Package 'grpSLOPE' reference manual

Title:	Group Sorted L1 Penalized Estimation
Description:	Group SLOPE (Group Sorted L1 Penalized Estimation) is a penalized linear regression method that is used for adaptive selection of groups of significant predictors in a high-dimensional linear model. The Group SLOPE method can control the (group) false discovery rate at a user-specified level (i.e., control the expected proportion of irrelevant among all selected groups of predictors). For additional information about the implemented methods please see Brzyski, Gossmann, Su, Bogdan (2018) <doi:10.1080/01621459.2017.1411269>.
Authors:	Alexej Gossmann [aut, cre], Damian Brzyski [aut], Weijie Su [aut], Malgorzata Bogdan [aut], Ewout van den Berg [ctb] (Code adapted from 'SLOPE' version 0.1.3, as well as from http://statweb.stanford.edu/~candes/SortedL1/software.html under GNU GPL-3), Emmanuel Candes [ctb] (Code adapted from 'SLOPE' version 0.1.3, as well as from http://statweb.stanford.edu/~candes/SortedL1/software.html under GNU GPL-3), Chiara Sabatti [ctb] (Code adapted from 'SLOPE' version 0.1.3, as well as from http://statweb.stanford.edu/~candes/SortedL1/software.html under GNU GPL-3), Evan Patterson [ctb] (Code adapted from 'SLOPE' version 0.1.3, as well as from http://statweb.stanford.edu/~candes/SortedL1/software.html under GNU GPL-3)
Maintainer:	Alexej Gossmann <[email protected]>
License:	GPL-3
Version:	0.3.3.9000
Built:	2025-03-03 04:03:07 UTC
Source:	https://github.com/agisga/grpslope

Alternating direction method of multipliers

Description

Compute the coefficient estimates for the Group SLOPE problem.

Usage

admmSolverGroupSLOPE(
  y,
  A,
  group,
  wt,
  lambda,
  rho = NULL,
  max.iter = 10000,
  verbose = FALSE,
  absolute.tol = 1e-04,
  relative.tol = 1e-04,
  z.init = NULL,
  u.init = NULL,
  ...
)
admmSolverGroupSLOPE(
  y,
  A,
  group,
  wt,
  lambda,
  rho = NULL,
  max.iter = 10000,
  verbose = FALSE,
  absolute.tol = 1e-04,
  relative.tol = 1e-04,
  z.init = NULL,
  u.init = NULL,
  ...
)

Arguments

`y`	the response vector
`A`	the model matrix
`group`	A vector describing the grouping structure. It should contain a group id for each predictor variable.
`wt`	A vector of weights (per coefficient)
`lambda`	A decreasing sequence of regularization parameters $\lambda$
`rho`	Penalty parameter in the augmented Lagrangian (see Boyd et al., 2011)
`max.iter`	Maximal number of iterations to carry out
`verbose`	A `logical` specifying whether to print output or not
`absolute.tol`	The absolute tolerance used in the stopping criteria for the primal and dual feasibility conditions (see Boyd et al., 2011, Sec. 3.3.1)
`relative.tol`	The relative tolerance used in the stopping criteria for the primal and dual feasibility conditions (see Boyd et al., 2011, Sec. 3.3.1)
`z.init`	An optional initial value for the iterative algorithm
`u.init`	An optional initial value for the iterative algorithm
`...`	Options passed to `prox_sorted_L1`

Details

admmSolverGroupSLOPE computes the coefficient estimates for the Group SLOPE model. The employed optimization algorithm is the alternating direction method of multipliers (ADMM).

Value

A list with the entries:

x: Solution (n-by-1 matrix)
status: Convergence status: 1 if optimal, 2 if iteration limit reached
iter: Number of iterations of the ADMM method

References

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein (2011) Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning 3 (1).

