# Copula generation (Gaussian, t and Gumbel) with the help of correlation matrix using R

I have a set of data of 2 variates. I have generated correlation matrix between the variates. Using copula package of R, I computed t-copula using correlation matrix. I used the following technique for that:

1. With the help of correlation matrix for example let it be corr_mat, I computed parameter vector, lets say param_vect.
2. Then I computed tcopula object, let it be t_object, using tCopula(param_vect, dim =..., dispstr="un", df =...)

3. Lastly I simulated 1000 sample from the copula using rcopula(1000,t_object).

The above procedure works fine for t-copula. Can someone please suggest me amethod to simulate samples from the Gaussian and Gumbell copulas? I tried with the gumbelcopula function, but it fails with errors. Could anyone one suggest the method to simulate copula using correlation matrix?

## 1 Answer

Correlation is problematic because it depends on the marginal distribution.

Change the margins (which doesn't alter the copula) and the correlation changes. A specific correlation is not associated with a given copula and vice-versa.

The correlation matrix that is an input to a t-copula doesn't define the correlation of the resulting variables that have that copula, so you may not be achieving what you think you are when you use the t-copula function. Similarly with the correlation in a Gaussian copula - you can specify the value of the parameter $\rho$, but that doesn't mean that the correlation of the variables you generate with that Gaussian copula is $\rho$. If they also have Gaussian margins, then the correlation is $\rho$, otherwise it will generally not be $\rho$, but some function of it that depends on what the margins are.

This is why many people focus on nonparametric correlations like Kendall's tau or Spearman's rho. Those are specified by a given copula, and apply to any (continuous) margin.

For a number of specific copula families there are often relatively simple relationships between the parameters of the family and one or both of those common measures of nonparametric correlation.

• Hi Glen You have provided with the correct conceptual flaw in my approach. But as I am newbie in this environment so I tried to take the very simple approach of going through correlation matrix route. It would nice of you if you can share the algorithm of your approach. – user45232 May 14 '14 at 9:52
• I didn't mention any approach. You could use the relationship between the copula and the nonparametric correlation to infer a parameter value, but that's not necessarily the best choice. So for example, with the Gumbel, the parameter $\alpha = \frac{1}{1-\tau}$, so the sample value $\hat\tau$ could be used to back out an estimate of $\alpha$, but I believe that's not the most common way to do it with the Gumbel. – Glen_b -Reinstate Monica May 14 '14 at 10:21