ISSN : 1411-5735

Volume 10 Nomor 2

Jurnal ILMU DASAR Vol. 10 No.2 Hlm. 109-244Jember

Jul i 2009

ISSN

14',11-5735

Atas

Bawah

lssN 1411- 57:5

, Jurnal ILMU DASARVolume 10 Nomor 2 Jul i 2009

Pimpinan EditorI Made Tirta

SekretarisKartika Senjarini

Editor PelaksanaEva Tyas UtamiEdy Supriyanto

Dewan EditorMoh. Hasan

SujitoWuryanti Handayani

Sattya Arimurti

Editor TeknikKusbudiono

Adminisffi:'g,,ilfffansanNur Syamsiyah Harpanti

Jurnal llmu Dasar diterbitkan oleh :Fakultas Matematika dan llmu Pengetahuan Alam, Universitas Jember.

Terbit sejak Janqpri 2000 dengan frekuensi penerbitan dua kali setahun.Terakreditasi berdasarkan SK Dirjen Dikti NOMOR: 65a/DlKTl/Kep/2008

tanggal 15 Desember 2008

Alamat Editor/Penerbit:Jl. Kajimantan 37 Kampus Tegalboto, Jember 68'12'l

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http://www.unej.ac.id/fakultaslmipa/ atau http:l/r,vww.mipa.unei.ac.id

Keterangan sampul :

: Cormel explants with shoot intiation of three gladiolus cultivars; (a) Nabila,(b) Clara and (c) Kaifa in 18 days" after planting on MS free hormone media.

: Plantlet performance of gladiolus cultivars: (a) cv. Nabila on 2 mgil BAr(b) cv.Clara and (c) cv. Kaifa on 3 mg/l BA supplemented in MS + 0.5 mg/lNAA after 60 days culture.

UCAPAN TERIMA KASIH

Diucapkan terima kasih kepada MITRA BESTARI atas kontribusinya pada penerbitan JurnalILMU DASAR Volume 10 tahun 2009 :

1. Prof. Agus Subekti, M.Sc., Ph.D.

2. Prof. Prof. Dra. Susanti Linuwih, M.Stats., ph.D.

3. Prof. Dr.rer.nat. Karna Wijaya, M.Eng.

4. Prof. Dr. Wiwik TriWahyuni, 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.

10. Prof. Endang Semiart i , M.Sc., Ph.D.

11. Prof. Dr. Mammed Sagi, M.S.

12. Prof. Endang Soetarto, M.Sc., ph.D.

13. Pepen Arif in, Ph.D.

14. Prof. Dr. Bambang Sugiharto, M.Sc.Agr.15. Prof. Dr. I Nyoman Budiantara, M.Si.16. Drs. Siswoyo, M.Sc., Ph.D.

17. Drs. Achmad Sjaiful lah, M.Sc., ph.D

18. Drs. Zulf ikar, Ph.D.

19. Drs. Bambang Kuswandi, M.sc., ph.D.

20. TriAtmojo Kusmayadi, M.Sc., ph.D.

21. Dr. Sekartedjo

22. Dr. Hidayat Teguh Wiyono, M.pd.23. Drs. Budi Lestari, M.Si.

24. Dr. dr. Supangat, M.Kes.

25. Dra. HariSulistyowati, M.Sc.

26. Dr. Wiwik SriWahyuni, M.Kes.27. Dr. M. lsa lrawan, M.T.

__*_/

Volume 10 Nomor 2 Jul i 2009 lssN 1411-5735

Jurnal ILMU DASAR

DAFTAR ISI

ln Vitro Regeneration of Three Gladiolus Cultivars Using Cormel Explants by Kurniawan Budiarto(109-1 13).

Solusi Analitik Persamaan Schrddinger Sistem Osilator Harmonik 1 Dimensi dengan Massa BergantungPosisi menggunakan Metode Transformasi (Anatyticat Sotution ol Schrodinger Eq-uation oi tne UarmonicOscillator System with Position Dependent Mass Using Transformation Method")ofefr'suiisna (114-120).