Examples

set.seed(1)
A   <- matrix(runif(100, 0, 1), 10, 10)
grp <- c(0, 0, 1, 1, 2, 2, 2, 2, 2, 3)
wt  <- c(2, 2, 2, 2, 5, 5, 5, 5, 5, 1)
x   <- c(0, 0, 5, 1, 0, 0, 0, 1, 0, 3)
y   <- A %*% x
lam <- 0.1 * (10:7)
result <- admmSolverGroupSLOPE(y = y, A = A, group = grp, wt = wt,
                               lambda=lam, rho = 1, verbose = FALSE)
result$x
#           [,1]
#  [1,] 0.000000
#  [2,] 0.000000
#  [3,] 3.856002
#  [4,] 2.080742
#  [5,] 0.000000
#  [6,] 0.000000
#  [7,] 0.000000
#  [8,] 0.000000
#  [9,] 0.000000
# [10,] 3.512829

set.seed(1)
A   <- matrix(runif(100, 0, 1), 10, 10)
grp <- c(0, 0, 1, 1, 2, 2, 2, 2, 2, 3)
wt  <- c(2, 2, 2, 2, 5, 5, 5, 5, 5, 1)
x   <- c(0, 0, 5, 1, 0, 0, 0, 1, 0, 3)
y   <- A %*% x
lam <- 0.1 * (10:7)
result <- admmSolverGroupSLOPE(y = y, A = A, group = grp, wt = wt,
                               lambda=lam, rho = 1, verbose = FALSE)
result$x
#           [,1]
#  [1,] 0.000000
#  [2,] 0.000000
#  [3,] 3.856002
#  [4,] 2.080742
#  [5,] 0.000000
#  [6,] 0.000000
#  [7,] 0.000000
#  [8,] 0.000000
#  [9,] 0.000000
# [10,] 3.512829

Extract model coefficients

Description

Extract the regression coefficients from a grpSLOPE object, either on the scale of the normalized design matrix (i.e., columns centered and scaled to unit norm), or on the original scale.

Usage

## S3 method for class 'grpSLOPE'
coef(object, scaled = TRUE, ...)
## S3 method for class 'grpSLOPE'
coef(object, scaled = TRUE, ...)

Arguments

`object`	A `grpSLOPE` object
`scaled`	Should the coefficients be returned for the normalized version of the design matrix?
`...`	Potentially further arguments passed to and from methods

Details

If scaled is set to TRUE, then the coefficients are returned for the normalized version of the design matrix, which is the scale on which they were computed. If scaled is set to FALSE, then the coefficients are transformed to correspond to the original (unaltered) design matrix. In case that scaled = FALSE, an estimate for the intercept term is returned with the other coefficients. In case that scaled = TRUE, the estimate of the intercept is always equal to zero, and is not explicitly provided.

Value

A named vector of regression coefficients where the names signify the group that each entry belongs to

Examples

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(letters[1:20]), each=5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
result <- grpSLOPE(X=A, y=y, group=grp, fdr=0.1)
head(coef(result), 8)
#       a_1       a_2       a_3       a_4       a_5       b_1       b_2       b_3 
#  7.942177  7.979269  8.667013  8.514861 10.026664  8.963364 10.037355 10.448692 
head(coef(result, scaled = FALSE), 8)
# (Intercept)         a_1         a_2         a_3         a_4         a_5         b_1         b_2 
#  -0.4418113   0.8886878   0.8372108   0.8422089   0.8629597   0.8615827   0.9323849   0.9333445 

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(letters[1:20]), each=5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
result <- grpSLOPE(X=A, y=y, group=grp, fdr=0.1)
head(coef(result), 8)
#       a_1       a_2       a_3       a_4       a_5       b_1       b_2       b_3 
#  7.942177  7.979269  8.667013  8.514861 10.026664  8.963364 10.037355 10.448692 
head(coef(result, scaled = FALSE), 8)
# (Intercept)         a_1         a_2         a_3         a_4         a_5         b_1         b_2 
#  -0.4418113   0.8886878   0.8372108   0.8422089   0.8629597   0.8615827   0.9323849   0.9333445

Get a groupID object

Description

Mostly intended for internal use.

Usage

getGroupID(group)
getGroupID(group)

Arguments

group

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

Value

An object of class groupID, which is a list, whose members are vectors of indices corresponding to each group. The names of the list members are the corresponding group names.

