Seasonal Time Series and Transfer Function Modelling for Natural Rubber Forecasting in India

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1 Inernaonal Journal of Comuer Trend and Technology (IJCTT) - volume4 Iue5 May 03 Seaonal Tme Sere and Tranfer Funcon Modellng for Naural Ruer Forecang n Inda P. Arumugam #, V. Anhakumar * # Reader Dearmen of Sac, Manonmanam Sundaranar Unvery, Trunelvel-670, Inda * Reearch Scholar Dearmen of Sac, Manonmanam Sundaranar Unvery, Trunelvel-670, Inda Arac Tme ere analy a owerful ool o deermne dynamc model amng a defnng and conrollng mo arorae varale of a yem. Tranfer funcon model one of he oular echnue n he me ere modellng for forecang. When here an ouu ere whch nclned y an nu ere, he ojecve of he ranfer funcon modellng o denfy he role of nu ere n deermnng he varale of nere. In h aer, he Tranfer Funcon model fed o he Naural Ruer roducon n Inda. The Tranfer funcon Model ha een ued o denfy a model and emae arameer for forecang of ruer roducon. Keyword Tme ere, Tranfer funcon, Auoregreve Inegraed Movng Average Model, Cro correlaon Funcon. I. INTRODUCTION In forecang and analy of me ere daa, well demonraed ha auoregreve negraed movng average (ARIMA), nervenon and ranfer funcon model are very effecve n handlng raccal alcaon. Modellng and forecang of mul varale me ere o emloy ranfer funcon model. Tranfer funcon model [3] can e regarded a exenon of clacal regreon model, and are ueful n many alcaon. Unverae model ue a ngle deenden or ouu varale a a funcon of own hory and revou error. Tranfer funcon model ngle or Mulle nu ha may oly affec he yem. The dynamc characerc of a yem are fully underood exlcly only hough a ranfer funcon model. The dynamc naure of he ranfer funcon relaonh le n aly o accoun for he nananeou and lagged effec of an nu varale on he ouu varale. amang Wdjanarkon Ook e al.,[5] ued he ox Jenkn mehodology o uld Tranfer funcon model for ranfall ndex daa n Indonea y comarng he foreca accuracy among ARIMA, ASTAR, Sngle nu Tranfer funcon, and mulle nu Tranfer funcon model. Three locaon of ran fall daa a e Java are ued a cae udy,.e. Ngale, Karangja, and Manngan. In h reearch, Seaonal ARIMA a he arorae ye of ranfall ndex daa ued. Khm e al.,[7] have rooed ranfer funcon model o redc elecrcy rce aed on oh a elecrcy rce and demand and dcued he raonal o uld. In h aer, he Tranfer funcon model fed o he naural ruer roducon whch moly nfluenced y ale. The dynamc relaonh eween he ale uded hrough he Tranfer funcon model. Fnally he fed Tranfer Funcon model can alo e ued for forecang. Tranfer funcon [6] ued he wo echnue o accuraely redc me ere daa of naural feld laex rce. The me ere foreca aed on ranfer funcon mehod wa comared o neural nework model acro he erod ahead n he foreca horzon. The reul of he udy mly ha neural nework forecang mehod a eer alernave aroach for redcng naural feld laex rce. II. DATA Ruer an amorhou, elac maeral oaned from he laex or a or varou rocal lan. There are wo ye of ruer naural and ynhec. Inda he fourh large roducer of naural ruer n he world and he ffh large conumer. Though Inda one of he leadng roducer of ruer ll mor ruer from oher counre. Ruer-roducng area n Inda are dvded no wo zone radonal and non-radonal. Under radonal zone we have Kanyakumar n Taml Nadu and ome drc of Kerala wherea under non-radonal zone, ruer roduced n coaal regon of Karnaaka, Goa, Andhra Pradeh, Ora, ome area of Konkan regon of Maharahra, Trura, and Andaman and Ncoar Iland. Ruer roducon value (Tonne) are recorded y