Forecasting of Areca Nut (Areca catechu) Yield Using Arima Model for Uttara Kannada District of Karnataka

Transcription

1 Ieraioal Joural of Sciece ad Research (IJSR) Impac Facor (0): Forecasig of Areca Nu (Areca caechu) Yield Usig Arima Model for Uara Kaada Disric of Karaaa Sriah Reddy A. B., Havaldar Y. N., Pava Kumar S.T 3, Adam Kamei 4 Research Associae A.R.S Aahapur, Adhra Pradesh Associae Professor of Agriculure Saisics, Uiversiy of Agriculural Scieces, Dharwad , 4 Ph.D Research Scholar, Deparme of Agriculural Saisics, Bida Chadra Krishi Vishwavidyala, Mohapur, Wes Begal-745 Absrac: Arecau (Areca caechu) also popular by ame supari or beelu. I is oe of impora commercial crop i Uara Kaada disric, coribuig aroud 9 perce of area ad perce of producio o he Karaaa sae oal. The prese sudy is based o he secodary daa of over 30 years colleced from Direcorae of Ecoomics ad Saisics. The predicio of arecau producio o a yearly ime scales has bee aemped by various research groups usig differe echiques models. Amog he mos effecive approaches for aalyzig ime series daa is he model iroduced by Box ad Jeis, ARIMA (Auoregressive Iegraed Movig Average). The selecio of models was doe based o heir R value ad roo mea square error (RMSE) value afer aalyzig he daa, ad furher hese models were used for predicio of arecau producio. From he differe (p.d.q) models, ARIMA (,, 5) was seleced based o RMSE (0.35) ad ormalized BIC (-.39) values for forecasig he producio of arecau i Uara Kaada disric. The model parameers were esimaed usig SPSS sofware ad hece i was ae as bes fied model ad forecasig has bee doe Keywords: Mea square error, Noliear, ARIMA, SPSS, AIC, BIC. Iroducio May mehods ad approaches for formulaig forecasig models are available i he lieraure. This research exclusively deals wih ime series forecasig model, i paricular, he Auo Regressive Iegraed Movig Average (ARIMA). These models were described by Box ad Jeis ad furher discussed i some oher resources such as Waler. The predicio of producio of arecau o yearly ime scales is o oly scieifically challegig bu is also impora for plaig ad devisig agriculural sraegies. Time series aalysis ad forecasig has become a major ool i differe applicaios i agriculure, horiculure, hydrology ad eviromeal maageme fields. I Karaaa, as per he Sae Horiculure Deparme source, aroud 4.55 lah acres (.84 lah hecares) is uder arecau culivaio which forms aroud 46 perce of all Idia oal. Is coribuio o oal producio is aroud.4 lah a o ha forms 47 perce of all Idia producio i I is impora o oe ha arecau culivaio is uderae wih varyig Exe i almos 8 ou of 30 disrics i Karaaa. Amog which, Chimagalur disric sads firs i boh area (0 %) ad producio (7%), Shimoga sads secod followed by Davaagere disric. The op 7 disrics viz. Chimagalur, Shimoga, Davagere, Dashia Kaada, Tumur, Chiradurga ad Uar Kaada occupy 89 per ce of he area uder arecau ad coribue aroud 9 per ce of areca produced i he sae. Where Uar Kaada coribues 7 perce ou of 89 perce o ha Karaaa sae. Small variaios i he imig ad he quaiy of arecau producio have he poeial o impac o agriculural oupu. Prior owledge of producio behavior will help Uara Kaada farmers ad also policy maers. So, he prese sudy was ae o forecas he producio well i advace.. Maerials ad Mehods The sudy was uderae i Uar Kada disric which is siuaed roughly i he mid Norh Weser par of he Sae. The disric lies bewee ad orh laiude ad bewee ad easer