Core concepts

Copula scale

For continuous margins, Sklar's theorem writes a joint distribution as

VineCopulas.jl models the copula , not the marginal distributions. Fitting therefore expects observations on the copula scale, typically pseudo-observations in .

A common rank transformation is

Marginal modeling can instead be handled with tools such as SklarDist in Copulas.jl before the vine model is fitted.

Pair-copula decomposition

A vine decomposes a multivariate copula density into bivariate copula densities evaluated at recursively computed conditional probabilities. For a D-vine with order ,

The conditional arguments are propagated with pair-copula h-functions. This same recursion drives density evaluation, Rosenblatt transforms, simulation, and sequential fitting.

Simplifying assumption

The current fitting and evaluation layer implements simplified vines: a conditional pair-copula may depend on which variables are conditioned upon, but its parameters do not vary with the realized values of those conditioning variables.

Truncation

A vine truncated after tree    treats higher-tree conditional dependence as independence. This is different from the finite candidate parameter domains used by automatic family selection. Vine truncation changes the model structure; candidate bounds only limit an optimizer's search space.