Fourier-PARMA Models and Their Application to River Flows
Publication: Journal of Hydrologic Engineering
Volume 12, Issue 5
Abstract
For analysis and design of water resource systems, it is sometimes useful to generate high-resolution (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 overfitting. 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 modeled, first as a monthly series and, second, 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.
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Acknowledgments
The second writer was partially supported by National Science Foundation Grant No. NSFDMS-0139927. The third writer was partially supported by National Science Foundation Grant Nos. NSFDMS-0139927, NSFDMS-0417819, and NSFDMS-0706440.
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Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 12 • Issue 5 • September 2007
Pages: 462 - 472
Copyright
© 2007 ASCE.
History
Received: Aug 26, 2005
Accepted: Feb 22, 2007
Published online: Sep 1, 2007
Published in print: Sep 2007
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