< List of probability distributions
Copulas (from the Latin link) are invaluable tools for understanding complex variables and their interrelationships. Copula distributions allow us to better identify dependencies between random variables in multivariate settings by combining independently specified marginal probability functions with copula densities. In other words, a copula helps to isolate joint or marginal probabilities of two variables in a multivariate system.
More specifically, any multivariate distribution can be constructed with a marginal distribution and a copula function, which isolates a multivariate distribution’s dependency structure. The formal definition is:
A d-dimensional copula, C : [0, 1]d : → [0, 1] is a cumulative distribution function (CDF) with uniform marginals .
When the individual marginals all have a uniform distribution over U(0, 1), various specialists can then use them in fields such as finance (pricing securities), risk analysis (model selection). All of this paves the way for using more advanced methods like Copula models to produce new insights from data sets.
In linguistics, grammatical copulas link parts of a sentence. They are not related to the copula distribution, despite sharing the same name.
Copula distribution types
The Copula family of distributions covers a vast range of functions, each with unique tail behavior. Among these are the Binary Clayton, Frank, and Gumbel distributions . These can be used to study correlation between variables; the Binary Clayton is best for finding lower-tail correlations while its asymmetrical counterpart -the Binary Gumbel – excels at identifying upper-tailed ones.
A third type, the Binary Frank, has a symmetrical and independent tails with 0 correlation coefficients for both tails:
The Binary t-student variant has a thick symmetrical tail, thus is sensitive to changes in random variables that affect tails.
Through the use of copulas, we can construct joint distributions in two steps: first, decide on appropriate marginals and then add a dependence structure. Once selected for both marginal and copula distribution models, estimating their respective parameters often occurs independently from each other. It’s important to note that different combinations of marginals may produce completely distinct joint distributions with only unchanged dependence structures via the same selection of Copulas.
One downside to using copulas is that they are somewhat esoteric in nature, which means that they can be difficult to understand and use — and easily misapplied. For most real world applications (such as in securities), sophisticated algorithms and computers must be used.
Image: Matteo Zandi, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons
 Haugh, M. (2016). An Introduction to Copulas. IEOR E4602: Quantitative Risk Management.
 Xue et al. (2019). Proceedings of PURPLE MOUNTAIN FORUM 2019-International Forum on Smart Grid Protection and Control, Volume 2. Springer.