Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods

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1 International Journal of Scientific Research in Knowledge, 2(7), pp , 2014 Available online at ISSN: ; 2014 IJSRPUB Full Length Research Paper Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods Ette Harrison Etuk 1*, Tariq Mahgoub Mohamed 2 1 Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Port Harcourt, NIGERIA 2 Department of Civil Engineering, Sudan University of Science and Technology, SUDAN *Corresponding Author: ettetuk@yahoo.com, ettehetuk@gmail.com Received 17 May 2014; Accepted 22 June 2014 Abstract. The time series being rainfall data is a typical seasonal series of one-year period. The time-plot of the realization herein called GASR and its correlogram are as expected, reflecting seasonality of period 12. For instance, the autocorrelation function is oscillatory of period 12. A 12-point differencing yields a series called SDGASR with a generally horizontal secular trend. It is adjudged stationary by the Augmented Dickey Fuller unit root test. Its correlogram gives an indication of stationarity as well as an involvement of the presence of a seasonal moving average component of order one and a seasonal autoregressive component of order two. This autocorrelation structure suggests three multiplicative SARIMA models, namely: (0, 0, 0)x(0, 1, 1) 12, (0, 0, 1)x(0, 1, 1) 12 and (0, 0, 1)x(2, 1, 1) 12. The first model is adjudged the most adequate. Its residuals have been observed to be uncorrelated. It may be the basis for the forecasting of rain in the region for planning purposes. Keywords: Sudan, Gadaref station, rainfall, Sarima models, time series analysis 1. INTRODUCTION Sudan is one of the countries whose economy is highly dependent on rain-fed agriculture and also facing recurring cycles of drought. Rainfall is considered as the most important climatic element that influences agriculture. Therefore monthly rainfall forecasting plays an important role in the planning and management of agricultural scheme and management of water resource systems. In this study, linear stochastic models known as multiplicative seasonal autoregressive integrated moving average (SARIMA) models were used to model monthly rainfall in Gadaref station. The region was selected as a result of its being the most important agricultural productive area, under rain-fed, in Sudan. The physical area considered in this study is a portion of the Gadaref region. Gadaref region lies in East Central part of Sudan, at the border with Ethiopia. The region experiences very hot summer and temperature in the region reaches up to 45 C in May. Generally the dry periods are accompanied with high temperatures, which lead to higher evaporation affecting natural vegetation and the agriculture of the region along with larger water resources sectors. Annual potential evapotranspiration exceeds annual precipitation in this region. The rainfall exceeds evapotranspiration only in August and September. This Gadaref station boundary coincides with 550 mm annual rainfall isoyhets. The climate in the Gedaref is semi-arid with mean annual temperature near 30 C (Elagib and Mansell, 2000). Gadaref region has a good fertile soil and relatively high rainfall intensities all over the region. Farming of sorghum and sesame covers much of the region land. The region is very important for the economy of Sudan. More than 70% of sorghum, which is one of the main food crops in the country, is grown in the rain-fed subsector. Seasonal time series are often modeled by SARIMA techniques. Rainfall the world over is a seasonal phenomenon with period 12 months. A few researchers who have modeled rainfall using SARIMA methods in recent times are Nirmala and Sundaram (2010), Rahman (2011), Ibrahim and Dauda (2012), Yusuf and Kane (2012), Osarumwese (2013), Abdul Aziz et al. (2013), Ali (2013), Wang et al. (2013) and Etuk et al. (2013). For instance, Nimarla and Sundaram (2010) fitted a SARIMA(0, 1, 1)x(0, 1, 1) 12 model to monthly rainfall in Tamilnadu, India. Abdul-Aziz et al. (2013) examined rainfall data pattern in Ashanti region of Ghana and fitted a SARIMA(0, 0, 0)x(2, 1, 0) 12 to it. Osarumwese (2013) modeled quarterly rainfall in Port Harcourt, Nigeria, 320