Peningkatan Kualitas Minyak Jelantah Menggunakan Adsorben Hs-NZA dalam Reaktor sistem Fluid fixedbed [he lmprovement of_Waste Cooking Oil Quatity using H5-NZA Adsorbent in Ftuid Fixed Bed Reactor)oleh Donatus setyawan p. Handoko, Triyono, Narsito, tutlr owi (121-1g2).

Konsistensi dan Asimtotik Normalitas Model Multivariale Adaptive Regression Spline (Mars) Respon Biner(Consistency and Asymptotic Normatity of Maximum Liketihood Estimior in MniS ainiry iisponse Modeloleh Bambang Widjanarko Otok (133-140).

Analisis Reservoar Daerah Potensi Panasbumi Gunung Rajabasa Kalianda dengan Metode Tahanan Jenisdan Geotermometer (Geothermat Reservoir Anatysis- of ilount Rajabasa Kaianda pitency Area usingResistivity Method and Geothermometer) oleh Nandi Haerudin, Vina Jaya parOeOe,' SyamsurijaiRasimeng (141-146).

Aplikasi Sistem Informasi Geografi (SlG) untuk-Mempelajari-Keragaman Struktur Habitat Laba-laba padaLansekap Pertanian di Daerah Aliran Sungai (DI\q). cialJul (AppticZtion of Geographical tnfomation System(!-tp) to S_tudy Diyersity of Habitat Structure of Spider i; Ag'ri;itturat Landscape in Cianjur Watershed) otehI Wayan Suana dan Yaherwandi (147-152).

Modified Newton Kantorovich Methods for Solving Microwave Inverse Scattering problems by Agung TjahjoNugroho (153,159).

ldentifikasi Fraksi Aktif Bakterisida pada Rimpang Lempuyang (Zingiber ramineumBlume) (tdentification ofThe Bactericide Active Fraction on Zinqiber qiamineum'8iumd earxio'leh I Made oira Swairtara (160-170).

Embe.dding Grat Kz,s,n pada Torus (Embedding K23,, Graphs on Torus) oleh Liliek Susilowati, NencyRosyida Y, Yayuk Wahyuni (171-180).

Uji Sitotoksisitas Ekstrak Metanol Buah Buni (An.tidesma bunius (L) Spreng) terhadap Set Hela (CytotoxicityEffect of Methanolic Extract.of. Buni-'s Fruits (Anldcsmg bunius 1Ly'sjreng)'against Hiti ceirg oten EndahPuspitasari & Evi Umayah Utfa (181-185).

Laser Induced Thin Film Production (LITFP) Using Nitrogen (N2) Laser Deposition by Syamsir Dewang (186-189).

Constructing Fuzzy Time Series Model Using Combination of Table Lookup and Singutar valueDecomposition Methods gnd lts Application to Foiecasting Inflation Rate by Agus Maman Abadi, Subanar,Widodo,Samsubar Sateh (190-i9S).

Simulasi Model Jaringan Selular melalui Pemrograman- Inte-ger (Simutation of Ceilular Network Modet byI nteg e r Prog ram m i ng) oteh Agustina pradjanin gsih (1 99-206j.

rl

Jurnal ILMU DASAR

DAFTAR ISI

Efek Protektif Propolis Dalam Mencegah stres oksidatif Akibat Aktifitas Fisik Berat (swrm ming stress)(Propotis Protective Effect to Prevent-oxidarive^ stress cai;;J-oy' itrenous pnyiiiit \iiii,ty (swimmingSfress)) ofeh Hairrudin dan Dina Hetianti (207-211).

Pusat dari Beberapa Gelanggang Polinom Miring (rhe centre of some skew potynomial Rings) oleh AmirKamaf Amir (212-2181.

Efek Kondisi Hiperglikemik terhadap struktur ovarium dan siklus Estrus Mencit (Mus musculusL) (Effect ofHyperglikemic conditions on ovarian structure.and estpui)icte i'ui"r(Mus musculus L)) oteh Eva TyasUtami, Rizka Fitrianti, Mahriani, Susantin Fajariyah tZtg-iiil. - '

Generalized Reduced Gradient Untuk optimasi Amunisi Kaliber 57 mm c-60 Het (Generalized ReducedGradient optimization for Ammunition caiiber 57 mm c-oo ieo-olii Muhammad sjahid Akbar, BambangWidjanarko Otok, dan Lesti Anggraini (225-23s) ev vre"rv ̂ ^vr

Beberapa senyawa Hasil lsolasi dari Kulit Batang Tumbuhan Kedoya (Dysoxylum gaudichaudianum (A.Juss.) Miq.) (Metiaceae) .(p9v9r7t compoundi tsotated rroi-'dtem Bark of Kedoya erqvluroaudichaudianum (A. JussJ.Miq ) (Meliaceie)) oleh ruliran, i"yutr" Hamdani, Rosyid Mahyudi, sriHidayati Syarief dan Nurut Hidayati (296-244i.

