Simulation of Nonstationary Spring Discharge Using Time Series Models

We present a detailed analysis and comparison of two time series models, i.e., ARIMA and ARIMA-GARCH, to simulate the discharge of a karst spring (Niangziguan Springs (NS) complex) in the northern China. Statistical tests for the residuals are applied to examine the reasonability of the models. Statistically, both models are reasonably good to simulate the mean value of the discharge of the NS complex. The statistical test shows that the residual discharge data have conditional time-varying variance and volatility clustering, known as heteroscedasticity of the data. Calibration test shows that the ARIMA-GARCH model gives a varying confidence interval, which can more effectively capture the heteroscedasticity of the data, comparing with a constant confidence interval in the ARIMA model. In the validation and application process, we applied two approaches to simulate the discharge data: (1) fixed models, and (2) evolving models. The confidence interval width monotonically increases in both fixed models, and the fixed ARIMA-GARCH model has faster increasing confidence interval width than the fixed ARIMA model. This suggests that the fixed time series models are only suitable for short-term prediction. However, we found that this drawback can be compensated by updating the model once new data become available. Our evolving models show more reasonable confidence interval width for both models. In addition, the application shows that the ARIMA-GARCH model is very sensitive to the data fluctuation. We also found the evolving ARIMA-GARCH model was able to return to the narrow confidence interval width once the fluctuation diminished. Hence, we conclude that the ARIMA-GARCH model is more suitable for the sequences with strong heteroscedasticity.