bspcov (version 1.0.0)

bmspcov: Bayesian Sparse Covariance Estimation

Description

Provides a Bayesian sparse and positive definite estimate of a covariance matrix via the beta-mixture shrinkage prior.

Usage

bmspcov(X, Sigma, prior = list(), nsample = list())

Value

Sigma

a nmc \(\times\) p(p+1)/2 matrix including lower triangular elements of covariance matrix.

Phi

a nmc \(\times\) p(p+1)/2 matrix including shrinkage parameters corresponding to lower triangular elements of covariance matrix.

p

dimension of covariance matrix.

Arguments

X

a n \(\times\) p data matrix with column mean zero.

Sigma

an initial guess for Sigma.

prior

a list giving the prior information. The list includes the following parameters (with default values in parentheses): a (1/2) and b (1/2) giving the shape parameters for beta distribution, lambda (1) giving the hyperparameter for the diagonal elements, tau1sq (10000/(n*p^4)) giving the hyperparameter for the shrinkage prior of covariance.

nsample

a list giving the MCMC parameters. The list includes the following integers (with default values in parentheses): burnin (1000) giving the number of MCMC samples in transition period, nmc (1000) giving the number of MCMC samples for analysis.

Author

Kyoungjae Lee and Seongil Jo

Details

Lee, Jo and Lee (2022) proposed the beta-mixture shrinkage prior for estimating a sparse and positive definite covariance matrix. The beta-mixture shrinkage prior for \(\Sigma = (\sigma_{jk})\) is defined as $$ \pi(\Sigma) = \frac{\pi^{u}(\Sigma)I(\Sigma \in C_p)}{\pi^{u}(\Sigma \in C_p)}, ~ C_p = \{\mbox{ all } p \times p \mbox{ positive definite matrices }\}, $$ where \(\pi^{u}(\cdot)\) is the unconstrained prior given by $$ \pi^{u}(\sigma_{jk} \mid \rho_{jk}) = N\left(\sigma_{jk} \mid 0, \frac{\rho_{jk}}{1 - \rho_{jk}}\tau_1^2\right)$$ $$ \pi^{u}(\rho_{jk}) = Beta(\rho_{jk} \mid a, b), ~ \rho_{jk} = 1 - 1/(1 + \phi_{jk})$$ $$ \pi^{u}(\sigma_{jj}) = Exp(\sigma_{jj} \mid \lambda). $$ For more details, see Lee, Jo and Lee (2022).

References

Lee, K., Jo, S., and Lee, J. (2022), "The beta-mixture shrinkage prior for sparse covariances with near-minimax posterior convergence rate", Journal of Multivariate Analysis, 192, 105067.

See Also

sbmspcov

Examples

set.seed(1)
n <- 20
p <- 5

# generate a sparse covariance matrix:
True.Sigma <- matrix(0, nrow = p, ncol = p)
diag(True.Sigma) <- 1
Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8)
nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1)
zeroIND = (1:(p*(p-1)/2))[-nonzeroIND]
Values[zeroIND] <- 0
True.Sigma[lower.tri(True.Sigma)] <- Values
True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)]
if(min(eigen(True.Sigma)$values) <= 0){
  delta <- -min(eigen(True.Sigma)$values) + 1.0e-5
  True.Sigma <- True.Sigma + delta*diag(p)
}

# generate a data
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma)

# compute sparse, positive covariance estimator:
fout <- bspcov::bmspcov(X = X, Sigma = diag(diag(cov(X))))
post.est.m <- bspcov::estimate(fout)
sqrt(mean((post.est.m - True.Sigma)^2))
sqrt(mean((cov(X) - True.Sigma)^2))