190

I

i

:

Il

C onstructing F uzzy Time............. (A gus M aman Abadi dkk)

ConstructingFazzy Time Series Model Using Combination of Table Lookupand Singular Value Decomposition Methods and

Its Application to Forecasting Inflation Rate

Agus Maman Abadir), Subanar2), widodo2), Samsubar Saleh3)

1)Department of Mathematics Education, Faculty of Mathematics and Natural Sciences,Yo gyakarta Stat e Univ e rsity

t)Ph.D 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

ABSTRACT

Fuzry time series is a dynamic process with linguistic values as its observations. Modelling fuzzy time sedesdata develo@ by some researchers used discrete membership functions and table lookup method furm tainingdata. This paper presents a new mettrod to modelling fuzzy time series data combining table lookup and singularvalue decomposition methods using continuous membership functions. Table lookup method is used toconstuct fuzzy relations from fraining data. Singular value decomposition of fuing strength mahix and QRfactorization are used to rcduce fuzz] relations. Furthermore, this method is applied to forecast inflation rate inIndonesia based on six-factors one-order fuz4r time series. This result is compared with neural nefivork rnethodand the proposed method gets a higherforecasting accuracy rate than the neural network method.

Keywords: Fuzry time series, tablelookup, singularvaluedecompositioq inflation rate.

INTRODUCTION some researchers (Sah & Degtiarev 2004, Chen& Hsu 2004).

Fuzzy time series is a dynamic process with Forecasting inflation rate in Indonesia bylinguistic values as its observations. Many fuzzy model resulted more accuracy than thatresearchers have developed fizzy time series by regression method (Abadi et al. 2006).model. Song & Chissom (1993a) developed Following the above paper, Abadi et al. (2007)fuzzy time series model based on fuzzy also constructed flzzy time series model usingrelational equation using Mamdani's method. table lookup method to forecast interest rate ofIn their paper, determination of the fuzzy Bank Indonesia certificate and the result gaverelation needs large computation. Meanwhile, high accuracy. Abadi et al. (2OO8a, 2008b)Song & Chissom (1993b, 1994) also showed that forecasting inflation rate usingconstructed first order fuzzy time series for singular value method had a higher accuracytime invariant and time variant cases. This thanthatusingWang'smethod.model needs complex computation for fuzzy Abadi et al. (2008c) constructed arelational equation. Furthermore, to overcome generalization of table lookup method usingthe weakness of the model, Chen (1996) firing strength of rules and applied it indesigned fuzzy time series model by clustering financial problems. Fuzzy time series modelof fuzzy relations. based on a generalized Wang's method was

Hwang et al. (1998) constructed fuzzy time designed and it was applied to forecast interestseries model to forecast the enrollment in rate of Bank Indonesia certificate that gave aAlabama University. Fuzzy time series modei higher prediction accuracy than using Wang'sbased on heuristic model gives more accuracy method (Abadi et al. 2009a). Furthermore,than its model designed by some previous forecasting interest rate of Bank Indonesiaresearchers (Huamg 2001). Forecasting for certificate based on multivariate fuzzy timeenrollment in Alabama University based on series data was done by Abadi et al. (2O09b).high order fuzzy time series resulted high Kustono et aI. (2O06) applied neural networkprediction accuracy (Chen 2002). First order method to forecast interest rate of Bankfuzzy titme series model is also developed by Indonesia certificate.

Jurnal ILMU DASAR. Vol. I0 No. 2. Juli 2009 : j,90-198

In this paper, we will design fuzzy timeseries model combining table lookup methodand singular value decomposition usingcontinuous membership functions to improvethe prediction accuracy. This method is used toforecast inflation rate in Indonesia.

