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In addition to the above approach based on the ACF and PACF plots of the first-differenced series, several alternative models were also examined for model selection (Table 2). At this stage, statistics calculated from the one-ahead forecast errors were considered. Better ARIMA models have smaller root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) values, which measure the variability of the forecasting errors. If the forecasts are not biased, mean error (ME) and mean percentage error (MPE) should be close to zero. ARIMA (3,1,0) and ARIMA (1,1,1) were found to be suitable models for turbidity and pH, respectively. The number of differences (d) is 1 for both models, since the data were first differenced to render stationarity.

Table 2

Model comparison

ParameterModelRMSEMAEMAPEMEMPE
Turbidity ARIMA (3,1,0) 0.1720 0.1278 16.0378 − 0.0012 − 3.1232 
ARIMA (2,1,0) 0.1775 0.1292 16.4177 − 0.0004 − 3.0140 
ARIMA (3,2,0) 0.2029 0.1534 18.5821 0.0001 − 1.5897 
ARIMA (3,1,1) 0.1729 0.1284 16.1178 − 0.0013 − 3.1543 
ARIMA (0,1,2) 0.1726 0.1305 16.4740 − 0.0015 − 3.5570 
pH ARIMA (1,1,1) 0.0844 0.0536 0.6713 0.0016 0.0085 
ARIMA (1,1,2) 0.0845 0.0536 0.6712 0.0017 0.0104 
ARIMA (3,1,1) 0.0847 0.0540 0.6752 0.0056 0.0584 
ARIMA (0,0,1) 0.0993 0.0746 0.9334 0.0003 − 0.0137 
ARIMA (1,0,0) 0.0848 0.0529 0.6625 − 0.0001 − 0.0123 
ParameterModelRMSEMAEMAPEMEMPE
Turbidity ARIMA (3,1,0) 0.1720 0.1278 16.0378 − 0.0012 − 3.1232 
ARIMA (2,1,0) 0.1775 0.1292 16.4177 − 0.0004 − 3.0140 
ARIMA (3,2,0) 0.2029 0.1534 18.5821 0.0001 − 1.5897 
ARIMA (3,1,1) 0.1729 0.1284 16.1178 − 0.0013 − 3.1543 
ARIMA (0,1,2) 0.1726 0.1305 16.4740 − 0.0015 − 3.5570 
pH ARIMA (1,1,1) 0.0844 0.0536 0.6713 0.0016 0.0085 
ARIMA (1,1,2) 0.0845 0.0536 0.6712 0.0017 0.0104 
ARIMA (3,1,1) 0.0847 0.0540 0.6752 0.0056 0.0584 
ARIMA (0,0,1) 0.0993 0.0746 0.9334 0.0003 − 0.0137 
ARIMA (1,0,0) 0.0848 0.0529 0.6625 − 0.0001 − 0.0123 

RMSE, root mean squared error; MAE, mean absolute error; MAPE, mean absolute percentage error; ME, mean error; MPE, mean percentage error.

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