American Journal of Innovative Research and Applied Sciences. ISSN 24295396 I www.americanjiras.com
ORIGINAL ARTICLE
 Lama Nizar Shiban 1  Ali Mohamad Alasaad 2  and  Abbas Abdulrahman Abdulrahman 3 
1. Tishreen University  department of Water Engineering and Irrigation  Lattakia  Syria 
2. Tishreen University  department of Water Engineering and Irrigation  Lattakia  Syria 
3. Tishreen University  department of Water Engineering and Irrigation  Lattakia  Syria 
 Received  16 December 2018   Accepted 21 January 2019   Published 27 January 2019   ID Article  ShibanManuscriptRef.1ajira160119Seriya1 
ABSTRACT
Background: Water is a basic element of life and it has become increasingly needed nowadays, especially in areas with low rainfall, due to the great development in various aspects of life. Objectives: The objective of this research is to study the rainfall in ALHWAIZ basin to explain the behavior of the time series and applying one model of BoxJenkins models to predict the future amounts. Methods: The Statistical analysis study was based on annual rainfall data for 59 years, and Through it we found the best model to represent the data after doing the wanted tests on model residuals. also, we dividing the data to 53 years to build the model and 6 years to test it, and finding the best forecasted model depending on the smallest of weighted mean of criteria RMSE, MAP, MAE. Results: This research finds that Autoregressive Integrated Moving Average Model (ARIMA Model) ARIMA (1,1,3) model is a good representation of the data, and ARIMA (2,1,0) model is the right model to forecast future rainfall which was decreasing amounted to 6.13 mm per year during the studied period. Conclusions: The study reveals that ARIMA model can be used as a good forecasting tool in forecasting the annual rainfall in ALHWAIZ basin.
Keywords: ALHWAIZ basin, ARIMA Model, Rainfall.
1. INTRODUCTION
The analysis of time series is one of the most important statistical methods, because it enables to know the changes in the studied phenomenon over time and forecasting the future values of it [1]. That important because it decrease the effects of extreme events such as floods and drought [2]. The time series can be stationary and nonstationary, and the series is considered stable if it has constant (mean and variance) [3]. Many researchers have under taken many studies to forecast rainfall using different techniques [4]. Because it is considered one of the most difficult elements of the hydrological cycle due to the complexity of the air operations that contribute to the formation of rainfall [5]. We will present in this research process of building a forecasting model for the rainfall of ALHWAIZ basin by using BoxJenkins method, which is one of the best methods that give good results.
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<sect1><title>2. MATERIALS AND METHODS 2. MATERIALS AND METHODS 2.1 Study site & data available ALHWAIZ basin is located on the eastern coast of the Mediterranean Sea in northwest of Syria. The annual rainfall data from 1959 until 2017 is measured for three rain stations that are covering the whole basin (Skhaba, Hmemem, Ein sharqiyeh).2.2 ARIMA Model: The Autoregressive Integrated Moving Average Model (ARIMA Model) is a time series model used for shortterm forecasting. General form of ARIMA (p,d,q) model is following as Eq.(1). [6]:
∅
B
(
1
−
B
)
d
Z
t
=
δ
+
θ
(
B
)
a
t
{∅ B left (1B right )} ^ {d} {Z} rsub {t} =δ+θ left (B right ) {a} rsub {t}
(1)Where
:
∅
B
=
1
−
∅
1
B
−
∅
2
B
2
−
…
∅
p
B
p
: ∅ B =1 {∅} rsub {1} B {∅} rsub {2} {B} ^ {2} … {∅} rsub {p} {B} ^ {p}
:
θ
B
=
1
−
θ
1
B
−
θ
2
B
2
−
θ
q
B
q
:θ B =1 {θ} rsub {1} B  {θ} rsub {2} {B} ^ {2}  {θ} rsub {q} {B} ^ {q}
∅
p
,
θ
q
:
{∅} rsub {p} , {θ} rsub {q} :
are parameters;
B
:
B:
backward operator;
d
d
: order of differences;
Z
t
:
{Z} rsub {t} :
the actual value;
δ
:
δ:
model parameter;
a
t
:
{a} rsub {t} :
independent random variable represent the error term at time t;
p
p
: the order of autoregressive process;
q
q
: the order of moving average [6, 7]. 2.3 BoxJenkins methodology: It was used to analyze the rainfall time series by using Minitab software [8]. This method can identify the order of ARIMA model, estimation, diagnostic checking and forecasting [9]. 2.4 Modeling procedure: We plot the data and transform it if necessary, whereas BoxJenkins method is only occasion for stationary time series data [10]. Models can be determined by the plots of Autocorrelation function (ACF) and Partial autocorrelation function (PACF), where it gives more information about behavior of time series [11]. We use Akaike's Information Criterion (AIC) in Eq.(2) to search for the best model [12]. After that we select min (AIC) as the best model that determines a model by how close its fitted values tend to be to the natural values in terms of a specific expected value [13].
