bdsvd {bdsvd}

Block Detection Using Singular Vectors (BD-SVD).

Description

Performs BD-SVD iteratively to reveal the block structure. Splits the data matrix into one (i.e., no split) or two submatrices, depending on the structure of the first sparse loading v (which is a sparse approximation of the first right singular vector, i.e., a vector with many zero values) that mirrors the shape of the covariance matrix. This procedure is continued iteratively until the block diagonal structure has been revealed.

The data matrix ordered according to this revealed block diagonal structure can be obtained by bdsvd.structure.

Usage

bdsvd(X, dof.lim, anp = "2", standardize = TRUE, max.iter, trace = FALSE)

Arguments

X

Data matrix of dimension n x p with possibly p >> n.

dof.lim

Interval limits for the number of non-zero components in the sparse loading (degrees of freedom). If S denotes the support of v, then the cardinality of the support, |S|, corresponds to the degrees of freedom. Default is dof.lim <- c(0, p-1) which is highly recommended to check for all levels of sparsity.

anp

Which regularization function should be used for the HBIC. anp = "1" implements a_{np} = 1 which corresponds to the BIC, anp = "2" implements a_{np} = 1/2 log(np) which corresponds to the regularization used by Bauer (202Xa), and anp = "3" implements a_{np} = log(log(np)) which corresponds to the regularization used by Wang et al. (2009) and Wang et al. (2013).

standardize

Standardize the data to have unit variance. Default is TRUE.

max.iter

How many iterations should be performed for computing the sparse loading. Default is 200.

trace

Print out progress as iterations are performed. Default is TRUE.

Details

The sparse loadings are computed using the method by Shen & Huang (2008), implemented in the irlba package.

Value

A list containing the feature names of the submatrices of X. The length of the list equals the number of submatrices.

References

Bauer, J.O. (202Xa). High-dimensional block diagonal covariance structure detection using singular vectors.

Wang, H., B. Li, and C. Leng (2009). Shrinkage tuning parameter selection with a diverging number of parameters, J. R. Stat. Soc. B 71 (3), 671–683.

Wang, L., Y. Kim, and R. Li (2013). Calibrating nonconvex penalized regression in ultra-high dimension, Ann. Stat. 41 (5), 2505–2536.

See Also

bdsvd.structure, bdsvd.ht, single.bdsvd

Examples

#Replicate the simulation study (c) from Bauer (202Xa).

## Not run: 
p <- 500 #Number of variables
n <- 250 #Number of observations
b <- 10  #Number of blocks
design <- "c" #Simulation design "a", "b", "c", or "d".

#Simulate data matrix X
set.seed(1)
Sigma <- bdsvd.cov.sim(p = p, b = b, design = design)
X <- mvtnorm::rmvnorm(n, mean=rep(0, p), sigma=Sigma)
colnames(X) <- seq_len(p)

bdsvd(X, standardize = FALSE)

## End(Not run)