Gaussian Copula
As a member of an Elliptical-Contoured Distribution
For details, please refer to Fang et al.
Let \(Z\) be a \(d\)-dimensional random vector that follows a stochastic representation \(\bm{\mu} + r\mathbf{A}\mathbf{u}\) such that
- \(\bm{\mu} \in \mathbb{R}^{d\times 1}\) (location vector)
- \(r \geq 0\), \(r\) is a (univariate) random variable of some density (independent radial part)
- \(\mathbf{A}\mathbf{A}^T = \boldsymbol{\Sigma}, \boldsymbol{\Sigma}\in\mathbb{R}^{d\times d}\) (positive-definite matrix modelling the covariance)
- \(\mathbf{u}\) is uniformly distributed on the unit sphere in \(\mathbb{R}^d, i.e. \mathbf{u}\sim U(\{ \mathbf{x} \in \mathbb{R}^d: \lVert \mathbf{x} \rVert=1 \})\)
One can think of \(\mathbf{A}\), the Cholesky factor of some covariance matrix, as some multivariate linear transformation acting on \(r\mathbf{u}\), a spherical distribution.
The probability density of \(Z \sim \text{EC}_d(\bm{\mu}, \boldsymbol{\Sigma}, g)\) can be written in the form
\[\begin{align} \lvert \boldsymbol{\Sigma} \rvert^{-1/2} g( (Z-\bm{\mu})^T \boldsymbol{\Sigma}^{-1} (Z-\bm{\mu}) ), \end{align}\]where \(g(\cdot)\) is some scale function uniquely determined by the distribution of \(r\).
When
\[\begin{align} g(x) := \frac{1}{(2\pi)^{d/2}} \exp (\frac{-x}{2}), \end{align}\]\(Z\) has a \(d\)-dimensional Gaussian probability distribution (also denoted as \(\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})\)):
\[\begin{align} \frac{1}{ ((2\pi)^{d} \lvert \boldsymbol{\Sigma} \rvert)^{1/2} } \exp \bigg(\frac{-1}{2} (Z-\bm{\mu})^T \boldsymbol{\Sigma}^{-1} (Z-\bm{\mu}) \bigg) \end{align}\]where \(\boldsymbol{\Sigma}: \{ \rho_{ij} \}\) is some postive definite (correlation matrix) such that for \(i,j = 1,\dots, d\), \(\rho_{ij} = \rho_{ji}\), and
\[\begin{align} \rho_{ij} = \begin{cases} 1 &\text{ if } i=j,\\ 0 < \rho_{ij} < 1 &\text{ if } i\neq j \end{cases} \end{align}\]Conditional Multivariate Gaussian Distribution
Consider a multivariate random vector \(\mathbf{X}\in\mathbb{R}^{d\times n}\), \(\mathbf{X}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})\). We want to compute the conditional joint distribution of \(\mathbf{X_1}\) given \(\mathbf{X_2}=\mathbf{x}_2\), such that \(\mathbf{X_1}\in\mathbb{R}^{d_1\times n}\), \(\mathbf{X_2}\in\mathbb{R}^{d_2\times n}\), and \(d=d_1+d_2\).
We first partition all relevant matrices as follows:
\[\begin{align*} \mathbf{X} = \begin{bmatrix} \mathbf{X_1}\\\mathbf{X_2} \end{bmatrix}, \bm{\mu} = \begin{bmatrix} \bm{\mu_1}\\\bm{\mu_2} \end{bmatrix}, \boldsymbol{\Sigma} = \begin{bmatrix} \boldsymbol{\Sigma_{11}} & \boldsymbol{\Sigma_{12}} \\ \boldsymbol{\Sigma_{21}} & \boldsymbol{\Sigma_{22}} & \end{bmatrix}, \htmlId{eq:test}{\tag{1}} \end{align*}\]where \(\boldsymbol{\Sigma_{11}} \in \mathbb{R}^{d_1\times d_1}\), \(\boldsymbol{\Sigma_{22}} \in \mathbb{R}^{d_2\times d_2}\), and \(\boldsymbol{\Sigma_{21}} = \boldsymbol{\Sigma_{12}}^T \in \mathbb{R}^{d_2\times d_1}\).
