## Thursday, December 21, 2017

### Copulas

So finally, I was obliged to try to understand what Copulas are. Put it simply. They are functions that connect marginal distributions to their multivariate distribution. Sklar (1959) theorem shows that the copula exists and is unique for any pair. Restricting to bivariate distribution function for sake of simplicity, Sklar theorem establish that the joint cumulative distribution $H(x,y)$ of any pair of continuous random variable (X,Y) may be written as as:
$$H(x,y) = C(F(x),G(y)) \ \ \ \forall x, y \in \mathbb{R}$$
where $F(x)$ and $G(y)$ are the marginal distributions of $X$ and $Y$. $C$, the copula, can be though as a function such that:
$$C:\, [0,1]^2 \mapsto [0,1]$$

Obviosly just in the Platonic world where you know $H$,$F$ and $G$, you can determine the unknown $C$. In practice you work with much less information, where you have the marginals, $F$ and $G$ and, maybe some information about the correlation of the random variables they describe.
So you have to infer the multivariate distribution, by selecting, as usual I would say, the copulas among a large set of copulas templates.
This is what is written for instance in Genest and Favre 2007, a paper particularly directed to hydrologists. So, for your introduction to Copulas you can probably start from that paper. However, I came to it, by its citation in Ebrechts, 2009. Traditional references on the subject are the books by Joe (1997) and Nelsen (1999) but a nice review paper (therefore more synthetic) is Frees and Valdez 1998.
Particularly relevant copulas are useful to understand correlations among variables, and the so called empirical copulas (e.g. Genest and Favre, 2007) can be used to this scope.
Among Hydrological application, I can mention, among others, De Michele and Salvadori (2002), Salvadori and De Michele (2004), Grimaldi and Serinaldi (2006), and Serinaldi et al., 2009.

References

• De Michele, C. (2003). A Generalized Pareto intensity-duration model of storm rainfall exploiting 2-Copulas. Journal of Geophysical Research, 108(D2), 225–11. http://doi.org/10.1029/2002JD002534
• Embrechts, P. (2009). Copulas: a personal view, 1–18.
• Frees, E. W., & Valdez, E. A. (1999). Understanding Relationships Using Copulas. North American Actuarial Journal, 2(1), 1–25.
• Genest, C., & Favre, A. A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 347–368. http://doi.org/10.1061/ASCE1084-0699200712:4347
• Genest, C., & Nešlehová, J. (2007). A primer on copulad for count data. Astin Bulletin, 37(02), 475–515. http://doi.org/10.1017/S0515036100014963
• Grimaldi, S., & Serinaldi, F. (2006). Asymmetric copula in multivariate flood frequency analysis. Advances in Water Resources, 29(8), 1155–1167. http://doi.org/10.1016/j.advwatres.2005.09.005
• Joe, H. 1997 . Multivariate models and dependence concepts, Chapmanand Hall, London.
• Mikosch, T. (2006). Copulas: Tales and facts. Extremes, 9(1), 3–20. http://doi.org/10.1007/s10687-006-0015-x
• Nelsen, R. B. 1999 . An introduction to copulas, Springer, New York.
• Salvadori, G., & De Michele, C. (2004). Frequency analysis via copulas: Theoretical aspects and applications to hydrological events. Water Resources Research, 40(12), 194–17. http://doi.org/10.1029/2004WR003133
• Serinaldi, F., Bonaccorso, B., Cancelliere, A., & Grimaldi, S. (2009). Probabilistic characterization of drought properties through copulas. Physics and Chemistry of the Earth, 34(10-12), 596–605. http://doi.org/10.1016/j.pce.2008.09.004