Journal of Quantitative Methods, Vol. 5, No. 2, December 2009

voLUME 5, I\UMBER 2

DECEMBER 2OO9

JOURNAL OF

rssN 1693-5098

|I UA]IIITAIIUE

M

ETH

|| II S

QUANTITATIVB METHODS

QUANTITATIVE METHODS provide a forum for communication between statisticians and

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with the users of applied of statistical

and mathematical

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Editor in Chief:

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,,,,?1?ffffii;"j#ffiffi

'1ii#lJl'.1'ii:J$?,HrJifr

'.*"

Editorial Assistant:

Dr. Jamal I Daoud

Department

of Sciences,

Faculty of Engineering

International

Islamic Universitv Malavsia

Editors:

Dr.ZhenbngZeng

Dr.FaizA.M. Elfaki

Shanghai

Institute for Biological Sciences

Departrnent

of Sciences

Chinese

Academy

of Science

Intemational

Islamic University,

Malaysia

Department

of Mathematics

Faculty of Planning

and Management

University Malaysia Trengganu,

Malaysia

Al Balqa, Applied University, Al-Salt, Jordan

Chen-ju Lin, Ph.D

Dr. Hon Wai Leong

Dept. of Industrial

Engineering&Management

Department

of Mathematics

YuanZe University, Taiwan

National University of Singapore,

Singapore

Dr.Raihan

Othman

Department

of Science

International

Islamic University,

Malaysia

Dr. Ismail Bin Mohd

Dr.Pieter

N.Juho

Department

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Natal University, South Africa

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Brunei Darussalam

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Department

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Dr. Andreas

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Dr.Mustofa

Usman

Vol. 5, No.2 December

2009

QUANTITATIVE METHODS

ISSN 1693-5098

TABLE OF CONTENTS

Data Envelopment

Analysis For Stocks

Selection

On Bursa Malaysia...

Ong Poay Ling, Anton Abdulbasah

Karnil, Mohd Nain Hj. Awang

On Group Method Of Data Handling (GIDID ForData Mining...

Iing Lukman, Noor Akma lbrahim, Indah Lia Puspita

and Eka Sariningsih

Stem Biomass Estimation of Roystonea

Regia Using Multiple Regression

Noraini Abdullah, ZainodinHj. Jubok, Amran Ahmed.

Continued Fraction and Diophantine

Equation...

Yusuf Hartono

Risk Analysis of Traffic And Revenue

Forecasts

For Toll Road Investment

In Indonesia

...

I Rudy Hermawan

K., Ade Sjafruddin,

dan Weka Indra Dharmawan

Use Of Ranks For Testing Fixed Treatment Effects In Basic Latin Square Design...

Sigit Nugroho

Analysis of Good corporate Governance

structure Effects on Bond Rating...

Einde Evana,

Agrianti KSA, and R. Weddie Andryanto

A New Method for Generating

Fuzry Rules from Training Data and Its Application To

Forecasting

Inflation Rate and Interest Rate of Bank fndonesia

Certificate.

Agus Maman

Abadi, Subanar,

Widodoand

Samsubar

Saleh

Mean-var Portfolio Optimization Under Capm with Lagged, Non Constant Volatilty

and

rhe Long

Memory

Erfect

....i;k;;;ffi;;;;.ilffi;ilr;dt

26

39

49

54

6 1

69

78

84

JOURNAL OF QUANTITATIVE METHODS

Vol. 5, No. 2, Dec€mber 2009

rssN 1693-5098

decq there

A IYEW METHOD FOR GEI\TERATING WZZ,V RTJLES FROM TRAINING DATA

AND ITS APPLICA.TION TO FORECASTING II\TFLATION RATE AIYI} INTEREST

RATE OF BAhiK INilX}IYESIA CERTIFICATE

Agus Maman Abadir, subapaf, widodq?and slmsubar sarehr

'Departnent of Mathematics _{Fsculty of Mathemstics and Nahnal Sciences,} Yogyakata State LJniversity, Indonesia

2Departrnentof

ffiffi13:r:ffiffi ffiffi *t*rii"o'ulsciences.

