Fourier-PARMA Models and Their Application to River Flows

For analysis and design of water resource systems, it is sometimes useful to generate highresolution (e.g., weekly) synthetic river flows. Periodic autoregressive moving average (PARMA) time series models provide a powerful tool for generating synthetic flows. Periodically stationary models are indicated when the basic statistics (mean, variance, and autocorrelation) of the time series exhibit significant seasonal variations. Parameter estimation for high-resolution PARMA models involves numerous parameters, which can lead to over-fitting. Thus, this paper develops a parsimonious method of parameter fitting for high-resolution PARMA models, using discrete Fourier transforms to represent the set of periodic autoregressive and moving average model coefficients. Model parameters are computed via the innovations algorithm, and the asymptotic distributions of the discrete Fourier transform coefficients are obtained. Those asymptotic results are useful to determine the statistically significant Fourier coefficients to include in the model. Effectiveness of the technique is shown using simulated data from different PARMA models. Discharge measurements from the Fraser River in British Columbia are then modelled, firstly as a monthly series and secondly as a weekly series. Diagnostic checks are used to ensure adequacy of the models. Finally, a careful statistical analysis of the PARMA model residuals, including a novel truncated Pareto model for the extreme tails, is combined with the Fourier-PARMA time series model to generate realistic synthetic flows. A key finding is that the Fourier-PARMA method produces superior results as compared to a conventional PARMA model, despite using far fewer parameters.