamin.
Ph.D.
B.
Drs.Moh.Hasan,M.Sc.,Ph.D.
L
Prof.Kusno,DEA.,Ph.D.
3.
10. Prof.Endang
Semiarti,
M.Sc.,Ph.D.
11. Prof.Dr.Mammed
Sagi,M.S.
12. Prof.EndangSoetarto,
M.Sc.,ph.D.
13. PepenArifin,Ph.D.
14.
15.
Prof.Dr.Bambang
Sugiharto,
M.Sc.Agr.
Prof.Dr.I NyomanBudiantara,
M.Si.
16. Drs.Siswoyo,
M.Sc.,Ph.D.
17. Drs.AchmadSjaifullah,
M.Sc.,ph.D
18. Drs.Zulfikar,
Ph.D.
19. Drs.Bambang
Kuswandi,
M.sc.,ph.D.
20. TriAtmojo
Kusmayadi,
M.Sc.,ph.D.
21.
Dr.Sekartedjo
22.
Dr.HidayatTeguhWiyono,M.pd.
23.
Drs.BudiLestari,
M.Si.
24.
Dr.dr. Supangat,
M.Kes.
25.
Dra.HariSulistyowati,
M.Sc.
26.
Dr.WiwikSriWahyuni,
M.Kes.
27.
Dr.M. lsa lrawan,M.T.
Volume10 Nomor2 Juli 2009
lssN 1411-5735
Jurnal ILMU DASAR
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ln Vitro Regeneration
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(109-1
13).
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(AnatyticatSotutionol Schrodinger
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olehDonatussetyawanp. Handoko,Triyono,Narsito,tutlr owi (121-1g2).
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and AsymptoticNormatityof MaximumLiketihoodEstimior in MniS ainiry iisponse Model
olehBambangWidjanarko
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dan Geotermometer(GeothermatReservoir Anatysisof ilount RajabasaKaianda pitency Area using
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(!-tp) to S_tudy
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ModifiedNewtonKantorovich
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problemsby AgungTjahjo
InverseScattering
Nugroho(153,159).
ldentifikasi
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pada RimpangLempuyang
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Grat Kz,s,npada Torus (EmbeddingK23,, Graphson Torus)oleh Liliek Susilowati, Nency
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by SyamsirDewang(186189).
ConstructingFuzzy Time Series Model Using Combinationof Table Lookup and Singutarvalue
Decomposition
Methodsgnd lts Application
to Foiecasting
InflationRateby Agus MamanAbadi,Subanar,
Widodo,Samsubar
Sateh(190-i9S).
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of CeilularNetworkModetby
Integer Programming)otehAgustinapradjaningsih (199-206j.
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(Effectof
Hyperglikemic
conditionson ovarianstructure.andestpui)icte i'ui"r(Mus musculus
L))
oteh
'
Eva
Tyas
Utami,RizkaFitrianti, Mahriani,SusantinFajariyah
tZtg-iiil.
Generalized
ReducedGradientUntukoptimasiAmunisiKaliber57 mm c-60 Het (Generalized
Reduced
Gradientoptimizationfor Ammunitioncaiiber57 mm c-oo ieo-olii Muhammad
ev vre"rv
sjahid Akbar,
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^^vr
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.(p9v9r7t
erqvlur
oaudichaudianum
(A. JussJ.Miq
oleh ruliran, i"yutr" Hamdani,RosyidMahyudi,sri
) (Meliaceie))
HidayatiSyariefdan NurutHidayati(296-244i.
I
(AgusM amanAbadi dkk)
ConstructingF uzzyTime.............
190
ConstructingFazzy Time SeriesModel Using Combination of Table Lookup
and Singular Value DecompositionMethods and
Its Application to ForecastingInflation Rate
Agus Maman Abadir),Subanar2),
widodo2),SamsubarSaleh3)
1)Department
andNaturalSciences,
of Mathematics
Education,
Facultyof Mathematics
Yogyakarta Stat e Univ ersity
Student, Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Gadjah Mada University
2)Department
of Mathematics, Faculty of Mathematics and Natural Sciences,
Gadjah Mada University
3)Department
of Economics, Faculty of Economics and Business, Gadjah Mada University
t)Ph.D
ABSTRACT
Modellingfuzzy time sedes
Fuzry time seriesis a dynamicprocesswith linguisticvaluesasits observations.
datadevelo@ by someresearchers
useddiscretemembership
functionsandtablelookupmethodfurm taining
data.Thispaperpresents
a newmettrodto modellingfuzzytime seriesdatacombiningtablelookupandsingular
value decompositionmethodsusing continuousmembershipfunctions.Table lookup methodis used to
of fuing strengthmahix and QR
constuctfuzzy relationsfrom frainingdata.Singularvaluedecomposition
this methodis appliedto forecastinflationratein
factorizationareusedto rcducefuzz] relations.Furthermore,
Indonesiabasedon six-factors
one-orderfuz4r time series.Thisresultis comparedwith neuralnefivorkrnethod
methodgetsa higherforecasting
ratethantheneuralnetworkmethod.
andtheproposed
accuracy
inflationrate.