Gedjah Mads Univeaslty, Indonesia

3Department

of

,.*"ff*rYffi

tdl#ffi

":Hrffitr;*j;$ff ifrfi u*"^ity,

rndonesia

JL Humaniora" Bulaksumur _{yoryakarh 552g1, Iodonesia}

Email: rmamanabadi@.vmail.corn- 2subanar@vahoo.conr. 2widodo-math@vahoo,conr. shumas(opaue.uern.ac,id

(Received l0 April 2@9, accepted 20 Octobsr 2009, published Decernber 20@)

Abstract' Table tookup scheme is a simple method to construct fuzzy rules at fuzzy model. That can be used to overcome the conflicting

ryle by determining each rule degree. The weakness of'furzy model based on table lookup scheme is that the fuzzy rules may not be complete so the fuzzy rules can not cover all values in the domain' In this paper a new method to generate fuzzy rules from traininj oata wiil k proposed. In this method, all complete fuzzy rules are identified by firing strength of each possibie fuzzy rule."rhen, the resulged furzy rules are.used to dosignfuzzy mo&l- Applicati,ons oittr" p.opor"o method to predict the Indonesian inflation rale and interest rate of Bank Indonesia certificate {BIC) wilf be discussed. Thb predictioas of the Indonesian inflation rate and interest rate of BIC using the proposed method have a higher accuracy than those using the table lookup scheme.

Keywords: fi'uzy rule, fuzzy model, firing strength of rule, inflation rate, interest rate of BIC.

l. Introduetion

Designing fuz4 rule base is one _{,of some steps in fuzzy modelin g. Fuzzy rule base is the heart of ruzzy model.} Recently, fuzzy model was developed by some researchers. _{waig, L.x. (1997) created fuzzy model based on} table lookup scheme, gradient descent haining, recursive least squires and clustering methods. The weakness of the fuzzy model based on table tookup scheme is that the fuz4 rulebase may not b€ complete so the fuzzy rule can not cover all values in the domain' wu, T-P, and Cben, -S.M. _{(ts9S) &sigred mem'terstrip functions and} fuzzy rules from training data using a -cut of fuzzy sets. In wu and Chen's .Ltto4 determining membership functions and fvzzy rules. needs large cromputations. To reduce the complexity of computation, iu*, y., et at (1999) built a method to decrease _{fuzzy rules using singular value decomposition.}

fen, J', et al (1998) developed fuzzy model by combining globat and local learning to improve the interprelability. New form of fuzry inference systems that was h-igily interpretable based on-a ludlcious choice of membership function was presented by Bikdash, M. (1999). C".."., LJ., et al (2005) constructed Takagi-Sugeno-Kang _{models having high interpretability using Taylor series approximation. Thin, pomares, H., et al} (2002) identified structure of fuzzy systems based on

"o*pi*" rules thai can decide which inpui uariattes .-nust be taken into account in the fuzzy system and how many membenhip functions are needed in every selected input variable- Abadi' A.M., et al (200sa) designed complete fuzzy rutes using singular value decomposition. In this method, the prediction enor of training data depended only on taking the n-umdr of singular naiues. Fuzzy models have been applled to many fields such as in communications, economics, engineering medicine, etc. Sp€cially in economics, Abadi, A.M., et al (2006) showed that forecasting the lndoneJian inflat*ion *t" by: fuzzy model resultsd more accuracy than that by regression mcthod. Then, Abii, A.M., et al (2007) constucted fuzzy time series model using tablc lookup scheme to forecast the interest rate of Bank Indonesia certificate (BIC) and its result gave high accuraoy. Then, Abadi, A.M., et al (2008b) showed that forecasting Indonesian inflation rate based on fuzzy timc series data using combination or talte lookup scheme and singular value

accutt 90t!!d Indoo Wang The n fiEr rule h intfi! rate N

2.w|

In thir Suppo r , , €[ by thc Step L For er Simih Stcp 2. For a sets 4 variaUr determi

