Modeling of Sokoto Daily Average Temperature: A Fractional Integration Approach

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1 Available online a hp:// Nigerian Journal of Basic and Applied Science (), 9():- ISSN Modeling of Sokoo Daily Average Temperaure: A Fracional Inegraion Approach * L.K. Ibrahim, B.K. Asare, M. Yakubu and U. Dauda Deparmen of Mahemaics and Compuer Science, Umaru Musa Yar adua Universiy, Kasina, Nigeria. Deparmen of mahemaics Usmanu Danfodiyo Universiy, Sokoo [Corresponding Auhor: ykmm@yahoo.com] ABSTRACT: Auoregressive fracional inegraed moving average modeling sraegy was used o model he daily average emperaure (DAT) series of Sokoo meropolis for he period of //3 o 3/4/7. The ime plo suggess ha here is persisence dependence in he series. The order of fracional inegraion was found o be The correc model for he daily average emperaure daa (DAT) of Sokoo meropolis was buil. Two models were found o be more adequae for describing, explaining and forecasing he emperaure, ARFIMA (3, 3884, ) and ARFIMA (, 3884, 3). Bu by checking he forecasabiliy, ARFIMA (3, 3884, ) model was found o be he bes opimal model ha will bes forecas Sokoo meropolis emperaure. The fied model should be used for fuure forecas of emperaure of Sokoo meropolis. Forecasing emperaure is imporan o Agriculuris, Geographers and Hydrologis. Air emperaure deermines he rae of evaporanspiraion. Keywords: Fracional Inegraion, ARFIMA, Temperaure and Sokoo. INTRODUCTION Sokoo is a ciy locaed in he exreme norhwes of Nigeria. The locaion of Sokoo in Nigeria is a Laiude 3 N and Longiude 5 5 E. Sokoo Sae is in he dry Sahel, surrounded by sandy Savannah and isolaed hills. Sokoo as a whole is very ho area. The warmes monhs are February o April when dayime emperaure is rising. The raining season is from June o Ocober during which showers are a daily occurrence. From lae Ocober o February, during he cold season, he climae is dominaed by he Hamaan wind blowing Sahara dus over he land. The dus dims he sunligh here by lowering emperaures significanly and also leading o he inconvenience of dus everywhere in houses. Research on long memory and fracionally inegraed processes has coninued a an acceleraing rae since he iniial publicaion of he work of Granger (9); Granger and Joyeux (9) and Hosking (98) which parameerized he processes of Hurs (95) on he ime series wih hyperbolically decaying auocorrelaions (Baillie, 7). The long memory or long erm dependence propery describes he high-order correlaion srucure of a series. If a series exhibis long memory, here is persisen emporal dependence even beween disan observaions. Such series are characerized by disinc bu nonperiodic cyclical paerns. Fracionally inegraed processes can give rise o long memory (Beran, 994). The ARFIMA (p,d,q) model was inroduced by Granger and Joyeux (9). Since hen here has been grea srides in he esimaion of long memory modelling, Granger and Joyeux (9); Hoskin (98); Geweke and Porer-Hudack (983); Sowel (99) and Mayoral (7). Fracional Inegraion is par of he larger classificaion of ime series, commonly referred o as long memory models. However, recen empirical evidence suggess ha emperaure series may be well described in erms of fracionally inegraed processes (Gil-Alana, 9). Fracionally inegraed I(d) processes have araced growing aenion among empirical researchers. In fac his is because I(d) processes provide an exension o he classical dichoomy of I() and I() ime series and equip us wih more general alernaives in long range dependence (Shimosu, ). Empirical research coninues o find evidence ha I(d) processes can provide a suiable descripion of cerain long range characerisics (Henry and Zaffaroni, ). The class of fracionally inegraed processes is an

2 Ibrahim e al.: Modeling of Sokoo Daily Average Temperaure: A Fracional Inegraion Approach exension of he class of ARIMA processes semming from Box and Jenkins mehodology. One of heir originaliies is he explici modeling of he long erm correlaion srucure (Diebol and Guiraud, ). Auoregressive fracionally inegraed moving average models gran social scienis greaer freedom and flexibiliy in modeling long memory processes, for example, modeling daa as fracionally inegraed allows he researchers o model slower raes of decay han wih oher common echniques such as ARMA or ARIMA models (Baillie, 996). The daily maximum and minimum emperaures in Melbourne, Ausralia, for he period 9899 were examined by means of fracional inegraion echniques. Using a parameric esing procedure (Robinson 994), he resuls show ha he ime series for boh daases can be specified in erms of fracionally inegraed saisical models: he maximum emperaures wih orders of inegraion slighly smaller han.5 and he minimum emperaures wih values slighly higher. Thus, he emperaures seem o be saionary in he case of he maximum values and non-saionary for he minimum ones, bu mean revering in boh cases. Moreover, no significan rends in he model were found, implying ha here is no evidence of climaic change in he daa (Gil-Alana, 4). However, in Africa and more especially in Nigeria lile or no much has been done in his area. The objecives of his sudy is, herefore, o use a baery of saisical echniques o sudy he daily average emperaure of Sokoo meropolis and use auoregressive fracionally inegraed moving average (AFRIMA) processes o build a model for he Sokoo daily average emperaure. MATERIALS AND METHODS Arfima (p, d, q) Model: The general Fracionally Inegraed Auoregressive Moving Average [ARFIMA (p, d, q)] model is: d Φ( B)( B) X ( ) = Θ ( B) ε ( ) Where B is he lag operaors Φ( B) and Θ ( B) are polynomials of orders p and q respecively wih Φ = - and Θ =. Box and Jenkins (976) cied ha he model should be parsimonious. Therefore, hey recommended he need o use as few model parameers as possible so ha he model fulfils all he diagnosic checks. Akaike (974) suggess a mahemaical formulaion of he parsimony crierion of model building as AIC (Akaike Informaion Crierion) for he purpose of selecing an opimal model fis o a given daa. Mahemaical formulaion of AIC is defined as: AIC (M) = lnσ + M ε Where M is he number of AR and MA parameers o esimae. The model ha gives he minimum AIC is seleced as a parsimonious model (Yurekli, 7). Fracionally Inegraed Processes The class of fracional inegraed processes is an exension of he class of ARIMA processes, semming from Box and Jenkins mehodology. One of heir originaliies is he explici modeling of he long erm correlaion srucure. According o he values of parameers, hese processes will possess he long range dependence propery or long memory inroduced by Hurs (95) and Mandelbro and Van Ness. (968). Le X, =,,.. n be a ime series and be ρ(k) is auocorrelaion funcion: ρ( k) = E( X, X k ) The saionary propery is verified if: k = ρ ( k ) < In his case, i is said ha memory propery if: k = ( k ) / ρ / < X has he long A way of represening such correlaion srucures is he use of fracional inegraed processes. These models are defined from he fracional differeniaion operaor[ β ] d. The fracional operaor is broken down using a binomial series: ( ) [ β d k d k k ] d β Γ =... ( ) ( k ) ( d) β + Ο β + Γ + Γ This operaor makes i possible o define fracional inegraed processes. I is assumed ha process x =...n (assumed o be cenered for he purpose of simpliciy, E[x] = ), follows an

3 Nigerian Journal of Basic and Applied Science (), 9(): - Auo Regressive Fracional Inegraed Moving Average (ARFIMA) process if: [ β ] d X = µ () Where µ is a usual ARMA (p, q) process, φ ( β ) µ = θ ( β ) q ε, () p ε is a whie noise wih zero mean and variance σ.we assume ha ε µ verifies he saionariy and inveribiliy condiions. This assumpion is necessary o esablish he following properies. One can demonsrae ha: The process X is saionary if d <, The process X is inverible if d > -. Odaki (993) spread his inerval o d > - by using a weak inveribiliy concep. The saionary process X will have long memory if < d <. The degree of persisence of he series can be measured hrough a fracional differencing parameer (Gil-Alana, 9). Tes and esimaion of order of Inegraion: There exis several procedures for esimaing he fracional differencing parameer in semi parameric conexs. Of hese, he logperiodogram regression esimae proposed by Geweke and Porer-Hudak (983) has been he mos widely used (Shimosu e al., ). Given a fracional inegraed process {Y }, is specral densiy is given by d f ( ω) = [sin( ω / )] f ( ω) Where ω is he Fourier frequency, f ( ω ) is he specral densiy corresponding o u and u is a saionary shor memory disurbance wih zero mean. The fracional differencing parameer d can be esimaed by he regression equaions consruced from X j (3) j= p( ω X ) = + ( 4 X j ) e GPH showed ha using a periodogram esimae of f ( ω j ), if he number of frequencies m used is a funcion g(n) (a posiive ineger) of he sample size n where m= g( n) = n α wih < α <, i can be demonsraed ha he leas squares u esimae d using he above regression is asympoically normally disribued in large samples. d ~ N π d, g( n) 6 ( U j U ) j= (4) Where U sample of j = and U is he ln[4sin ( ω j / )] U j, j = g(n). Saionariy/Uni Roo Tes: The paern and general behaviour of he series is examined from he ime plo. The series was examined for saionariy, lineariy and gaussianiy. The ess for uni roos/saionariy are: Augmened Dickey Fuller es or he Phillips-Perron es and KPSS es proposed by Kwiakowski e al., (99) o es for he saionariy of a series. The ADF es has he following model represenaion: y = α y + x δ + β y + β y + + β y... p p Where, y is he differenced series. y is he immediae previous observaion. x is he opional exogenous regressor which may be consan, or a consan and rend. α and δ are parameers o be esimaed. β,..., p β is he coefficiens of he lagged difference erm up o lag p. Tes saisic: α = α / se( α ) The hypohesis esing: H : α = (The series conains uni roos) H : α < (The series is saionary) Model Checking: Afer esimaion of he model, he BoxJenkins model building sraegy enails a diagnosis of he adequacy of he model. More specifically, i is necessary o ascerain in wha way he model is adequae and in wha way i is inadequae. This sage of he modeling sraegy involves several seps Kendall and Ord (99). A good way o check model adequacy of an overall Box-Jenkins model is o analyze he 3

4 Ibrahim e al.: Modeling of Sokoo Daily Average Temperaure: A Fracional Inegraion Approach residual obained from he model. Two saisic(s) have suggesed deermining wheher he firs K sample auocorrelaion indicae adequacy of he model, ha are, he Box-Pierce saisic and Ljung-Box saisic (pormaneau es). In spie of his, we can also check he model adequacy by examining he sample auocorrelaion funcion of he residual (ACF) and sample parial auocorrelaion funcion of he residual (PACF). We can conclude ha he model is adequae if here are no spikes in he ACF and PACF (Azami, 9). We can also use he Jarque-Bera es o es for normaliy of residual. Sofwares for he Analysis: The R, MINITAB and Grel sofwares ware used for he analysis. Daa and Daa Analysis: The daily average emperaure (DAT) series of Sokoo meropolis for he period //3 o 3/4/7 was used (Table ). The daa ses are obained from Energy Research Cener, Usmanu Danfodiyo Universiy, Sokoo. Table : Resuls of Summary Saisics for DAT in C Parameer Value Mean 8.86 Median 8. Minimum. Maximum. Sd. Dev CV 6687 Skewness Ex. Kurosis -978 The DAT and is graphical properies Visual inspecion of he plo of daily average emperaure series in Figure reveals ha shor erm random flucuaions, bu long flucuaions of unequal duraion are readily apparen. Long erm or persisen dependence observed. The series is assumed o be non saionary. Auocorrelaion Funcion (ACF): A ime series exhibis long memory when here is significan dependence beween observaions ha are separaed by a long period of ime. Characerisic of a long memory ime series is an auocorrelaion funcion ha decays hyperbolically o zero. Figure shows he plo of he sample auocorrelaion funcion (ACF) of he daily average emperaure series (DAT). Non periodic cycles, persisence and slow decay of ACF are observed, wih relaively high concenraion of mass a high lags which naurally is suggesing long memory model. Daily mean emperaure Figure: Daily average emperaure of Sokoo meropolis from //3-3/4/7 measured in C ACF for Daily average emperaure 6 lag PACF for Daily average emperaure +-.96/T^.5 6 lag +-.96/T^.5 Figure : ACF and PACF for daily average emperaure of Sokoo meropolis Specral Densiy Funcion: The frequency domain definiion of long memory saes ha he specral densiy funcion is unbounded a some frequency in he inerval [, ]. Persisence would be refleced in he specral densiy funcion wih a relaively high concenraion of mass around zero frequency (Labao, 997). Figure 3 gives he specral densiy plo of he daily average emperaure series of Sokoo meropolis wih relaively high concenraion of mass around zero frequency. 4

5 Nigerian Journal of Basic and Applied Science (), 9(): Specrum of Daily average emperaure periods 6 scaled frequency Figure.3:Specrum of he daily average emperaure ( C) of Sokoo meropolis Saionariy and Uni Roo Tes: The KPSS es saisic ess he null hypohesis of saionariy agains he alernaive of uni roo and he decision rule is o accep he null hypohesis when he value of he es saisic is less han he criical value. The ADF saisic es he null hypohesis of presence of uni roo agains he alernaive of no uni roo and he decision rule is o rejec he null hypohesis when he value of es saisic is less han he criical value. The resul is shown in Table. According o he ADF es and KPSS es for he daily average emperaure series, he resuls have shown ha he ime series is neiher I () nor I (). Furher ess should be done o assume he order of differeniaion. Table : Resuls for ADF and KPSS Tess for DAT VARIABLE CRITICAL LEVEL ADF DAT CRITICAL LEVEL KPSS.5% 5% %.5% 5% % s 4 4 TEST STAT Tes and esimaion of order of Inegraion: There exis many approaches for esimaing and esing he fracional differencing parameer. Some of hem are parameric and some semi parameric (Gil-Alana, 4). In his research we implemen one of he semi parameric echniques proposed by Geweke e al., (983). The applicaion of his es on he series allows us o es he null hypohesis of a uni roo (d = ) agains he alernaive of fracional inegraion (d < ). The fracional differencing parameer d is esimaed by using he regression of equaion (5). The value can be adjused using he bandwidh parameer. Tradiionally he number of bandwidh is chosen from he inerval [T /, T 4/5 ] (Robinson, 994). However, Hurvich and Deo (998) showed ha he opimal bandwidh is (T ).The bandwidh of is chosen in his work and he resul of he esimaed fracional parameer d is given in Table 3. Table 3: Resuls of GPH esimae for d parameer Variable d Sd.as Sd.reg DAT Here d is he fracional differencing parameer, sd.as is he sandard deviaion and sd.reg is he sandard error. The value of he differencing parameer d is non ineger and i is showing non saionary and mean revering. The value of he esimaed d parameer is used in equaion () o fracionally difference he daa. Similar ess were performed on he fracionally differenced series, and observed ha he new series is saionary. The resuls of he ess are given in Table 4. The newly obained se of daa can be used o perform he long memory analysis, since all he fracionally inegraed par has been removed. 5

6 Ibrahim e al.: Modeling of Sokoo Daily Average Temperaure: A Fracional Inegraion Approach Table 4: Resuls of ADF and KPSS ess for he fracionally difference series CRITICAL LEVEL ADF CRITICAL LEVEL KPSS VARIABLES % 5%.5% % 5%.5% TEST STAT ARFIMA Model Idenificaion The value of he esimaed fracional differencing parameer d is used o fracionally difference he daa. The newly obained se of daa is now ( β ) = ( β ), which is φ µ p θ q ε ARMA (p, q) series, and is aken and modeled as an ARMA (p, q) process since all he inegraed par have been removed. The resuls of he models are in Table 5. We seleced hree models wih low AIC which is a common procedure in ARFIMA modeling (Laurini e al, 3) and find he bes among hem. The models seleced are ARFIMA (3, 3884, ), ARFIMA (, 3884, 3) and ARFIMA (, 3884, ) Esimaion of he Models: We esimae he parameers of ARFIMA (p, d, q) as he opimal model. Parameers are esimaed by exac maximum likelihood mehod and he order of ARFIMA parameers are seleced from Akaike informaion crieria and are given in Table 6. Table 5: Resuls of Model idenificaion for DAT Model AIC HQC BIC ARFIMA(, 3884, ) ARFIMA(, 3884, ) ARFIMA(, 3884, ) ARFIMA(, 3884, ) ARFIMA(, 3884, ) ARFIMA(, 3884, 3) ARFIMA(3, 3884, ) ARFIMA(3, 3884, ) Table 6: Resuls of Esimaed parameers of ARFIMA (p, d, q) models PARAMETERS ARFIMA (, d, ) ARFIMA (, d, 3) ARFIMA (3, d, ) φ φ φ θ θ θ µ

7 Nigerian Journal of Basic and Applied Science (), 9(): - Model Checking: Before he inerpreaion and use of he model, we are o look a some ess o check wheher he model is specified correcly. The following ess are applied o he residuals: (i) Tes for auocorrelaion and parial auocorrelaion. (ii) Pormaneau es for residual auocorrelaion. (iii) Jarque Bera es for non normaliy. Auocorrelaion and Parial Auocorrelaion of Residuals: In esing he auocorrelaion and parial auocorrelaion, if he residual is uncorrelaed, hen he resul is adequae. Figures 4-6 shows ha here is no serial correlaion observed in he residuals of he variables, since all he series are wihin he 95% confidence inervals excep in ARFIMA (, 3884, ).. Auocorrelaion Parial Auocorrelaion Figure 5: ARFIMA (, 3884, ) Residual ACF and PACF 6 6 Auocorrelaion Parial Auocorrelaion Parial A uocorrelaion Figure 4: ARFIMA (3, 3884, ) Residual ACF and PACF 6 Auocorrelaion Figure 6: ARFIMA(, 3884, 3) Residual ACF and PACF 6 7

8 Ibrahim e al.: Modeling of Sokoo Daily Average Temperaure: A Fracional Inegraion Approach We formally confirm he presence and absence of auocorrelaion in he variables by pormaneau es. The resuls for he es are presened in Tables 7-8. The null hypohesis of no auocorrelaion can be acceped in Tables 7-8. We conclude ha here is no correlaion in ARFIMA (, 3884, 3) and ARFIMA (3, 3884, ) models residual. Tes for Normaliy of Residual: The normaliy es of he residuals was performed o check if he residuals are normally disribued. The mos popular mehod of checking normaliy is he Jarque-Bera es. The es saisic measures he difference of he skewness and kurosis of he series wih hose from he normal disribuion. Under he null, he Jarque-Bera saisic is disribued as χ () (Luhkepohl and Krazig, 4). Tables 9 - indicae ha he residuals of he models are normally disribued because all he p- values are greaer han he criical value of 5 Table 7: Resuls of ARFIMA (, 3884, ) pormaneau es for auocorrelaion of residual s Chi-square DF P-value Table 8: Resuls of ARFIMA (,3884,3) pormaneau es for auocorrelaion of residual s Chi-square DF p-value Table 9: Resul for ARFIMA (3, 3884, ) model Jarque-Bera es for normaliy of residual Variables Tes saisic DF P-value DAT Table : Resul for ARFIMA(,3884,3) model Jarque-Bera es for normaliy of residual Variables Tes saisic DF P-value DAT Forecas: Afer good ARFIMA (p, d, q) model have been goen, he nex is o see is abiliy o forecas. The abiliy o do so will furher esify he validiy of he model. Wha we inend o do is o compare he forecas and he acual values and choose among, he one wih he minimum error as our opimal model. The smaller he values of he error, he beer he forecasing performance of he model (Olanrewaju and Olaoluwa, 9). Tables - are he en days inerval forecas for he models. Table : Forecas for ARFIMA (, 3884, 3) Period Forecas Lower Upper Sd.error Acual

9 Nigerian Journal of Basic and Applied Science (), 9(): - Table : Forecas for ARFIMA (3, 3884, ) Period Forecas Lower Upper Sd.error Acual From he resuls in Tables -, ARFIMA (3, 3884, ) is seleced as he opimal model because by comparing he forecas and he acual values, i was observed ha he model ARFIMA (3, 3884, ) has minimum error. CONCLUSION In his research, he sochasic srucure of he daily average emperaure daa of Sokoo (DAT) has been analyzed by using fracionally inegraed processes. In doing so, a much richer dynamic behavior of he series has been capured, no achieved by he classical I()/I() represenaions bu by fracional inegraion. We implemen he mehod proposed by Geweke and Porer-Hudak (983), o es and esimae d under few prior assumpions concerning he specral densiy of a ime series. A number of heurisic mehods o esimae self similariy parameer H have been presened and applied o he daa and observed he esimaed H. We are able o show ha he emperaure series of Sokoo meropolis is fracionally inegraed and has persisence dependence, herefore exhibis long memory. Finally, he fracionally differenced DAT is aken and modeled using he Box-Jenkins approach. The correc model for he daily average emperaure daa (DAT) of Sokoo meropolis was buil. Two models were found o be more adequae for describing, explaining and forecasing he emperaure ARFIMA (3, 3884, ) and ARFIMA (, 3884, 3). Bu by checking he forecasabiliy, ARFIMA (3, 3884, ) model is he bes opimal model ha will bes forecas Sokoo meropolis emperaure. The fied model should be used for fuure forecas of emperaure of Sokoo meropolis. Forecasing emperaure is imporan o Agriculuris, Geographers and Hydrologis. Air emperaure deermines he rae of evaporranspiraion. REFERENCES Akaike, H. (974). Maximum Likelihood Idenicaion of Gaussian Auoregressive Moving Average Models".Biomerika, 6(): Baillie, R.T. (996). Long memory processes and fracional inegraion in economerics. J. Economerics 73: 59. Baillie, R.T. and Kapeanios, G. (7). Semi parameric esimaion of long memory: The Holy grail or poisoned chalice. Inernaional Symposium on Financial Economerics. Beran, J. (994). Saisics for Long-memory processes. London: Chapman & Hall. Box, G.E.P. and Jenkins, G.M. (976). Time series analysis: forecasing and conrol, nd Edn. San fransisco, CA: Holden-Day Diebol, C. and Guiraud, V. (). Long memory ime series and fracional inegraion. A cliomeric conribuion o French and German economic and social hisory. Hisorical Social Res. 5: 4 -. Geweke, J. and Porer-Hudak, S. (983). The esimaion and applicaion of long memory ime series models. Time Series Anal, 4: 37 Gil-Alana, L.A. (9). Persisence and ime rends in he emperaures in Spain. Advances in Meeorology, -. 9

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