Keywords:Fuzrytimeseries,tablelookup,singularvaluedecompositioq
(Sah& Degtiarev2004, Chen
someresearchers
& Hsu 2004).
Fuzzy time series is a dynamic processwith
Forecastinginflation rate in Indonesia by
linguistic values as its observations.Many fuzzy model resultedmore accuracythan that
researchershave developedfizzy time series by regression method (Abadi et al. 2006).
model. Song & Chissom (1993a) developed Following the abovepaper,Abadi et al. (2007)
fuzzy time series model based on fuzzy also constructedflzzy time seriesmodel using
relational equation using Mamdani's method. table lookup methodto forecastinterestrate of
In their paper, determination of the fuzzy Bank Indonesiacertificate and the result gave
relation needslarge computation.Meanwhile, high accuracy. Abadi et al. (2OO8a,2008b)
Song & Chissom (1993b, 1994) also showed that forecastinginflation rate using
constructedfirst order fuzzy time series for
singular value method had a higher accuracy
time invariant and time variant cases.This thanthatusingWang'smethod.
model needs complex computation for fuzzy
Abadi et al. (2008c) constructed a
relationalequation. Furthermore,to overcome generalizationof table lookup method using
the weakness of the model, Chen (1996) firing strength of rules and applied it in
designedfuzzy time seriesmodel by clustering financial problems. Fuzzy time series model
of fuzzyrelations.
basedon a generalizedWang's method was
Hwang et al. (1998) constructedfuzzy time designedand it was appliedto forecastinterest
series model to forecast the enrollment in
rate of Bank Indonesiacertificate that gave a
Alabama University. Fuzzy time seriesmodei higher prediction accuracythan using Wang's
basedon heuristic model gives more accuracy method (Abadi et al. 2009a). Furthermore,
than its model designed by some previous forecasting interest rate of Bank Indonesia
researchers(Huamg 2001). Forecasting for certificate based on multivariate fuzzy time
enrollment in Alabama University based on seriesdata was done by Abadi et al. (2O09b).
high order fuzzy time series resulted high Kustono et aI. (2O06)applied neural network
prediction accuracy (Chen 2002). First order method to forecast interest rate of Bank
fuzzy titmeseriesmodel is also developedby
Indonesiacertificate.
INTRODUCTION
i
:
I
l
Jurnal ILMU DASAR.Vol. I0 No. 2. Juli 2009 : j,90-198
In this paper, we will design fuzzy time
series model combining table lookup method
and singular value decomposition using
continuous membershipfunctions to improve
the predictionaccuracy.This methodis usedto
forecastinflation rate in Indonesia.
METHODS
are defined.If F(r) is the collectionof/, (r), i = l,
2, 3,..., then F(t)is called fuzzy time serieson
I (t) . Based on this definition, fuzzy time series
F(t) can be consideredas a linguistic variable and
/, (r) as the possiblelinguistic valuesof f. (r) . The
value of F(r) can be different dependingon time t.
Therefore F(r) is function of time r. Let{(r) be
QR factorization and singular value
decomposition
In this section, we will introduce QR factoizatiot
and singular value decompositionof matrix and its
propertiesreferredfrom Scheick(1997). l,et B be m
x n matrix and suppose mSn.
The QR
factorizationof B is given by B =QR, where Q is
m x m orthogonalmatrix and R = [R,, R,r) is m x n
matrix with m x m uppet triangularmatrix R,, . The
QR factorizationof matrix B always exists and can
be computed by Gram-Schmidt orthogonalization.
Any m x n matrixA can be expressedas
A =USV, ,
( l)
where U and V are orthogonal matrices of
dimensionsm x tn, n x /r respectively,and S is n x n
matrix whose entries are 0 except s,; = o,
i =1 ,2,...,r
wit h
6, 2 o, 2. . . > o. > 0,
r <min(m,n). Equation (l) is called a singular
value decomposition(SVD.1of A and the numbers
q are called singular values of A. If U , V are
columns of U and V respectively,then equation(l)
fizzy tme serieson f (r). If {(/) is causedby
(F,(t-r),F,(t-1)), (4 Q -Z),r"Q -z)),
(F,(t -n),F,(t -n)),
then a fuzzy logical
relationship
is
presented
by
(F,(t -n),F,(t -n)),... ,(F,(t -2),F,(t -2)),
(F,(t -1), F,(t - l)) -+ 4 (t ) . The fuzzy logical
relationshipis called two-factorsn-order fuzzy time
series forecastingmodel, where {(r),{(r)
are
called the main factor and the secondaryfactor fuzzy
time series respectively. If a fuzzy logical
relationship is presented as
(F,(r- n), F,(t - n),...,F.(t - n)),...,
(F,(t -2),F,(t -2),...,F^(t -2)),
(F,(t -L),F,(t -r),...,F_(t -l)) + 4(r) ,
(2)
then the fuzzy logical relationship is called mfactors n-order fuzzy time seriesforecastingmodel,
where {(t) is called the main factor fuzzy time
seriesand F,(t),...,F,,(t)are called the secondary
factor fuzzy time series. The application of
multivariate high order fuzzy time series can be
found in l*e et al. (2006)and Jilani et aI. (2007).