!*0,

Stcp 3. From sl possiHr but diff degree pair(4,

Step 4. { The nrk with anl from hu Stcp 5. ( We can fivzy m defined I

much les

that the I fuzzf'rd

3. Ner

Given d

Jownal of Quan|itdive Metho&, Va5 No'2, December 2009, 7833 79

decompcition methods had a higher accufttcy than that using Wang's method- Based on the previous research' there are irreresting topics in frzzy model especially in determining fuuy rules that give good prediction accuracy. To oveircome the weakness of tabll bokup scheme (Wang's method), in tbis paper, we design

;*ui;

n rw *rer tI {wq *oa"l *ing-ruing strength

of rule Ta.thu! its rcsult is uspd to predict thg

Indonesian inflation rate andinterest rat€ ;f BIC. The pmposed method has higher prediction accuracy than W*g', method in its applications to predicting the lndonesian inflation rate and interest rate of BIC'

The rest of this pper is organized as follorrs. In section 2, we briefly review the Wang's me{hod to coostruct n

'"y *oa.f. In'xfoon 3, we pr65ent a new method to constmct complete fuzzy rules

using.filng strenglh of

rule bssed ur training OaL fn section 4, we apply tlre proposed_'n-.9d to forecasting the inflation rate and inter€st rate of BIC. We also compare the propoied mettroO wittr the Wang's method in the forecasting inflation rate and intp.rest 1-3tp of BIC, FiqAlly, Some conclusions q1p dispussod in section 5'

2. Wang's method fordesigning fuzzy rules

In this sestior," we will introduce the wang's method to consEuct fuzzy rules referred from wang, L.X. (1997). Suppose tlrat we are given the following l{ input<nrtput dulai (x,o,x2r,--,x,ri!o), p=loZ'3,"',il where

x,, efa,, F,lc R arrd % e1ur, FrlcR , i: 1, 2, "', n'Designing fuzzy model using Wang's method is given by the folloudng stePs:

Siep l, Oefins &zzJ sets to gorer the input and output domains'

F o r e a c h s p a c e [ a , , g , l , i : 1 , 2 , . . . , n , d e f i n e N , f i t z r z y s e t s l , i , i : l r 2 , . . . , N i w h i c h a r e c o m p l e t e i n f a , , p , l ' Similarly, define N, fuzy ss{s Br ,i = l,2o .-.,Nv which are complek in lar, f ,7'

Stcp 2. Generate one rule from one input-output pair'

For each input-ou9ut pair(r,r,rrr,. ..,x,.;!r). determine the membership value of x,r, I : 1,2' "', n in fiz4

sets 4 , j = l,2, ..., Ntand membership value of yrin fuzzy sets Bi ,i = 1,2' "', N"' Then, for each input variabler,r, i=l,2,...,n,determinethe fuzzysetinwhichr,, hasthe largcsmembershipvalue. Indherwor4 determine t such that tto(t,)2lt,(",), 'i =lo?,-.N,. Similarly, ddermine Bn such that

pd,(Jr)>lto(t),1=1,2,"-Nr'Finally,wcconstructafrvzyIF-THENrule:

I F 4 i s 4andx,is.l' and'.-andr,isli,THEN yisftr ( l ) Stcp 3. Compute degree of each rule dqsigred in step 2'

frol step 2,'one rui-e is gener*ed by onJinput-ouput pair. If the number of input-output-data is large, then it is possible itrat there ur" tti" conflicting rules. Two rules be.come conflicting rules if the rules have same IF parts but different THEN parts. To resolie this pnoblem, we assigt a degree tro each rule designed in step 2' The degree of rule is 'defined as follows:- s-uppase the rules (l) is consuucted by the input-output pair(x,o,xr",. .-x,rilr)

'then its degrre is defined as

D(rute)

=

l] tt o G, o)

a n (t,)

Stcp 4, Conshuct the fuzzy rule base.

