Pair-copula decomposition

Vines express a multivariate copula density as a product of bivariate copula densities evaluated at recursively computed conditional probabilities.

D-vine density

For the order , a full D-vine density can be written as

Here is the pair-copula density assigned to tree and edge . In the package this corresponds to:

edges[k][i]

with coordinates (left, right).

C-vine density

For a C-vine order , tree has root and edges connecting the root to variables . A full C-vine density is

In the package this is also stored as edges[k][i], but the interpretation is different: the first coordinate is the current root and the second coordinate is the child.

Truncated vines

A truncated vine stops after tree   . Pair-copulas above tree are omitted, which is equivalent to using independence copulas for higher-order conditional pairs in the simplified construction.

using VineCopulas

edges = [
    [GaussianCopula([1.0 0.4; 0.4 1.0]), ClaytonCopula(2, 1.5), FrankCopula(2, 2.0)],
    [GumbelCopula(2, 1.3), JoeCopula(2, 1.4)],
]

cv = CVineCopula([4, 3, 2, 1], edges; trunc=2)
truncation(cv)

2