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Water Resources
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Groundwater Level Forecasting with Time Series Analysis
Table 3. Results of estimations for ARIMA models
| Model
|
Parameter
|
Coefficient
|
St. Dev.
|
t-value
|
| ARIMA(0,1,1)(1,1,0)12
|
MA1
|
0.0028
|
0.1144
|
0.02
|
| SAR12
|
-0.5768
|
0.1065
|
-5.41
|
| CONSTANT
|
0.0008
|
0.2594
|
0.00
|
| ARIMA(1,1,0)(1,1,0)12
|
AR1
|
-0.0021
|
0.1144
|
-0.02
|
| SAR12
|
-0.5727
|
0.1066
|
-5.37
|
| CONSTANT
|
0.0006
|
0.2602
|
0.00
|
| ARIMA(1,1,1)(1,1,1)12
|
AR1
|
-0.1016
|
0.8447
|
-0.12
|
| MA1
|
-0.2270
|
0.8162
|
-0.28
|
| SAR1
|
-0.3225
|
0.1149
|
-2.81
|
| SMA1
|
0.9192
|
0.0850
|
10.82
|
| CONSTANT
|
-0.03478
|
0.03617
|
-0.96
|
| ARIMA(1,1,1)(1,1,0)12
|
AR1
|
0.8036
|
0.0765
|
10.50
|
| MA1
|
0.9794
|
0.0431
|
22.74
|
| SAR12
|
-0.6026
|
0.1058
|
-5.70
|
| CONSTANT
|
0.00479
|
0.01086
|
0.44
|
| ARIMA(1,1,0)(0,1,1)12
|
AR1
|
0.1014
|
0.1173
|
0.86
|
| SMA12
|
0.8729
|
0.0980
|
8.91
|
| CONSTANT
|
-0.00998
|
0.05413
|
-0.18
|
| ARIMA(0,1,1)(0,1,1)12
|
MA1
|
-0.1158
|
0.1157
|
-1.00
|
| SMA12
|
0.8631
|
0.0969
|
8.90
|
| CONSTANT
|
-0.00907
|
0.06239
|
-0.15
|
3.3 Diagnostic checking
The statistical adequacy of the estimated models is then verified. The ACF function for the residuals resulting from a good ARIMA model will have statistically zero autocorrelation coefficients. Figure 5 shows a plot of the residuals for ARIMA(1,1,0)(1,1,0)12 model. The residual plot shows small variations around the zero mean. The plot of the estimated residual ACF in Figure 5 indicates that there is no significant autocorrelation, and the model adopted will be acceptable.
3.4 Forecasting
Two ARIMA models were applied to forecast the 21 water level values from January 1998 to September 1999. The forecasts are then compared with the measured data. The forecasted time series and its 95% confidence level error bound are plotted in Figure 5 for both models. It is observed that all measured monthly values fall within the error bound, and the forecasts track the seasonal pattern reasonably well.
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