Examples

group  <- c("A", "A", 2, 9, "A", 9, 9, 2, "A")
group.id <- getGroupID(group)
group.id
# $A
# [1] 1 2 5 9
# 
# $`2`
# [1] 3 8
# 
# $`9`
# [1] 4 6 7
# 
# attr(,"class")
# [1] "groupID"

group  <- c("A", "A", 2, 9, "A", 9, 9, 2, "A")
group.id <- getGroupID(group)
group.id
# $A
# [1] 1 2 5 9
# 
# $`2`
# [1] 3 8
# 
# $`9`
# [1] 4 6 7
# 
# attr(,"class")
# [1] "groupID"

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,
  ...
)
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 $i$ th 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 

# 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

Regularizing sequence for Group SLOPE

Description

Generate the regularizing sequence lambda for the Group SLOPE problem according to one of multiple methods (see Details).

Usage

lambdaGroupSLOPE(method, fdr, group, wt, n.obs = NULL)
lambdaGroupSLOPE(method, fdr, group, wt, n.obs = NULL)

Arguments

`method`	Possible values are "max", "mean", and "corrected". See under Details.
`fdr`	Target group false discovery rate (gFDR)
`group`	A vector describing the grouping structure. It should contain a group id for each predictor variable.
`wt`	A named vector of weights, one weight per group of predictors (named according to names as in vector `group`)
`n.obs`	Number of observations (i.e., number of rows in `A`); required only if method is "corrected"

Details

Multiple methods are available to generate the regularizing sequence lambda:

"max" – lambdas as in Theorem 2.5 in Brzyski et. al. (2016). Provalby controls gFDR in orthogonal designs.
"mean" – lambdas of equation (2.16) in Brzyski et. al. (2016). Applicable for gFDR control in orthogonal designs. Less conservative than "max".
"corrected" – lambdas of Procedure 1 in Brzyski et. al. (2016); in the special case that all group sizes are equal and wt is a constant vector, Procedure 6 of Brzyski et. al. (2016) is applied. Applicable for gFDR control when predictors from different groups are stochastically independent.

Value

A vector containing the calculated lambda values.

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

# specify 6 groups of sizes 2, 3, and 4
group <- c(1, 1, 2, 2, 2, 3, 3, 3, 3,
           4, 4, 5, 5, 5, 6, 6, 6, 6)
# set the weight for each group to the square root of the group's size
wt <- rep(c(sqrt(2), sqrt(3), sqrt(4)), 2)
names(wt) <- 1:6
# compute different lambda sequences
lambda.max <- lambdaGroupSLOPE(method="max", fdr=0.1, group=group, wt=wt) 
lambda.mean <- lambdaGroupSLOPE(method="mean", fdr=0.1, group=group, wt=wt) 
lambda.corrected <- lambdaGroupSLOPE(method="corrected", fdr=0.1,
                                     group=group, wt=wt, n.obs=1000)
rbind(lambda.max, lambda.mean, lambda.corrected)
#                      [,1]     [,2]     [,3]     [,4]     [,5]     [,6]
# lambda.max       2.023449 1.844234 1.730818 1.645615 1.576359 1.517427
# lambda.mean      1.880540 1.723559 1.626517 1.554561 1.496603 1.447609
# lambda.corrected 1.880540 1.729811 1.637290 1.568971 1.514028 1.467551

# specify 6 groups of sizes 2, 3, and 4
group <- c(1, 1, 2, 2, 2, 3, 3, 3, 3,
           4, 4, 5, 5, 5, 6, 6, 6, 6)
# set the weight for each group to the square root of the group's size
wt <- rep(c(sqrt(2), sqrt(3), sqrt(4)), 2)
names(wt) <- 1:6
# compute different lambda sequences
lambda.max <- lambdaGroupSLOPE(method="max", fdr=0.1, group=group, wt=wt) 
lambda.mean <- lambdaGroupSLOPE(method="mean", fdr=0.1, group=group, wt=wt) 
lambda.corrected <- lambdaGroupSLOPE(method="corrected", fdr=0.1,
                                     group=group, wt=wt, n.obs=1000)
rbind(lambda.max, lambda.mean, lambda.corrected)
#                      [,1]     [,2]     [,3]     [,4]     [,5]     [,6]
# lambda.max       2.023449 1.844234 1.730818 1.645615 1.576359 1.517427
# lambda.mean      1.880540 1.723559 1.626517 1.554561 1.496603 1.447609
# lambda.corrected 1.880540 1.729811 1.637290 1.568971 1.514028 1.467551

Obtain predictions

Description

Obtain predictions from a grpSLOPE model on new data

Usage

## S3 method for class 'grpSLOPE'
predict(object, newdata, ...)
## S3 method for class 'grpSLOPE'
predict(object, newdata, ...)

