grpSLOPE {grpSLOPE}

Group SLOPE (Group Sorted L-One Penalized Estimation)

Description

Performs selection of significant groups of predictors and estimation of the corresponding coefficients using the Group SLOPE method (see Brzyski et. al., 2016).

Usage

grpSLOPE(
  X,
  y,
  group,
  fdr,
  lambda = "corrected",
  sigma = NULL,
  verbose = FALSE,
  orthogonalize = NULL,
  normalize = TRUE,
  max.iter = 10000,
  dual.gap.tol = 1e-06,
  infeas.tol = 1e-06,
  x.init = NULL,
  ...
)

Arguments

X

The model matrix

y

The response variable

group

A vector describing the grouping structure. It should contain a group id for each predictor variable.

fdr

Target group false discovery rate (gFDR)

lambda

Method used to obtain the regularizing sequence lambda. Possible values are "max", "mean", and "corrected" (default). See lambdaGroupSLOPE for detail. Alternatively, any non-increasing sequence of the correct length can be passed.

sigma

Noise level. If ommited, estimated from the data, using Procedure 2 in Brzyski et. al. (2016).

verbose

A logical specifying whether to print output or not

orthogonalize

Whether to orthogonalize the model matrix within each group. Do not set manually unless you are certain that your data is appropriately pre-processed.

normalize

Whether to center the input data and re-scale the columns of the design matrix to have unit norms. Do not disable this unless you are certain that your data are appropriately pre-processed.

max.iter

See proximalGradientSolverGroupSLOPE.

dual.gap.tol

See proximalGradientSolverGroupSLOPE.

infeas.tol

See proximalGradientSolverGroupSLOPE.

x.init

See proximalGradientSolverGroupSLOPE.

...

Options passed to prox_sorted_L1

Details

Multiple methods are available to generate the regularizing sequence lambda, see lambdaGroupSLOPE for detail. The model matrix is transformed by orthogonalization within each group (see Section 2.1 in Brzyski et. al., 2016), and penalization is imposed on \| X_{I_i} \beta_{I_i} \|. When orthogonalize = TRUE, due to within group orthogonalization, the solution vector beta cannot be computed, if a group submatrix does not have full column rank (e.g., if there are more predictors in a selected group than there are observations). In that case only the solution vector c of the transformed (orthogonalized) model is returned. Additionally, in any case the vector group.norms is returned with its ith entry being \| X_{I_i} \beta_{I_i} \|, i.e., the overall effect of each group. Note that all of these results are returned on the scale of the normalized versions of X and y. However, original.scale contains the regression coefficients transformed to correspond to the original (unaltered) X and y. In that case, an estimate for the intercept term is also returned with the other coefficients in original.scale (while on the normalized scale the estimate of the intercept is always equal to zero, and is not explicitly provided in the grpSLOPE output).

Value

A list with members:

beta

Solution vector. See Details.

c

Solution vector of the transformed model. See Details.

group.norms

Overall effect of each group. See Details.

selected

Names of selected groups (i.e., groups of predictors with at least one non-zero coefficient estimate)

optimal

Convergence status

iter

Iterations of the proximal gradient method

lambda

Regularizing sequence

lambda.method

Method used to construct the regularizing sequence

sigma

(Estimated) noise level

group

The provided grouping structure (corresponding to beta)

group.c

Grouping structure of the transformed model (corresponding to c)

original.scale

A list containing the estimated intercept and regression coefficients on the original scale. See Details.

References

D. Brzyski, A. Gossmann, W. Su, and M. Bogdan (2016) Group SLOPE – adaptive selection of groups of predictors, https://arxiv.org/abs/1610.04960

D. Brzyski, A. Gossmann, W. Su, and M. Bogdan (2019) Group SLOPE – adaptive selection of groups of predictors. Journal of the American Statistical Association 114 (525): 419–33.

Examples

# generate some data
set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(1:20), each=5)
b   <- c(runif(20), rep(0, 80))
# (i.e., groups 1, 2, 3, 4, are truly significant)
y   <- A %*% b + rnorm(10) 
fdr <- 0.1 # target false discovery rate
# fit a Group SLOPE model
result <- grpSLOPE(X=A, y=y, group=grp, fdr=fdr)
result$selected
# [1] "1"  "2"  "3"  "4"  "14"
result$sigma
# [1] 0.7968632
head(result$group.norms)
#         1         2         3         4         5         6 
#  2.905449  5.516103  8.964201 10.253792  0.000000  0.000000