Getting Started

Installation

The TwinCopulas package is available through the Julia package system by running Pkg.add("TwinCopulas"). Throughout, we assume that you have installed the package.

Starting With a Bivariate Gaussian Copula

We start by drawing 1000 observations from a Gaussian Copula random variable.

The first step is to set up the environment:

julia> using Random, Distributions, TwinCopulas

julia> Random.seed!(123) # Setting the seed

Then, we create a bivariate Gaussian Copula distribution G and obtain samples using rand:

julia> G = GaussianCopula(0.5)
GaussianCopula{Float64}(θ=0.5)

The object G represents a probability distribution, in our case the bivariate Gaussian Copula. One can query its properties such as the Kendall's tau ($\tau$, we need de letter \tau) :

julia> TwinCopulas.τ(G)
0.33333333333333337

We can also draw samples from G with rand.

julia> x = rand(G,1000)
2×1000 Matrix{Float64}:
 0.842684  0.151638  0.488056  0.0362257  0.474858  …  0.525896  0.640167  0.965114  0.0298152  0.632266
 0.195875  0.189835  0.777506  0.0273238  0.425615     0.41442   0.467543  0.533448  0.750643   0.359143

You can easily obtain the pdf, cdf, and many other functions for a distribution. For instance, the $\rho_s$ (Spearman's Rho), $\lambda_u, \lambda_l$ (Upper and lower tail dependence), $\beta$ (Blomqvist $\beta$) and $\gamma$ (Ginni's coefficient):

julia> cdf(G, [0.5,0.6])
0.3804363762931904

julia> pdf(G, [0.5,0.6])
1.1424140110638463

Note:

Functions such as β (Blomqvist's $\beta$), τ (Kendall's tau), ρ_s (Spearman's Rho), and tail dependence measures (λ_u, λ_l) must be prefixed with TwinCopulas because they are not directly exported by the module. For example, use TwinCopulas.β(copula) to access Blomqvist's $\beta$.

Using Other Copulas

The package contains a large number of additional Copulas of two main types:

bicopula == ArrayLikeVariate{1}
SklarDist == ArrayLikeVariate{1}

Each type splits further into bivariate Continuous.

For instance, you can define the following Copulas (among many others):

julia> T = tCopula(ν, ρ) # bicopula (Elliptical)
julia> A = ClaytonCopula(θ)  # bicopula (Archimedean)
julia> E = GalambosCopula(θ) # bicopula (Extreme Value)
julia> R = ArchimaxCopula(A, E) # bicopula (Archimax)

In addition, you can create Copulas from univariate distributions (univariate margins):

julia> SklarDist(GaussianCopula(θ), [Normal(mu, sigma), Beta(α, β)])

To find out which parameters are appropriate for a given Copula C, you can use fieldnames(D):

julia> fieldnames(tCopula)
(:θ, :ν)
julia> fieldnames(ArchimaxCopula)
(:Archimedean, :Extreme)
julia> fieldnames(SklarDist)
(:copula, :margins)

This tells you that a t Copula is initialized with dregree of freddom ν and θ, ArchimaxCopula with an Archimedean Copula and Extreme value Copula, SklarDist need a bivariate copula (bicopula) and univariate distributions margins.