logiude. The prese sudy is based o he secodary daa of over 30 years colleced from Direcorae of Ecoomics ad Saisics, Bagalore. Amog he differe ime series aalysis mehods Box ad Jeis, ARIMA (Auoregressive Iegraed Movig Average) was seleced. ARIMA ime-series models radiioally expressed as ARIMA (p,d,q) combie as may as 3 ypes of processes, viz. auoregressio (AR) of order p; differecig d imes o mae a series saioary ad movig average (MA) of order q. I may be poied ou ha his mehodology applies oly o saioary daa, a characerisic of which is ha he mea ad variace are cosa over ime. Three sages for carryig ou he aalysis are i) Ideificaio ii) Esimaio ad iii) Diagosic checig. A he ideificaio sage, wo graphical devices, viz. esimaed fucio (PACF) are used wih a view o eaively selec oe or more cadidae ARIMA models. Deoig he umber of observaios by ad he umber of compuable lags by (l,c), Licesed Uder Creaive Commos Aribuio CC BY Paper ID:

2 r ad [ r ( Z Z)( Z Z) / ( Z Z) i j Ieraioal Joural of Sciece ad Research (IJSR) Impac Facor (0): ˆ ' jr j ]/[ ' jrj ];,,3... j Where ˆ ˆ ˆ ˆ ˆ r, j i, j ' j; 3,4... j,3 C Y Y Y Y Y K = 0,,,... T =,,... - Y Y = Legh of ime period If he r values ail off o zero rapidly, i idicaes ha he origial series is saioary. Oherwise successive differeces, are compued ad he above procedure is coiued ill saioary i is achieved. Paers based o spies i ad r values are used o selec he appropriae values of p ad q which are, respecively, he orders of AR ad MA. Fially, a he diagosic checig sage, a appropriae model is seleced based o he followig goodess of fi saisics i) Aaie s Iformaio crierio (AIC): AIC = l V*(p,q) + (/)(p + q), Where V* is a esimae of whie oise variace obaied by fiig he correspodig ARIMA model. Mea Bayesia Iformaio crierio (BIC): absolue error (MAE): Roo mea squared error (RMSE): BIC = lv* (p,q) + (p + q) [l()/] RMSE [ ( Z ˆ Z ) / ] / MAE Z Zˆ / The lower he values of above saisics, he beer are he model. A saisically adequae model is oe whose radom shocs are idepede. Mai sages i seig up a Box- Jeis forecasig model are as follows Ideificaio, Esimaig he parameers, diagosic checig ad Forecasig 3. Ideificaio of Models A good sarig poi for ime series aalysis is a graphical plo of he daa. I helps o ideify he presece of reds. Before esimaig he parameer (p, q) of model, he daa are o examied o decide abou he model which bes explais he daa. This is doe by examiig he sample ACF (Auocorrelaio fucio) ad PACF (Parial Auocorrelaio fucio) of differeced series YB B. The sample auo correlaios for ime lags ca be foud ad deoed by rb B as follows. ˆ Y r Y 3. C Y C Y 0 Boh ACF ad PACF are used as he aid i he ideificaio of appropriae models. There are several ways of deermiig he order ype of process, bu sill here was o exac procedure for ideifyig he model. 4. Esimaio of Parameers Afer eaively ideifyig he suiable model, ex sep is o obai Leas Square Esimaes of he parameers such ha he error sum of squares is miimum. S (, ) = eb B ² (,) 3. =,, 3... There are fudameally wo ways of geig esimaes for such parameers. Trial ad error: Examie may differe values ad choose se of values ha miimizes he sum of squares residual Ieracive mehod: Choose a prelimiary esimae ad le a compuer programme refie he esimae ieracively. The laer mehod is used i our aalysis for esimaig he parameers. 