Arguments

`object`	A `grpSLOPE` object
`newdata`	Predictor variables arranged in a matrix
`...`	Potentially further arguments passed to and from methods

Details

Note that newdata must contain the same predictor variables as columns in the same order as the design matrix X that was used for the grpSLOPE model fit.

Value

A vector of length nrow(newdata) containing the resulting predictions.

Examples

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(1:20), each = 5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1)
newdata <- matrix(rnorm(800), 8, 100)
# group SLOPE predictions:
predict(result, newdata)
# [1] -5.283385 -6.313938 -3.173068  1.901488  9.796677 -0.144516 -0.611164 -5.167620
# true mean values:
as.vector(newdata %*% b)
# [1] -5.0937160 -6.5814111 -3.5776124  2.7877449 11.0668777  1.0253236 -0.4261076 -4.8622940

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(1:20), each = 5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1)
newdata <- matrix(rnorm(800), 8, 100)
# group SLOPE predictions:
predict(result, newdata)
# [1] -5.283385 -6.313938 -3.173068  1.901488  9.796677 -0.144516 -0.611164 -5.167620
# true mean values:
as.vector(newdata %*% b)
# [1] -5.0937160 -6.5814111 -3.5776124  2.7877449 11.0668777  1.0253236 -0.4261076 -4.8622940

Prox for sorted L1 norm

Description

Compute the prox for the sorted L1 norm. That is, given a vector $x$ and a decreasing vector $\lambda$ , compute the unique value of $y$ minimizing

$\frac{1}{2} \Vert x - y \Vert_2^2 + \sum_{i=1}^n \lambda_i |x|_{(i)}.$

At present, two methods for computing the sorted L1 prox are supported. By default, we use a fast custom C implementation. Since SLOPE can be viewed as an isotonic regression problem, the prox can also be computed using the isotone package. This option is provided primarily for testing.

Usage

prox_sorted_L1(x, lambda, method = c("c", "isotone"))
prox_sorted_L1(x, lambda, method = c("c", "isotone"))

Arguments

`x`	input vector
`lambda`	vector of $\lambda$ 's, sorted in decreasing order
`method`	underlying prox implementation, either 'c' or 'isotone' (see Details)

Details

This function has been adapted (with only cosmetic changes) from the R package SLOPE version 0.1.3, due to this function being deprecated and defunct in SLOPE versions which are newer than 0.1.3.

Value

The solution vector $y$ .

Prox for group SLOPE

Description

Evaluate the proximal mapping for the group SLOPE problem.

Usage

proxGroupSortedL1(y, group, lambda, ...)
proxGroupSortedL1(y, group, lambda, ...)

Arguments

`y`	The response vector
`group`	Either a vector or an object of class `groupID` (e.g., as produced by `getGroupID`), which is describing the grouping structure. If it is a vector, then it should contain a group id for each predictor variable.
`lambda`	A decreasing sequence of regularization parameters $\lambda$
`...`	Options passed to `prox_sorted_L1`

Details

proxGroupSortedL1 evaluates the proximal mapping of the group SLOPE problem by reducing it to the prox for the (regular) SLOPE and then applying the fast prox algorithm for the Sorted L1 norm.

Value

The solution vector.

References

M. Bogdan, E. van den Berg, C. Sabatti, W. Su, E. Candes (2015), SLOPE – Adaptive variable selection via convex optimization, https://arxiv.org/abs/1407.3824

Examples

grp <- c(0,0,0,1,1,0,2,1,0,2)
proxGroupSortedL1(y = 1:10, group = grp, lambda = c(10, 9, 8))
#  [1] 0.2032270 0.4064540 0.6096810 0.8771198 1.0963997 1.2193620 1.3338960
#  [8] 1.7542395 1.8290430 1.9055657

grp <- c(0,0,0,1,1,0,2,1,0,2)
proxGroupSortedL1(y = 1:10, group = grp, lambda = c(10, 9, 8))
#  [1] 0.2032270 0.4064540 0.6096810 0.8771198 1.0963997 1.2193620 1.3338960
#  [8] 1.7542395 1.8290430 1.9055657

Proximal gradient method for Group SLOPE

Description

Compute the coefficient estimates for the Group SLOPE problem.