METHODS

QR factorization and singular valuedecompositionIn this section, we will introduce QR factoizatiotand singular value decomposition of matrix and itsproperties referred from Scheick (1997). l,et B be mx n matrix and suppose mSn. The QRfactorization of B is given by B =QR, where Q ism x m orthogonal matrix and R = [R,, R,r) is m x n

matrix with m x m uppet triangular matrix R,, . The

QR factorization of matrix B always exists and canbe computed by Gram-Schmidt orthogonalization.Any m x n matrix A can be expressed asA =USV, , ( l )

where U and V are orthogonal matrices ofdimensions m x tn, n x /r respectively, and S is n x nmatrix whose entries are 0 except s,; = o,

i =1,2, . . . , r wi th 6,2 o,2. . . > o. > 0,

r <min(m,n). Equation (l) is called a singularvalue decomposition (SVD.1 of A and the numbersq are called singular values of A. If U , V arecolumns of U and V respectively, then equation (l)

can be wri t ten 6 I =2oJJ V' .

I-et ff a ll', =7rt; be the Frobenius norm of A.

Since U and V are orthogonal matrices, then

l lu, l l=1 and l lv, l l=t. Hencei l , 12 ,

l le l l . = l l lo l lv ' l l =2o' . Let A =USV'bel l , r l lF

SVD of A. For given p < r , the optimal rank p

approximation ofA is given by A,, =io,IJ,V, ' .

Thusl t , , i l r

l lA -A" l l , =ll>.o,u v; -1o,u ,v;l l , .=l lz" ' , '1 l l= t o '

i l ' -P, l l

this means that Ao is the best rank p approximationof A and the approximation error depend only onthe summation of the square of the rest singularvalues.

Designing fuzzy time series model using tablelookup methodLet Y (t) cR, r = . . . , -1, 0, l ,2, . . . , be rhe universe

of discourse in which fuzzy sets "f, (l ) (i = 1, 2, 3,...)

are defined. If F(r) is the collection of/, (r), i = l,

2, 3,..., then F(t)is called fuzzy time series onI (t) . Based on this definition, fuzzy time seriesF(t) can be considered as a linguistic variable and

/, (r) as the possible linguistic values of f. (r) . The

value of F(r) can be different depending on time t.

Therefore F(r) is function of time r. Let{(r) be

fizzy tme series on f (r). If {(/) is caused by

(F,(t -r),F,(t -1)), (4 Q -Z),r"Q -z)),(F,(t -n),F,(t -n)), then a fuzzy logicalrelationship 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 logicalrelationship is called two-factors n-order fuzzy timeseries forecasting model, where {(r),{(r) arecalled the main factor and the secondary factor fuzzytime series respectively. If a fuzzy logicalrelationship 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 m-factors n-order fuzzy time series forecasting model,where {(t) is called the main factor fuzzy time

series and F,(t),...,F,,(t)are called the secondaryfactor fuzzy time series. The application ofmultivariate high order fuzzy time series can befound in l*e et al. (2006) and Jilani et aI. (2007).

Like in modeling traditional time series data,training data are used to set up the relationshipamong data values at different times. In fuzzy timeseries, the relationship is different from that intraditional time series. In fuzzy time series, weexploit the past exp€rience knowledge into themodel. The experience knowledge has form "IF ...THEN ...". This form is called fuzzy rules. So fuzzyrules is the heart of fuzzy time series model.Furthermore, main step to modeling fitzzy timeseries data is to identify the training data using fuzzyrules.

LetA,,(t - i) , . . . ,An,.,( l - i )be Ni fuzzy sets in

fuzzy t ime ser ies { ( t - i ) , i=0,1,2,3, . . . ,n,k=1,2, . .., m and the membership functions of the fuzzysets are continuous, normal and complete. The rule:

R' i IF (x , ( , - n) is A:. , ( t - n) md . . .