2 Etuk and Mohamed Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods as a SARIMA(0, 0, 0)x(2, 1, 0) 4 model. Yusuf and Kane (2012) fitted the SARIMA models of orders (1, 1, 2)x(1, 1, 1) 12 and (4, 0, 2)x(1, 0, 1) 12 respectively for monthly rainfall in Malaaca and Kuantan in Malaysia. Etuk et al (2013) modeled monthly rainfall in Port Harcourt, Nigeria as SARIMA(5, 1, 0)x(0, 1, 1) 12. Fig. 1: GASR 2. MATERIALS AND METHODS 2.1. Data For this study, a Gadaref rainfall gauge was considered and 480 monthly rainfall data was procured for the period from 1971 to Wei (1990) states that a minimum number of 50 observations are needed to build reasonable autoregressive integrated moving average (ARIMA) model. The monthly rainfall records for Gadaref station show most of the rain falls in the period from June to September, and reaches its peak in August. The maximum intensity of rain is in the range of mm/h usually in the form of convective showers and thunderstorms of short duration, small aerial extent and high intensity Modelling by Sarima Methods A stationary time series {Xt} is said to follow an autoregressive moving average model of orders p and q, denoted by ARMA(p, q) if it satisfies the difference equation X t - 1 X t-1-2 X t p X t-p = t + 1 t t-2 + q t-q (1) Here the sequence of random variables { t } is a white noise process. Moreover the s and the s are constants such that the model is both stationary and invertible. Model (1) may be written as A(L)X t = B(L) t (2) where A(L) is called the autoregressive (AR) operator and given by A(L) = 1-1 L - 2 L p L p and B(L) is called the moving average (MA) operator and defined as B(L) = L + 2 L q L q. Here L is the backshift operator defined by L k X t = X t-k. For stationarity, the zeros of A(L) = 0 must lie outside the unit circle. Similarly, for invertibility, the zeros of B(L) = 0 must lie outside the unit circle. If the time series {X t } is non-stationary as is often the case, Box and Jenkins (1976) made a proposal that differencing to an appropriate degree could make the series to be stationary. If the minimum degree to which the series is differenced to attain stationarity is d then if the diferenced series denoted by { d X t } satisfies (1), the original series is said to follow an autoregressive integrated moving average model or orders p, d and q and designated ARIMA(p, d, q). Here the difference operator = 1 L. Seasonality shall be tested by the Augmented Dickey Fuller (ADF) test. If the series {X t } is seasonal of period s, Box and Jenkins (1976) further proposed that it could be modeled as 321

3 International Journal of Scientific Research in Knowledge, 2(7), pp , 2014 A(L) (L s ) d D sx t = B(L) (L s ) t (3) where (L) and (L) are called the seasonal AR and MA operators respectively. Suppose they are respectively polynomials of order P and Q in L, and the coefficients are such that the model (3) is both stationary and invertible, the time series {X t } is said to follow a seasonal autoregressive integrated moving average of orders p, d, q, P, D, Q and s designated SARIMA(p, d, q)x(p, D, Q) s model. The operator s is the seasonal difference operator defined by s = 1 L s and D is the seasonal differencing order. Fig. 2: Correlogram of Gasr The fitting of the model (3) begins with order determination. The seasonality period s may be obvious from the nature or time-plot of the series. For instance as mentioned in section 1, rainfall is a seasonal time series with s = 12 months. If s is not that obvious from the time plot the autocorrelation function (ACF) could reveal the value of s, as the lag where the function is significant. The differencing operators d and D are often chosen to be at most equal to 1 each. The nonseasonal and seasonal AR orders p and P are estimated by the nonseasonal and the seasonal cut-off lags of the partial autocorrelation function (PACF) respectively. Similarly the nonseasonal and the seasonal MA orders q and Q are estimated respectively by the nonseasonal and seasonal cut-off points of the ACF. Once the orders have been determined model fitting invariably involves the application of nonlinear optimization techniques like the least squares procedure or the maximum likelihood procedure. A fitted model must be subjected to some residual analysis to ascertain its goodness-of-fit to the data. In this work the statistical and econometric software Eviews was used for all analytical work. 322

4 Etuk and Mohamed Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods Fig. 3: SDGASR Table 1: Estimation Of The Sarima(0, 0, 0)X(0, 1, 1) 12 Model 3. RESULTS AND DISCUSSION The time-plot of the realization which we call GASR in Figure 1 shows as expected seasonality of period 12 months. Compared to Port Harcourt which lies in the rainfall belt where rainfall falls virtually every month of the year (see for example, Etuk et al (2013)), the rainfall in Gadaref is such that long seasons of drought separate seasons of rainfall. The ADF test adjudges GASR as stationary. However the ACF in Figure 2 of GASR shows clearly that the stationarity hypothesis cannot be true. The ACF exhibits oscillatory movements of period 12 months. This shows that GASR cannot be stationary but seasonal of period 12. The ACF is oscillatory of period 12, an indication of non-stationarity. A seasonal differencing yields SDGASR which exhibits a horizontal secular trend as evident in Figure 3. Both the ADF test and the ACF in Figure 4 show that SDGASR is stationarity. Moreover the ACF shows up seasonality of order 12 and the existence of a seasonal MA component of order 1. The PACF shows 323