canbe writte n6 I = 2oJ J V' .
Like in modeling traditional time series data,
training data are used to set up the relationship
I-et ffa ll', =7rt; be the Frobenius norm of A.
among data values at different times. In fuzzy time
series, the relationship is different from that in
Since U and V are orthogonal matrices, then
traditional time series. In fuzzy time series, we
and
Henceexploit the past exp€rience knowledge into the
llu,ll=1
llv,ll=t.
il ,
12
,
model. The experienceknowledgehas form "IF ...
' l l = 2 o ' . L e t A = U S V' b e
lle ll. = lllollv
THEN ...". This form is called fuzzy rules.So fuzzy
ll,r
llF
rules is the heart of fuzzy time series model.
SVD of A. For given p < r , the optimal rank p
Furthermore, main step to modeling fitzzy time
approximation
ofA is givenby A,, =io,IJ,V,' .
seriesdata is to identify the training datausing fuzzy
rules.
Thus
LetA,,(t -i),...,An,.,(l-i)be Ni fuzzy setsin
lt ,
,
ilr
v; -1o,u,v;ll,
llA-A"ll,=ll>.o,u
.
=llz"','1ll=
t o'
il'-P,
ll
this meansthat Ao is the best rank p approximation
of A and the approximationerror dependonly on
the summationof the square of the rest singular
values.
Designing fuzzy time series model using table
lookup method
Let Y (t) cR, r = ...,-1, 0, l,2, ...,be rheuniverse
of discoursein which fuzzy sets (l ) (i = 1, 2, 3,...)
"f,
L
f u z z yt i m es e r i e s{ ( t - i ) , i =0 , 1 , 2 , 3 , . . . , n , k =1 ,
2, . .., m and the membershipfunctionsof the fuzzy
setsare continuous,normal and complete.The rule:
R ' i IF ( x ,( , - n) i s A:.,( t - n) m d ...
and r_ (t - n ) is A,' _ (r - n)) and ...
and (x, (r - l) is Ai , ( - t) and ...
and.r,,(r - l) is Ai . (r - t)) and...
and(x,(r - l) is Af.,(r - l) and...
andx (r - l) is A,:,,(r - t)), THEN.r,(r) is A,1,(r)
",
(3)
r92
is equivalentto the fuzzy logical relationship(2) and
vice versa.So (3) can be viewed as fuzzy relation in
U x V wher e U = U, x . . . x U, ", c R' '',
V cR
with A (rr(r -n),...,x,(t-l),...,.r,(r- n),...,x,,(/
-l))=
pA,,(x t(t - n))...lrA
- l))...|to,. ,,(t - n)...pA
-'t) ,
.(x
,,Q '(t
_.,,,(t
where
A = Airt(t -r)x...xA-,.,
(r - l)x...xA-..,n (t - n)x...xA-,,,.. (r - l).
Let F,(t -I),F,(t -1),...,4,(r -l) -+ 4(r) be mfactors one-order fuzzy time series forecasting
model. so{(r -1),{(t -r),...,F,(r -l) -+ 4(t)
can be viewed as fuzzy time series forecasting
model with m inputs and one output. In this paper,
we will design m-factors one-order time invariant
fuzzy time series model using table lookup and
singular value decompositionmethods.This method
can be generalizedto m-factors n-order fuzzy lime
series model. Table lookup method is used to
constructfuzzy logical relationshipsand the singular
value decomposition method is used to reduce
unimportantfuzzy logical relationships.
Supposewe are given the following N training
data:(x,,,(/ - l), x,,,(r - l),...,x,, (t - l);x,, (r)),
p =1,2,3,...,N. Constructing fuzzy logical
relationships from training data using the table
lookup methodis presentedasfollows:
Step 1. Define the universesof discoursefor main
factor
and
secondary
factor.
Let
U =[a,, f ,]. n be universeof discoursefor main
factor, xtpt -l),xto(t1ela,, Bl
= 2, 3, . . . , m , be
Ia, , f , lc R, i
and
V
=
universe of
discourse
for
secondary
factors.
xv G - 1) e[ q, , F, | .
