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I am new to Copulas and am trying to conceptually understand that differences between the two main types of Copulas: Regular Copulas and Vine Copulas. Both are used to simulate data from correlated multivariate probability distributions (which is a difficult problem) - especially in situations where the marginal distributions are not of the same type (e.g. a joint distribution of normal and exponential distributions).

As I see, Vine Copulas seem to be more complicated than Regular Copulas. This makes me wonder : What advantages can be gained by using Vine Copulas over Regular Copulas?

Here my own attempt to answer this question:

Definition: A Copula is a function that links marginal distributions to their multivariate distribution. If we have two random variables $X$ and $Y$ with cumulative distribution functions $F_X(x)$ and $F_Y(y)$ , and a copula function $C(u, v)$, then the joint CDF is given by:

$$ F(x, y) = C[F_X(x), F_Y(y)] $$

To illustrate the added advantage of using a Vine Copula over a Regular Copula, I tried to create the following example:

Example: Consider the following 3 dimensional probability distribution: $$f(x, y, z)$$

Using the chain rule https://en.wikipedia.org/wiki/Chain_rule_(probability) - We can break this into two parts:

$$ f(x, y, z) = f_X(x) f_{Y|X}(y|x) f_{Z|XY}(z|x, y) $$

Now, to compare the advantage of using a Vine Copula over a Regular Copula:

  • If we use a Regular Copula, we can represent this probability distribution using a single Copula function (i.e. there is 1 copula function which is a function of individual distributions):

$$ F(x, y, z) = C[F_X(x) , F_{Y|X}(y|x) , F_{Z|XY}(z|x, y)] $$

  • Using a Vine Copula, I think we can represent this probability distribution using multiple Copula functions (i.e. we multiply 3 different Copulas by each other):

$$ F(x, y, z) = C_{12}[F_X(x), F_{Y|X}(y|x)] \cdot C_{13}[(F_X(x), F_{Z|XY}(z|x, y)] \cdot C_{23|1}[F_{Y|X}(y|x), F_{Z|XY}(z|x, y)|F_X(x)] $$

Thus, it appears that the advantage of using a Vine Copula over a Regular Copula is that you have even more flexibility in deciding which Copula functions you want to fit between individual sets of distributions.

Conclusion: Thus, is this the main advantage of Vine Copulas over Regular Copulas? We have the ability to combine different Copula Functions (e.g. Gaussian, Archimedean) together allowing us to capture even more complex relationships in the data?

Is my understanding correct?

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  • $\begingroup$ Because a copula is fully general -- every finite multivariate distribution is representable as a copula (Sklar's Theorem) -- your conclusion about being able to "capture even more complex relationships" cannot be correct. Vine copulas, which express a copula as a sequence of relatively simple copulas, effectively are less complicated than a general multivariate copula; but there is a concomitant cost in constructing the sequence. In effect, a vine copula is a method of analyzing a general copula into a sequence of simpler parts. $\endgroup$
    – whuber
    Commented Jan 31 at 15:49
  • $\begingroup$ thanks for this insight ... why use a vine copula over a regular copula? is it because its simpler to do so? $\endgroup$ Commented Jan 31 at 16:22
  • $\begingroup$ Because it provides a structured mechanism to model something that otherwise is so formidably complicated it would be impossible even to begin. How, for instance, would you go about fitting a general copula to (say) data with 100 variables? $\endgroup$
    – whuber
    Commented Jan 31 at 16:26
  • $\begingroup$ I made notes on Copulas here - are they correct? stats.stackexchange.com/questions/638554/… $\endgroup$ Commented Feb 4 at 23:01

1 Answer 1

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Copula:

  1. Very complicated in multivariate cases. Not all copula can be extended into multivariate copula.
  2. It imposes the same dependency among all variables, not the case for many real-life data.

Vine copula:

  1. Allow to model two variables at a time. Hence, there is no limitation on the type of copula used. In the vine copula model, where the number of variables, d, is $\geq 3$, we connected the variables into trees. In each tree, we connect only two variables. Hence, we only fit bivariate copulas for each pair of variables. Which then results in the second advantage below.
  2. Different construction types based on the correlations between variables in the data.
  3. Extending mixture copula into a mixture of vine copulas to provide even more accurate results and flexibility. The mixture copula model allows for only one type of mixture component for all the variables. For example, we fit a mixture of Frank and Clayton copulas to all the variables. Meanwhile, we can fit a different mixture of copulas for each pair of variables in the vine copula.

I have added some good papers for mixture of vines:

  1. Mixture of D-vine copula (allow for one type of copula) mixture of D-vines

  2. Mixture of R-vines (allow to vary the type of the copulas) enter link description here

Your understanding is correct, plus the added points.

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  • $\begingroup$ Is it possible to flesh 3 out a bit more? $\endgroup$ Commented Feb 3 at 10:34
  • $\begingroup$ is it possible to also flesh 1 out a bit more please? :) $\endgroup$ Commented Feb 3 at 17:12
  • $\begingroup$ @statsplease done. $\endgroup$
    – Maryam
    Commented Feb 4 at 6:31
  • $\begingroup$ @user123945 done $\endgroup$
    – Maryam
    Commented Feb 4 at 6:31
  • $\begingroup$ Your characterizations (1) and (2) of copulas look incorrect to me. It's especially not the case that a copula "imposes the same dependency among all variables" in any sense of that phrase. $\endgroup$
    – whuber
    Commented Feb 4 at 15:05

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