5. Diagosic Checig Afer havig esimaed he parameers of a eaively ideified ARIMA model, i is ecessary o do diagosic checig o verify ha he model is adequae. Examiig ACF ad PACF of residuals may show a adequacy or iadequacy of he model. If i shows radom residuals, he i idicaes ha he eaively ideified model was adequae. Whe a iadequacy is deeced, he checs should give a idicaio of how he model eed be modified, afer which furher fiig ad checig aes place. Oe of he procedures for diagosic checig meioed by Box-Jeis is called over fiig i.e. usig more parameers ha ecessary. Bu he mai difficuly i he correc ideificaio is o geig eough clues from he ACF because of iappropriae level of differecig. The residuals of ACF ad PACF cosidered radom whe all heir ACF were wihi he limis..96 ( ) The miimum Aie Iformaio Coefficie (AIC) crierio is used o deermie boh he differecig order (d, D) Licesed Uder Creaive Commos Aribuio CC BY Paper ID:

3 Ieraioal Joural of Sciece ad Research (IJSR) Impac Facor (0): required aaiig saioariy ad he appropriae umber of AR ad MA parameers, i ca be compued as follows. AIC p q ( log ) log m = Esimaed MSE = Number of observaios m = p + q + P + Q YB + B = YB + 3B = YB + B + YB -9B - YB -0B + eb + 3B - eb -B (H) eb -9B + (H) eb -0B Taig codiioal expecaios a ime, YB B () = YB B (3) = YB ()B + yb -9B - YB -0B +0- (0)-(H) (YB -9B -YB - 0B)+(H) (YB -0B YB -B ) This diagosic checig helps us o ideify he differeces i he model, so ha he model could be subjeced o modificaio, if eed be. 6. Forecasig Afer saisfyig abou he adequacy of he fied model, i ca be used for forecasig. Forecass based o he model. (-B) (-B)P sp YB B = (-B) (-(H)P sp B) eb B were compued for upo 4 mohs (m) ahead. The above model gives he forecasig equaio as YB B = YB -B + YB -B - YB -3B + eb B -eb -B (H) eb -B + (H) eb -3B Give daa upo ime he opioal forecas of Y (also called Ex-Ae forecas) model a he is he codiioal expecaio of YB + B. I follows, i paricular, ha e Y Y The errors eb B i model are i fac ha forecas errors for ui lead ime. Tha for a opimal forecas hese oe sep ahead forecas errors ough o form a ucorrelaed series is oherwise obvious. Suppose, if hese forecas errors were auocorrelaed ad he i could be possible o forecas he ex forecas error i which case i could o be opimal. The required expecaios are easily foud because E Y Y m, Ee 0 m m =,, 3... E m Y m Y mee m am Y m Y m m = 0,,... For isace, o deermie he hree moh ahead (-3) forecas for series YB B (use equaio Because, E ( e ˆ ) 0, E( e ) Y Y e i.e. YB B (3) = 0 YB B () The forecas YB B () ca be obaied i a similar way i erms of YB B () from E (YB + B ). Similarly YB B () ca be obaied from E (YB + B ). I pracice i is very easy o compue he forecas YB B (), YB B (), YB B (3) ec. recursively usig he forecas fucio. E (YB + B ) = E(YB + B - + QB + B - eb + B -) - eb + B - (H) eb + B - + (H) eb + B -3 ad usig 3.8 ad 3.9. However, usig hese mehods, Ex-pos forecass ca also be calculaed for comparig wih he value acually realized. The accuracy of forecass for boh Ex-ae ad Ex-pos were esed usig he followig ess MAPE ad RMSE. 7. Resuls ad Discussio As Box-Jei model was preferred o he muliplicaive ime series model for forecasig purpose. I was used for forecasig of arecau producio. The resuls were preseed below. The deailed aalysis of forecasig of producio of arecau i Uara Kaada disric has bee preseed as uder. The eaive models were firs ideified based o he Auo Correlaio Fucio (ACF) ad Parial Auo Correlaio Fucio (PACF) he he suiable model was seleced. The compued values