pense (version 1.2.1)

pensem: Perform an M-step after the EN S-Estimator

Description

Compute the PENSEM estimate, an efficient and robust elastic net estimator for linear regression.

Refine an already computed PENSE with an additional M-step

Usage

pensem(x, ...)
# S3 method for default
pensem(x, y, alpha = 0.5, nlambda = 50, lambda, lambda_s,
  lambda_min_ratio, standardize = TRUE, initial = c("warm", "cold"),
  warm_reset = 10, cv_k = 5, cv_objective, ncores = getOption("mc.cores",
  1L), cl = NULL, s_options = pense_options(),
  mm_options = mstep_options(), init_options = initest_options(),
  en_options = en_options_aug_lars(), ...)
# S3 method for pense
pensem(x, alpha, scale, nlambda = 50, lambda,
  lambda_min_ratio, standardize, cv_k = 5, cv_objective,
  ncores = getOption("mc.cores", 1L), cl = NULL,
  mm_options = mstep_options(), en_options, x_train, y_train, ...)

Arguments

x

either a numeric data matrix or a fitted PENSE estimate obtained from pense.

...

currently ignored.

y

numeric response vector.

alpha

elastic net mixing parameter with \(0 \le \alpha \le 1\). alpha = 1 is the LASSO penalty, and alpha = 0 the Ridge penalty. If a pense object is supplied as first argument,

nlambda

if lambda is not given, a grid of nlambda lambda values is generated based on the data.

lambda

a single value or a grid of values for the regularization parameter of the M-step. Assumed to be on the same scale as the data. If missing, a grid of lambda values is automatically generated (see parameter nlambda). If supplied and standardize = TRUE, the lambda values will be adjusted accordingly.

lambda_s

regularization parameter for the S-estimator. If missing, a grid of lambda values is chosen automatically. If standardize = TRUE, the lambda values will be adjusted accordingly.

lambda_min_ratio

If the grid should be chosen automatically, the ratio of the smallest lambda to the (computed) largest lambda.

standardize

should the data be standardized robustly? Estimates are returned on the original scale. Defaults to TRUE.

initial

how to initialize the estimator at a new lambda in the grid. The default, "warm", computes a cold initial estimator at several lambda values and uses the PENSE coefficient to warm-start the estimator at the next larger lambda value. At the largest value in the lambda grid, PENSE will be initialized with the 0-vector. "cold" computes the full initial estimator at every lambda value.

warm_reset

if initial = "warm" (default), how many cold initial estimates be computed?

cv_k

perform k-fold CV to choose the optimal lambda for prediction.

cv_objective

a function (name) to compute the CV performance. By default, the robust tau-scale is used.

ncores, cl

use multiple cores or the supplied cluster for the cross-validation. See pense for more details.

s_options

additional options for the PENSE algorithm. See pense_options for details.

mm_options

additional options for the M-step.

init_options

additional options for computing the cold initial estimates. Ignored if initial = "warm" and warm_reset = 0. See initest_options for details.

en_options

additional options for the EN algorithm. See elnet and en_options for details.

scale

initial scale estimate for the M step. By default the S-scale from the initial estimator (x) is used.

x_train, y_train

override arguments provided to the original call to pense.

Value

An object of class "pensem". All elements as an object of class pense as well as the following:

init_scale

the initial scale estimate used in the M step.

sest

the PENSE estimate used to initialize the M step.

bdp

breakdown point of the MM-estimator.

Details

Performs an M-step using the S-estimator at the optimal penalty parameter as returned from pense as the initial estimate. For "fat" datasets, the initial scale as returned by the S-estimate is adjusted according to Maronna & Yohai (2010).

References

Maronna, R. and Yohai, V. (2010). Correcting MM estimates for "fat" data sets. Computational Statistics & Data Analysis, 54:31683173.

See Also

pense to compute only the S-estimator.

Examples

# NOT RUN {
##
## A very simple example on artificial data
##

# Generate some dummy data
set.seed(12345)
n <- 30
p <- 15
x <- 1 + matrix(rnorm(n * p), ncol = p)
y <- x %*% c(2:5, numeric(p - 4)) + rnorm(n)

x_test <- 1 + matrix(rnorm(10 * n * p), ncol = p)
y_test <- x_test %*% c(2:5, numeric(p - 4)) + rnorm(n)

# Compute the MM-estimator with an EN penalty for 30 lambda values
# (Note: In real applications, warm_reset should be at least 5)
set.seed(1234)
est_mm <- pensem(
    x, y,
    alpha = 0.7,
    nlambda = 20,
    warm_reset = 1L,
    cv_k = 3
)

# We can plot the CV prediction error curve
plot(est_mm)

# What is the RMSPE on test data
(rmspe <- sqrt(mean((y_test - predict(est_mm, newdata = x_test))^2)))

##
## This is the same as computing first the S-estimator and adding the
## M-step afterwards
##
set.seed(1234)
est_s <- pense(
    x, y,
    alpha = 0.7,
    nlambda = 20,
    warm_reset = 1L,
    cv_k = 3
)

est_mm_2 <- pensem(
    est_s,
    nlambda = 20,
    cv_k = 3
)

## The initial S-estimate is the same used in both `pensem` calls
## because the seed which governs the CV to select the optimal lambda was the
## same
sum(abs(est_s$coefficients - est_mm$sest$coefficients))
## Therefore, the MM-estimate at each lambda is also the same
sum(abs(est_mm_2$coefficients - est_mm$coefficients))
# }