Step 2. Define fuzzy sets on the universes of
discourse.
LetA,.oQ-i),...,A*,.n(r-i )be Ni fuzzy
setsin fuzzy time series{ (t - I ) . The fuzzy setsare
continuous,normaland completein [ar, p, ] c R, i
=0 , 1,k = 1, 2, 3, . . . , m.
Step 3. Set up fuzzy relationshipsusing training
data.
For
each
input-output
pair
(x ,,(t -l),x ,o(t -l),...,x ,
,(t - 1);,r,,(/ )) ,
determinethe membershipvalues of x,,(t -1) in
the same antecedentpart but different consequent
part, then the fuzzy logical relationshipsare called
conflicting fuzzy relations.So we must chooseone
fuzzy logical relationshipof conflicting group that
hasthe maximum degree.
For a fuzzy logical relationship generatedby the
input-output
pair
(x,,,(t - l), x .,,(t - 1),...,
x,,,,(t - l); x,,,(r)), we define
its degreeas
(p o.,,,_,,(*,,(t - 1))lt ..,,,
- l)...
_,,(x,,,(t
^
po,.
-71)p^..,,,,(x,,(t)).
.^r,r(x,,,,(t
From this step we have the following M collections
of fuzzy logical relationshipsdesignedfrom training
data:
R ' : ( A '. ( r - l ) , A '. ( t - t ) , . . . , A '. ( / - l ) ) - +A ' ( r ) ,
ti,
ti'
t;."
l = 1,2,3,...,M.
'i.'
(4)
Step 4. Determinethe membershipfunction for each
fuzzy logical relationshipresultedin rhe Step 3. If
eachfuzzy logical relationshipis viewed as a fuzzy
r e l a t i o n i n U x V w i t h U =U , x . . . x U , , c R '',
V c R , then the membershipfunction for the fuzzy
logical relationship(4) is definedby
/-t (x,o(t - l), x,, (t - l),...,x,,"(r- l); x,, (r))
^,
=.p^
..,r.,(x,,(t -l))/t^,r,u,,("r,,,(t - l))...
po.
_,(t -l))lt^,,.,,,,(x
,,(t))
.^,,_,(*
Step 5. For given fuzzy set input A'(t -l)in
input
space U, establish the fuzzy set output A:(t)it
output spaceV for eachflzzy logical relationship(4)
defined
as
F (x, (t )) = sup(p (x (t - l))p (x ( - 1);x, (r )))),
^;
^.
^,
where x (t - l) = (x ,(t -l),...,x ,,,(r- 1)).
Step 6. Find out fuzzy set A'(t) as the combination
of
M
flzzy
sets e,161,e',6ye',Q),...,A:
()
defined
as
(x, (t ),...,p
(t )))
1t (x, (t )) = max(po-,,,
^,,,,
^.,,,(x,
= m a x ( s u p ( l ^Q ( - l ) ) p " ( x ( r - l ) : x , ( r ) ) )
= n,i,ax
(x,(r)))).
tsun(a.
GQ- D)fr!^,,,u,,(x,(r- 1))4,,,
4,,.r(t -l) and membershipvalues of x,,,(t) in
4,,.,(/).
Step 7. Calculatethe forecastingoutputs.Basedon
determine the Step6, if fuzzy setinput A'(t -l) is given,then
the membershipfunction of the forecastingoutput
that
A '( r ) i s
x , , ( -t t ) , i = I ,
> po, , ( , - , . , (0.
p ^ . , , , ( x , ( t )=)
Then, for eachx*,,(r-i),
A...0(t -i) such
Fo. , r , , r ( r o. r ( t- r )
2, ..., N*. Finally,for eachinput-outputpair, obtaina
fuzzy
logical
relationship
as
(A
j. .,( t
- l) , A j.
.,( t
- l) ,...,4 .
( r - l) ) + A ,;,,(r ) .
.,"
If there are some fuzzy logical relationships having
max(sup(p.(x (r - l))l-l /.,,,, _,,(x
r e - | D1",,(.r,(/ )))).
(s)
Step 8. Defuzzify the output of the model.If the aim
of output of the model is fuzzy set, then we stop at
Jurnal ILMU DASAR,Vol. I0 No.2, JuIi 2009 : 190-198
the Step 7. We use this step if we want the real
output.For example,if ftzzy set input A'(r -l)is
given with Gaussian membership function
.q (x, (r _ I) _x, (r _ l))' .