5 International Journal of Scientific Research in Knowledge, 2(7), pp , 2014 evidence of the involvement of a seasonal AR component of order 2. Based on this autocorrelation structure three models are proposed and fitted: 1) A SARIMA(0, 0, 0)x(0, 1, 1) 12 model estimated in Table 1 by SDGASR t = t-12 + t (4) 2) A SARIMA(0, 0, 1)x(0, 1, 1) 12 model estimated in Table 2 by SDGASR t = t t t-13 + t (5) 3) A SARIMA(0, 0, 1)x(2, 1, 1) 12 model estimated in Table 3 by R 2 for models (4), (5) and (6) are 46.62%, 46.59% and 40.31% respectively. This means that model (4) accounts the data the most. Also, of the three models, (4) has the lowest Akaike Information Criterion (AIC). The correlogram of its residuals in Figure 5 shows that the residuals are uncorrelated. Hence the model is adequate. Model (4) is MA model whereby the current value of SDGASR depends on the unobserved current value and the 12-month earlier values of the white noise or random shocks. 4. CONCLUSION It may be concluded that the monthly rainfall in Gadaref, Sudan follows a SARIMA(0, 0, 0)x(0, 0, 1) 12 model. It may be used as the basis for forecasting, planning and management of the rainfall in this region. SDGASR t SDGASR t SDGASR t-24 = t t t t-13 (6) Table 2: The Estimation Of Sarima(0, 0, 1)X(0, 1, 1) 12 Model 324

6 Etuk and Mohamed Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods Fig. 4: Correlogram of Sdgasr Table 3: The Estimation Of Sarima(0, 0, 1)X(2, 1, 1) 12 Model REFERENCES Abdul-Aziz AM, Kwame A, Munyakazi L, Nsowa- Nuamah NNN (2013). Modelling and Forecasting Rainfall Pattern in Ghana as a Seasonal Arima Process: The Case of Ashanti Region. International Journal of Humanities and Social Science, 3(3): Ali SM (2013). Time Series Analysis of Baghdad Rainfall Using ARIMA method. Iraqi Journal of Science, 54(4):

7 International Journal of Scientific Research in Knowledge, 2(7), pp , 2014 Box GEP, Jenkins GM (1976). Time Series Analysis, Forecasting and Control, Holden-Day: San Francisco. Elagib NA, Mansell MG (2000). Recent trends and anomalies in mean seasonal and annual temperatures over Sudan. Journal of Arid Environments, 45(3): Etuk EH, Moffat IU, Chims BE (2013). Modelling Monthly Rainfall data of Port Harcourt, by Seasonal Box-Jenkins Methods. International Journal of Science, 2: Ibrahim LK, Dauda U (2012). Modeling Monthly Rainfall Time Series Using Ets State Space and Sarima Models. International Journal of Physics and Mathematical Research, 1(1): Nirmala M, Sundaram SM (2010). A Seasonal Arima Model for Forecasting monthly rainfall in Tamilnadu. National Journal on Advances in Building Sciences and Mechanics, 1(2): Osarumwense O (2013). Applicability of Box Jenkins SARIMA Model in Rainfall Forecasting: A Case study of Port Harcourt south south Nigeria. Canadian journal in Computing Mathematics, Natural Sciences, Engineering and Medicine, 4(1): 1 4. Rahman MA (2011). Forecasting and Modelling Rainfall Data in Bangladesh: By Seasonal Auto Regressive Integrated Moving Average (SARIMA), Lambert Academic Publishing (LAP). Wang S, Feng J, Liu G (2013). Application of Seasonal Time Series Model in the precipitation forecast. Mathematical and Computer Modelling, 58(3&4): Wei WWS (1990). Time Series Analysis. Addison- Wesley Publishing, Reading, MA, USA. Yusuf F, Kane IL (2012). Modeling Monthly Rainfall Time Series using ETS state space and Sarima models. International Journal of Current Research, 4(9): Fig. 5: Correlogram Of Sarima(0, 0, 0)X(0, 1, 1) Residuals 326

8 Etuk and Mohamed Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station, Sudan, by Sarima Methods Dr Ette Harrison Etuk is an Associate Professor of Statistics in the Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Port Harcourt, Nigeria. He has produced many graduates in both undergraduate and graduate levels in his many years of experience in University teaching and administration. He has published extensively in reputable journals. His research interests are in the areas of Time Series Analysis, Operations Research and Experimental Designs. Tariq Mahgoub Mohamed was born on January 1, He is a Sudanese by nationality. He has B. Sc. Degree in Water Resources Engineering from the University of Khartoum, Khartoum, Sudan in 1998, M. Sc. Degree in Water Resources Engineering from the same University in Currently he is doing his Ph. D. in Civil Engineering Hydrology in Sudan University of Science and Technology, Sudan. 327