hdi (version 0.1-9)

groupBound: Lower bound on the l1-norm of groups of regression variables

Description

Computes a lower bound that forms a one-sided confidence interval for the group l1-norm of a specified group of regression parameters. It is assumed that errors have a Gaussian distribution with unknown noise level. The underlying vector that inference is made about is the l1-sparsest approximation to the noiseless data.

Usage

groupBound(x, y, group, alpha = 0.05, eps = 0.1, nsplit = 11,
           s = min(10, ncol(x) - 1), setseed = TRUE,
           silent = FALSE, lpSolve = TRUE, parallel = FALSE,
           ncores = getOption("mc.cores", 2L))

Arguments

x

numeric design matrix of the regression \(n \times p\) with \(p\) columns for \(p\) predictor variables and \(n\) rows corresponding to \(n\) observations.

y

numeric response variable of length \(n\).

group

either a numeric vector with entries in \(\{1,...,p\}\) or a list with such numeric vectors. If group is a numeric vector, this is the group of variables for which a lower bound is computed. If group is a list, the lower bound is computed for each group in the list.

alpha

numeric level in \((0,1)\) at which the test / confidence interval is computed.

eps

a level of eps * alpha is used and the values of different splits are aggregated using the (1 - eps) quantile. See reference below for more details.

nsplit

the number of data splits used.

s

the dimensionality of the projection that is used. Lower values lead to faster computation and if \(n > 50\), then s is set to 50 if left unspecified, to avoid lengthy computations.

setseed

a logical; if this is true (recommended), then the same random seeds are used for all groups, which makes the confidence intervals simultaneously valid over all groups of variables tested.

silent

logical enabling progress output.

lpSolve

logical; only set it to false if lpSolve() is not working on the current machine: setting it to false will result in much slower computations; only use on small problems.

parallel

should parallelization be used? (logical)

ncores

number of cores used for parallelization.

Value

If group is a single numeric vector, a scalar containg the lower bound for this group of variables is returned. If group is a list, a numeric vector is retuned where each entry corresponds to the group of variables defined in the same order in group.

Details

The data are split since the noise level is unknown. On the first part of the random split, a cross-validated lasso solution is computed, using the glmnet implementation. This estimator is used as an initial estimator on the second half of the data. Results at level alpha are aggregated over nsplit splits via the median of results at levels alpha/2.

References

Meinshausen, N. (2015) Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design. Journal of the Royal Statistical Society: Series B, 77, 923--945; 10.1111/rssb.12094.

See Also

Use clusterGroupBound to test all groups in a hierarchical clustering tree.

Examples

# NOT RUN {
## Create a regression problem with correlated design: p = 6, n = 50,
## block size B = 3 and within-block correlation of rho = 0.99
p   <- 6
n   <- 50
B   <- 3
rho <- 0.99

ind   <- rep(1:ceiling(p / B), each = B)[1:p]
Sigma <- diag(p)

for (ii in unique(ind)){
  id <- which(ind == ii)
  Sigma[id, id] <- rho
}
diag(Sigma) <- 1

x <- matrix(rnorm(n * p), nrow = n) %*% chol(Sigma)

## Create response with active variable 1
beta    <- rep(0, p)
beta[1] <- 5

y  <- as.numeric(x %*% beta + rnorm(n))

## Compute lower bounds:

## Lower bound for the L1-norm of *all* variables 1-6 of the sparsest
## optimal vector
nsplit <- 4  ## to make example run fast (use larger value)
lowerBoundAll <- groupBound(x, y, 1:p, nsplit = nsplit)
cat("\nlower bound for all variables 1-6: ", lowerBoundAll, "\n")

## Compute additional lower bounds:
q()## Lower bounds for variable 1 itself, then group {1,3}, 1-2, 1-3, 2-6,
lowerBound <- groupBound(x, y, list(1, c(1,3), 1:2, 1:3, 2:6),
                         nsplit = nsplit)
cat("lower bound for the groups\n\t {1}, {1,3}, {1,2}, {1..3}, {2..6}:\n\t",
    format(formatC(c(lowerBound))), "\n")
# }