It^ ,,_,,(x(t - I)) = exp(-):--------------------r--1,
then the forecaiting ."d Ju,pu, urinjrn" Step 7 and
centeraveragedefuzzifieris
x,(t) = f ( x , ( t - l) , . . . , x _( r - l) ) =
- l) - x ; ' ( r - l ) ) ',
$. . exP(-.1-------------:
^- - , E( x , ( r
Z),
. -)
t-'
!=t
a; +O;.,
RESULTS AND DISCUSSION
s (x, (r - l) --r ,' (r - l))'
Iexp( I
ai +oi,
(6)
where y is centerof the fuzzy set Ai., (r ) .
The procedure to design fuzzy time series model
using tablelookup methodis shownin Figure l.
Reducing unimportant fuzzy logical relationships
using SVD-QR factorization method
If the number of training data is large, then the
number of flzzy logical relationshipsmay be large
too. So increasing the number of fuzzy logical
relationships will
add the complexity of
computation. To overcome the complexity of
computation, we will apply singular value
decompositionmethod to reduce the fuzzy logical
relationships
usingthe followingsteps.
Step l. Set up the firing strengthofthe fuzzy logical
relationship (4) for each training datum (x;y) =
(x,(t -l),x,(t -l),...,.r,,,(t -t);x, (r)) asfollows
Lt @;y)=
l[l /t^,, ,,-,,{,
, (t -t))p ^, (x,(t))
Step2. ConstructNx M matrix
L ,(t) L
r, (l )
( L ,0
)
L ,(2 ) L ,(2 ) L
L ,(2 )
L=l
|
t =t l =l
lM
M
M
Ml
L, ( N) L
L, ( N) )
\ r , ( lr )
Step 3. Computesingular value decompositionof L
a s L =U SV ' , wher e I J andVar e Nx Nand M x M
onhogonal matricesrespectively,S is N x M matrix
who se ent r ies
s a = 0, i * j, s ii= 6
i =1,2,...,r
with q ) o, > . . . > 6, > _0, r < m in( N, M ) .
Step 4. Determine the number of fuzzy logical
relationships that will be taken as k
with
k < rank(L\ .
step 5. Partition u u, , =(',"
Y''
I
V,, V,,)
, *h.r. v,, i.
t x t matrix, V,,is (M-k) x ft matrix, and construct
v' = (v ,i v :,).
Step6. Apply QR-factorization
to V-' andfind M x
M permutationmatrix E such that f, E =eR ,
where Q is t x k orthogonal
matrix,R = [RrrRrz],
R1I is /<x /<uppertriangular
matrix.
Step7. Assignthe positionof entriesone'sin the
first k columnsof matrixE thatindicatetheposition
of thek mostimportantfuzzylogicalrelationihips.
Step 8. Constructfuzzy time seriesforecaiting
model(5) or (6) usingthe k mosrimportant
fuzz!
logicalrelationships.
Figure 2 shows the procedureto design fuzzy
time seriesmodel using combinationof tablb
lookup and SVD methods. The method is
applied to forecast the inflation rate in
Indonesiabasedon six-factorsone-orderfuzzv
time seriesmodel. The main factor is inflation
rate and the secondaryfactors are the interest
rate of Bank Indonesiacertificate,interestrate
of deposit,money supply, total of depositand
exchangerate.The dataofthe factorsare taken
from January1999to February2003. The data
from January1999to January20O2areusedto
training and the data from February 2002 to
February2003areusedto testins.
In this paper, we will preOi-ctthe inflation
rate of f'month using data of inflation rate,
interest rate of Bank Indonesiacertificate.
interestrate of deposit,moneysupply,total of
depositandexchangerateof (k-l)'h month.The
universesof discourseof interestrate of Bank
Indonesia certificate, interest rate of deposit,
exchangerate, total of deposit,money supply,
inflation rate are definedas [10,40], tl0;401,
[6000, 12000], t360000, 4600001, 40000,
900001,[-2, 4] respectively.
We define fuzzy sets that are continuous.
complete and normal on the universe of
discoursesuchthat thefuzzy setscan cover the
input spaces. We define sixteen fuzzv
setsB,, B,,..,,B,u,
sixteen
fuzzy
sets
C,,Cr,...,C,u, twenty
five
fuzzy
setsDr, 4,..., 4s ,
twenty
one
fuzzy
setsE,,Er,...,Er,,
twenty
one
fuzzy
sets4 , 4,..., 4, ,
thirteen fuzzy sets
4,4,...,4ron the universesof discourseof
the interestrate of Bank Indonesiacertificate.
interestrate of deposit,exchangerate, total of
deposit, money supply, inflation rate
respectively. All fuzzy sets are defined by
Gaussianmembershipfunction. Then, we set
r93
194
(AgusM amanAbadidkk)
ConstructingF uzzyTime.............
up fnzzylogicalrelationships
basedon training
dataresulting36fuzzyrelationsin theform:
(B',r,Q
-D,C"(t -r),D",(t-r),E'r,Q-D,
F,l"(t-l),A',,(r- l)) -+ A'. (r)
Tabfe l. Six-factors one-order fuzzy logical
relationshipgroupsfor inflation rate using
tablelookup method.
