Compute elastic net S-estimates (PENSE estimates) along a grid of penalization levels with optional penalty loadings for adaptive elastic net.
pense( x, y, alpha, nlambda = 50, nlambda_enpy = 10, lambda, lambda_min_ratio, enpy_lambda, penalty_loadings, intercept = TRUE, bdp = 0.25, cc, add_zero_based = TRUE, enpy_specific = FALSE, other_starts, eps = 1e-06, explore_solutions = 10, explore_tol = 0.1, explore_it = 20, max_solutions = 10, comparison_tol = sqrt(eps), sparse = FALSE, ncores = 1, standardize = TRUE, algorithm_opts = mm_algorithm_options(), mscale_opts = mscale_algorithm_options(), enpy_opts = enpy_options(), cv_k = deprecated(), cv_objective = deprecated(), ... )
vector of response values of length
elastic net penalty mixing parameter with \(0 \le \alpha \le 1\).
number of penalization levels.
number of penalization levels where the EN-PY initial estimate is computed.
optional user-supplied sequence of penalization levels. If given and not
Smallest value of the penalization level as a fraction of the largest
level (i.e., the smallest value for which all coefficients are zero). The default depends on
the sample size relative to the number of variables and
optional user-supplied sequence of penalization levels at which EN-PY
initial estimates are computed. If given and not
a vector of positive penalty loadings (a.k.a. weights) for different
penalization of each coefficient. Only allowed for
include an intercept in the model.
desired breakdown point of the estimator, between 0 and 0.5. The actual breakdown point may be slightly larger/smaller to avoid instabilities of the S-loss.
tuning constant for the S-estimator. Default is to chosen based on the breakdown
also consider the 0-based regularization path. See details for a description.
use the EN-PY initial estimates only at the penalization level they are computed for. See details for a description.
a list of other staring points, created by
number of solutions to compute up to the desired precision
numerical tolerance and maximum number of iterations for
exploring possible solutions. The tolerance should be (much) looser than
only retain up to
numeric tolerance to determine if two solutions are equal.
The comparison is first done on the absolute difference in the value of the objective
function at the solution If this is less than
use sparse coefficient vectors.
number of CPU cores to use in parallel. By default, only one CPU core is used. Not supported on all platforms, in which case a warning is given.
logical flag to standardize the
options for the MM algorithm to compute the estimates.
options for the M-scale estimation. See
deprecated and ignored. See
ignored. See the section on deprecated parameters below.
a list-like object with the following items
the sequence of
a list of sequences of penalization levels, one per
a list of estimates. Each estimate contains the following information:
beta (slope) estimate.
penalization level at which the estimate is computed.
alpha hyper-parameter at which the estimate is computed.
value of the objective function at the solution.
> 0 the algorithm experienced issues when
computing the estimate.
optional status message from the algorithm.
the actual breakdown point used.
the original call.
The function supports several different strategies to compute, and use the provided starting points for optimizing the PENSE objective function.
Starting points are computed internally but can also be supplied via
By default, starting points are computed internally by the EN-PY procedure for penalization
levels supplied in
enpy_lambda (or the automatically generated grid of length
By default, starting points computed by the EN-PY procedure are shared for all penalization
lambda (or the automatically generated grid of length
If the starting points should be specific to the penalization level the starting points'
penalization level, set the
enpy_specific argument to
In addition to EN-PY initial estimates, the algorithm can also use the "0-based" strategy if
add_zero_based = TRUE (by default). Here, the 0-vector is used to start the optimization at
the largest penalization level in
lambda. At subsequent penalization levels, the solution at
the previous penalization level is also used as starting point.
At every penalization level, all starting points are explored using the loose numerical
explore_tol. Only the best
explore_solutions are computed to the stringent
Finally, only the best
max_solutions are retained and carried forward as starting points for
the subsequent penalization level.
Starting with version 2.0.0, cross-validation is performed by separate function
Arguments related cross-validation cause an error when supplied to
Furthermore, the following arguments are deprecated as of version 2.0.0:
pense() is called with any of these arguments, warnings detail how to replace them.
pense_cv() for selecting hyper-parameters via cross-validation.
coef.pense_fit() for extracting coefficient estimates.
plot.pense_fit() for plotting the regularization path.
Other functions to compute robust estimates:
# Compute the PENSE regularization path for Freeny's revenue data # (see ?freeny) data(freeny) x <- as.matrix(freeny[ , 2:5]) regpath <- pense(x, freeny$y, alpha = 0.5) plot(regpath)# Extract the coefficients at a certain penalization level coef(regpath, lambda = regpath$lambda[][])#> (Intercept) lag.quarterly.revenue price.index #> -6.6475338 0.2411667 -0.6985229 #> income.level market.potential #> 0.7098337 0.9619783# What penalization level leads to good prediction performance? set.seed(123) cv_results <- pense_cv(x, freeny$y, alpha = 0.5, cv_repl = 2, cv_k = 4) plot(cv_results, se_mult = 1)# Extract the coefficients at the penalization level with # smallest prediction error ... coef(cv_results)#> (Intercept) lag.quarterly.revenue price.index #> -8.5228825 0.2072828 -0.6946405 #> income.level market.potential #> 0.6778202 1.1430756# ... or at the penalization level with prediction error # statistically indistinguishable from the minimum. coef(cv_results, lambda = '1-se')#> (Intercept) lag.quarterly.revenue price.index #> -8.9377554 0.2066104 -0.6851005 #> income.level market.potential #> 0.6654687 1.1777421