VineCopulas.jl

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Pair-copula conditionals ​

Conditional distributions are the main recursive building blocks of vine copulas. For a bivariate copula ,

and

VineCopulas.jl exposes these operations through

julia
hfunc1(C, u, v)
hfunc2(C, u, v)
hinv1(C, q, v)
hinv2(C, q, u)

The inverse functions solve for the conditioned coordinate. In particular,

while

These quantities are used repeatedly in vine density evaluation, simulation, Rosenblatt transforms, inverse Rosenblatt transforms, and the recursive pseudo-observations generated during sequential fitting.

Relation with Copulas.jl ​

Copulas.jl defines the mathematical representation of the underlying bivariate copula and provides its conditional-distribution interface.

For a compatible C <: Copulas.Copula{2}, the generic VineCopulas.jl conditional implementation is equivalent to

julia
hfunc1(C, u, v) =
    cdf(condition(C, 2, v), u)

hfunc2(C, u, v) =
    cdf(condition(C, 1, u), v)

hinv1(C, q, v) =
    quantile(condition(C, 2, v), q)

hinv2(C, q, u) =
    quantile(condition(C, 1, u), q)

The index passed to condition denotes the coordinate whose value is fixed. Therefore,

julia
condition(C, 2, v)

represents the conditional distribution of the first coordinate given that the second coordinate is equal to v.

Similarly,

julia
condition(C, 1, u)

represents the conditional distribution of the second coordinate given that the first coordinate is equal to u.

This convention is important because interchanging the conditioning coordinate changes hfunc1 and hfunc2.

Generic fallback ​

A bivariate copula implemented in Copulas.jl does not need a duplicated conditional implementation in VineCopulas.jl merely to be evaluated inside a vine.

When the required conditional distribution supports cdf and quantile, the generic pair-copula fallback delegates these operations to Copulas.condition.

This keeps the mathematical conditional-distribution machinery in Copulas.jl while allowing VineCopulas.jl to focus on vine-specific algorithms.

Specialized kernels ​

Some pair-copula families provide specialized hfunc1, hfunc2, hinv1, or hinv2 methods.

These methods must preserve exactly the same conditional convention as the generic fallback, but may be useful when they provide:

  • closed-form expressions;

  • fewer allocations;

  • fewer expensive transformations;

  • improved numerical or tail stability;

  • specialized handling of singular copulas or generalized inverses.

The generic Copulas.condition implementation therefore defines the common semantics, while specialized pair-copula kernels provide optional fast or numerically robust paths.

Evaluation support versus fitting support ​

A family being usable as a pair-copula edge does not imply that it is automatically available for family selection.

Generic evaluation requires a reliable bivariate density and conditional distribution interface.

Automatic fitting and selection additionally require fitting support, parameter-domain information, rotation handling where applicable, candidate metadata, and dedicated correctness tests.

In short,

text
pair-copula evaluation support ≠ automatic selection support