σ
(
2
)
+
2
∗
m
(
2
)
AIC
(
m
)
=
n
∗
ln
AIC(m)=n*ln {(σ} ^ {2} )+2*m (2)
Where m: number of parameters in the model; n: number of observation;
σ
2
:
{σ} ^ {2} :
variation of residuals.Where m: number of parameters in the model; n: number of observation;
σ
2
:
{σ} ^ {2} :
variation of residuals.The most important part is to make sure that the residuals of the model random, and the estimated parameters are statistically significant [14]. So, we check the residuals from our chosen model by plotting (ACF) and (PACF) of the residuals and doing many tests to analysis model's residuals [15]. If the model is not suitable, the cycle repeats from the identification of the model [16]. It is not necessary that representation model is the same as the future forecasting model, so we depend on forecasting model to prediction [1].Finally, we generate forecasts according to the final forecasting model, and depending on forecasting criteria which are: Root Mean Squared Error (RMSE) Eq.(3), Mean Absolute Deviation (MAD) Eq.(4), and Mean Absolute Percentage Error (MAPE) Eq.(5), and calculating the smallest of weighted mean of them which is given by Eq.(6) to Eq.(8) [2].
RMSE
=
∑
i
=
1
n
(
Y
t
−
Y
^
)
2
n
(
3
)
RMSE = sqrt {{sum from {i =1} to {n} {{( {Y} rsub {t}  widehat {Y} )} ^ {2}}} over {n}} ( 3 )
MAD
=
1
n
∑
t
=
1
n

Y
t
−
Y
^
t

(
4
)
MAD= {1} over {n} sum from {t=1} to {n} {left lline {Y} rsub {t}  {widehat {Y}} rsub {t} right rline} ( 4 )
MAPE
=
1
n
∑
t
=
1
n

Y
t
−
Y
^
t
Y
t

(
5
)
MAPE= {1} over {n} sum from {t=1} to {n} {left lline {{Y} rsub {t}  {widehat {Y}} rsub {t}} over {{Y} rsub {t}} right rline} ( 5 )
Where:
Y
t
{Y} rsub {t}
is the actual value;
Y
^
t
{widehat {Y}} rsub {t}
: is the simulated value.Weighted Mean 1 = (RMSE +4 MAPE+MAD)/6 (6)Weighted Mean 2 = (4RMSE + MAPE+MAD)/6 (7)Weighted Mean 3 = (RMSE + MAPE+4MAD)/6 (8)
3. RESULTS 3. RESULTS 3.1 Build Representation Model:We used Minitab software to analyze annual rainfall (19592017). In figure 1 the graph trend of original data showed nonstationary data, because the data is decreasing along the axis of time, and it showed the value of decreasing=6.13mm/yr. the graph of original data of (ACF) showed nonseasonal and nonstatic data. In figure 2 we test stability of series in variance by applying normality test (Kolmogorov Smirnov test) that showed the data is stable in variance. We take the first regular difference (d1) to stabilize the variance in mean and remove the trend. A graph of time plot of annual rainfall after doing the difference shows stationary on the value of the data center, whereas the trend is decreasing at a fairly constant rate.Figure 1: The figure presents trend of original data and the plot of Autocorrelation Function.Figure 2: The figure presents normality test and the plot of trend data after in deference.To determine the best model, we use min(AIC) and choose ARIMA (1,1,3) that shown in table (1), and considers it the best model due to the residuals passed the required tests. Table 1: The table presents the AKAKI test results.ARIMA Model
σ
2
{σ} ^ {2}
m
m
AICARIMA Model
σ
2
{σ} ^ {2}
m
m
AIC
ARIMA (0,0,1)35939.12622.8853ARIMA (0,1,1)27255.82606.5683
ARIMA (1,0,0)33575.22618.871ARIMA (1,1,0)35680.42622.4591
ARIMA(2,0,0)31189.03616.5214ARIMA(2,1,0)36317.83625.5037
ARIMA(3,0,0)31085.64618.3255ARIMA(3,1,0)33276.14622.3431
ARIMA(4,0,0)30892.05619.9569ARIMA(4,1,0)32263.65622.52
ARIMA(1,0,2)31475.94619.0617ARIMA(1,1,2)28125.84612.4221
ARIMA(2,1,3)23374.16605.5036ARIMA(1,1,3)24130.55605.3827
ARIMA(4,1,5)24449.010616.1563ARIMA(5,2,3)35170.69635.61
ARIMA(5,1,1)26198.57614.234ARIMA(5,2,4)29880.910627.9935
ARIMA(1,2,0)63770.62656.7198ARIMA(4,2,1)56264.96657.3317
ARIMA(2,2,0)59753.63654.8811ARIMA(4,2,2)55696.27658.7324
ARIMA(3,2,0)54499.84651.4512ARIMA(4,2,3)34413.18632.3254
ARIMA Model: Autoregressive Integrated Moving Average Model.In tables (2) and (3), we test the residuals of the selected model by using 1Sample Z and Runs Test, whereas Pvalue >0.05. The figure 3 explained that the residuals look like white noise (WN) by observation the plots of (ACF, PACF), and we draw Histogram of residuals and apply normality test of them. Table 2: The table presents the oneSample Z results.NMeanSt DevSE MeanPValue
58315031680.999
Table 3: The table presents the runs Test results.NNumber of observations> MeanNumber of observations< MeanPValue
5831270.818
Figure 3: The figure presents the plots of (ACF, PACF) of residuals, Histogram of residuals and normality test of them.By equal 1, we write the final shape of representation model that shown in Eq.(9).
