Using Copulas in Hydrology: Benefits, Cautions, and Issues
Publication: Journal of Hydrologic Engineering
Volume 12, Issue 4
Abstract
This paper discusses the bivariate modeling of extreme tails of correlated hydrological random variables. We take a copula approach and model the dependence structure independently of the marginal distributions. We apply results from the classical extreme value theory to choose marginal distributions for excesses of high thresholds and consider six copula families to capture the dependence structure of these excesses. While copulas can differ somewhat in the degree of association that they provide, differences in which part of the distribution this association is more pronounced can be substantial. We discuss certain pertinent properties of copulas and give some insight to assist the practitioner in their selection. We examine the effects of model misspecification and the impact of the chosen method of estimation, targeting the estimated quantities frequently used by hydrologists. A simulation study shows not only the dangers of improper copula selection, but also the possible benefits of using a bivariate approach to estimate univariate quantities. We apply the methodology to the study of low-flow events and analyze two Canadian hydrometric data sets.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writer acknowledges ongoing grant support from the Natural Sciences and Engineering Research Council of Canada. The writer also thanks two referees for valuable comments that improved the presentation.
References
Abdous, B., Genest, C., and Rémillard, B. (2005). “Dependence properties of metaelliptical distributions.” Statistical modeling and analysis for complex data problems, P. Duchesne and B. Rémillard, eds., Springer, Academic, New York, 1–15.
Ashkar, F., El Jabi, N., and Issa, M. (1998). “A bivariate analysis ofthe volume and duration of low-flow events.” Stochastic Hydrol. Hydraul., 12(2), 97–116.
Barão, M. I., and Tawn, J. A. (1999). “Extremal analysis of short series with outliers: Sea-levels and athletic records.” Appl. Stat., 48(4), 469–487.
Charpentier, A., and Juri, A. (2004). “Limiting dependence structure for credit default.” Rep. No. 2004-16, Institut National de la Statistique et des Études Économiques.
Chen, X., Fan, Y., and Patton, A. (2004). “Simple tests for models of dependence between multiple financial time series, with applications to U.S. equity returns and exchange rates.” Financial Markets Group, London School of Economics, Discussion Paper No. 483.
Davison, A. C., and Smith, R. L. (1990). “Models for exceedances over high thresholds.” J. R. Stat. Soc. Ser. B (Methodol.), 52(3), 393–442.
Demarta, S., and McNeil, A. J. (2005). “The copula and related copulas.” Int. Statist. Rev., 73(1), 111–129.
Dupuis, D. J., and Tawn, J. A. (2001). “Effects of misspecification in bivariate extreme value problems.” Extremes, 4(4), 315–330.
Favre, A.-C., El Adlouni, S., Perreault, L., Thiémonge, N., and Bobée, B. (2004). “Multivariate hydrological frequency analysis using copulas.” Water Resour. Res., 40(1), W01101 .
Fermanian, J.-D. (2005). “Goodness-of-fit tests for copulas.” J. Multivariate Anal., 95(1), 119–152.
Frahm, G., Junker, M., and Szimayer, I. (2003). “Elliptical copulas: Applicability and limitations.” Stat. Probab. Lett., 63(3), 275–286.
Genest, C., and MacKay, R. J. (1986). “Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données.” Can. J. Stat., 14(2), 145–159.
Genest, C., Quessy, J. F., and Rémillard, B. (2006). “Goodness-of-fit procedures for copula models based on the probability integral transformation.” Scand. J. Stat., 33(2), 337–366.
Genest, C., and Rivest, L.-P. (1993). “Statistical inference procedures for bivariate Archimedean copulas.” J. Am. Stat. Assoc., 88(423), 1034–1043.
Hosking, J. R. M., and Wallis, J. R. (1987). “Parameter and quantile estimation for the generalized Pareto distribution.” Technometrics, 29(3), 339–349.
Joe, H. (1997). Multivariate models and dependence concepts, Chapman and Hall, London.
Juri, A., and Wüthrich, M. (2002). “Copula convergence theorems for tail events.” Insur. Math. Econ., 30(3), 405–420.
Pickands, J. (1975). “Statistical inference using extreme order statistics.” Ann. Stat., 3(1), 119–131.
Pickands, J. (1981). “Multivariate extreme value distributions.” Proc., 43rd Session Int. Statist. Inst., Buenos Aires, 859–878.
Sklar, A. (1959). “Fonctions de répartition à dimensions et leurs marges.” Publ. Inst. Stat. Univ. Paris, 8, 229–231.
Todorovic, P., and Rousselle, J. (1971). “Some problems of flood analysis.” Water Resour. Res., 7(5), 1144–1150.
Todorovic, P., and Zelenhasic, E. (1970). “A stochastic model for flood analysis.” Water Resour. Res., 6(6), 1641–1648.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 12 • Issue 4 • July 2007
Pages: 381 - 393
Copyright
© 2007 ASCE.
History
Received: Aug 29, 2006
Accepted: Aug 29, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.