estR: estR

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

estR

Description

This function computes the sample Q-scores rank correlation matrix. A ridge penalization is possible.

Usage

estR(
  sample,
  omega = 1,
  Q = function(t) {
     stats::qnorm(t)
 }
)

Arguments

sample	A sample from a q-dimensional random vector \mathbf{X} (n \times q matrix with observations in rows, variables in columns).
omega	The penalty parameter for ridge penalization (default = 1, meaning no penalization).
Q	The quantile function to be applied to the copula pseudo-observations (default = qnorm()).

Details

Given a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1},...,\mathbf{X}_{k}) with \mathbf{X}_{i} = (X_{i1}, \dots, X_{id_{i}}) a d_{i} dimensional random vector, i.e., q = d_{1} + ... + d_{k}, the sample Q-scores rank correlation matrix is given as

\widehat{\mathbf{R}}_{n} = \begin{pmatrix} \widehat{\mathbf{R}}_{11} & \widehat{\mathbf{R}}_{12} & \cdots & \widehat{\mathbf{R}}_{1k} \\ \widehat{\mathbf{R}}_{12}^{\text{T}} & \widehat{\mathbf{R}}_{22} & \cdots & \widehat{\mathbf{R}}_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ \widehat{\mathbf{R}}_{1k}^{\text{T}} & \widehat{\mathbf{R}}_{2k}^{\text{T}} & \cdots & \widehat{\mathbf{R}}_{kk} \end{pmatrix} \hspace{0.2cm} \text{with} \hspace{0.2cm} \left (\widehat{\mathbf{R}}_{im} \right )_{jt} = \widehat{\rho}_{ij,mt} = \frac{\frac{1}{n} \sum_{\ell = 1}^{n} \widehat{Z}_{ij}^{(\ell)} \widehat{Z}_{mt}^{(\ell)}}{\frac{1}{n} \sum_{\ell = 1}^{n} \left [Q \left (\frac{\ell}{n+1} \right ) \right ]^{2}},

for i,m = 1, \dots, k, j = 1, \dots, d_{i}, and t = 1, \dots, d_{m}, based on the observed Q-scores

\widehat{Z}_{ij}^{(\ell)} = Q \left (\frac{n}{n+1} \widehat{F}_{ij} \left (X_{ij}^{(\ell)} \right )\right ) = Q \left (\frac{1}{n+1} \sum_{t = 1}^{n} 1 \left \{X_{ij}^{(t)} \leq X_{ij}^{(\ell)} \right \} \right ),

for \ell = 1, \dots, n, where \widehat{F}_{ij} is the empirical cdf of the sample X_{ij}^{(1)},\dots,X_{ij}^{(n)} for i = 1, \dots, k and j = 1, \dots, d_{i}. The underlying assumption is that the copula of \mathbf{X} is meta-elliptical. The default for Q is the standard normal quantile function (corresponding to the assumption of a Gaussian copula). Ridge penalization (especially in the Gaussian copula setting) with penalty parameter omega = \omega boils down to computing

\omega \widehat{\mathbf{R}}_{n} + (1-\omega) \mathbf{I}_{q},

where \mathbf{I}_{q} stands for the identity matrix.

Value

The (ridge penalized) sample Q-scores rank correlation matrix.

References

De Keyser, S. & Gijbels, I. (2024). Some new tests for independence among continuous random vectors.

Warton, D.I. (2008). Penalized normal likelihood and ridge regularization of correlation and covariance matrices. Journal of the American Statistical Association 103(481):340-349.
doi: https://doi.org/10.1198/016214508000000021.

See Also

cvomega for selecting omega using K-fold cross-validation in case of a Gaussian copula.

Examples

# Multivariate normal copula setting

q = 10
n = 50

# AR(1) correlation matrix with correlation 0.5
R = 0.5^(abs(matrix(1:q-1,nrow = q, ncol = q, byrow = TRUE) - (1:q-1)))

# Sample from multivariate normal distribution
sample = mvtnorm::rmvnorm(n,rep(0,q),R,method = "chol")

# 5-fold cross-validation with Gaussian likelihood as loss for selecting omega
omega = cvomega(sample = sample,omegas = seq(0.01,0.999,len = 50),K = 5)

R_est = estR(sample,omega = omega)

# Multivariate Student-t copula setting

q = 10
n = 500

# Degrees of freedom
nu = 7

# Density of R^2, with R the radius of the elliptical distribution
# Identifiability contraint is that R is the correlation matrix
R2 = function(t,q){(gamma((q+nu)/2)/(((nu-2)^(q/2)) * gamma(nu/2) * gamma(q/2))) *
                   (t^((q/2)-1)) * ((1+(t/(nu-2)))^(-(q+nu)/2))}

# Univariate quantile function, with unit variance
Q = function(t){extraDistr::qlst(t,nu,0,sqrt((nu-2)/nu))}

# AR(1) correlation matrix with correlation 0.5
R = 0.5^(abs(matrix(1:q-1,nrow = q, ncol = q, byrow = TRUE) - (1:q-1)))

# Sample from multivariate Student-t distribution
# with correlation matrix R and nu degrees of freedom
sample = ElliptCopulas::EllDistrSim(n,q,t(chol(R)),density_R2 = function(t){R2(t,q)})

R_est = estR(sample,Q = Q)

VecDep documentation built on April 4, 2025, 5:14 a.m.