z
t
=
(
1
−
0.737
)
Z
t
−
1
+
0.737
Z
t
−
2
−
13.809
+
a
t
−
0.134
∗
a
t
−
1
−
0.55
∗
a
t
−
2
−
0.416
∗
a
t
−
3
:
a
t
≈
WN
(
0,24130.5
)
{z} rsub {t} = left (1 0.737 right ) {Z} rsub {t1} +0.737 {Z} rsub {t2} 13.809+ {a} rsub {t} 0.134* {a} rsub {t1} 0.55* {a} rsub {t2} 0.416* {a} rsub {t3} : {a} rsub {t} ≈WN left (0,24130.5 right )
(9) 3.1 Build a Forecasting Model:We used the data of annual rainfall (19592011) to build a forecasting model, and the data from 2012 to 2017 to verify the quality of model. The figure 4 shown the comparing between measured values and forecasted values after proposing several models to find the best model. Calculating of forecasting criteria as in table (4). Table (5) contains a conclusion of weighted Mean values of criteria to find ARIMA (2,1,0) as the best forecasting model that have the least values (6.41, 21.408, 10.627) and it gives us forecasted rainfall for (201820192020) with (651.18644.1 644) mm values. Table (6) shows the error ratios between original data and forecasting values of ARIMA model.
/
/
;
;
/
/
;
;
/
/
;
;
Figure 4: The figure presents the comparing between actual and forecasted values for many models. Table 4: The table presents the MAPE, RMSE, MAD Criteria.ModelMAPERMSEMAD
ARIMA (3,1,1)0.013935.0279.581
ARIMA (4,1,2)0.01844.32312.514
ARIMA (2,1,0)0.012308.438
ARIMA Model: Autoregressive Integrated Moving Average Model; RMSE: Root Mean Squared Error; MAD: Mean Absolute Deviation; MAPE: Mean Absolute Percentage. Table 5: The table presents the Weighted Mean of (MAPE, RMSE, MAD) Criteria.ModelWeighted Mean 1Weighted Mean 2Weighted Mean 3
ARIMA (3,1,1)7.4424.9512.227
ARIMA (4,1,2)9.4831.63715.732
ARIMA (2,1,0)6.4121.40810.627
ARIMA Model: Autoregressive Integrated Moving Average Model; RMSE: Root Mean Squared Error; MAD: Mean Absolute Deviation; MAPE: Mean Absolute Percentage. Table 6: The table presents the error ratios between original and forecasting data.Original (mm)Forecast (mm)Error Ratio (%)Year
836749.7110.322012
617612.640.72013
551686.2724.52014
775.26640.417.392015
713662.397.092016
610645.855.872017
4. DISCUSSION We use BoxJenkins method to predict annual rainfall for 20182019 and 2020 by analyzing the annual rainfall series from 1959 to 2017 which was annually decreasing during the studied period by 6.13 mm. We leaving the last 6 years as a modeltesting set. We found Autoregressive Integrated Moving Average Model (ARIMA Model) (1,1,3) model as the best model to represent the data after applying the least value of AIC and doing the required tests on the model residuals and passing them successfully, whereas the autocorrelation does not exist between series residuals, and P>Pvalue (random residuals) and taken the natural shape and following the normal distribution.Also, we concluded that ARIMA (2,1,0) is the best forecasted model depending on the smallest of weighted mean of forecasting criteria which were equal: RMSE=30, MAP=0.012 and MAD=8.438 for the chosen model. The forecasted values were very close to actual values; therefore, we recorded the rainfall values for years (2018 to 2020) that we obtained from the model whereas these predictions are reliable and ready to use. 5. CONCLUSION There are many methods for forecast time series, but BoxJenkins models is adopting to analysis the time series data, due to its high accuracy, and their compatibility with the characteristics of each region and therefore cannot be directly applied to a different area. Especially there is a lot of important effect for the natural characteristics of the areas in changing the amount of forecasting rainfall from year to year. If we used the model that we finding it, we always had to update it, because ARIMA is a shortterm forecast method.6. REFERENCES 1. Hayek S., Hammad M. Predict of the monthly water volumes incoming in ALROOS River in the Syrian Coast by using the time series analysis. 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