The rule base consists otitre following thre€ sets of rules: (1) The rules designed in Step 2 that do not conflict with any other rules; tii rrt"

-r"

froir a conflicting group that has the maximum d"se"; (3) Linguistic rules from human exPerts.

Step 5. Consnuct the fuzzy model using thefiuzy rule base'

We can use any nofi"ri t*ty inferelnce engine and defuzzifiet combined with the fuzzy rule base to desigrr fuzzy model. ti ttt" nurnU"t of raining data I N and the number of all possible combinations of the fuzzy sets defined for the input variables is l-t N, , then the number of fuzzy rules generated by Wang's method may be

much less than both

^/ and fl{

. Then, the fuzzy rule base generated by this method may not be complete so

that the fuzzy rules can not cover all values in the input spaces. To overcome this weakness, we will design the fu2ry des covering all values in input sipaces'

3. New method for constructing

fury rules

Jownal of euantitative Metho*, vo.s No.2, Deceuber 2009, zg{.3 g0 Step 1. Dsfine fuzzy sets to cover the input and output space&

For each space [4,, f ,1, i = 1,2, ..., r, defrne N, fiEq *ts A! , j = 1,2,..., /{ which are complete and normal infa,,f ,1. Similarly,define t{, fuzzysets Br,j=1,2,...,N, whicharecompleteandnormal infa,,frl. Step 2. Determine al! po. ssible antecedents of fuqy rulg candidates.

Based on the Step I' there ut" nlf, antecedents of fuz4 rule candidates. The antecedent has form "4 is 4n and x. is 4 nd... and x" is {' " simplified by * 4 and A! nd ...nd 4 ". For example, if we have two inpul and wp deline two fuzzy scts A, A. for first input spoee 4nd f,,, C, for second inpql spaee" then all possibleantecedcnts _{of fitzy rule candidates}

ue A,andC,;.1 atdCr 4*dC,; \wrdCr. Step 3. Determine consequence of each fuzzy rule candidate.

For each antecedent 4 nd At and ... and A!, the consequence of fuzzy rule is determined by firing strength of the rule uo&,)u*Qr)...P+Q,,)ttn(l)based on the training data- Choosing the consequence is done as follows: _{For any Eaining data(xrr,xr,,...,x"ri-)rr)and for any tuzzy setgr, choosegr such that}

It*(x'e)lt*(x,r)...F4(x,".)tr",(ln)2 _{uo(\,)tro(x,,)...!t+(x^")ltn(l),forsome (4o.,rr,,,...,x,,.;yo.).}

If there are at least tu'oBr such that uo(xrn\,.pe(x,,.)tturo) > p4!,(q)...rt*(x",)rtu,(%), then choose one of 8'' . From this step, we have the fuzzy rule:

iF -r, is A( and x, is A{ and... and x. is lj. , THEN y is B" so if we continue this step for every antecedent, we get n N, complete fuzzy rules. Step 4. Construct fuzzy rule base.

The fu:zy rule base is constructed by the fl l', fuzzy rules designed by Step 3. S*p 5. Design fuzzy model using fuzzy rule hse.

Fulv model is designed by combining the fuzzy rule base and any fuzzifier, fuzzy inference engine and defuzzifier- For example, if we use singleton fuzzifier, produc,t inference engine and ont", uu"nag" defuzzifier, then the fuzzy modcl has form:

M ,

Lb,llp.,(x,)

f (x,xr,...,x,)=

'n

"j='

-Lp,ttr@,)

rvhere M is the number of rules.

lrom Step 3, the sel of fuzzy rules constructed by this me&od contains fuzzy rules designed by the Wang's method.'Iherefore the proposed method is the generalization _{of the wang's method.}

4. Applications

of the proposed

method

In this section' we apply the proposed method to formast the Indorresian inflation rate and inter€st rate of BIC. The proposed method is implemented using Matlab 6.5.1 .

