cvomega: cvomega

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

cvomega

Description

This functions selects the omega tuning parameter for ridge penalization of the empirical Gaussian copula correlation matrix via cross-validation. The objective function is the Gaussian log-likelihood, and a grid search is performed using K folds.

Usage

cvomega(sample, omegas, K)

Arguments

sample	A sample from a q-dimensional random vector \mathbf{X} (n \times q matrix with observations in rows, variables in columns).
omegas	A grid of candidate penalty parameters in [0,1].
K	The number of folds to be used.

Details

The loss function is the Gaussian log-likelihood, i.e., given an estimated (penalized) Gaussian copula correlation matrix (normal scores rank correlation matrix) \widehat{\mathbf{R}}_{n}^{(-j)} computed on a training set leaving out fold j, and \widehat{\mathbf{R}}_{n}^{(j)} the empirical (non-penalized) Gaussian copula correlation matrix computed on test fold j, we search for the tuning parameter that minimizes

\sum_{j = 1}^{K} \left [\ln \left ( \left | \widehat{\mathbf{R}}_{n}^{(-j)} \right | \right ) + \text{tr} \left \{\widehat{\mathbf{R}}_{n}^{(j)} \left (\widehat{\mathbf{R}}_{n}^{(-j)} \right )^{-1} \right \} \right ].

The underlying assumption is that the copula of \mathbf{X} is Gaussian.

Value

The optimal ridge penalty parameter minimizing the cross-validation error.

References

De Keyser, S. & Gijbels, I. (2024). High-dimensional copula-based Wasserstein dependence. doi: https://doi.org/10.48550/arXiv.2404.07141.

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

estR for computing the (Ridge penalized) empirical Gaussian copula correlation matrix.

Examples

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)

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