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 ...

and x ", (r - l) is A,:,, (r - t)), THEN.r, (r ) is A,1, (r )

(3)

L

r92

is equivalent to the fuzzy logical relationship (2) andvice versa. So (3) can be viewed as fuzzy relation inUxV where U =U,x. . .xU," , cR' ' ' , V cR

with A (rr(r -n),. . . ,x,(t - l) , . . . , .r ,(r - n),. . . ,x,,( / - l ))=

pA,,(x t(t - n))... lrA ,,Q '(t

- l))...|to,. .(x ,,(t - n)...pA _.,,,(t - ' t) ,

where

A = Air t ( 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 m-factors one-order fuzzy time series forecastingmodel. so{(r -1),{(t -r), . . . ,F,(r - l) -+ 4(t)can be viewed as fuzzy time series forecastingmodel with m inputs and one output. In this paper,we will design m-factors one-order time invariantfuzzy time series model using table lookup andsingular value decomposition methods. This methodcan be generalized to m-factors n-order fuzzy limeseries model. Table lookup method is used toconstruct fuzzy logical relationships and the singularvalue decomposition method is used to reduceunimportant fuzzy logical relationships.

Suppose we are given the following N trainingdata: (x,, , ( / - l ) , x,, , (r - l ) , . . . , x,, ( t - l ) ;x,, (r )),p =1,2,3,.. . ,N . Constructing fuzzy logicalrelationships from training data using the tablelookup method is presented as follows:Step 1. Define the universes of discourse for mainfactor and secondary factor. LetU =[a,, f ,]. n be universe of discourse for main

factor, xtpt - l) ,xto(t1ela,, Bl and V =

Ia, , f , lcR, i =2,3, . . . ,m, be universe ofdiscourse for secondary factors.xvG-1)e[q, ,F, | .

Step 2. Define fuzzy sets on the universes ofdiscourse. LetA,.oQ -i) , . . . ,A*,.n(r - i )be Ni fuzzy

sets in fuzzy time series { (t - I ) . The fuzzy sets are

continuous, normal and complete in [ar, p, ] c R, i=0,1, k =1,2,3, . . . ,m .Step 3. Set up fuzzy relationships using trainingdata. For each input-output pair(x , ,( t - l ) ,x ,o(t - l) , . . . ,x , ,( t - 1);,r , , ( / )) ,

determine the membership values of x,,(t -1) in

4,,.r(t -l) and membership values of x,,,(t) in

4,,. , ( /) . Then, for eachx*,,(r - i) , determine

A...0 (t - i) such that

Fo.,r , , r ( r o.r( t - r ) > po,,( , - , . , (x 0. , , ( t - t ) , i = I ,

2, ..., N*. Finally, for each input-output pair, obtain afuzzy logical relationship as(A

j . . , ( t - l ) , A j . . , ( t

- l ) , . . . , 4 . . , " ( r - l ) ) + A,; , , ( r ) .

If there are some fuzzy logical relationships having

the same antecedent part but different consequentpart, then the fuzzy logical relationships are calledconflicting fuzzy relations. So we must choose onefuzzy logical relationship of conflicting group thathas the maximum degree.For a fuzzy logical relationship generated by theinput-output pair(x,,, (t - l), x .,, (t - 1),..., x,,,, (t - l); x,,, (r )), we define

its degree as(p o ., , , _,,(*, , ( t - 1)) l t ̂ . . , , , _,,(x,, ,( t - l ) . . .

po,. .^r,r(x ,,,,(t

-71)p^ ..,,,,(x ,,(t)).

From this step we have the following M collectionsof fuzzy logical relationships designed from trainingdata:

R' : (A' . ( r - l ) ,A' . ( t - t ) , . . . ,A ' . ( / - l ) ) -+A' ( r ) ,t i , t i '

l=1,2,3, . . . ,M.t ; . "

' i . '(4)

Step 4. Determine the membership function for eachfuzzy logical relationship resulted in rhe Step 3. Ifeach fuzzy logical relationship is viewed as a fuzzyrelat ion in UxV with U =U,x. . .xU,, cR' ' ,

V c R , then the membership function for the fuzzylogical relationship (4) is defined by

/-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)) l t^,, . , , , ,(x , ,( t))

Step 5. For given fuzzy set input A'(t -l)in input

space U, establish the fuzzy set output A:(t)itoutput space V for each flzzy 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 f lzzy sets e,161,e',6ye',Q),. . . ,A: ()

defined as

1t ^,, , ,

(x, ( t )) = max (po-,, , (x, ( t ) , . . . , p ^., , ,(x,

( t )))

= max(sup( l^ Q( - l ) )p" (x(r - l ) :x,(r) ) )

= n,i,ax tsun(a. G Q - D)fr !^,,, u,,(x, (r - 1))4,,, (x, (r )))).

