Compute (Adaptive) Elastic Net S-Estimates of Regression

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(),
  ...
)

Arguments

x	`n` by `p` matrix of numeric predictors.
y	vector of response values of length `n`. For binary classification, `y` should be a factor with 2 levels.
alpha	elastic net penalty mixing parameter with \(0 \le \alpha \le 1\). `alpha = 1` is the LASSO penalty, and `alpha = 0` the Ridge penalty.
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 `NULL`, `nlambda` and `lambda_min_ratio` are ignored.
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 `alpha`. If more observations than variables are available, the default is `1e-3 * alpha`, otherwise `1e-2 * alpha`.
enpy_lambda	optional user-supplied sequence of penalization levels at which EN-PY initial estimates are computed. If given and not `NULL`, `nlambda_enpy` is ignored.
penalty_loadings	a vector of positive penalty loadings (a.k.a. weights) for different penalization of each coefficient. Only allowed for `alpha` > 0.
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 `bdp`. Does not affect the estimated coefficients, only the estimated scale of the residuals.
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 `starting_point()`. If the output of `enpy_initial_estimates()` is given, the starting points will be shared among all penalization levels. Note that if a the starting point is specific to a penalization level, this penalization level is added to the grid of penalization levels (either the manually specified grid in `lambda` or the automatically generated grid of size `nlambda`). If `standardize = TRUE`, the starting points are also scaled.
eps	numerical tolerance.
explore_solutions	number of solutions to compute up to the desired precision `eps`.
explore_tol	numerical tolerance for exploring possible solutions. Should be (much) looser than `eps` to be useful.
max_solutions	only retain up to `max_solutions` unique solutions per penalization level.
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 `comparison_tol`, two solutions are deemed equal if the squared difference of the intercepts is less than `comparison_tol` and the squared \(L_2\) norm of the difference vector is less than `comparison_tol`.
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 `x` variables prior to fitting the PENSE estimates. Coefficients are always returned on the original scale. This can fail for variables with a large proportion of a single value (e.g., zero-inflated data). In this case, either compute with `standardize = FALSE` or standardize the data manually.
algorithm_opts	options for the MM algorithm to compute the estimates. See `mm_algorithm_options()` for details.
mscale_opts	options for the M-scale estimation. See `mscale_algorithm_options()` for details.
enpy_opts	options for the ENPY initial estimates, created with the `enpy_options()` function. See `enpy_initial_estimates()` for details.
cv_k, cv_objective	deprecated and ignored. See `pense_cv()` for estimating prediction performance via cross-validation.
...	ignored. See the section on deprecated parameters below.

Value

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.

Strategies for Using Starting Points

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.

Deprecated Arguments

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.

Examples

# 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