How Cholesky Sampling Differs from Copula Modeling

Summary

This explanation distinguishes a numerical sampling technique from a model of dependence. Cholesky decomposition factors a covariance or correlation matrix so that independent draws can be transformed into correlated draws from a specified multivariate distribution, such as a multivariate normal.

A copula instead describes how marginal distributions are joined into a joint distribution, allowing dependence to be modeled separately from the behavior of each variable on its own. This can represent dependence structures that are not captured by a direct multivariate distribution with a convenient correlation description. Cholesky may be used within a simulation procedure for a copula, but it does not define the copula itself. The discussion is conceptual and brief; it gives no worked example or guidance on selecting or estimating a copula.

Key ideas

• Cholesky decomposition is a matrix factorization used to generate correlated draws from a specified multivariate distribution.
• A copula links marginal distributions through a model of their dependence.
• Cholesky can support sampling from a copula-based model without being a copula.
• The choice of dependence model matters when ordinary multivariate distributions do not capture the relationships of interest.

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# Why is Cholesky decomposition not a copula?


# Why is Cholesky decomposition not a copula?












I've been told that Cholesky decomposition can be used as part of a process involving copulas to generate correlated random variables, but it is not a copula itself. From my understanding, both of these can be used to simulate dependent random variables. So why the distinction, what's the difference? Please help me understand.

## Answer by lukas kiss (score 2, accepted)

https://quant.stackexchange.com/a/81381

Cholesky decomposition is used in Monte Carlo methods to easily simulate correlated random variables, where the correlation is well defined/easily defined. Like multivariate distribution normal.

In some cases, you can have two random variables with complex correlational structure, which can not be modeled directly by a multivariate distribution. In that case you will use a copula. They allow you to describe complex correlation structure by a simple multivariate distribution and two marginal distributions.

In nutshell:

- Cholesky decomposition is a method used in Monte Carlo simulations to sample from multivariate distribution.

- Copula is a model to model complex correlations between random variables. When you sample from copula, you can use Cholesky decomposition.

For more information on copulas, I recommend:

- An intuitive, visual guide to copulas

- Bayesian copula estimation: Describing correlated joint distributions

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.