Documentation

Type hierarchy

The very top of the type hierarchy is type Copula, which subsumes all other types of copulas. It splits up into bivariate pair copulas of type PairCop on the one side, and type PCC on the other side.

PCC represents pair copula constructions. This is a flexible way of constructing higher-dimensional copulas, where any subtype of type PairCop can be used as a bivariate building block.

Furthermore, there are different types of PairCop copulas. One fundamental type of pair copulas is the family of parametric copulas, represented by type AbstractParamPC. Parametric copulas also can be used as starting point for the creation of new pair copulas.

One way to create new pair copulas is through application of a copula modification, which for example could be a rotation or a reflection. These types of modified copulas are subsumed in type ModPC, and their type hierarchy is shown more elaborately below.

A different way to create new pair copulas is through convex combination of existing pair copulas, called mixture copulas, and is implemented as type MixPC.

Note that we face a reflexive structure at this point: ModPC and MixPC types both are built on pair copulas, but they are pair copulas themselves, too.

Parametric pair copulas build the very core of this package, as they are required both as starting point for copula modifications and for pair copula constructions. However, most parametric copula families are not yet implemented in core Julia yet. Hence, in the meantime, there exist two different representations of parametric copulas: ParamPC, which is a pure Julia implementation for a few parametric families, and ParamPC_Cpp, which makes use of the C++ copula package VineCopulaCPP.

Every abstraction on top of parametric copulas, however, is written in pure Julia.

Possible extensions in the type hierarchy are:

• multivariate parametric copulas
• multivariate non-parametric copulas
• bivariate non-parametric copulas

Pair copula construction: related types

A pair copula basically consists of two things: a description of the density decomposition (called vine), and copulas to capture the dependency between individual pairwise relations.

The conventional way to represent a density decomposition is through an R-Vine matrix. However, a vine evenly could be described as an array of individual conditioning trees, one for each variable.

There are two different representations of conditioning trees: types CTreePaths and CTreeParRef. For reasons of efficiency, however, a Vine is not stored as Array of AbstractCTree, but as a single two-dimensional matrix, where each column is a conditioning tree in parent referencing notation.

Copula modifications

A copula modification consists of a given pair copula together with some modification operation. The most prominent such modifications are rotations and reflections.

As there has not yet evolved any standard of whether rotations are defined clockwise or counter-clockwise, both ways are implemented.

Copula theory

For details on the underlying copula theory take a look at my research notes.

Acknowledgements

In its current version, Copulas.jl basically is only a wrapper to the fabulous VineCopulaCPP c++ library of Malte Kurz. VineCopulaCPP is a well tested library comprising parametric copulas with all their respective functions (pdf, cdf,…), as well as simulation routines and optimization algorithms for copula estimation.