The distribution of \(\mathbf{X_1}\) conditional on \(\mathbf{X_2}=\mathbf{x}_2\) is a multivariate normal \((\mathbf{X_1}\mid\mathbf{X_2=\mathbf{x}_2})\sim \mathcal{N}(\bar{\bm{\mu}},\bar{\boldsymbol{\Sigma}})\), where
\[\begin{align*} \bar{\bm{\mu}} &= \bm{\mu_1} + \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}(\mathbf{x_2}-\bm{\mu_2})\\ \bar{\boldsymbol{\Sigma}} &= \boldsymbol{\Sigma}_{11} - \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}\boldsymbol{\Sigma}_{21} \end{align*}\]Proof
We can show this using a trick by creating some linear combination, \(\mathbf{Z}\), using \(\mathbf{X_1}\) and \(\mathbf{X_2}\), such that \(\mathbf{Z}\) has zero correlation with \(\mathbf{X_2}\). Since any linear combination of normally distributed random variables is also normally distributed, \(\mathbf{Z}\) and \(\mathbf{X_2}\) have a joint normal distribution, and will therefore be independent. This gives us the following expressions:
\[\begin{align*} \mathbf{Z} &= c_1\mathbf{X_1} + c_2\mathbf{X_2} \\ \text{cov}(\mathbf{Z}, \mathbf{X_2}) &= 0 \\ \text{var}(\mathbf{Z}\mid\mathbf{X_2}) &= \text{var}(\mathbf{Z})\\ \mathbb{E}(\mathbf{Z}\mid\mathbf{X_2}) &= \mathbb{E}(\mathbf{Z}) = c_1\bm{\mu_1} + c_2\bm{\mu_2} \end{align*}\]We have
\[\begin{align*} \mathbb{E}(\mathbf{Z}\mid\mathbf{X_2}) = \mathbb{E}(c_1\mathbf{X_1} + c_2\mathbf{X_2} \mid \mathbf{X_2}) &= c_1\bm{\mu_1} + c_2\bm{\mu_2}\\ c_1\mathbb{E}(\mathbf{X_1} \mid \mathbf{X_2} ) + c_2\mathbf{x_2} &= c_1\bm{\mu_1} + c_2\bm{\mu_2}\\ \mathbb{E}(\mathbf{X_1} \mid \mathbf{X_2} ) &= \bm{\mu_1} + \frac{c_2}{c_1}( \bm{\mu_2} - \mathbf{x_2} ) \\ \end{align*}\]Let \(A:=\frac{c_2}{c_1}\). Then
\[\begin{align*} \mathbb{E}(\mathbf{X_1} \mid \mathbf{X_2} ) &= \bm{\mu_1} + A ( \bm{\mu_2} - \mathbf{x_2} ) \\ \end{align*}\]Then
\[\begin{align*} \text{var}(\mathbf{X_1}\mid\mathbf{X_2}) &= \text{var}\bigg(\frac{1}{c_1}(\mathbf{Z}-c_2\mathbf{X_2})\mid\mathbf{X_2} \bigg)\\ &=\text{var}\bigg(\frac{1}{c_1}(\mathbf{Z})\mid\mathbf{X_2} \bigg)\\ &= \text{var}\bigg(\frac{1}{c_1}(\mathbf{Z}) \bigg)\\ &= \text{var}\bigg(\mathbf{X_1} + A(\mathbf{X_2}) \bigg)\\ &= \text{var}(\mathbf{X_1}) + A\text{var}(\mathbf{X_2})A^T + A\text{cov}(\mathbf{X_1},\mathbf{X_2}) + \text{cov}(\mathbf{X_2},\mathbf{X_1})A^T \\ &= \boldsymbol{\Sigma}_{11} + A\boldsymbol{\Sigma}_{22}A^T + A\boldsymbol{\Sigma}_{21} + \boldsymbol{\Sigma}_{12}A^T\\ &= \boldsymbol{\Sigma}_{11} + A\boldsymbol{\Sigma}_{22}A^T + 2A\boldsymbol{\Sigma}_{21} \\ &=\boldsymbol{\Sigma}_{11} + A ( \boldsymbol{\Sigma}_{22}A^T + 2\boldsymbol{\Sigma}_{21} ) \end{align*}\]We are left with the problem of determining \(A\). Using
\[\begin{align*} \text{cov}(\mathbf{Z}, \mathbf{X_2}) = c_1\text{cov}(\mathbf{X_1}, \mathbf{X_2}) + c_2 \text{cov}(\mathbf{X_2}, \mathbf{X_2})&=0\\ c_1\text{cov}(\mathbf{X_1}, \mathbf{X_2}) + c_2 \text{var}(\mathbf{X_2}) &= 0\\ \text{cov}(\mathbf{X_1}, \mathbf{X_2}) &= - \frac{c_2}{c_1} \text{var}(\mathbf{X_2})\\ \boldsymbol{\Sigma_{12}} &= - A \boldsymbol{\Sigma_{22}}\\ A &= -\boldsymbol{\Sigma_{12}}\boldsymbol{\Sigma_{22}}^{-1}\\ \end{align*}\]This gives us the expressions for mean and covariance respectively as follows:
\[\begin{align*} \mathbb{E}(\mathbf{X_1} \mid \mathbf{X_2} ) &= \bm{\mu_1} + \boldsymbol{\Sigma_{12}}\boldsymbol{\Sigma_{22}}^{-1} ( \mathbf{x_2} - \bm{\mu_2}) \\ \end{align*}\] \[\begin{align*} \text{var}(\mathbf{X_1}\mid\mathbf{X_2}) &= \boldsymbol{\Sigma}_{11} + A ( \boldsymbol{\Sigma}_{22}A^T + 2\boldsymbol{\Sigma}_{21} )\\ &= \boldsymbol{\Sigma}_{11} -\boldsymbol{\Sigma_{12}}\boldsymbol{\Sigma_{22}}^{-1} ( -\boldsymbol{\Sigma}_{22}\boldsymbol{\Sigma_{22}}^{-1}\boldsymbol{\Sigma_{21}} + 2\boldsymbol{\Sigma}_{21} ) \\ &= \boldsymbol{\Sigma}_{11} -\boldsymbol{\Sigma_{12}}\boldsymbol{\Sigma_{22}}^{-1} ( -\boldsymbol{\Sigma_{21}} + 2\boldsymbol{\Sigma}_{21} ) \\ &= \boldsymbol{\Sigma}_{11} -\boldsymbol{\Sigma_{12}}\boldsymbol{\Sigma_{22}}^{-1}\boldsymbol{\Sigma_{21}} \\ \end{align*}\]Copulas from Meta-Elliptical Distributions
Let \(\mathbf{X} = (X_1, X_2, \dots, X_d)^T\) be a random vector with each component \(X_i\) having a given continuous marginal probability density \(f_i(x_i)\) and corresponding cumulative distribution \(F_i(x_i)\). Without loss of generality, let \(Z:=(Z_1, Z_2, \dots, Z_d)^T \sim \text{EC}_d(\bm{0}, \boldsymbol{\Sigma}, g)\) where \(g\) is given by
\[\begin{align} g(x) := \frac{1}{(2\pi)^{d/2}} \exp (\frac{-x}{2}). \end{align}\]Then
\[\begin{align} Z_i = \Phi_g^{-1}( F_i(X_i) ), \end{align}\]where \(\Phi_g^{-1}\) is the inverse of \(\Phi_g\), a univariate standard Gaussian distribution \(\mathcal{N}(0,1)\).
\(\mathbf{X}\) is said to have a meta-elliptical distribution denoted as \(\text{ME}_d(\mathbf{0}, \boldsymbol{\Sigma}, g; F_1, \dots, F_d)\). The density function of \(\mathbf{X}\) is then
\[\begin{align} h(x_1, x_2, \cdots, x_d) = \phi(\Phi_g^{-1}( F_1(x_1) ), \Phi_g^{-1}( F_2(x_2) ), \dots, \Phi_g^{-1}( F_d(x_d) ) ) \prod^d_{i=1} f_x(x_i), \end{align}\]where \(\phi\) is the \(d\)-dimensional multivariate weighted Gaussian density function and the copula density of an elliptically-contoured distribution. Then, denoting \(\Phi\) as the \(d\)-dimensional multivariate weighted Gaussian CDF, the corresponding Gaussian copula can be defined as:
\[\begin{align} C^{\text{Gauss}} ( F_1(x_1), F_2(x_2), \dots, F_d(x_d) ) = \Phi(\Phi_g^{-1}( F_1(x_1) ), \Phi_g^{-1}( F_2(x_2) ), \dots, \Phi_g^{-1}( F_d(x_d) ) ) \end{align}\]Kendall’s Correlation Coefficient
Let \(\mathbf{X}=(X_1, X_2, \dots, X_d)\) have a meta-elliptical distribution \(\text{ME}_d(\mathbf{0}, \boldsymbol{\Sigma}, g; F_1, \dots, F_d)\). Then the Kendall’s tau of \(\mathbf{X}\) is given by
\[\begin{align} \tau = \frac{2}{\pi} \text{arcsin} (\rho), \end{align}\]and it depends only on \(\rho\) and is invariant in the class of meta-elliptical distributions \(\text{ME}_d(\mathbf{0}, \boldsymbol{\Sigma}, g; F_1, \dots, F_d)\).