he ruer ndury ung Monhly daa for he erod from January 99 o Decemer 0 (64 monh). The Naural Ruer roducon are colleced from ( for 64 monh (January, 99 o Decemer, 0). A. ARIMA Model III. METHODOLOGIES One of me ere model whch oular and moly ued ARIMA model. aed on Auoregreve (AR) model how ha here a relaon eween value n he reen (Z ) and value n he a (Z -k ), added y random value. Movng Average (MA) model how ha here a relaon eween a value n he reen (Z ) and redual n he a (a -k wh k=,, ). ARIMA (, d, ) model a mxure of AR () and MA (), wh a non aonary daa aern and d dfferencng order. The form of ARIMA (, d, ) ISSN: h:// Page 366

2 Inernaonal Journal of Comuer Trend and Technology (IJCTT) - volume4 Iue5 May 03 d Z a Where AR model order, MA model order, d Dfferencng order, and Generalzaon of ARIMA model for a eaonal aern daa Whch wren a ARIMA (, d, )(P,D,Q), ( ) Where eaonal erod Q D d Z Q a Q. Tranfer Funcon Q and Tranfer funcon model dfferen from ARIMA model. ARIMA model unverae me ere model u ranfer funcon mulvarae me ere model. Th mean ha ARIMA model relae he ere only o a. ede he a ere, ranfer funcon model alo relae he ere o oher me ere. Tranfer funcon model can e ued o model ngle ouu and mulle ouu yem, In he cae of ngle ouu model, only one euaon reured o decre he yem, I referred o a a ngle euaon ranfer model. A mulle ouu ranfer funcon model referred o a a mul euaon ranfer funcon model or a mulaneou ranfer funcon model funcon (STF) model. A more comlee decron of modellng and forecang ung mul euaon model can e found. A ngle euaon ranfer funcon model may conan more han one nu varale a n mulle regreon model. The ngle nu ranfer funcon model, r Z N c r a Where, X N.. r r Mul Inu Tranfer Funcon Model Z C r X r X Where he raonal ranfer funcon m... rm ( ) r( ) XmN, for each nu varale ha he form n ngle ranfer funcon model. ) Idenfcaon of a model decrng η and of a fnal ranfer funcon model: I neceary o check wheher he relmnary model adeuae y analyng he redual (η ) wh he value of he nu x. The model decre η deermned y he followng euaon: ˆ n ˆ n Where η a durance erm ha follow an ARIMA model, Hence, an arorae fnal ranfer funcon model of he form: C x ˆ n Z Z ˆ n.) Prewhenng of x and y : Once an arorae model decrng x can e denfed he relaonh eween x and y wll e emaed. The rewhened x and y value can e calculaed y euaon reecvely: ˆ x x z for x x ˆ x ˆ ˆ x x z x Where Z (x) and Z rereen nu and ouu daa ha are ranformed o e aonary form..) Calculaon of he amle cro correlaon funcon (CCF) and denfcaon of a relmnary ranfer funcon model : In order o denfy a relmnary ranfer funcon model decrng he relaonh eween y and x, he amle cro correlaon funcon (CCF) eween he α value and he β value mu e comued from he followng euaon: for y ISSN: h:// Page 367

3 Inernaonal Journal of Comuer Trend and Technology (IJCTT) - volume4 Iue5 May 03 r k, n k k n n ˆ The general relmnary ranfer funcon model comued a: Z C x ˆ n Z ˆ n In elecng he form of he model he value of, r and mu e deermned from he correlogram of r k (β, α ). The valued of he numer of erod efore he nu daa (x ) egn o nfluence ouu daa (Y ). I eual o he lag where he fr ke n he SCC encouned or he numer of wegh ha are no gnfcanly from zero. The value of r rereen he numer own a value z. The value of rereen he numer of a z (x) value nfluencng z. ) Parameer Emaon: In e afer all funcon of he model ha een rucured he arameer of hee funcon wll e emaed. Good emaor of he arameer can e found ung lea uare y aumng ha hoe daa are aonary..) Tranfer