((r.(, - l), r,(, - l), {(, - l), r,(, - l), r"(r - l),x (r - l))
+r ,( t)
(8r4.
ct4,
Dt3.
Er2,
n,
Ai l )
2
(8r5,
ct4,
Dt2.
Er3,
n,
A8)
-)
3
(815,
cr4,
Dtz,
Et3,
F3,
A5)
-)p
4
(Bt4.
cr3,
DlO.
Er5,
E.
A4)
5
(Bto.
cl I,
D9,
Et6,
P.
44)
6
(B?,
c8,
D4.
EB,
P.
A4)
1
(84,
c5.
D5,
Er3,
P,
A3)
8
(Bl.
c3.
D7,
El l,
Fr.
A3)
I
(Bl.
c2.
Dr r.
ElO,
(83,
c2_
D5.
U.
F4,
45)
J*
(ts3,
c2.
D7.
E9,
F4,
45)
-)ru
(82,
c2,
D5,
E6,
F8,
A8)
Jet
(82,
c2.
D7.
E7.
F5.
A8)
Ju
(82,
c2,
D1,
E7.
F5,
A5)
t5
(82,
cf,
D7,
E1,
F5,
A4)
t6
(Bl.
cl,
D9.
E1,
F5,
A6)
t'l
(82,
cl,
Dr I,
88,
F6,
A7)
IE
(82,
CI,
DI2.
B.
(83.
Cr,
Dl3.
E3.
n,
A8)
(Bl,
c2,
Dl0,
E2,
n.
A6)
Dr2.
83,
F8,
44)
F8,
A1j
t0
tl
t2
t4
Jrr
Jm
Jer
Jas
I9
20
es
-t
--)
2l
22
23
(83,
C2,
Dr5.
86.
(83,
C2.
Dt5.
E1,
m,
A8)
(81,
c2,
Dr5,
87,
Fl4.
AC)
(Bt,
c2,
Dr5,
E9,
B,
A6)
(B3.
C3,
Dt6,
Elr,
B,
A1'
(84.
Cl,
DtS,
EB.
B,
47)
--)
L,(t)
L
L,(2)
L
MM
Thereare thirty four nonzerosingularvaluesof
Z. The distributionof the singularvaluesof L
can be seenin Figure3. The singularvaluesof
L decreasestrictly after the first twenty nine
singularvalues(Figure3). Basedon properties
of SVD, the error of training data can be
decreasedby taking more singular valuesof l,
but this may causeincreasingerror of testing
data.
The error of training data dependson the
summation of the squareof the rest singular
values. So we must choosethe appropriateft
singularvalues.In this paper,we choosethe
first eight, twenty, and twenty nine singular
values.To get permutationmatrix E, we apply
the QR factorization and then assign the
position of entriesone's in the first ft columns
of matrix E that indicate the position of the ft
most important fvzzy logical relationship.The
mean square error (MSE) of training and
testing data from the different number of
reducedfuzzy logical relationshipsare shown
in Table2.
The mean squareerror (MSE) of training
and testing data from the different number of
reducedfuzzy logical relationshipsare shown
in Table2.
+
--)
--)
21
Table2. Comparisonof MSE of training and
testing data using the different
methods.
28
29
30
3l
32
33
l5
cl,
D24,
El5.
FtO.
A6)
(84,
Cl,
Dzt,
Et4.
Fro,
A7)
(84.
Cl,
D23,
EI4.
Ft r.
A8)
(B5,
C3,
Dt5,
Er0,
Fr2,
A9)
(85.
Cl,
Dl2,
Et0,
Fl3.
45)
(85,
C5.
Dr6.
Et2,
Fl3,
46)
(85,
C5.
Dl9.
El6.
Fr2,
A6)
(85,
C5,
Dr9,
Er7,
Fr4,
A8)
relations
8
Proposed
method
zo
0.312380
29
0.l9 1000
0.66290
0.30173
o.zlL62
Table
lookup
method
Neural
network
36
0.063906
0.30645
0.757744
0.42400
J
-)
-..
l6
Table I presentsallfuzzy logical relationships.
To know the ft most important fuzzy logical
relationships,we apply the singular value
decompositionmethod to matrix L of firing
strength of the fuzzy logical relationshipsin
Table I for eachtraining datum,
Numberof
MSE of
training
data
0.485210
Method
MSE of
testing
data
Basedon Table 2, fuzzy time seriesmodel(5)
or (6) designedby table lookup methodgives
the smallesterrorof trainingdata. If we use29
fuzzy logical relationships,then we have the
smallesterror of testinedata.
