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, 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(), ... )
x |
|
---|---|
y | vector of response values of length |
alpha | elastic net penalty mixing parameter with \(0 \le \alpha \le 1\). |
nlambda | number of penalization levels. |
nlambda_enpy | number of penalization levels where the EN-PY initial estimate is computed. |
lambda | optional user-supplied sequence of penalization levels. If given and not |
lambda_min_ratio | 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 |
enpy_lambda | optional user-supplied sequence of penalization levels at which EN-PY initial estimates are
computed. If given and not |
penalty_loadings | a vector of positive penalty loadings (a.k.a. weights) for different penalization of each
coefficient. Only allowed for |
intercept | include an intercept in the model. |
bdp | desired breakdown point of the estimator, between 0 and 0.5. |
cc | tuning constant for the S-estimator. Default is to chosen based on the breakdown point |
add_zero_based | also consider the 0-based regularization path. See details for a description. |
enpy_specific | use the EN-PY initial estimates only at the penalization level they are computed for. See details for a description. |
other_starts | a list of other staring points, created by |
eps | numerical tolerance. |
explore_solutions | number of solutions to compute up to the desired precision |
explore_tol | numerical tolerance for exploring possible solutions. Should be (much) looser than |
max_solutions | only retain up to |
comparison_tol | 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 |
sparse | use sparse coefficient vectors. |
ncores | number of CPU cores to use in parallel. By default, only one CPU core is used. May not be supported on your platform, in which case a warning is given. |
standardize | logical flag to standardize the |
algorithm_opts | options for the MM algorithm to compute the estimates. See |
mscale_opts | options for the M-scale estimation. See |
enpy_opts | options for the ENPY initial estimates, created with the
|
cv_k, cv_objective | deprecated and ignored. See |
... | ignored. See the section on deprecated parameters below. |
a list-like object with the following items
lambda
the sequence of penalization levels.
estimates
a list of estimates. Each estimate contains the following information:
intercept
intercept estimate.
beta
beta (slope) estimate.
lambda
penalization level at which the estimate is computed.
alpha
alpha hyper-parameter at which the estimate is computed.
objf_value
value of the objective function at the solution.
statuscode
if > 0
the algorithm experienced issues when computing the estimate.
status
optional status message from the algorithm.
call
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 other_starts
.
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 nlambda_enpy
).
By default, starting points computed by the EN-PY procedure are shared for all penalization levels in lambda
(or the automatically generated grid of length nlambda
).
If the starting points should be specific to the penalization level the starting points' penalization level,
set the enpy_specific
argument to TRUE
.
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 tolerance explore_tol
.
Only the best explore_solutions
are computed to the stringent numerical tolerance eps
.
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 pense_cv()
.
Arguments related cross-validation cause an error when supplied to pense()
.
Furthermore, the following arguments are deprecated as of version 2.0.0:
initial
, warm_reset
, cl
, options
, init_options
, en_options
.
If 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:
regmest()
# 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[40])#> (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? 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 #> -7.8411184 0.2115870 -0.7062462 #> income.level market.potential #> 0.6956763 1.0835638# ... or at the penalization level with prediction error # statistically indistinguishable from the minimum. coef(cv_results, lambda = 'se')#> (Intercept) lag.quarterly.revenue price.index #> -10.2047807 0.2111069 -0.6464577 #> income.level market.potential #> 0.6199943 1.2792788