Gaussian Copula

The name copula comes from the Latin for “link” or “tie”, as it literally means this model is used to describe the inter-correlation between random variables. More specifically, it is a multivariate CDF for which the marginal probability distribution of each variable is uniform on the interval [0, 1].

It’s widely used in estimating high-dimensional applications, since any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula that describes the dependence structure between the variables, according to Sklar’s theorem.

1. Consider a random vector \((X_{1},X_{2},\dots ,X_{d})\), where marginal CDFs \({\displaystyle F_{i}(x)=\Pr[X_{i}\leq x]}\) are continuous.

   Probability integral transform: Suppose that a random variable \(X\) has a continuous distribution for which the CDF is \({\displaystyle F_{X}.}\) Then the random variable \(U\) defined as \({\displaystyle U:=F_{X}(X)\,,}\) has a standard uniform distribution \(U[0,1]\).

we can make a random vector \((U_{1},U_{2},\dots ,U_{d})=\left(F_{1}(X_{1}),F_{2}(X_{2}),\dots ,F_{d}(X_{d})\right)\), which has marginals be U[0,1]. The CDF of this random vector(aka The copula C) can be written as \({C=\displaystyle C(u_{1},u_{2},\dots ,u_{d})=\Pr[U_{1}\leq u_{1},U_{2}\leq u_{2},\dots ,U_{d}\leq u_{d}].}\).

simulator

The reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions.

1. Generate a sample \((U_{1},U_{2},\dots ,U_{d})\) from the copula function.
2. Constructed \((X_{1},X_{2},\dots ,X_{d})=\left(F_{1}^{-1}(U_{1}),F_{2}^{-1}(U_{2}),\dots ,F_{d}^{-1}(U_{d})\right)\)

Rewrite the above formula for the copula function with X:\({\displaystyle C(u_{1},u_{2},\dots ,u_{d})=\Pr[X_{1}\leq F_{1}^{-1}(u_{1}),X_{2}\leq F_{2}^{-1}(u_{2}),\dots ,X_{d}\leq F_{d}^{-1}(u_{d})].}\)

Gaussian Copula

For a given correlation matrix \({\displaystyle R\in [-1,1]^{d\times d}}\), the Gaussian copula with parameter matrix \(R\) can be written as \(C_{R}^{\text{Gauss}}(u)=\Phi _{R}\left(\Phi ^{-1}(u_{1}),\dots ,\Phi ^{-1}(u_{d})\right)\)

The Gaussian copula is a distribution over the unit = ${\displaystyle c_{R}^{\text{Gauss}}(u)={\frac {1}{\sqrt {\det {R}}}}\exp \left(-{\frac {1}{2}}{\begin{pmatrix}\Phi ^{-1}(u_{1})\\vdots \\Phi ^{-1}(u_{d})\end{pmatrix}}^{T}\cdot \left(R^{-1}-I\right)\cdot {\begin{pmatrix}\Phi ^{-1}(u_{1})\\vdots \\Phi ^{-1}(u_{d})\end{pmatrix}}\right),}$