Jurnal ILMU DASAR,Vol. I0 No.2, Juli 2009: 190_I9g
Input : x111-1;,
x2(t_t,tt
... t
Determineuniverseof discourseof
main and secondaryfactors
Constructfuzzy setsNr,..., N.
Constructfuzzy logical relations
ConstructM fuzzy logical relations
Determinemembershipfunction of
eachfuzzy logical relation
Determine outputfuzzy setAr(t) of
eachfuzzy logical relation
Determine outputfuzzy setA(t) of union of
A(t)
Figure l. The procedureof forecastingfuzzy time
seriesdatausing tablelookup method.
195
g F uzzyTime.............(AgusM amanAbadi dkk)
Constructin
Input: fuzzy logical relations
generatedfrom table lookup method
Determinefiring strengthof fuzzy logical relations
Constructfiring strengthmatrix L
DNSt = USf
Identifysingularvaluesd, 2 or 2... ) q t 0
Choosethe k largestsingularvalues,with k ( r
Construct
f'=(r,I
V{,)
DetermineQR factorizationon V'
Constructpermutationmatrix E w'th Vr E = QR
Determineposition of entry I on the first ft columnsof E
Constructfuzzy model using the t most importantfuzzy relations
Figure 2. The procedureof forecastingfuzzy time seriesdatausing combinationof tablelookup
and SVD- QRfactorj^zation
methods.
Jumal ILIUIUDASAR,Vol. 10 No.2, Juli 2009 : 190-198
t97
Figure 3. Distribution of singularvaluesof matrix L.
(a)
(b)
r i i ;i l i -i
i f i .!i fi i l i ri 1
3r
jii :
--iIi-,-1.-.
,{i
r
ir'r- ft*
it
i lr
ft
iY
(c)
,iiiiAil,*i i,Ji i
l ii+
Ii llli,lti ii;Ji
r!.i
ili*i
li'i
I
\l
i\ii: $'
i i { r.
I
,t
,"]
(d)
Figure4. Prediction and true values of inflation rate using proposedmethod: (a) eight fuzzy
logical relationships,(b). twenty fuzzy logical relationships,(c) twenty nine fuzzy
logical relationships,(d) thirty six fuzzy logical relationships.
So to forecasting inflation rate, fuzzy time
seriesmodel (5) or (6) designedby 29 fuzzy
logical relationshipsgives a better accuracy
than that designed by 8 and 2O fuzzy logical
relationships.Furthermore,Table 2 showsthat
in predicting inflation rate, the proposed
methodresultsa betteraccuracythan the neural
networkmethod.
Figure 4(c) illustrates that prediction
accuracy is improved by choosing 29 fuzzy
logical relationships.The plotting of prediction
and true valuesof inflation rate using different
numberof fuzzy logical relationshipsis shown
in Figure4.
CONCLUSION
In this paper,we havepresenteda new method
to designfuzzy time seriesmodel. The method
combines the table lookup and SVD-QR
factorization methods. Based on the training
data, the table lookup method is used to
construct fuzzy logical relationshipsand we
apply the singular value decompositionand
QR-factorizationmethods to the firing strength
matrix of the fuzzy logical relationships to
remove the less important fuzzy logical
relationships.The proposedmethod is applied
to forecast the inflation rate. Furthermore.
198
ConstructingF uuy Time.............
(AgusM amanAbadi dkk)
predicting inflation rate using the proposed Abadi AM, Subanar,Widodo & Saleh S. 2009b.
PeramalanTingkat Suku Bunga Sertifikat Bank
methodgives more accuracythan that using the
IndonesiaBerdasarkanData Fuzzy Time Series
table lookup and neural network methods.The
Multivariat. Proceeding Seminar Nasional
precisionof forecastingdependsalso to taking
Matematika. FMIPA Universitas Jember. pp:
factors as input variables and the number of
462-4',74.
definedfuzzy sets.In the next work, we will
Chen SM. 1996.Forecasting
EnrollmentsBasedon
design how to select the important input
Fuzzy Time Series.Fasy Setsand Systems.81..
variablesto improvepredictionaccuracy.
3l l-3t9.
Chen SM. 2002.ForecastingEnrollmentsBasedon
Acknowledgements
High-order Fuzzy Time Series.Cyberneticsand
The authors would like to thank to Dp2M
Systems
Journal.33: l-16.
DIKTI, Departmentof MathematicsEducation Chen SM & Hsu CC. 2004. A New Method to
Forecasting Enrollments Using Fuzzy Time
Yogyakarta State University, Department of
Seies. InternationalJournal of Applied Sciences
Mathematics Gadjah Mada University,
and Eng ineer ing. 2(3): 234-244.
