Modeling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis.

Transcription

1 Journal of Mahemaical Modelling and Applicaion 01, Vol. 1, No.5, ISSN: Modeling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis. Md. Mahsin Insiue of Saisical Research and Training (ISRT), Universiy of Dhaka, Yesmin Akher Insiue of Saisical Research and Training (ISRT), Universiy of Dhaka Monira Begum Insiue of Saisical Research and Training (ISRT), Universiy of Dhaka, Absrac Time series analysis and forecasing has become a major ool in differen applicaions in meeorological phenomena, such as rainfall, humidiy, emperaure, draugh ec. and environmenal managemen fields. Among he mos effecive approaches for analyzing ime series daa is he model inroduced by Box and Jenkins, ARIMA (Auoregressive Inegraed Moving Average). In his sudy we used Box-Jenkins mehodology o build seasonal ARIMA model for monhly rainfall daa aken for Dhaka saion for he period from (June) wih a oal of 354 readings. In his paper, ARIMA (0, 0, 1) (0, 1, 1) 1 model was found adequae and his model is used o forecasing he monhly rainfall for he upcoming wo years o help decision makers o esablish prioriies in erms of waer demand managemen. Keywords: Time series analysis, Rainfall forecasing, Box-Jenkins mehodology, SARIMA model. Mehodology The ARIMA model is an imporan forecasing ool, and is he basis of many fundamenal ideas in ime-series analysis. The acronym ARIMA sands for auoregressive inegraed moving average, and i is someimes called Box-Jenkins models. An auoregressive model of order p is convenionally classified as AR (p) and a moving average model wih q erms is known as MA (q). A combined model ha conains p auoregressive erms and q moving average erms is called ARMA (p, q) (Gujarai, 1995). If he objec series is differenced d imes o achieve saionariy, he model is classified as ARIMA (p, d, q), where he symbol I signifies inegraed. Thus, an ARIMA model is a combinaion of an auoregressive (AR) process and a moving average (MA) process applied o a non-saionary daa series. So he general non-seasonal ARIMA (p, d, q) model is as: AR: p = order of he auoregressive par, I: d = degree of differencing involved and MA: q = order of he moving average par. The equaion for he simples ARIMA (p, d, q) model is as follows: Y = c + φ Y + φ Y θ e φpy p + e θ1e 1 θe p p (1) or in backshif noaion (1 φ p 1B φ B... φ p B ) Y = c + (1 θ1b θ B... θ q B q ) e () Where, c = consan erm, i φ = i h auoregressive parameer, j = j h moving average parameer, e = he error erm a ime and B k = k h order backward shif operaor. In addiion o he non-seasonal ARIMA (p, d, q) model, inroduced above, we could idenify seasonal ARIMA (P, D, Q) parameers for our daa. These parameers are: Seasonal auoregressive (P),

2 68 Md. Mashin, Yesmin,Akher, Monira Begum seasonal Differencing (D) and seasonal moving average (Q). The general form of he Seasonal ARIMA (p, d, q) (P, D, Q) S model using backshif noaion is given by, φ ( φ = θ θ e... (3) AR s d s D s B ) SAR ( B )(1 B) (1 B ) Y MA ( B) SMA ( B ) Where, s = number of periods per season, φ AR = non-seasonal auoregressive parameer, θ = non-seasonal moving average parameer, θ = seasonal moving average parameer. MA The Box-Jenkins (BJ) Mehodology SMA In ime series analysis, he Box Jenkins mehodology, named afer he saisicians George Box and Gwilym Jenkins, applies auoregressive moving average ARMA or ARIMA models o find he bes fi of a ime series o pas values of his ime series, in order o make forecass. This approach possesses many appealing feaures. To idenify a perfec ARIMA model for a paricular ime series daa, Box and Jenkins (1976) proposed a mehodology ha consiss of four phases viz. i) Model idenificaion; ii) Esimaion of model parameers; iii) Diagnosic checking for he idenified model appropriaeness for modeling and iv) Applicaion of he model (i.e. forecasing). The firs sep in developing a Box Jenkins model is o deermine if he ime series is saionary and if here is any significan seasonaliy ha needs o be modeled. The daa was examined o check for he mos appropriae class of ARIMA processes hrough selecing he order of he consecuive and seasonal differencing required making series saionary, as well as specifying he order of he regular and seasonal auoregressive and moving average polynomials necessary o adequaely represen he ime series model. The Auocorrelaion Funcion (ACF) and he Parial Auocorrelaion Funcion (PACF) are he mos imporan elemens of ime series analysis and forecasing. The ACF measures he amoun of linear dependence beween observaions in a ime series ha are separaed by a lag k. The PACF plo helps o deermine how many auo regressive erms are necessary o reveal one or more of he following characerisics: ime lags where high correlaions appear, seasonaliy of he series, rend eiher in he mean level or in he variance of he series. Saionary of a daa se can also be idenified by performing Pormaneau es, which is used o es wheher a daa se is significanly differen from a zero se. A common pormaneau es is he Box-Pierce es, designed by Box and Pierce (1970). This residual from a forecas model es is based on he Box-Pierce saisic: Where, h is he maximum lag being considered, n is he number of observaions in he series and r k is he auocorrelaion a lag k. If he residuals are whie noise, he saisic Q has a chi-square ( χ ) disribuion wih degrees of freedom (h - m) where m is he number of parameers in he model which has been fied o he daa. An alernaive pormaneau es is he Ljung-Box due o Ljung and Box [9]. They argued ha he alernaive saisic: n * rk Q = n( n + )... ] (4) n k k= 1 Q = n h r k k = 1 has a disribuion closer o he chi-square disribuion wih (h - m) degrees of freedom han does he Q saisic. I is normal o conclude ha he daa are no whie noise if he value of Q or Q* lies in he exreme 5% of he righ-hand ail of he ( χ ) disribuion. To choose he bes model among he class of plausible models we use he Akaike's Informaion Crierion (AIC), proposed by Akaike (1974). The model which has he minimum AIC value is our model of ineres.

