Module vine 

Vine copulas module.

Vine copulas (also called pair-copula constructions) are a flexible way to model high-dimensional dependence structures by decomposing them into bivariate copulas.

This module implements:

• C-vine (Canonical vine) - star-shaped structure
• D-vine (Drawable vine) - path-shaped structure

Vine Copulas Overview

A d-dimensional density can be decomposed into:

• d marginal densities
• d(d-1)/2 bivariate copulas (pair-copulas)

The vine structure determines which variables are coupled and in which order.

Tree layout

Both constructors take trees, where trees[l] holds the pair-copulas of tree l + 1. Sampling uses each pair-copula’s position in trees; the var1, var2, and conditioning_set labels of a PairCopula are descriptive only. With 0-based variable indices:

• C-vine: trees[l][e] couples variables l and l + e + 1, given variables 0..l.
• D-vine: trees[l][e] couples variables e and e + l + 1, given variables e + 1..=e + l.

All pair-copula families in CopulaType are exchangeable, so the h-functions do not depend on the argument order within a pair. Gaussian and Student-t pair-copulas use closed-form h-functions and inverses (Aas et al., 2009); the other families differentiate the copula CDF numerically and invert by bisection.

Bibliography

• Aas, K., et al. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics.
• Bedford, T., & Cooke, R. M. (2002). Vines - A new graphical model for dependent random variables.
• Joe, H. (2014). Dependence Modeling with Copulas. CRC Press.

Structs

CVineCopula

C-vine copula (Canonical vine).

DVineCopula

D-vine copula (Drawable vine).

PairCopula

A pair-copula element in the vine structure.

Enums

CopulaType

Enum representing different copula types for vine constructions.