funcon Order: For numeraor and denomnaor comonen he value rereen he maxmum order. Order 0 alway ncluded for numeraor comonen. Order numeraor he model nclude order, and 0.Order 3 denomnaor, he model nclude order 3,and.The numeraor under of he ranfer funcon ecfe whch revou value from he eleced ndeenden (redcor) ere are ued o redc curren value of he deenden ere. The denomnaor order of he ranfer funcon ecfe how devaon from he ere mean for revou value of he eleced ndeenden (redcor) ere are ued o redc curren value of he deenden ere. Two mehod have een uggeed for removng auocorrelaon efore calculang he CCF Pre-when (fler) he ere recommended y ox and Jenkn. F earae ARIMA model o x and y and hen Calculae he cro correlaon funcon aed on he redual. The fr mehod more wdely ued alhough he ue no comleely reolved; re whenng nvolve fve conceual e. Fr dfference each ere unl aonary aou mean. d X Z d Y The degree of dfferencng need no e he ame for he X and Y ere. Second,f an ARIMA model o he dfferenced redcor ere ω a Thrd, ue he nvere of he model o fler ω (leavng he redual a ) a Fourh, ue he ame nvere o he fler ω (or re when) Z. Yeldng he re whened Z Ffh, cro correlaon he redual a and he re whened. Th CCF reflec he mule wegh and herefore may e ued o denfy ranfer funcon. The econd mehod of removng auocorrelaon o f earae ARIMA model o he redcor and ouu ere and hen fler each ere alone o denfy he ranfer funcon. 3) Dagnoc checkng: A dagnoc checkng emloyed o valdae he model aumon and o check wheher he model adeuae. I neceary o do dagnoc checkng even f he eleced model may erform o e he e among oher. I check wheher he hyohe made on he redual are rue or no. Thee redual mu e a whe noe ere. Zero mean and conan varance, uncorrelaed roce and normal druon. Thee reuremen can e nvegaed y necng he auocorrelaon funcon (ACF) and aral auocorrelaon funcon (PACF) lo of he redual and akng e for randomne uch a Ljung ox ac. 4) Forecang wh ranfer funcon: If he hyohee on he redual from e 3 are afed, he foreca rce of he fnal model are hen comued and comared he reul wh he e daa. Tranfer funcon model whch are exenon of Famlar lnear regreon model have een wdely ued n varou feld of reearch. Tranfer funcon model can e ued o udy he Dynamc nerrelaonh among he varale n an economc yem. The funcon denfcaon mehod can e ued n he Same manner no maer f he ranfer funcon model ha ngle nu or mulle nu varale. Th mehod more raccal and eaer o ue hen he cro correlaon funcon. Tranfer funcon model can e ued o model can e ued o model only one euaon reured o decre he yem. I referred reured o ISSN: h:// Page 368

4 Inernaonal Journal of Comuer Trend and Technology (IJCTT) - volume4 Iue5 May 03 decre he yem. I referred o a a ngle euaon ranfer funcon model. A mulle ouu ranfer funcon model referred o a a mul euaon ranfer funcon model or a mulaneou ranfer funcon model. The ranfer funcon only reen f ndeenden Varale are ecfed. C) Seaonal Tranfer funcon: Seaonal Tranfer funcon noaon for h redcor me ere X, wh eaonal facor where, Df d D lagk N D ouu daa of ranfer funcon model, reecvely. Thee me ere daa con of n oervaon of a of he ranfer funcon aroach o modellng me ere con of four e. The arorae ranfer funcon model wll e denfed. I aumed ha nu and ouu me ere mu e oh aonary. If no neceary o ranform hoe daa no aonary form. The lo grah of oh he me ere of ruer roducon and ale were examned and hee how ha he eaonal me ere of ruer were no aonary * * D he eaonal order of he dfferencng for he h redcor me ere. P he eaonal order of