3 Modeling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis 69 Afer choosing he mos appropriae model (sep 1 above) he model parameers are esimaed (sep ) by using he leas square mehod. In his sep, values of he parameers are chosen o make he Sum of he Squared Residuals (SSR) beween he real daa and he esimaed values as small as possible. In general, nonlinear esimaion mehod is used o esimae he above idenified parameers o maximize he likelihood (probabiliy) of he observed series given he parameer values. Maximum likelihood esimaion is generally he preferred echnique. In diagnose checking sep (sep hree), he residuals from he fied model shall be examined agains adequacy. This is usually done by correlaion analysis hrough he residual ACF plos. If he residuals are correlaed, hen he model should be refined as in sep one above. Oherwise, he auocorrelaions are whie noise and he model is adequae o represen our ime series. Afer he applicaion of he previous procedure for a given ime series, a calibraed model will be developed which has enclosed he basic saisical properies of he ime series ino is parameers (sep four). Resuls and Discussion Since he daa is a monhly rainfall, Fig. 1, shows ha here is a seasonal cycle of he series and he series is no saionary. The ACF and PACF of he original daa, as shown in Fig., show ha he rainfall daa is no saionary. In order o fi an ARIMA model saionary daa in boh variance and mean are needed. We could aain saionariy in he variance by having log ransformaion and differencing of he original daa o aain saionary in he mean. For our daa, we need o have seasonal firs difference, D = 1, of he original daa in order o have saionary series. Afer ha, we need o es he ACF and PACF for he differenced series o check saionary. As shown in Fig. 3, he ACF and PACF for he differenced and de-seasonalized rainfall daa are almos sable which suppor he assumpion ha he series is saionary in boh he mean and he variance afer having 1s order non seasonal difference. Therefore, an ARIMA (p, 0, q) (P, 1, Q) 1 model could be idenified for he differenced and de-seasonalized rainfall daa. Afer ARIMA model was idenified above, he p, q, P and Q parameers need o be idenified for our model. Rainfall Year Figure 1: Monhly rainfall daa for Dhaka saion (1981-June, 010).

4 70 Md. Mashin, Yesmin,Akher, Monira Begum ACF Series: PACF Figure. : Plo of ACF (op) and PACF (boom) for original rainfall daa. The main ask in auomaic ARIMA forecasing (Rob J. Hyndman, Yeasmin Khandakar, 008) is selecing an appropriae model order, ha is he values p, q, P, Q, D, d. If d and D are known, we can selec he orders p, q, P and Q via an informaion crierion such as he AIC: AIC = log (L) + (p + q + P + Q + k) where k = 1 if c 0 and 0 oherwise, and L is he maximized likelihood of he model fied o he s D d ( 1 B ) (1 B) Y differenced daa. On he basis of he auomaic ARIMA forecasing our seleced model ARIMA (0, 0, 1) (0, 1, 1) 1 is adequae o represen our daa and could be used o forecas he upcoming rainfall daa. ACF PACF Figure. 3: Auocorrelaion (ACF) and Parial Auocorrelaion (PACF) for firs order seasonal differencing and de-seasonalized original rainfall daa. As discussed before ARIMA (0, 0, 1) (0, 1, 1) 1 model could be wrien in he following form: 1 1 (1 B ) Y = (1 θ1 MAB)(1 θ1 SMAB ) e...(5) This equaion can be muliplied ou and wrien in a form ha is used in forecasing as shown in Eq. 6: Y...(6) = Y 1 + e θ 1MAe 1 θ1 SMAe 1 θ1maθ1 SMAe 13