Step 7. Calculate the forecasting outputs. Based onthe Step 6, if fuzzy set input A'(t -l) is given, thenthe membership function of the forecasting outputA'(r) is

p^. , , , (x,( t ) ) =

max (sup(p. (x (r - l )) l- l / . , , , , _,,(x r e - | D1",, ( .r , ( / )))).

(s)Step 8. Defuzzify the output of the model. If the aimof 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 realoutput. For example, if ftzzy set input A'(r -l)isgiven 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 andcenter average defuzzifier isx,( t ) = f (x,( t - l ) , . . . ,x _(r - l ) ) =

Step 6. Apply QR-factorization to V-' and find M xM permutation matrix E such that f, E =eR ,where Q is t x k orthogonal matrix, R = [Rrr Rrz],R1 I is /< x /< upper triangular matrix.Step 7. Assign the position of entries one's in thefirst k columns of matrix E that indicate the positionof the k most important fuzzy logical relationihips.Step 8. Construct fuzzy time series forecaitingmodel (5) or (6) using the k mosr important fuzz!logical relationships.

RESULTS AND DISCUSSION

Figure 2 shows the procedure to design fuzzytime series model using combination of tablblookup and SVD methods. The method isapplied to forecast the inflation rate inIndonesia based on six-factors one-order fuzzvtime series model. The main factor is inflationrate and the secondary factors are the interestrate of Bank Indonesia certificate, interest rateof deposit, money supply, total of deposit andexchange rate. The data ofthe factors are takenfrom January 1999 to February 2003. The datafrom January 1999 to January 20O2 are used totraining and the data from February 2002 toFebruary 2003 are used to testins.

In this paper, we will preOi-ct the inflationrate of f'month using data of inflation rate,interest rate of Bank Indonesia certificate.interest rate of deposit, money supply, total ofdeposit and exchange rate of (k-l)'h month. Theuniverses of discourse of interest rate of BankIndonesia certificate, interest rate of deposit,exchange rate, total of deposit, money supply,inflation rate are defined as [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 ofdiscourse such that thefuzzy sets can cover theinput spaces. We define sixteen fuzzvsets B, , B,,..,, B,u , sixteen fuzzy setsC,,Cr,...,C,u, twenty five fuzzysets Dr , 4,..., 4s , twenty one fuzzysets E,,Er,...,Er,, twenty one fuzzysets 4 , 4,..., 4, , thirteen fuzzy sets

4,4,...,4ron the universes of discourse ofthe interest rate of Bank Indonesia certificate.interest rate of deposit, exchange rate, total ofdeposit, money supply, inflation raterespectively. All fuzzy sets are defined byGaussian membership function. Then, we set

r93

$.. ^-- , E(x,(r - l ) -x; ' ( r - l ) ) ' ,Z), exP(-.1-------------: . -)!=t t- ' a; +O;.,

Iexp( s (x, (r - l) --r , ' (r - l)) 'ai +oi,

(6)where y is center of the fuzzy set Ai., (r ) .The procedure to design fuzzy time series modelusing table lookup method is shown in Figure l.

Reducing unimportant fuzzy logical relationshipsusing SVD-QR factorization methodIf the number of training data is large, then thenumber of flzzy logical relationships may be largetoo. So increasing the number of fuzzy logicalrelationships will add the complexity ofcomputation. To overcome the complexity ofcomputation, we will apply singular valuedecomposition method to reduce the fuzzy logicalrelationships using the following steps.Step l. Set up the firing strength ofthe fuzzy logicalrelationship (4) for each training datum (x;y) =(x,(t - l) ,x,(t - l) , . . . , .r , , , ( t - t) ;x, (r)) asfol lows

Lt @;y) =

l [ l /t^, , ,,-,,{, , (t -t))p ̂ , (x ,(t ))t=t l= l

Step 2. Construct Nx M matrix

( L,0 L,( t ) L r , ( l ) )

L=l L,(2) L,(2) L L,(2)

|

lM M M Ml\ r , ( l r ) L,(N) L L,(N))

Step 3. Compute singular value decomposition of Las L=USV ' ,where IJandVare NxNandMxMonhogonal matrices respectively, S is N x M matrixwhoseentr ies sa =0, i * j ,s i i= 6 i =1,2, . . . , r

wi th q ) o,>. . .>6,>_0, r <min(N,M).

