Copula-Based Markov Process for Forecasting and Analyzing Risk of Water Quality Time Series

This study applies the copula-based Markov process to model water quality time series. The bivariate copula is applied to investigate the first-order Markov processes. The D-Vine copula is applied to investigate the more complicated higher-order ($k\ge 2$) Markov processes. The Value-at-Risk (VaR), computed using the best-fitted copula-based Markov process, is applied for the risk analysis. Using water quality time series at the Snohomish River watershed (Washington) and the Chattahoochee River watershed (Georgia), the results show that the copula-based Markov processes (1) are able to properly model the temporal dependence for dissolved oxygen (DO) series [i.e., forecast root-mean-square error (RMSE) $<1\mathrm{mg}/\mathrm{L}$ at both watersheds] and temperature (T) series (i.e., forecast RMSE = 1.5°C at the Chattahoochee watershed); and (2) can only predict the overall trend for nitrate and conductivity series, due to the fact that these two series also depend heavily on other factors (e.g., runoff). Overall, the study indicates that the copula-based Markov process may be an efficient tool in the assessment of water quality and the associated risks with the following advantages: (1) constructing the transitional probability explicitly and properly; (2) studying the temporal dependence independently from the marginal distributions; (3) avoiding the strict assumptions of the classic time series modeling approach (e.g., the time series belonging to the Gaussian process); and (4) providing a reasonable risk measure through the VaR.