bandPPP {bspcov}

Bayesian Estimation of a Banded Covariance Matrix

Description

Provides a post-processed posterior for Bayesian inference of a banded covariance matrix.

Usage

bandPPP(X, k, eps, prior = list(), nsample = 2000)

Arguments

X

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

k

a scalar value (natural number) specifying the bandwidth of covariance matrix.

eps

a small positive number decreasing to 0 with default value (log(k))^2 * (k + log(p))/n.

prior

a list giving the prior information. The list includes the following parameters (with default values in parentheses): A (I) giving the positive definite scale matrix for the inverse-Wishart prior, nu (p + k) giving the degree of freedom of the inverse-Wishar prior.

nsample

a scalar value giving the number of the post-processed posterior samples.

Details

Lee, Lee, and Lee (2023+) proposed a two-step procedure generating samples from the post-processed posterior for Bayesian inference of a banded covariance matrix:

• Initial posterior computing step: Generate random samples from the following initial posterior obtained by using the inverse-Wishart prior IW_p(B_0, \nu_0)

  \Sigma \mid X_1, \ldots, X_n \sim IW_p(B_0 + nS_n, \nu_0 + n),

  where S_n = n^{-1}\sum_{i=1}^{n}X_iX_i^\top.

• Post-processing step: Post-process the samples generated from the initial samples

  \Sigma_{(i)} := \left\{\begin{array}{ll}B_{k}(\Sigma^{(i)}) + \left[\epsilon_n - \lambda_{\min}\{B_{k}(\Sigma^{(i)})\}\right]I_p, & \mbox{ if } \lambda_{\min}\{B_{k}(\Sigma^{(i)})\} < \epsilon_n, \\ B_{k}(\Sigma^{(i)}), & \mbox{ otherwise }, \end{array}\right.

where \Sigma^{(1)}, \ldots, \Sigma^{(N)} are the initial posterior samples, \epsilon_n is a small positive number decreasing to 0 as n \rightarrow \infty, and B_k(B) denotes the k-band operation given as

B_{k}(B) = \{b_{ij}I(|i - j| \le k)\} \mbox{ for any } B = (b_{ij}) \in R^{p \times p}.

For more details, see Lee, Lee and Lee (2023+).

Value

Sigma

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

p

dimension of covariance matrix.

Author(s)

Kwangmin Lee

References

Lee, K., Lee, K., and Lee, J. (2023+), "Post-processes posteriors for banded covariances", Bayesian Analysis, DOI: 10.1214/22-BA1333.

See Also

cv.bandPPP estimate

Examples

n <- 25
p <- 50
Sigma0 <- diag(1, p)
X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = Sigma0)
res <- bspcov::bandPPP(X,2,0.01,nsample=100)