Step 4. Determine the number of fuzzy logicalrelationships that will be taken as k withk < rank(L\ .

step 5. Partitio n u u, , =('," Y''

I , *h.r. v,, i.V,, V,, )

t x t matrix, V,,is (M-k) x ft matrix, and construct

v' =(v, i v: , ) .

I

194

up fnzzy logical relationships based on trainingdata resulting 36 fuzzy relations in the form:

(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 logicalrelationship groups for inflation rate usingtable lookup method.

((r.(, - l), r,(, - l), {(, - l), r,(, - l), r"(r - l),x (r - l))

+r,( t )

C onstructing F uzzy Time............. ( A gus M aman Abadi dkk)

2

3

4

5

6

1

8

I

t0

t l

t2

t4

t5

t6

t'l

IEI9

20

2l

22

23

21

28

29

30

3l

32

33

l5

l6

(8r4.

(8r5,

(815,

(Bt4.

(Bto.

(B?,

(84,

(Bl.

(Bl .

(83,

(ts3,

(82,

(82,

(82,

(82,

(Bl .

(82,

(82,

(83. Cr, Dl3. E3. n, A8)

(Bl, c2, Dl0, E2, n. A6)

Dr2. 83, F8, 44)

There are thirty four nonzero singular values ofZ. The distribution of the singular values of Lcan be seen in Figure 3. The singular values ofL decrease strictly after the first twenty ninesingular values (Figure 3). Based on propertiesof SVD, the error of training data can bedecreased by taking more singular values of l,but this may cause increasing error of testingdata.

The error of training data depends on thesummation of the square of the rest singularvalues. So we must choose the appropriate ftsingular values. In this paper, we choose thefirst eight, twenty, and twenty nine singularvalues. To get permutation matrix E, we applythe QR factorization and then assign theposition of entries one's in the first ft columnsof matrix E that indicate the position of the ftmost important fvzzy logical relationship. Themean square error (MSE) of training andtesting data from the different number ofreduced fuzzy logical relationships are shownin Table 2.

The mean square error (MSE) of trainingand testing data from the different number ofreduced fuzzy logical relationships are shownin Table 2.

Table2. Comparison of MSE of training andtesting data using the differentmethods.

Method Number of

relations

MSE oftraining

data

MSE oftestingdata

Proposedmethod

8 0.485210 0.66290zo 0.3 12380 0.3017329 0. l9 1000 o.zlL62

Tablelookupmethod

36 0.063906 0.30645

Neuralnetwork

0.757744 0.42400

Based on Table 2, fuzzy time series model (5)or (6) designed by table lookup method givesthe smallest error of training data. If we use 29fuzzy logical relationships, then we have thesmallest error of testine data.

ct4, Dt3. Er2,

ct4, Dt2. Er3,

cr4, Dtz, Et3,

cr3, DlO. Er5,

cl I , D9, Et6,

c8, D4. EB,

c5. D5, Er3,

c3. D7, El l ,

c2. Dr r . ElO,

c2_ D5. U.

c2. D7. E9,

c2, D5, E6,

c2. D7. E7.

c2, D1, E7.

cf, D7, E1,

cl , D9. E1,

cl , Dr I , 88,

CI, DI2. B.

n, Ai l )

n, A8)

F3, A5)

E. A4)

P. 44)

P. A4)

P, A3)

Fr. A3)

F4, 45)

F4, 45)

F8, A8)

F5. A8)

F5, A5)

F5, A4)

F5, A6)

F6, A7)

-) es

-)p

Jrr

Jm

J*

-)ru

Jet

Ju

Jer

Jas

-t

--)

--)+

--)

--)

J

-)

- . .