5 Modeling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis 71 The following able gives he maximum likelihood esimaes and heir sandard errors for he ARIMA (1, 0, 0) (0, 1, 1) 1 model. Table 1: Parameer Esimaion for a ARIMA (0, 0, 1) (0, 1, 1) 1 model Coefficiens Esimae Sandard error P-Value θ 1MA θ 1SMA ˆσ e = 1871: log likelihood = , AIC = , AICc = , BIC = Figure 4 displays a plo of he sandardized residuals, he ACF of he residuals, heir normal Q-Q plo and he p-values associaed wih he Q-saisic, (4), for his model and i appears his model fis he daa well. Inspecion of he ime plo of he sandardized residuals in Figure 4 shows few ouliers in he series. The ACF of he sandardized residuals shows no apparen deparure from he model assumpions, and he Q-saisic is never significan a he lags shown. The Ljung-Box es for his model gives a chi-squared value of 3.7 wih 34 degrees of freedom, leading o a p- value of 0.9 a furher indicaion ha he model has capured he dependence in he ime series. The normal Q-Q plo of he residuals shows deparure from normaliy a he ails due o he ouliers bu he Shapiro-Wilk es of normaliy has a es saisic of W = 0.998, leading o a p-value of 7.565e-1 ( ), and normaliy is no rejeced a any of he usual significance levels. Sandardized Residuals Time ACF of Residuals Normal Q-Q Plo of Sd Residuals ACF Sample Quaniles Theoreical Quaniles p values for Ljung-Box saisic p value lag Figure 4: Diagnosics for he ARIMA (0, 0, 1) (0, 1, 1) 1 fi on he rainfall daa. Figure 5 shows a comparison beween he real values and he ones resuled from he developed ARIMA model for he period beween 1981 and 010. Using he ARIMA (0, 0, 1) (0, 1, 1) 1 model, we obain he graphical plo for he acual pich series versus he prediced pich series and from he visual inspecion of he plo i is quie eviden ha he chosen model is good enough as he prediced

6 7 Md. Mashin, Yesmin,Akher, Monira Begum series is very close o he observed series and his model could be used o forecas he upcoming rainfall in Dhaka, Bangladesh. Acual Vs Fied values of he Rainfall daa Rainfall Acual Values Fied Values Year Figure 5: Comparison graph of acual vs. fied values of he Rainfall daa. Table : Forecased amoun of rainfall (& CI) in Dhaka, Bangladesh during he period July 010- June 01. Year Forecased Rainfall(millimee Confidence Inerval (95%) of Forecased Rainfall (mm) r) LCL UCL July Aug Sep Oc Nov Dec Jan Feb Mar Apr May June July Aug Sep Oc Nov Dec Jan Feb Mar Apr May June Forecasing or predicing fuure is one of he main reasons for developing ime series models. On he basis of he developed model Table displays he forecased average monhly rainfall ogeher wih 95% forecas limis for wo years.

7 Modeling Rainfall in Dhaka Division of Bangladesh Using Time Series Analysis 73 Conclusion In his paper, we aemped o forecas monhly rainfall using SARIMA model. As he RMSE values on es daa are comparaively less, he predicion model is reliable. By comparing he fied and acual values of rainfall daa using our deermined model he rainfall forecass are sufficienly accurae. Our seleced ARIMA (0, 0, 1) (0, 1, 1) 1 model give us wo years prediced monhly rainfall along wih heir 95% confidence inerval ha can help decision makers o esablish sraegies, prioriies and proper use of waer resources in Dhaka. References Akaike, Hirougu., A new look a he saisical model idenificaion. IEEE Transacions on Auomaic Conrol 19 (6): DOI: /TAC MR Banglapedia 003. Naional Encyclopedia of Bangladesh. Asiaic Sociey of Bangladesh: Dhaka. Box, G.E.P. and G.M. Jenkins, Time Series Analysis: Forecasing and Conrol. Revised Edn., Hoden-Day, San Francisco. Box, G. E. P. and Pierce, D. A., Disribuion of he Auocorrelaions in Auoregressive Moving Average Time Series Models, Journal of he American Saisical Associaion, 65 : Dore, MHI., 005. Climae change and changes in global precipiaion paerns: wha do we know? Environ. In. 31(8): Gujarai D. N., Basic Economerics, 5 h Edn., McGraw-Hill Book Co., New York. Hulme, M., Osborn, T. J. and Johns, T. C., Precipiaion sensiiviy o global warming: comparison of observaions wih HADCM simulaions. Geophysical Res. Leer 5: Kayano, M. T. and Sans ıgolo, C Inerannual o decadal variaions of precipiaion and daily maximum and daily minimum emperaures in souhern Brazil. Theoreical and App. Clim. 97(1- ): Ljung, G. M. and Box, G. E. P., On a measure of lack of fi in ime series models. Biomerika, 65 : [10] Rob J. Hyndman, Yeasmin Khandakar, 008. Auomaic Time Series Forecasing: The forecas Package for R, 7(3): 1-, DOI: hp:// Shahid S., 008. Spaial and emporal characerisics of droughs in he wesern par of Bangladesh. Hydrol. Processes (13): Waler Vandaele, Applied ime series and Box-Jenkins Models. Academic Press Inc., Orlando, Florida, ISBN: 10: , pp: 417.