4.1 Forecasting inflation rate

The data ofinflation rate arelaken from January 1999 to January 2003. The data fiom January 1999 to March 2002 are used to training and_the data from Aprit 2002 to January 2003 are used to testing. ihe procedure to forecasting _{inflation rate based on the proposed method is given byihe following steps:}

(l)' Define the universe of discourse of each input. In this paper, we will predict the inflation rate of month t using inflation rate data of months &-t and &-2 so we will builA fuzzy model using 2-input and l-output. The universo ofdiscourse ofeach input is defined as [-2, 4].

(2). Define fuzzy sets on universe of discourse of each input such that fuzzy sets can cover the input spaces. we dsfine thirteen fuzzy sets 4,4,...,1\, that are normal and complete on [-2, 4l of each input space with Gaussian membership function.

(3). Determine all possible antecedents of fuzz1 rule candidates. There are 169 antecedents of fuzzy rule candidates and the consequ€nce ofeach antecedent _{is chosen using Step 3 ofthe proposed method. So we have} 169 fu24 rules in the form:

R,: "lf -tr _2 is 4 *d .r*-, is 1b, tUeN xo is AJ.

wherr

(4).o

(5). 0

defrrz

defuz

Table t method constr[l tudfg fuzzifiG

gl

Josnal of gnntitdive Metho&, Yo.S No.2, December2h|g, Z8A3

gl

wherel

:1,2, ,.."169,

it,iz=

1,2,.,.,13

and

AJ e14,4,-.,4r1 .

(4). Construct

fuzy rule base

frun frrzzy rules designed

in St€p 3.

(5). Oesigrr fu2ry model combining the fuzy rule basc and cartain fizzifier, fiuzy inference engine and

defuzafier. In this paper, we use singleton fuzdfrer, minimum and produst infererce engines, center average

defuzzifier.

Teble l. Comparison

of mean $quare

€frors of f-orpeasting

inflation rate

using the Wang's method

and proposed

method

Methods

Numberof

fuzzyrules

MSE of training data MSE of testing data

Minimum

inference

ensine

Pmduct inference

eneine

Minimum

inference

ensine

Product inference

ensine Wang's

method

23

0.47t60

0.45780

0.35309

0.35286

Proposed method

169

0.43436

0.407E7

o26097

0.26734

Table I shows a comparison of the MSE of training and testing data based on the Wang's me{hod and proposed method. There are twenty three fuzzy rules generated by the Wang's method and there are 169 fuzy-mles constructed by the proposed rnethod, The predistion of inllation rat€ by the Wang's method with singl*on fuzzifier, product inference engine and center average defirzzifier has a higher accuracy than that with singleton fitnifier, minimum inference engine and c€nt€r average defiuzifier.

Table I illushates that the forecasting inflation raie by the proposed method with singleton fiszzifrer, minimum inference engine and center average dcfuzzifier has a higher accuracy than t$at with singleton fv.ifier, product inference engine ald oent€r average defuzzifier. The true values and prediction values ofthe inflation rate based on the Wang's method and the proposed method using minimum and product inference engines are shown in Figure I and Figure 2 resp@ively,

IE !rt

rd[

lor

B ! 5

h

"r,

le IN

f

k-f 5

x

bcrr

3 r f ,

. l c

a i

Tht

Ftg. l. The prediction and true values ofinflation rate based on the Wang's method using: (a) minimum inference engine; (b) product inference engine

Fig. 2. The prediction and fue values of inflation rate based on the proposed method using: (a) minimum inference engine; (b) product inference engine

lm.

NE

[image:7.612.71.525.20.331.2] [image:7.612.21.516.21.826.2] [image:7.612.51.492.412.784.2]

Journal of Quantitative Methods, Vo.S No.2, December 2009, 7843 tz 4.2 Forecasting interest rate of Bank Indonesia certificate

The data of the intercst rate of BIC fiom January 1999 to December ?00! afe us€d to Eerning flld th9 data from January 2002 to February 2003 are used to lesting. The interest rate of BIC of P month rvill be predicted by data of interest rate of BIC of (t-l)* and (t-2)h months so we will build fuzzy model using 2-input and l-output which universe ofdiscourse ofeach input is [10,401. Seven fuzzy sets are de{ined in dvery input space with Gaussian membership function. The olher st€ps to forecasting the interest rate of BIC Use thq same steps of the predicting inflation rate.

