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

International Journal of Scientific Research in Knowledge, 2(7), pp. 320-327, 2014

Available online at http://www.ijsrpub.com/ijsrk

ISSN: 2322-4541; 2014 IJSRPUB

http://dx.doi.org/10.12983/ijsrk-2014-p0320-0327

320

Full Length Research Paper

Time Series Analysis of Monthly Rainfall data for the Gadaref rainfall station,

Sudan, by Sarima Methods

Ette Harrison Etuk1*

, Tariq Mahgoub Mohamed2

1Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Port Harcourt, NIGERIA

2Department of Civil Engineering, Sudan University of Science and Technology, SUDAN

*Corresponding Author: [email protected], [email protected]

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 45C 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 30C (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,

Etuk and Mohamed

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

321

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 2010. 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 100

150 mm/h usually in the form of convective showers and thunderstorms of short duration, small aerial

extent and high intensity.

2.2. 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

Xt - 1Xt-1 - 2Xt-2 - - pXt-p = t + 1t-1 + 2t-2 +

qt-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)Xt = B(L)t (2) where A(L) is called the autoregressive (AR)

operator and given by A(L) = 1 - 1L - 2L2 - -

pLp and B(L) is called the moving average (MA)

operator and defined as B(L) = 1 + 1L + 2L2 + +

qLq. Here L is the backshift operator defined by L

kXt

= Xt-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 {Xt} 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 {dXt} 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 {Xt} is seasonal of period s, Box and

Jenkins (1976) further proposed that it could be

modeled as


322

A(L)(Ls)dDsXt = B(L)(Ls)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 {Xt} 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 non-

linear 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.

Etuk and Mohamed


323

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


324

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

SDGASRt = -0.8856 t-12 + t (4)


SDGASRt = -0.4303t-1 0.8798t-12 + 0.04344t-13 +

t (5)


SDGASRt + 0.1339SDGASRt-12 + 0.1652SDGASRt-24

= t 0.0706t-1 0.7579t-12 + 0.0788t-13 (6)

R2 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.

Table 2: The Estimation Of Sarima(0, 0, 1)X(0, 1, 1)12 Model

Etuk and Mohamed


325

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): 224 233. Ali SM (2013). Time Series Analysis of Baghdad

Rainfall Using ARIMA method. Iraqi Journal

of Science, 54(4): 1136 1142.


326

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): 263 288. Etuk EH, Moffat IU, Chims BE (2013). Modelling

Monthly Rainfall data of Port Harcourt, by

Seasonal Box-Jenkins Methods. International

Journal of Science, 2: 60 65. 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): 011 014. 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): 43 47.

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): 677 683. 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): 195 200.

Fig. 5: Correlogram Of Sarima(0, 0, 0)X(0, 1, 1) Residuals

Etuk and Mohamed


327

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, 1975. 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 2005. Currently

he is doing his Ph. D. in Civil Engineering Hydrology in Sudan University of Science and Technology, Sudan.

Documents

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