Usage

proximalGradientSolverGroupSLOPE(
  y,
  A,
  group,
  wt,
  lambda,
  max.iter = 10000,
  verbose = FALSE,
  dual.gap.tol = 1e-06,
  infeas.tol = 1e-06,
  x.init = NULL,
  ...
)
proximalGradientSolverGroupSLOPE(
  y,
  A,
  group,
  wt,
  lambda,
  max.iter = 10000,
  verbose = FALSE,
  dual.gap.tol = 1e-06,
  infeas.tol = 1e-06,
  x.init = NULL,
  ...
)

Arguments

`y`	the response vector
`A`	the model matrix
`group`	A vector describing the grouping structure. It should contain a group id for each predictor variable.
`wt`	A vector of weights (per coefficient)
`lambda`	A decreasing sequence of regularization parameters $\lambda$
`max.iter`	Maximal number of iterations to carry out
`verbose`	A `logical` specifying whether to print output or not
`dual.gap.tol`	The tolerance used in the stopping criteria for the duality gap
`infeas.tol`	The tolerance used in the stopping criteria for the infeasibility
`x.init`	An optional initial value for the iterative algorithm
`...`	Options passed to `prox_sorted_L1`

Details

proximalGradientSolverGroupSLOPE computes the coefficient estimates for the Group SLOPE model. The employed optimization algorithm is FISTA with backtracking Lipschitz search.

Value

A list with the entries:

x: Solution (n-by-1 matrix)
status: Convergence status: 1 if optimal, 2 if iteration limit reached
L: Approximation of the Lipschitz constant (step size)
iter: Iterations of the proximal gradient method
L.iter: Total number of iterations spent in Lipschitz search

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. doi:10.1080/01621459.2017.1411269

A. Gossmann, S. Cao, Y.-P. Wang (2015) Identification of Significant Genetic Variants via SLOPE, and Its Extension to Group SLOPE. In Proceedings of ACM BCB 2015. doi:10.1145/2808719.2808743

Examples

set.seed(1)
A   <- matrix(runif(100, 0, 1), 10, 10)
grp <- c(0, 0, 1, 1, 2, 2, 2, 2, 2, 3)
wt  <- c(2, 2, 2, 2, 5, 5, 5, 5, 5, 1)
x   <- c(0, 0, 5, 1, 0, 0, 0, 1, 0, 3)
y   <- A %*% x
lam <- 0.1 * (10:7)
result <- proximalGradientSolverGroupSLOPE(y=y, A=A, group=grp, wt=wt, lambda=lam, verbose=FALSE)
result$x
#           [,1]
#  [1,] 0.000000
#  [2,] 0.000000
#  [3,] 3.856005
#  [4,] 2.080736
#  [5,] 0.000000
#  [6,] 0.000000
#  [7,] 0.000000
#  [8,] 0.000000
#  [9,] 0.000000
# [10,] 3.512833

set.seed(1)
A   <- matrix(runif(100, 0, 1), 10, 10)
grp <- c(0, 0, 1, 1, 2, 2, 2, 2, 2, 3)
wt  <- c(2, 2, 2, 2, 5, 5, 5, 5, 5, 1)
x   <- c(0, 0, 5, 1, 0, 0, 0, 1, 0, 3)
y   <- A %*% x
lam <- 0.1 * (10:7)
result <- proximalGradientSolverGroupSLOPE(y=y, A=A, group=grp, wt=wt, lambda=lam, verbose=FALSE)
result$x
#           [,1]
#  [1,] 0.000000
#  [2,] 0.000000
#  [3,] 3.856005
#  [4,] 2.080736
#  [5,] 0.000000
#  [6,] 0.000000
#  [7,] 0.000000
#  [8,] 0.000000
#  [9,] 0.000000
# [10,] 3.512833

Extract (estimated) noise level

Description

Extract the noise level of the grpSLOPE model.