Teblc 2. Comparison of mean squar€ errors of predicting interest rate of BIC using thg Wang'$ melhod and proposgd method

Method

Number of fuzzy rules

MSE of trainins data MSE of testine dafa Average forecasting errors oftesting data

(o/o\ Minirnum inference engine Product inference engine Minimum inference engine Product inference engine Minimum int'erence ensine Product inference ensine Wang's method

12

0.98759

1.0687

0.46438 0.55075

3.8568

4.4393

hoposed method

[image:8.612.67.463.190.310.2]

49

0.9536r

0.91682

o.2E3Z:7 0.24098

2.q973

2.7813

[image:8.612.103.430.356.483.2]

Table 2 shows that there are twelve fuzzy rules generated by Wang's method and there are forty nine fuzzy nrles constructed by proposed method. The alerage forecasting erors of the prediction of the interest rate of BIC by Wang's method and proposed method are 3.8568 Vo and2.78l3 o/o respectively.

Fig. 3. the prediction and true values of interest rete of BIC based on the Wang's method using: (a) minimum inference engine; (b) product inference engine

From Table 2, forecasting the interest rate of BIC based on the Wang's method using minimurn inference engine has a good accuracy. Based on the proposed metho4 forecasting of the interest rate of BIC using product inference engine has a good accuracy. The MSE values and average forecasting errors based on the proposed method are smaller than those based on the Wang's method, Figure 3 and Figure 4 show the true values and prediction values of the interest rate of BIC based on the Wang's method and proposed mehod respectively.

5. Cr

In thb metiod values i d"sipc We ap for€cal than ft input r

otrimd

optirnd

6. Ach

The auf

this re*

Refera

lll Ah

qt

t2l Ah

beJ SFJ

t3l Ah

unr

,1p [4] Ah

t o l siq [5] Bft

!u

[6] Hcn 427 lzl Pq

fitza

l8l \r"r

[9] \r'rr

trai

nOl Yr

Iffi [ l l l Y o

coE

[image:8.612.105.416.589.703.2]

Jwmal of Suantitat ve Methods, Yo.S No.Z, December 2009, 7843 5. Conclusions

Jn this papcr, we t-ravg preseft€d a qew method f-or higtring csrnplet€ fu_zz-y rules _f,nq!r.l qaining dara. The msthod uses tlre firing strenglh of rule to construct complete fuzzy rules. The resulted fuzzry rules can cover all values in the input spaces. The set of the fuzzy rules generated by the proposed method contains all fuzzy nrtes designed by the Wang's method In other wod the proposed method is generalization of the Wang's method. We apply the pfqposed method b f-orecast the Indonesian inflalion rate and intgr€st ra& of BIC. The result is tha! fuecasting Indonesian inflation rde and interest rate of BIC using the proposed method has a htgher apcuracy dun that using the Wang's method. To increase the prediction accuracy, we san define more fuzzJ sets in the input and output spaces but defining more fuzzry set$ en imply complexity of computations. So determining the qpttmal t!um&.r of fivzy rules is irnpqlt4rlt !o get efficient c_ottlput4tiols. In the qsxt works, we will @ig1 the optimal number of fuzzy rules.

6. Acknowledgoment

The authors would like to thank DP2M DIKTI and LPFM Gadjah Mada University for the financial support of this ree€arch.