Departmentof Economicsand BusinessGadjah
Huamg K. 2001. HeuristicModels of Fuzzy Time
Mada University. This work is the par-tof our
Seriesfor Forecasting.Fuzzy Setsand Systems.
works supportedby DP2M-DIKTI Indonesia
123 369-386.
underGrantNo: 0 I 8/SP2FVPP
/DP2Ir41IJ12}OB. Hwang JR, Chen SM & Lee CH. 1998.Handling
REFERENCES
Abadi AM, Subanar,
Widodo& SalehS. 2006.
Fuzzy Model for EstimatingInflation Rate.
Procceedingsof The International Conference
on Mathematicsand Natural Sciences.Institut
TeknologiBandung:7 36-739.
Abadi AM, Subanar,Widodo & Saleh S. 2007.
Forecasting Interest Rate of Bank Indonesia
Certificate Based on Univariate Fuzzt Time
Series.InternationalConferenceon Mathematics
and Its applicationsSEAMS. Gadjah Mada
University.
Abadi AM, Subanar,Widodo & Saleh S. 2008a.
ConstructingCompleteFuzzy Rules of Fuzzy
Model Using Singular Value Decomposition.
Proceedingsof The International Conferenceon
Mathematics, Statistics and Applications
(ICMSA). SyiahKualaUniversity.1: 6l-66.
Abadi AM, Subanar,Widodo & Saleh S. 2008b.
Designing Fuzzy Time Series Model and lts
Application to Forecasting Inflation Rate. 7rh
World Congressin Probability and Statistics.
NationalUniversityof Singapore.
Abadi AM, Subanar,Widodo & SalehS. 2008c.A
New Method for GeneratingFuzzy Rule from
Training Data and Its Application in Finacial
Problems. The Proceedings of The 3'd
International Conferenceon Mathematics and
Statistics(ICoMS-3). Institut PertanianBogor:
655-66l.
Abadi AM, Subanar,Widodo & Saleh S. 2009a.
Designing Fuzzy Time Series Model Using
Generalized
Wang'sMethodand Its Application
to ForecastingInterest Rate of Bank Indonesia
Certificate. Proceedings of The International
Seminaron Scienceand Technology.Islamic
Universityoflndonesia.:I l-16.
ForecastingProblemsUsing Fuzzy Time Series.
FuzzySetsarul Systems.L00 217-228.
Jilani TA, Burney SMA & Ardil C. 2007.
Multivariate High Order Fuzzy Time Series
Forecasting for
Car Road Accidents.
International Journal
of
Computational
Intell igence. 4(l): | 5-20.
Kustono,Supriyadi& SukisnoT.2006. Peramalan
Suku Bunga Sertifikat Bank Indonesia dengan
MenggunakanJaringan Syaraf Tiruan. [Laporan
penelitian dosen muda, Universitas Negeri
Yogyakarta,Yogyakartal.
Lee LW, Wang LH, Chen SM & Leu YH. 2006.
Handling ForecastingProblemsBased on Zwofactors High Order Fu72y Time Series. IEEE
Transactions on Fuzzy Systems.l4(3): 468 477.
Sah M & Degtiarev KY. 2004. Forecasting
Enrollments Model Based on First-order Fuzzy
Time Series. Transaction on Engineering,
Computing and Technology VI. Enformatika.
Yl:375-378.
Scheick JT. 1997. Linear algebra with applications.
Singapore: McGraw-Hill.
Song Q & Chissom BS. 1993a. Forecasting
Enrollments with Fuzzy Time Series, Part I.
FuzzySetsand Systems.54: 1-9.
Song Q & ChissomBS. 1993b.Fuzzy Time Series
and Its Models. FuzzySetsand Systems.54.269277.
Song O & Chissom BS. 1994. Forecasting
Enrollments with Fuzzy Time Series, Part II.
Fuzzy Setsand Svstems.62; l-8.
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References:
Thesis,
Heather AF. 2000. The Magnetic and Electrical Properties of Permalloy-Carbon Thin Film Multilayer.[unpublished
Collageof Williamand Mary,Virginial.
Changing
Heatherif . zOOZ.The Magneiicanl Etectricaleroperties.-. Virginia.HoppeHG, GiesenhagenHC & Gocke K. 1998.
patternsof BacterialS;bstrate Decompositionin a Eutrophication
Gradient.AquaticMicrobialEcology15:1-13.
Hoppe HG, GiesenhagenHC & Gocke x. tgge. Changing Patternsof BacterialSubstrateDecompositionin a Eutrophication
Gradient. Aquatic MicrobialEcology. 15:1- 13.
Lindsey J.K. On h-tiketihood,random6ff ectand penatised likelihood.http://luc.be/-jlindsey/hglm.ps[25 Oktober 2007]
AquaticMicrobialEcology
StoeckerDK & GustafsonDE. 2003. Cell-surfaceproteolyticactivityof photosyntheticdinoflagellates.
30:175-183.