(83, C2, Dr5. 86. F8, A1j

(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)

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)

Table I presents allfuzzy logical relationships.To know the ft most important fuzzy logicalrelationships, we apply the singular valuedecomposition method to matrix L of firingstrength of the fuzzy logical relationships inTable I for each training datum,

L,( t ) L

L,(2) L

MM

Input : x111-1; , x2(t_t , t t . . . t

Determine universe of discourse ofmain and secondary factors

Construct fuzzy sets Nr,..., N.

Construct fuzzy logical relations

Construct M fuzzy logical relations

Determine membership function ofeach fuzzy logical relation

Determine output fuzzy set Ar(t) ofeach fuzzy logical relation

Determine output fuzzy set A(t) of union ofA(t)

Jurnal ILMU DASAR, Vol. I0 No. 2, Juli 2009 : 190_I9g195

Figure l. The procedure of forecasting fuzzy time series data using table lookup method.

Figure 2.

C onstructin g F uzzy Time.. ........... (A gus M aman Abadi dkk)

The procedure of forecasting fuzzy time series data using combination of table lookupand SVD- QRfactorj^zation methods.

Input: fuzzy logical relationsgenerated from table lookup method

Determine firing strength of fuzzy logical relations

Construct firing strength matrix L

DNS t = USf

Identify singular values d, 2 or 2... ) q t 0

Choose the k largest singular values, with k ( r

Construct f'=(r,I V{,)

Determine QR factorization on V'

Construct permutation matrix E w'th Vr E = QR

Determine position of entry I on the first ft columns of E

Construct fuzzy model using the t most important fuzzy relations

Jumal ILIUIU DASAR, Vol. 10 No.2, Juli 2009 : 190-198 t97

Figure 3. Distribution of singular values of matrix L.

(a)

3rj i i :

--iIi-,-1.-.,{i rir'r- it

i l riY

ft* ft

(c)

Figure 4. Prediction and true values of inflation rate using proposed method: (a) eight fuzzylogical relationships, (b). twenty fuzzy logical relationships, (c) twenty nine fuzzylogical relationships, (d) thirty six fuzzy logical relationships.

So to forecasting inflation rate, fuzzy timeseries model (5) or (6) designed by 29 fuzzylogical relationships gives a better accuracythan that designed by 8 and 2O fuzzy logicalrelationships. Furthermore, Table 2 shows thatin predicting inflation rate, the proposedmethod results a better accuracy than the neuralnetwork method.

Figure 4(c) illustrates that predictionaccuracy is improved by choosing 29 fuzzylogical relationships. The plotting of predictionand true values of inflation rate using differentnumber of fuzzy logical relationships is shownin Figure 4.

CONCLUSION

In this paper, we have presented a new methodto design fuzzy time series model. The methodcombines the table lookup and SVD-QRfactorization methods. Based on the trainingdata, the table lookup method is used toconstruct fuzzy logical relationships and weapply the singular value decomposition andQR-factorization methods to the firing strengthmatrix of the fuzzy logical relationships toremove the less important fuzzy logicalrelationships. The proposed method is appliedto forecast the inflation rate. Furthermore.

(b)

I\ l

r i i ; i l i - ii f i . ! i f i i l i r i1, i i i iAi l ,* i i , Ji iI i l l l i , l t i i i ;Ji lr ! . i l i ' i i l i * i

i i $ ' i i { r .\ i : I

ii+

, t,"]

(d)

198

predicting inflation rate using the proposedmethod gives more accuracy than that using thetable lookup and neural network methods. Theprecision of forecasting depends also to takingfactors as input variables and the number ofdefined fuzzy sets. In the next work, we willdesign how to select the important inputvariables to improve prediction accuracy.

AcknowledgementsThe authors would like to thank to Dp2MDIKTI, Department of Mathematics EducationYogyakarta State University, Department ofMathematics Gadjah Mada University,Department of Economics and Business GadjahMada University. This work is the par-t of ourworks supported by DP2M-DIKTI Indonesiaunder Grant No: 0 I 8/SP2FVPP /DP2Ir41IJ12}OB.

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1.2-

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Stoecker DK & Gustafson DE. 2003. Cell-surface proteolytic activity of photosynthetic dinoflagellates. Aquatic Microbial Ecology

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