bmspcov {bspcov}

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())

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.

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).

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.

Author(s)

Kyoungjae Lee and Seongil Jo

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))