he denomnaor for he h redcor me ere. Q he eaonal order of he numeraor for he k h redcor me ere S he lengh of he eaonal cycle. The mahemacal noaon ued o decre a eaonal ranform funcon. Fg. Tme ere lo of acual ruer roducon n Inda where, ( ), ( ) ( ), ( ) d k, he denomnaor eaonal olynomal of he ranfer funcon for he h redcor me ere D,,..,,,., he numeraor eaonal olynomal of he ranfer funcon for he h redcor me ere Q,,..,, Q,. ω, ( ) D) Reul and Dcuon: Tranfer funcon mehod a dynamc regreon model whch allow he exlanaory varale o e ncluded. The man ojecve of he model o redc wha haen o he foreca varale or ouu me ere, called y, f he exlanaory varale or nu me ere,called x change. Le x and y rereen nu and Fg. The lo of Acual roducon agan Foreca roducon y Seaonal Tranfer funcon (0,,)(,,0) Mehod TRANS FER FUNCTIO N MODEL AND MODEL STATISTI C F Sac TALE Mean Saonary R- uare.70 R- Suared.669 RMSE MAPE 8.35 MAE IC ISSN: h:// Page 369

5 Inernaonal Journal of Comuer Trend and Technology (IJCTT) - volume4 Iue5 May 03 Fg. 3 The amle cro correlaon funcon (SCC) and model ac TALE PARAMETER ESTIMATION OF TRANSFER FUNCTION MODEL Prod uc on Sale TF Em ae Conan 4. Delay Numerae Lag0.478 Dfference Denomnaer lag Numeraer eaonal lag Denomnaor eaonal lag SE Sg REFERENCES [] ova A. Seaonal Tme ere and Tranfer Funcon Modelng, Journal of une & Economc Sacvol vol.3,,no, 3(4), ,Oc985. [] Monca chogna,carlo Galan, Gude Maaroo, Auomac Idenfcaon of eaonal Tranfer model y mean of Ieraon ewe and Genec Algorhm.Dearmen of cence ac.journal of Tme ere Analy vol 9,No, ,007. [3] ox, G.E.P, and G.M. Jenkn and G.C.Renel, Tme ere analy Forecang and conrol, 4 h edon, John Wley and on, Inc., New Jerey, 998. [4] Mara emla camargo, Waler rene flo,angela do anoe Dullu, Tranfer funcon and nervenon model for he udy of razlan nflaonary roce,afrcan journal of une Managemeny Vol.4(5),PP ,May00,ISSN @00 [5] amang Wdjanarko Ook and Suharono, Develomen of Ranfall Forecang Model n Indonea y ung ASTAR, Tranfer Funcon and ARIMA Mehod, Euroean Journal of Scenfc Reearch, Vol.38 No. 3(009), [6] Walalak Ahr awong and Porn Chachaun, Tme ere Analy for Naural Feld Laex Prce Predcon, Kng Mongku Inue of Technology Ladkraang, angkok 050, Thaland 00. [7] A.A, Khn, Zanaladn M.and Mad.Nar.S, Comarave Forecang Model Accuracy of Shor- erm Naural Ruer Prce, Trend n Agrculural Economc4 ():-7, 0, ISSN /DOI: 0.393/ae [8] Chnye S. Meke, Shor erm forecang of Ngeran naural ruer exor, Wudecker journal of Agrculural Reearch, Vol. (0), ,(0). [9] Mad nar hamudn and Famah mohd arhad, Comoe Model for Shor Term Forecang for Naural Ruer Prce Peranka 3(),83-88(990), UPM Serdang, Selangor Darul Ehan, Malaya. [0] Lon Lmlu, Forecang redenal conumon of naural ga ung monhly and uarerly me ere, Inernaonal journal of forecang 7(99)3-6,.3-6 Norh Holland. IV CONCLUSION In h aer, we have reened effcen echnue o accuraely redc me ere daa of naural ruer roducon n Inda. The me ere foreca aed on ranfer funcon mehod wa comared o roducon and ale n he foreca horzon. The forecaed accuracy meaure of he denfed Tranfer Funcon model whch mall. From a dealed analy of he numercal reul, can e concluded ha he ualy of redcon ung he rooed echnue conderaly good comared wh oher andard me ere model when ouu ere are nfluenced y nu ere. ACKNOWLEDGMENT The auhor would lke o hank he reference for gven ha very helful commen and uggeon, her ngh and commen led o a eer reenaon of he dea exreed n h aer. Th reearch work wa uored y he unvery Gran Common, New Delh. ISSN: h:// Page 370