of ACF ad PACF of arecau producio were 3 lags. Sice he coefficie dropped o zero afer he firs or secod lag. Each idividual coefficie of ACF ad PACF were esed for heir sigificace usig es. The absece of pea a firs values clearly idicae suiabiliy of he choice of d =, o accomplish saioary series. Hece, based o ACF ad PACF may models were ried, fially model (,, 5) was eaively ideified based o R value (0.979) for producio of arecau. From he differe (p.d.q) models, ARIMA (,, 5) was seleced based o MAPE (.66) ad ormalised BIC (-.39) values for forecasig he producio of arecau i Uara Kaada disric. The model parameers were esimaed usig SPSS sofware ad hece i was ae as bes fied model ad forecasig has bee doe. The resuls were preseed i he Table. Resuls of ARIMA model o he producio of Arecau Fied Models R RMSE MAPE MAE MaxAPE MaxAE Normalized BIC ARIMA (,,5) ARIMA (0,,) Licesed Uder Creaive Commos Aribuio CC BY Paper ID:

4 Ieraioal Joural of Sciece ad Research (IJSR) Impac Facor (0): The prediced values of producio of arecau for he ex five years (0, 0, 03, 04 ad 05) were represeed graphically ad show i he figure. I his sudy we gave large scale compariso of differe models i order o ow he bes model for he forecasig of he arecau producio. For he model compariso, yearly producio of arecau was cosidered. The compariso of all he models was carried ou i he process based o he MAPE ad RMSE values which w.ere cosidered o be leas. Accordig o he Table 4. which represes he MAPE ad RMSE values, he ARIMA model wih leas MAPE value of.66 per ce ad RMSE value of 0.35 was cosidered as bes fi amog all he models cosidered. The suiable model was ARIMA(,,5) for forecasig he producio of arecau from (98-05).Based o ACF ad PACF may models were ried, fially model (,, 5) was eaively ideified based o R value (0.979) for forecasig of arecau producio i Uara Kaada disric. The resuls were preseed i he Table. Forecasig of producio was doe based o he model (,, 5). The prediced values of producio of arecau for he ex five years (0, 0, 03, 04 ad 05) were represeed graphically ad show i he figure Similar sudy was doe by Hamilo (). Predicio of arecau producio for he ex five years by usig ARIMA model Year Observed producio Prediced producio (i 000 oes) (i 000 oes) Licesed Uder Creaive Commos Aribuio CC BY Paper ID:

5 Ieraioal Joural of Sciece ad Research (IJSR) Impac Facor (0): ACF ad PACF of residuals of fied ARIMA model for arecau producio Refereces [] Hamilo, J.D., 994. Time Series Aalysis. Priceo Uiversiy Press, Priceo, pp: 7-7. [] Ahmad, B., A. Ghafoor ad H. Badar, 005. Forecasig ad growh reds of producio ad expor of iow from Paisa. J. Agric. Soc. Sci., : 0-4 [3] Box, G.E.P. ad G.M. Jei, 976. Time Series of Aalysis, Forecasig ad Corol, Sam Frascico, Holde-Day, Califoria. USA. [4] Grager, C. W. T. ad P. Newbold.986. Forecasig Ecoomics Time Series, Secod Ediio, Academic Press, Ic., Orlado, Florida. [5] Gujarai, D. N., 003. Basic Ecoomeric, Fourh Ediio, MeGraw-Hill Ic., New Yor. Pp.3-6. [6] Haque, M. E., M. I. Hossai ad K. M. M. Rahma Searchig for he bes fiig deermiisic model for iovaive growh aalysis ad forecasig of rice producio i Bagladesh. The Bagladesh Joural of Agriculural Ecoomics. 7(): [7] Masy, V., 00, The use of Box-Jeis mehodology o predic he price developme of agriculural commodiies. Aca Uierisaais Agriculurae e Silviculurae- Medeliaae-Bruesis, 49(): Licesed Uder Creaive Commos Aribuio CC BY Paper ID: