Inernaional Journal of Saisics and Applicaions 017, 7(6): 98-303 DOI: 10.593/j.saisics.0170706.04 Long Range Dependency of Apparen Temperaure in Birnin Kebbi Babayemi A. W. 1,*, Onwuka G. I. 1, Olorunpomi O. T. 1 Deparmen of Mahemaics, Kebbi Sae Universiy of Science and Technology, Aliero, Nigeria Deparmen of Mahemaics, Nigeria Police Academy, Wudil, Nigeria Absrac This research examined he long range dependency compormen of apparen emperaure in BirninKebbi meropolis, using hourly daa for a period of wo years from 013 o 014. We found ha he daa poins are highly correlaed, here is a sense of reurn of a paricular characerisic afer some hours and his depics seasonaliy rais and fracional inegraed d series suspeced. Auoregressive Fracional Inegraed Moving Average (ARFIMA) modelrevealed ha he series for he periods invesigaed is fracionally inegraed wih high-lag correlaion srucure, d=0.495. The long range dependency which describes he number of days for he reurn characerisics of persisence of he shock in a very long ime period afer i occurs was esimaed o be hree hundred and welve days (31) days approximaely a year; hus, once a shock is fel on apparen emperaure, i will go on for abou a year before he paern or srucure will be repeaed. Keywords Apparen emperaure, ARFIMA, High-lag correlaion srucure, Long range dependency 1. Background o he Sudy Apparen emperaure allows for he quaniaive assessmen of hea sress and is used o deermine he limi of hea exposure in order o preven hea relaed illnesses. The Hea Index was originally developed by Seadman (011) as an assessmen for sulriness, bu commonly referred o as he apparen emperaure. Parameers involved in deermining apparen emperaure are waer vapor pressure, surface area of skin, significan diameer of a human, clohing, core emperaure, aciviy, effecive wind speed, clohing resisance o hea ransfer, radiaion o and from he skin s surface, sweaing rae, venilaion rae, skin resisance o hea ransfer, and surface resisance o moisure ransfer. The hea index is he measure of how ho i really feels when relaive humidiy is facored wih he acual air emperaure (Rohfusz, 1990; Mendelsohn, 003; Epsein, 006; Rober, e al. 010; Anderson e al. 013; NOAA, 013; Qu, e al. 016). In esimaion of ARIMA (p,d,q) model he researchers ofen peg he value of d as an ineger value mos especially beween 0 and, however in esimaing d in pracice i is no always an ineger value here are ofen a imes ha he value of d lies beween -0.5 and 0.5. Many ime series exhibi oo much long range dependence o be classified as eiher I(0) or I(1) bu do no belong o eiher of * Corresponding auhor: whafsa@gmail.com (Babayemi A. W.) Published online a hp://journal.sapub.org/saisics Copyrigh 017 Scienific & Academic Publishing. All Righs Reserved hem. In he recen, majoriy of hese long range dependence models are ofen called long memory models are normally aken care of by he Auoregressive Fracional Inegraed Moving Average (ARFIMA) model. The series ha exhibis long memory process has auocorrelaion funcion ha damps hyperbolically more slowly han he geomeric damping exhibied by shor memory auoregressive moving-average, ARMA(p,q) process, which is predicable a long horizons(granger, e al. 1980; Hosking, 1981; Ding, e al. 1993; Taqqu, e al. 1995; Baillie, 006; Robinson, 003; Agosinelli, e al. 003; Larranga, 00; Nesrin, 006; Haldrup, e al 007; Baum, 013; Conreras-Reyes, e al. 013). Prior o his invesigaion on long memory of hea index in BIRNIN KEBBI meropolis, here have been previous sudies involving hea in oher fields such as Agric, Engineering, Physics, Chemisry or Biology in KEBBI Sae bu lile or no enquiry has been carried ou on long memory effec of apparen emperaure in KEBBI using ime series approach. Over he years, here ends o be a severe hea during a paricular period each year in he invesigaed area. This means we have higher values of hea index a a paricular poin in ime every year. The research aims a invesigaing he long range dependency characerisics of apparen emperaure in BIRNIN KEBBI meropolis wih a view o deermining: (i) he auocorrelaion paern for he long range dependency of he daa poins for he research; (ii) fracional value of d for he daa poins; and (iii) inerpre he long memory characerisics.
Inernaional Journal of Saisics and Applicaions 017, 7(6): 98-303 99. Mehodology We adoped ime series mehodology o long memory characerisic of apparen emperaure in BIRNIN KEBBI meropolis. I deals wih hourly daa of hea index colleced over a period of wo years from 013 o 014, and he se of daa would be subjeced o saisical esing procedures o confirm if he daa are ruly long memory, inegraed, or saionary processes. I is expeced o add o he exising lieraure on long memory behavior in hea index and will end o help he merology or praciioners in field of hydrology or dendrology. The daa are secondary and are colleced from Energy Research cener BIRNIN KEBBI..1. ARMA Process An auoregressive moving average (ARMA) process consiss of boh auoregressive and moving average erms. If he process has erms from boh an AR(p) and MA(q) process, hen he process is called ARMA(p, q) and can be expressed as: ( B)( y ) ( B) φ µ = θ ε (1) where εε ~WWWW(0, σσ εε ) is a whie noise disribuion erm, so ha for all, E ( ε ) = 0 ; Var ( ε ) = σε and ε are serially uncorrelaed. All ARMA models are saionary. If an observed ime series is non-saionary (e.g. upward rend), i mus be convered o saionary ime series (e.g. by differencing) (Anderson, e al 1979; Pelgrin, 011)... ARFIMA Process ARFIMA process allows for long memory series o be fracionally inegraed. The general form of fracional ARMA model or he ARFIMA (p,d,q) model is: d ( B)( 1 B) y ( B) φ = θ ε () Where d denoes non-ineger fracional differencing parameer, y is he dependen variable a ime, ε is disribued normally wih mean 0 and variance σ, ε ( ) 1 φ B = φ B φ B φ B 1 p and p ( ) 1 θ B = 1+ θ B p + θb + + θpb represens AR and MA componens respecively; ( B) Φ and ( B) Θ have no common roos, B is he backward shif operaor and ( 1 ) B d is he fracional differencing operaor given by he binomial expansion: ( 1 ) large j is ηη jj Γ ( j+ d) ( j ) ( d) d j j B = B = η B Γ + 1 Γ j j= 0 j= 0 jj dd 1 for ; d ( 0.5 / 0.5) and ( ) Γ(d) ε is a whie noise sequence wih zero mean and Innovaion variance σ. The properies of an ARFIMA process area bridged in he following heorem. Le hen: (a) (b) y be an ARFIMA (p,d,q) process y is saionary and inverible if d < 0.5 and all he roo of φ ( B) = 0 lie ouside he uni circle. y is non-saionary if d 0.5 and all he roo of θ ( B) = 0 lie ouside he uni circle. (c) if 0.5 < d < 0.5, he auocovariance of d y; γ = E yy Bk, as k ; where B k k is a funcion of d. ( ) 1 The auocovariance funcion of ARFIMA process decay hyperbolically o zero as k in conras o he faser geomeric decay of a saionary ARMA process..3. Pormaneau Tes I is a es used for invesigaing he presence of auocorrelaion in ime series, number of lags µ and h are predeermined. Hypohesis HH OO : ρρ μμ,1 =. = ρρ μμ,h = 0 (all he lags correlaing are zero) H i : ρρ μμ,1 0 i.e for a leas one i=1,..h is esed i.e a leas one lag wih non zero correlaion. Tes Saisic LLLL h = TT h ( 1 jj =1 TT JJ ) ρρ μμ,jj = T 1 jj μμ iiii μμ ρρ μμ,jj where 3. Analysis of Daa and Resuls Figure 3.1 reveals he embedded long range dependency characerisics of he hourly daa poins in KEBBI beween 013/14. The daa are highly correlaed, here is a sense of reurn of a paricular characerisic afer some hours and his depics seasonaliy rais and fracional inegraed d series suspeced. There was a drop afer he firs seven housand hours of which i now picks and falls afer fifeen hours, also he graph poin shows he endency of high correlaion beween he se of daa poin which amoun o he fac he series is sable over ime. The auocorrelaion funcion of he firs wo housand five hundred (500) observaions in he daa se displayed a rai of seasonaliy as i can be seen in he figure above ha for every lag muliple of welve (600) here appeared rough and picks, very srong correlaion was noiced. The figure 3.3 revealed ha when he number of he lag is double we have a longer urning poin, very srong correlaion was noiced. The figure 3.4 shows ha here is a very srong correlaion among he daa poins and he sinusoidal movemen imiaes a seasonal rai.
300 Babayemi A. W. e al.: Long Range Dependency of Apparen Temperaure in Birnin Kebbi Figure 3.1. Visual Represenaion of Hourly Hea Index in KEBBI beween 013 and 014 Figure 3.. Visual Represenaion of Firs Two Thousand Five Hundred Hours Observaion Figure 3.3. Visual Represenaion of Five Thousand Hours Observaions
Inernaional Journal of Saisics and Applicaions 017, 7(6): 98-303 301 Figure 3.4. Visual Represenaion of Ten Thousand Hours Observaions Table 3.1. Descripive Saisics for he Hea Index a Differen Hours NOB Mean Sd Median Min. Max. Skewness Kurosis Se 15418 7.7 11.31 8 1 59 0.1 0.87 0.09 500 37.61 8.789 37 13.0 61 0.17.46 0.178 5000 4.59 11.99 43.00 5.0 55.6 15.6 67. 15.6 10000 41.1 11. 41.0 5.0 55.6 9.64 440.0 9.6 Table 3.1 shows he descripive saisics of he hea index where he all samples are sub divided in hree caegories he firs wo housand five hundred observaions, five housand observaions and en housand observaions; in order o examine he characerisics of he inegraed order of a differen sample hours. The minimum value of hea index for he hall sample is 1 and he maximum is 59. The average mean of he hall sample is 7.7. Table 3.. Pormaneau Tes for Auocorrelaion S/N ACF PACF Q-STAT p-value 1 0.093 0.093 13.5 0.000 0.087 0.079 49.36 0.000 3 0.081 0.068 351.68 0.000 4 0.075 0.057 438.68 0.000 5 0.070 0.049 514.3 0.000 6 0.06 0.038 57.69 0.000 7 0.055 0.031 619.3 0.000 8 0.048 0.04 655.35 0.000 9 0.044 0.00 684.64 0.000 10 0.037 0.014 705.6 0.000 11 0.033 0.011 71.8 0.000 1 0.030 0.010 736.1 0.000 13 0.033 0.014 75.9 0.000 14 0.035 0.017 771.98 0.000 15 0.041 0.03 797.84 0.000 16 0.045 0.06 88.96 0.000 17 0.050 0.030 867.43 0.000 18 0.054 0.03 91.44 0.000 19 0.059 0.035 967.04 0.000 0 0.065 0.038 103.0 0.000 1 0.071 0.041 1108.9 0.000 The able 3. above shows he resuls of he Pormaneau es a lag inerval of 00 of which he firs 1 of hem are displayed. I reveals highly correlaed se of daa poins which connoes non saionary level of he series. Table 3.3. ADF Tes for he Long Memory Process Dickey-fuller Saisics Lag-order p-value 6.058 4 0.01 The resul from dickey fuller appears as if he series of apparen emperaure is saionary unforunaely however, i is a long memory process where he value of he inegraed order d in his case is no uni raher, a fracional value. Having searched he ARFIMA models in an inerval of [0,3] for boh orders of p and q wih fracional inegraed d ( 0.5,0.5) using Schwarz Informaion Crierion, he ARFIMA(3, 0.4950, 1) end o be he mos preferable model, he resuls are displayed in Table 3.4 below.
30 Babayemi A. W. e al.: Long Range Dependency of Apparen Temperaure in Birnin Kebbi Table 3.4. Selecion of ARFIMA Models Model Loglikelihood d Sd error p- value ARFIMA(3,1) 1394.57 0.4950 0.00000017 0.00 ARFIMA(3,) 1391.45 0.4991 0.00000049 0.00 ARFIMA(,) 13869.91 0.4998 0.000196000 0.00 ARFIMA(3,3) 1673.58 0.4999 0.011980000 0.00 ARFIMA(,3) 1460.11 0.9960 0.000014500 0.00 Esimaion of he ARFIMA(3, 0.4950, 1) Variable coefficien Sd Error p-value d 0.4950 0.00000017 0.0000 AR(3) 0.1858 0.008396000 0.0000 MA(1) 0.536 0.00867000 0.0000 R-squared=0.86433; Loglikelihood=1394.57; Invered MA roo=-0.5 Invered AR Roos= 0.57,-0.9+50i, 0.9-50i 4. Discussion of Resuls The research invesigaed he long range dependency characerisics of apparen emperaure in Birnin Kebbi using ARFIMA model. The plos and he original auocorrelaion srucure are carefully examined and i is eviden from he ADF es ha he series under invesigaion does no conain uni roo, neiher can i be associaed wih a saionary o a series. I is noiced from he preliminary invesigaion he series possess long range dependency characerisics of d lying beween 0 and 0.5 using Geweke, e al (008) esimaor and ARFIMA. ARFIMA model gives a beer esimae. Long range dependency end o reflec he persisence of shock in a very long ime period afer i occurs, meaning ha; once a shocks is fel on he apparen emperaure, i will go ono a very long ime period for which he paern or srucure will be repeaed. This can also be seen in he auocorrelaion srucure of all samples and when i is sub divided in o hree caegories, he average ime of reurn is hree hundred and welve days which is approximaely one year o all samples. Sample 500 showed asinusoidal paern in he srucure, while samples 500, 5000 and 10000 imiaed he repeiive naure of shock persisence afer a very long period of ime. This shows ha here is always a period where he apparen emperaure is a pick or a lowes poin. This research work preven he misleading inegraed of d = 1 for ARIMA process. 5. Conclusions The research was se ou o invesigae he long memory behaviour of apparen emperaure in KEBBI meropolis, KEBBI Sae using hourly daa for a period of wo years (013/14). Time series mehodology of Auoregressive Fracional Inegraed Moving Average model (ARFIMA) is used in he sudy. The resuls revealed ha he series for he period invesigaed is highly correlaed and fracionally inegraed wih he value of d is 0.495 and he number of days for he reurn characerisics of he persisence of he shock is esimaed o be 31 days approximaely a year. REFERENCES [1] Agosinelli, C. and L. Bisaglia (003). Robus esimaion of ARFIMA process. Technical Repor UniversiàCàFoscari di Venezia. [] Anderson, O. D. and De Gooijer, J. G. (1979). \On discriminaing beween IMA(1,1) and ARMA(1,1) processes: Some exensions o a paper by Wichern," The Saisician, 8,(), Pp. 119-133. [3] Baillie, R.T. (1996). Long memoery process and fracional inegraion in economerics. Journal of Economerics 73; Pp. 5-59. [4] Baum, C.F. (013). ARFIMA (long memory) models, Boson college, www.bc,edu/ec-c/s 013/83/Ec83,s013.nn08.sli de pgf. [5] Conreras-Reyes, J.E. and W. Palma (013). Saisical Analysis of Auoregressive Fracionally Inegraed Moving Average models in R. Springer-Verlag Berlin Heidelberg. arxiv: 108.178v1 [sa.co] 8 Aug 01; doi.10.1007/s00180-013-0408-7. [6] Dickey, D.A. and W.A Fuller (1979). Disribuion of he Esimaors for Auoregressive Time Series wih a Uni Roo. Journal of he American Saisical Associaion, 74, Pp. 47 431. [7] Ding, Z., Granger, C.W.J. and R.F. Engle (1993). A Long Memory Propery of Sock Reurns and a New Model, Journal of Empirical Finance, 1, Pp. 83-106. [8] Epsein,Y., and D. Moran (006). Thermal comfor and he hea sress indices. Indusrial Healh, Pp. 388 398. [9] Pelgrin, F. (011). Lecure : ARMA(p,q) model (par 3). Universiy of Lausanne, _Ecole des HEC Deparmen of mahemaics (IMEA-Nice).
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