Gamma‐Autoregressive Models for Stream‐Flow Simulation
Publication: Journal of Hydraulic Engineering
Volume 116, Issue 11
Abstract
Since hydrologic time series in general, and stream‐flow series, in particular, are dependent and not normally distributed, use of the classical autoregressive and moving average (ARMA) models to represent such series requires transformation of the original series into normal before applying the model. On the other hand, gamma‐autoregressive (GAR) models assume that the underlying series is dependent with a gamma marginal distribution and the models do not require variable transformation. However, the models require the estimation of certain statistics generally leading to biased estimates of the model parameters. This paper presents a procedure for bias correction, based on computer simulation studies, applicable for estimating parameters of GAR(l) models. Applications of the proposed procedure to annual stream‐flow series of several rivers are included. The GAR(l) model, when used in conjunction with the proposed estimation procedure, is an attractive alternative for synthetic stream‐flow simulation, is simple to use, and does not require any transformation of the original data.
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References
1.
Bobee, B., and Robitaille, R. (1975). “Correction of bias in the estimation of the coefficient of skewness.” Water Resour. Res., 11(6), 851–854.
2.
Box, G. E. P., and Cox, D. R. (1964). “An analysis of transformations.” J. Royal Statistics Society, Series B, 26, 211–252.
3.
Box, G. E. P., and Jenkins, G. M. (1976). Time series analysis. Forecasting and control. Holden Day Inc., San Francisco, Calif.
4.
Fiering, M. B., and Jackson, B. B. (1971). “Synthetic streamflows.” Water Resources Monograph No. 1, Am. Geophysical Union, Washington, D.C.
5.
Fernandez, B., and Salas, J. D. (1986). “Periodic gamma autoregressive processes for operational hydrology.” Water Resour. Res., 22(10), 1385–1396.
6.
Fisher, R. (1921). “On the mathematical foundations of theoretical statistics.” Philosophical Trans., Royal Soc. of London, Series A, 222, 309.
7.
Gaver, D. P., and Lewis, P. A. W. (1980). “First order autoregressive gamma sequences and point process.” Adv. Appl. Prob., 12(3), 727–745.
8.
Hirsch, R. M. (1979). “Synthetic hydrology and water supply reliability.” Water Resour. Res., 15(6), 1603–1615.
9.
Jacob, P. A., and Lewis, P. A. W. (1977). “A mixed autoregressive moving average exponential sequence and point process (EARMA 1,1).” Adv. Appl. Prob., 9(1), 87–104.
10.
Kendall, M. G. (1963). The advanced theory of statistics. Vol. 3, 2nd Ed., Charles Griffin and Co. Ltd., London, England.
11.
Kirby, W. (1974). “Algebraic boundness of sample statistics.” Water Resour. Res., 10(2), 220–222.
12.
Lall, W., and Beard, L. R. (1982). “Estimation of pearson type 3 moments.” Water Resour. Res., 18(5), 1563–1569.
13.
Lawrance, A. J. (1982). “The innovation distribution of a gamma distributed autoregressive process.” Scandinavian J. Statistics, 9(4), 234–236.
14.
Lawrance, A. J., and Lewis, P. A. W. (1981). “A new autoregressive time series model in exponential variables [NEAR(l)].” Adv. Appl. Prob., 13(4), 826–845.
15.
Lettenmaier, D. P., and Burges, S. (1977). “An operational approach to preserving skew in hydrologic models of log‐normal persistence.” Water Resour. Res., 13(2), 281–290.
16.
Matalas, N. C. (1966). “Time series analysis.” Water Resour. Res., 3(4), 817–829.
17.
Matalas, N. C. (1967). “Mathematical assessment of synthetic hydrology.” Water Resour. Res., 3(4), 937–945.
18.
Matalas, N. C., and Langbein, W. B. (1962). “Information content of the mean.” J. Geophysical Res., 67(9), 3441–3448.
19.
Mejia, J. M., and Rodriguez‐Iturbe, I. (1974). “Correlation links between normal and lognormal process.” Water Resour. Res., 10(4), 689–690.
20.
Obeysekera, J. T. B., and Yevjevich, V. (1985). “A note on simulation of samples of gamma autoregressive variables.” Water Resour. Res., 21(10), 1669–1572.
21.
O'Connell, P. E. (1977). “ARIMA Models in Synthetic Hydrology.” Mathematical models for surface water hydrology, T. A. Ciriani, V. Maione and J. R. Wallis, eds., John Wiley and Sons, New York, N.Y., 51–68.
22.
Sim, C. H. (1987). “A mixed gamma ARMA(1, 1) model for river flow time series.” Water Resour. Res., 23(1), 32–36.
23.
Stedinger, J. R., and Taylor, M. R. (1982). “Synthetic streamflow generation 1. Model verification and validation.” Water Resour. Res., 18(4), 909–918.
24.
Wallis, J. R., Matalas, N. C., and Slack, J. R. (1974). “Just a moment!.” Water Resour. Res., 1(2), 211–219.
25.
Wallis, J. R., and O'Connell, P. E. (1972). “Small sample estimation of .” Water Resour. Res., 8(3), 707–712.
26.
Weiss, G. (1977). “Shot noise models for the generation of synthetic streamflow data.” Water Resour. Res., 13(1), 101–108.
27.
Yevjevich, V. M. (1963). “Fluctuations of wet and dry years. Part 1. Research data assembly and mathematical models.” Hydrology Paper 1. Colorado State Univ., Fort Collins, Colo.
28.
Yevjevich, V. M. (1972a). Stochastic processes in hydrology. Water Resour. Publications, Littleton, Colo.
29.
Yevjevich, V. M. (1972b). “Structural analysis of hydrologic time series.” Hydrology Paper 56, Colorado State Univ., Fort Collins, Colo.
30.
Yevjevich, V. M., and Obeysekera, J. T. B. (1984). “Estimation of skewness of hydrologic variables.” Water Resour. Res., 20(7), 935–943.
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Published In
Journal of Hydraulic Engineering
Volume 116 • Issue 11 • November 1990
Pages: 1403 - 1414
Copyright
Copyright © 1990 ASCE.
History
Published online: Nov 1, 1990
Published in print: Nov 1990
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