Assessment of prediction performances of stochastic models: Monthly groundwater level prediction in Southern Italy
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Abstract: Stochastic modelling of hydrological time series with insufficient length and data gaps is a serious challenge since these problems significantly affect the reliability of statistical models predicting and forecasting skills. In this paper, we proposed a method for searching the seasonal autoregressive integrated moving average (SARIMA) model parameters to predict the behavior of groundwater time series affected by the issues mentioned. Based on the analysis of statistical indices, 8 stations among 44 available within the Campania region (Italy) have been selected as the highest quality measurements. Different SARIMA models, with different autoregressive, moving average and differentiation orders had been used. By reviewing the criteria used to determine the consistency and goodness-of-fit of the model, it is revealed that the model with specific combination of parameters, SARIMA (0,1,3) (0,1,2) 12, has a high R2 value, larger than 92%, for each of the 8 selected stations. The same model has also good performances for what concern the forecasting skills, with an average NSE of about 96%. Therefore, this study has the potential to provide a new horizon for the simulation and reconstruction of groundwater time series within the investigated area.
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Key words:
- Groundwater level forecast /
- Stochastic modelling /
- Southern Italy /
- Seasonality /
- Homogeneity
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Figure 2. ARIMA model prediction method (Chatfield et al.1973)
Table 1. Forecasting models Results based on AIC and BIC criteria
Name of station Parameters of model AIC BIC Acera Capomazzo (1,1,2)(0,1,2)12 514.44 542.39 Casamicciola (1,1,2)(0,1,1)12 −1228.75 −1209.5 Cassano di Sessa Aurunca (0,1,3)(0,1,2)12 914.41 942.54 Forio(Calitto) (1,1,2)(0,1,1)12 607.55 630.75 Forio(Pontone) (0,1,3)(0,1,2)12 −571.39 −547.92 Forio(Umberto I) (1,1,1)(0,1,1)12 −39.8 −24.01 Nocelleto di Carinola (1,1,1)(0,1,1)12 −377.44 −309.26 Parete(tre ponti) (1,1,2)(0,1,1)12 −702.58 −682.21 Table 2. The comparison of different SARIMA models based on accuracy measures
Parete
(tre ponti)Nocelleto di carinola Forio (Umberto I) Forio (Pontone) Forio (Calitto) Cassano di Sessa Aurunca Casamicciola Accera Copmazzo SARIMA (0,1,3)(0,1,2)12 NSE 0.944 0.996 0.920 0.949 0.985 0.999 0.932 0.984 MAE 0.065 0.110 0.131 0.067 0.217 0.255 0.026 0.214 RMSE 0.107 0.183 0.218 0.108 0.363 0.420 0.037 0.331 Pearson cоr. 0.971 0.935 0.960 0.974 0.962 0.930 0.965 0.983 MSE 0.011 0.032 0.047 0.011 0.121 0.176 0,001 0.109 d 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 BIAS$ −0.026 0.027 −0.006 −0.012 −0.046 −0.045 0.036 0.003 MSDE 4.7E-04 4.3E-06 1.5E-05 3.1E-09 0.006 0.101 2.2E-06 0.016 R2 0.944 0.874 0.922 0.949 0.9257 0.865 0.932 0.967 AIC −678.64 −321.22 −36.87 −571.61 625.55 914.41 −1203.63 532.08 BIC −654.2 −294.18 −13.18 −547.92 652.85 941.39 −1180.55 559.97 SARIMA (1,1,2)(0,1,1)12 NSE 0.943 0.996 0.920 0.947 0.689 0.999 0.930 0.984 MAE 0.064 0.109 0.129 0.068 0.529 0.255 0.026 0.211 RMSE 0.108 0.182 0.218 0.110 1625774 0.422 0.037 0.329 Pearson cor. 0.971 0.934 0.960 0.973 0.553 0.929 0.964 0.983 MSE 0.012 0.032 0.047 0.012 0.419 0.178 0.001 0.107 d 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 BIAS$ −0.018 0.032 −0.0088 −0.013 −2.340 −0.046 −0.117 0.057 MSDE 0.0001 2.6E-06 1.3E-05 3.1E-09 0.198 0.106 2.2E-06 0017 R2 0.943 0.873 0.921 0.948 0.306 0.863 0.931 0.967 AIC −702.58 −327.03 −37.95 −562.72 607.55 920.63 −1226.74 519.45 BIC −682.21 −304.5 −18.2 −542.97 630.3 943.12 −1207.5 542.69 SARIMA (1,1,2)(0,1,1)12 NSE 0.943 0.996 0.920 0.946 0.985 0.999 0.940 0.984 MAE 0.063 0.110 0.129 0.069 0.219 0.257 0.024 0.212 RMSE 0.108 0.183 0.218 0.111 0.365 0.424 0.034 0.330 Pearson cor. 0.971 0.934 0.960 0.973 0.962 0.928 0.970 0.983 MSE 0,011 0.032 0.047 0.012 0.122 0.180 0.001 0.107 d 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 BIAS$ −0.018 0.033 −0.008 −0.013 −0.042 −0.047 0.048 0.061 MSDE 0,0001 2.4E-06 1.3E-05 3.1E-09 0.005 0.109 2.2E-06 0017 R2 0.943 0.873 0.921 0.947 0.862 0.862 0.941 0.967 AIC −672.01 −327.44 −39.8 −561.6 629.36 924.5 −1194.32 518.6 BIC −655.71 −309.41 −24 −545.8 647.56 942.491 −1178.931 537.19 Table 3. Statistical index values for the selected SARIMA model (0,1,3) (0,1,2)12
NSE BIAS% R2 d r Accera Copmazzo 0.98 0.003 0.96 0.99 0.98 Casamicciola 0.93 0.03 0.93 0.99 0.96 Cassano di Sessa Aurunca 0.99 −0.04 0.86 0.99 0.93 Forio Calitto 0.98 −0.04 0.92 0.99 0.96 Forio Pontone 0.94 −0.012 0.94 0.99 0.97 Forio Umberto I 0.92 −0.006 0.92 0.99 0.96 Nocelleto di carinola 0.99 0.02 0.87 0.99 0.93 Parete tre ponti 0.94 −0.02 0.94 0.99 0.97 Model Quality (Very good) 0.75< NSE<1.00 PBIAS<±10 0.75 < R2≤ 1.0 1 r > 0.7 -
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