Modelling Hydrological Time Series Using Wakeby Distribution

Authors: Ilaria Lucrezia Amerise

Abstract:

The statistical modelling of precipitation data for a given portion of territory is fundamental for the monitoring of climatic conditions and for Hydrogeological Management Plans (HMP). This modelling is rendered particularly complex by the changes taking place in the frequency and intensity of precipitation, presumably to be attributed to the global climate change. This paper applies the Wakeby distribution (with 5 parameters) as a theoretical reference model. The number and the quality of the parameters indicate that this distribution may be the appropriate choice for the interpolations of the hydrological variables and, moreover, the Wakeby is particularly suitable for describing phenomena producing heavy tails. The proposed estimation methods for determining the value of the Wakeby parameters are the same as those used for density functions with heavy tails. The commonly used procedure is the classic method of moments weighed with probabilities (probability weighted moments, PWM) although this has often shown difficulty of convergence, or rather, convergence to a configuration of inappropriate parameters. In this paper, we analyze the problem of the likelihood estimation of a random variable expressed through its quantile function. The method of maximum likelihood, in this case, is more demanding than in the situations of more usual estimation. The reasons for this lie, in the sampling and asymptotic properties of the estimators of maximum likelihood which improve the estimates obtained with indications of their variability and, therefore, their accuracy and reliability. These features are highly appreciated in contexts where poor decisions, attributable to an inefficient or incomplete information base, can cause serious damages.

Keywords: Generalized extreme values (GEV), likelihood estimation, precipitation data, Wakeby distribution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.2702598

References:

[1] Brachetti P. and Ciccoli, M. and Di Pillo, G. and Lucidi, S. “A new version of the Price’s algorithm for global optimization” Journal of Global Optimization 10, 165 – 184 (1997).
[2] Dempster, A. P. and Laird, N. M. and Rubin, D. B. “Maximum likelihood from incomplete data via the EM Algorithm” Journal of the Royal Statistical Society. Series B (Methodological), 39, 1–38 (1977).
[3] Gilchrist, W. G., “Modeling and fitting quantile distributions and regressions”, Sheffield Hallam University (2006).
[4] Griffiths, G. A. “A theoretically based Wakeby distribution for annual flood series” Hydrological Sciences - Journal - des Sciences Hydrologiques 34, 231–248 (1989).
[5] Honaker, J. and King, G. and Blackwell, M. “Amelia II: A Program for Missing Data” Journal of Statistical Software 45, 1–47 (2011).
[6] Jones, R. A. and Scholz, F. W. and Ossiander, M. and Shorack G. R. “Tolerance bounds for log gamma regression models Technometrics 27, 109 – 118 (1985).
[7] Lawton, W. H. and Sylvestre, E. A., “Elimination of linear parameters in nonlinear regression” Technometrics 13, 461 – 467 (1971).
[8] Parzen, E. “Nonparametric statistical data modelling” Journal of the American Statistical Association 74,105 – 131 (1979).
[9] Su, B. and Kundzewicz, Z. W. and Jiang, T. “Simulation of extreme precipitation over the Yangtze River Basin using Wakeby distribution” Theoretical and Applied Climatology 96, 209–219 (2009).
[10] Tarsitano, A. “Fitting Wakeby model using maximum likelihood” Convegno intermedio SIS 2005: Statistica e Ambiente,Messina, 21-23 September, 2005, 253–256 (2005).