References

[1] Abadi, A-M., Suhnar, Widodo and SaleL S. (2006). Fuuy model for ctimaling inflation r&. Procceeding o!Internatiorul Confererce on Matlematics arrd Natural kiences- Institut Teknologi Bandrng, Indonesia [2] Abadi, A.M., Subanar, Widodo and Saleh, S. (2007). Fuecasting int€rest rate of Bank Indonesia certificate

based on univariate fuzzy time serics,, Internatiotul Conferetre on Matlematics atd lts applicatiotu SU MS, Gadjah Mada Univerplty, "ltrdsnesia

[3] Abadi, A.M., Subanar, Widodo and Saleh, S. (2008s). Constnrcting complete firzy rules of fuzzy modet using singular value decomposi6on. Preeeding of Internaional Cogference on Mathemdics, Staistics and Applicaions (ICMSA). Syiah Kuala University, Indonesia

[4] Abadi, A.M., Subonar, Widodo and Sale[ S. (2008b). Designing fuzzy time series model and its applicaticn to forecasting inflation rztr. 7'' World Congrets in Prabobility ctrd Stuistics. National University of Singapce, Singapore

[5] Bikdas]t, M. (1999). A highly interpretable furm of Sugeno inlbrrcnc€ systems, IEEE Trarxrctions onfuzry Esfenr, 1(6),68ffi96

[6] Henera, LJ., Pomareg H., Rojas, I., Valensuela, O. and Prieto, A. (2005). Fuzzy sets and systems,l53, {)3-427

[{ Pomares, H., Rojas, I., Gonzales, J. and Prieto, A. (2002). Strusture identification in complete rulebased firzzy systcms - IEEE Transqtions onfto4t SJ,sterrs, l0(3), 349-359

[8] Wang t)( (1997). A cource lnfuzzy systems andcontral. Prentice-Hall, Inc, Upper Saddle River

[9] Wq T.P. and Chen, S.M. (1999]. A new method for construc.ting membership fr.rnctions and rules from haining examples. IEEE Transactiotls on Slstems, Maa and Cyherrctics-Part B: Cybenretics,29(l), 2540 [10] Yam, Y., Baranyi, P. and Yan& C.T. (1999). Reduction of fuzzy rule base via singular value decomposition.

IEEE Trourctions on fuzry Slstems, 7 (2\, 120.132

[ll] Yen, J., Wang L. and Gillespie, C.W. (1998).Improvingthe interpretability of TSK fuzzy models byN combining global learning and local learning. IEEE Transrctions onfuzzlt Systems, 6{4),530-537

83

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should

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DeparEnent

of Science,

Faculty

ofEngineering

International

lslamic University

Malaysia

Email: Jamal5

8@iiu.edu.my

The manuscript

that contains

original research

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In addition to, survey or

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Roman.

STEP.STRESS

TEST USING THE NON.IIOMOGENEOUS POISSON PROCESS

FOR REPAIRABLE SYSTEMS AND ITS APPLICATION IN AGRICULTURE

Xiao Yang' and Imad Khamis'

'Department

of Mathematics,

Southern

Missouri State

University

One University Plaza.

Cape Girardeau,

MO-63701

Abstract. Most of the available

literature

on step-stress.

tests

is focused

on non repairable

systems

with weibull or exponential

distribution. In this paper an altemative

prospective

is presented

for

step-stress

tests on repairable

systems,

in which each experimental

unit can be repaired and replaced

into

tests after a failure. The basic life distribution involved is a non-homogeneous

Poisson

Process.

The

cumulative Exposure

Non-Homogeneous

Poisson

Process

(CENHPP) Model will be proposed.

Inferential

procedure

and optimum design

will be discussed

for simple step-stress

accelerated

test

plans with fixed shape parameter.

And several potential applications

of CENHPP models in

agriculture

will be represented.

Keywords: Accredited

life test; step-stress

test; Non-homogeneous

Poisson

Process;

repairable

systems;

maximum

likelihood;

optimum

desigrt

1. Introduction

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References

[1] Tang, L.C. (1990).Analysis

of step-stress

acceleratedlife-test

data: A new Approach.

IEEE

Traw actions

on Reliability,

45 (l), 60-7

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