# [ot-users] Normal copula

regis lebrun regis_anne.lebrun_dutfoy at yahoo.fr
Tue Jan 31 00:50:34 CET 2017

```Hi Anita,

The correlation matrix of a Normal copula is only a symmetric positive definite matrix with unit diagonal. It is neither the Spearman correlation of the copula nor its linear correlation. It is a shape matrix, ie a parameter that has an influence on the shape of the copula thus on the intensity of the dependence between the components of a random vector having this copula. To get a statistical understanding of this matrix, you have to choose adapted marginal distributions to the copula. If your marginals are all Normal, then the resulting composed distribution is a multivariate Normal distribution (by definition of the Normal copula) and the linear correlation of this multivariate Normal distribution is exactly the shape matrix of the Normal copula.
Concerning the choice of R, either you have statistical data to make the choice, or you rely on an expert judgement (you pick a value at your will). The interpretation of Rij is the following: the probability of having X_j large knowing that X_i is large is higher than if X_i and X_j were independent if Rij>0, and lower if Rij<0.

Concerning the link with Spearman correlation and Kendall association, there is an increasing relation between each of these measures of association and the corresponding component of Rij. The question is: if I have data, can I use the empirical value of the Spearman correlation or the Kendall association to estimate the value of the shape matrix of a Normal copula? The answer is yes, it will lead to a consistent estimator of R as soon as the Spearman correlation matrix can be obtained as a Spearman correlation matrix of a Normal copula, which is not always the case (it is ok in dimension 2, but the set of attainable Spearman correlation matrix decrease rapidly with the dimension, the same with Kendall association).

Finally, the effect of R on your probability of failure is case-dependent. If your failure event depends only on X1 and X2, you can take any value of R34 in [-1,1], it will make no difference on the probability of failure. If your failure event is maid mainly of points where both X1 and X2 are large, then a positive value of R12 will increase the probability of failure while a negative value of R12 will lower this value. In the specific case of the FORM approximation, the matrix R corresponds to a linear transformation step of the Nataf transformation which maps a distribution with Normal copula into a standard multidimensional Normal distribution (the standard space).

I hope that this explanation will help you in your understanding of these methods/concepts.

Best regards,

Régis

>________________________________
> De : Anita Laera <anita.laera87 at gmail.com>
>À : users at openturns.org
>Envoyé le : Lundi 30 janvier 2017 14h56
>Objet : [ot-users] Normal copula
>
>
>
>Hi all,
>I have a question on the Normal copula, available in OpenTURNS.
>I am not an expert in this topic, so I would really appreciate your help on this.
>For Normal copula, the argument is the correlation matrix R (symmetric, positive and definite).
>It is not really clear to me how to choose the correlation coefficients, when to use just a matrix with certain coefficients and when, from that matrix, transform it into the Spearmann or Kendall correlation matrix.
>And in the end, what is the effect on the parameters? I launched a FORM calculation with 4 variables, considering first an Independent Copula, and then a Normal Copula with R[0,1] = 0.2. I can see that the results are different but I can't really understand how this influences the value that is assigned to each variable at each iteration. If 2 of the 4 variables are correlated with a coefficient 0.2, what should I expect?
>
>
>Thank you in advance for you time.
>
>
>Kind regards,
>Anita
>
>
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```