Pair Copulas in Vines

Pair copulas are the bivariate building blocks on vine edges. In VineCopulas.jl, a pair copula is not a new family enum or a parallel mathematical hierarchy:

PairCopula === Copulas.Copula{2}

That design keeps the package close to Copulas.jl: individual copula families belong upstream, while vine-specific traversal and selection belong here.

Copulas.jl ownership

Copulas.jl owns the mathematical behavior of individual pair-copulas: parameter domains, cdf, pdf, logpdf, rand, conditioning, fitting methods, and distribution-level semantics.

VineCopulas.jl owns the vine use of those pair-copulas:

• attaching pair-copulas to edges;

• propagating conditional probabilities through trees;

• choosing pair families during vine selection;

• composing pair log-densities into a vine log-density;

• using fused kernels when traversal needs several pair quantities together.

Extension rule

A compatible Copulas.Copula{2} should be usable in a vine without defining a VineCopulas-specific family type. Extra VineCopulas.jl methods are performance hooks, not the semantic starting point.

h-functions

For a bivariate copula ,

The public API uses the common vine names:

using VineCopulas

C = GaussianCopula(2, 0.6)

(h1 = hfunc1(C, 0.25, 0.8),
 h2 = hfunc2(C, 0.25, 0.8))

(h1 = 0.07019663469687006, h2 = 0.9403707938799101)

Conceptually these agree with Copulas.condition:

cdf(condition(C, 2, v), u)
cdf(condition(C, 1, u), v)

The first call fixes the second coordinate at v and evaluates the conditional distribution of the first coordinate. The second call fixes the first coordinate at u.

Conditional inverses

Inverse h-functions map a conditional probability back to a copula-scale coordinate:

using VineCopulas

C = ClaytonCopula(2, 1.5)
v = 0.7
q = 0.35

u = hinv1(C, q, v)
(u = u, roundtrip = hfunc1(C, u, v))

(u = 0.5431513506786843, roundtrip = 0.3500000000000001)

They are used by inverse Rosenblatt transforms and simulation. For compatible pair copulas, the semantic fallback is the conditional quantile:

quantile(condition(C, 2, v), q)
quantile(condition(C, 1, u), q)

Generalized inverse semantics

Inverse h-functions should behave like conditional quantiles, not like an arbitrary root-finding branch. This matters near boundaries and for singular or nearly singular pair-copulas.

Rotations

Rotations reuse a base pair-copula with coordinate flips. They are useful because many one-sided tail-dependent families only represent one dependence orientation in their base form.

using VineCopulas
using Copulas

base = ClaytonCopula(2, 2.0)
rotated = SurvivalCopula(base, (false, true))

(base_tau = Copulas.τ(base), rotated_tau = Copulas.τ(rotated))

(base_tau = 0.5, rotated_tau = -0.5)

During automatic selection, allow_rotations=true lets the selector consider rotated versions where appropriate.

Note

Gaussian, Student-t, and Frank pair-copulas already support both positive and negative association in their base parameterization, so the default search does not need rotated duplicates for them.

Supported families

The default automatic-selection set is intentionally smaller than the complete catalog available through Copulas.jl.

Group	Families
Elliptical	Gaussian, Student-t
One-parameter Archimedean	Clayton, Gumbel, Frank, Joe
Two-parameter BB	BB1, BB6, BB7, BB8
Optional	Independence

ALL_PARAMETRIC_PAIR_FAMILIES additionally includes AMH, Gumbel-Barnett, inverse-Gaussian Archimedean, BB2, BB3, BB9, and BB10 families.

The bestiary describes the supported groups from a vine perspective. For the mathematics of each individual copula family, use the Copulas.jl documentation.

A custom-compatible pair

Explicit vines accept compatible bivariate copulas directly:

vine = DVineCopula([1, 2, 3], [[C12, C23], [C13_2]])

The pair objects may be different families on different edges. This is one of the main strengths of vine models.

Fitting needs more than evaluation

Manual construction only needs pair conditional operations for the vine tasks you call. Automatic selection also needs fitting support and parameter accounting. A pair-copula can therefore be valid for manual vines before it is a good automatic-selection candidate.