# Theory of Copula in Hydrology and Hydroclimatology

• Rajib Maity
Chapter
Part of the Springer Transactions in Civil and Environmental Engineering book series (STICEE)

## Abstract

This chapter starts with an introduction to copulas. The copula theory is relatively new to hydrology and hydroclimatology but has already established itself to be highly potential in frequency analysis, multivariate modeling, simulation and prediction. Development of joint distribution between multiple variables is the key to analyze utilizing the potential of copulas. The chapter starts with the mathematical theory of copulas and gradually move on to the application. If the readers are already aware of the background theory and look for application of copula theory, they can directly proceed to Sect. 10.8. Basic mathematical formulations for most commonly used copulas are discussed, and illustrative examples are provided. It will enable the readers to carry out applications to other problems. All the illustrative examples are designed with very few data points. This helps to show the calculation steps explicitly. Please note that any statistical analysis should be done with sufficiently long data. Once the readers understand the steps, computer codes can be written easily for large data sets. Example of MATLAB codes is also provided at the end.

## References

1. Bosq, Denis. 2012. Nonparametric statistics for stochastic processes: Estimation and prediction, vol. 110. New York: Springer Science & Business Media.
2. Genest, Christian, and Jock MacKay. 1986. The joy of copulas: Bivariate distributions with uniform marginals. The American Statistician 40 (4): 280–283.
3. Genest, Christian, and Louis-Paul Rivest. 1993. Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88 (423): 1034–1043.
4. Genest, Christian, and Anne-Catherine Favre. 2007. Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering 12 (4): 347–368.
5. Genest, Christian, Kilani Ghoudi, and L.-P. Rivest. 1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82 (3): 543–552.
6. Genest, Christian, Bruno Rémillard, and David Beaudoin. 2009. Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 44 (2): 199–213.
7. Joe, Harry. 1997. Multivariate models and multivariate dependence concepts. Boca Raton: CRC Press.
8. Kojadinovic, Ivan, and Jun Yan. 2011. A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems. Statistics and Computing 21 (1): 17–30.
9. Nelsen, Roger B. 1999. An introduction to copulas, 269. Berlin: Springer.
10. Salvadori, G., and C. De Michele. 2007. On the use of copulas in hydrology: Theory and practice. Journal of Hydrologic Engineering 12 (4): 369–380.
11. Sklar, A. 1959. Fonctions de réepartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universitée de Paris.Google Scholar