Usage

## S3 method for class 'grpSLOPE'
sigma(object, ...)
## S3 method for class 'grpSLOPE'
sigma(object, ...)

Arguments

`object`	A `grpSLOPE` object
`...`	Potentially further arguments passed to and from methods

Details

This basically obtains object$sigma. For R (>= 3.3.0) sigma is an S3 method with the default method coming from the stats package.

Value

The noise level of the given grpSLOPE model. A number.

Examples

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(1:20), each = 5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
# model with unknown noise level
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1)
sigma(result)
# [1] 0.6505558
# model with known noise level
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1, sigma = 1)
sigma(result)
# [1] 1

set.seed(1)
A   <- matrix(rnorm(100^2), 100, 100)
grp <- rep(rep(1:20), each = 5)
b   <- c(rep(1, 20), rep(0, 80))
y   <- A %*% b + rnorm(10) 
# model with unknown noise level
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1)
sigma(result)
# [1] 0.6505558
# model with known noise level
result <- grpSLOPE(X = A, y = y, group = grp, fdr = 0.1, sigma = 1)
sigma(result)
# [1] 1

Sorted L1 solver

Description

Solves the sorted L1 penalized regression problem: given a matrix $A$ , a vector $b$ , and a decreasing vector $\lambda$ , find the vector $x$ minimizing

$\frac{1}{2}\Vert Ax - b \Vert_2^2 + \sum_{i=1}^p \lambda_i |x|_{(i)}.$

This optimization problem is convex and is solved using an accelerated proximal gradient descent method.

Usage

SLOPE_solver(
  A,
  b,
  lambda,
  initial = NULL,
  prox = prox_sorted_L1,
  max_iter = 10000,
  grad_iter = 20,
  opt_iter = 1,
  tol_infeas = 1e-06,
  tol_rel_gap = 1e-06
)
SLOPE_solver(
  A,
  b,
  lambda,
  initial = NULL,
  prox = prox_sorted_L1,
  max_iter = 10000,
  grad_iter = 20,
  opt_iter = 1,
  tol_infeas = 1e-06,
  tol_rel_gap = 1e-06
)

Arguments

`A`	an $n$ -by- $p$ matrix
`b`	vector of length $n$
`lambda`	vector of length $p$ , sorted in decreasing order
`initial`	initial guess for $x$
`prox`	function that computes the sorted L1 prox
`max_iter`	maximum number of iterations in the gradient descent
`grad_iter`	number of iterations between gradient updates
`opt_iter`	number of iterations between checks for optimality
`tol_infeas`	tolerance for infeasibility
`tol_rel_gap`	tolerance for relative gap between primal and dual problems

Details

This function has been adapted (with only cosmetic changes) from the R package SLOPE version 0.1.3, due to this function being deprecated and defunct in SLOPE versions which are newer than 0.1.3.

Value

An object of class SLOPE_solver.result. This object is a list containing at least the following components:

`x`	solution vector $x$
`optimal`	logical: whether the solution is optimal
`iter`	number of iterations

Package 'grpSLOPE'

Help Index

Alternating direction method of multipliers

Description

Usage

Arguments

Details

Value

References

Examples

Extract model coefficients

Description

Usage

Arguments

Details

Value

Examples

Get a groupID object

Description

Usage

Arguments

Value

Examples

Group SLOPE (Group Sorted L-One Penalized Estimation)

Description

Usage

Arguments

Details

Value

References

Examples

Regularizing sequence for Group SLOPE

Description

Usage

Arguments

Details

Value

References

Examples

Obtain predictions

Description

Usage

Arguments

Details

Value

Examples

Prox for sorted L1 norm

Description

Usage

Arguments

Details

Value

Prox for group SLOPE

Description

Usage

Arguments

Details

Value

References

Examples

Proximal gradient method for Group SLOPE

Description

Usage

Arguments

Details

Value

References

Examples

Extract (estimated) noise level

Description

Usage

Arguments

Details

Value

Examples

Sorted L1 solver

Description

Usage

Arguments

Details