Run the search part and the evaluation part for a projection predictive
variable selection. The search part determines the predictor ranking (also
known as solution path), i.e., the best submodel for each submodel size
(number of predictor terms). The evaluation part determines the predictive
performance of the submodels along the predictor ranking. In contrast to
varsel(), cv_varsel() performs a cross-validation (CV) by running the
search part with the training data of each CV fold separately (an exception
is explained in section "Note" below) and by running the evaluation part on
the corresponding test set of each CV fold. A special method is
cv_varsel.vsel() because it re-uses the search results from an earlier
cv_varsel() (or varsel()) run, as illustrated in the main vignette.
cv_varsel(object, ...)
# S3 method for default
cv_varsel(object, ...)
# S3 method for vsel
cv_varsel(
object,
cv_method = object$cv_method %||% "LOO",
nloo = object$nloo,
K = object$K %||% if (!inherits(object, "datafit")) 5 else 10,
cvfits = object$cvfits,
validate_search = object$validate_search %||% TRUE,
...
)
# S3 method for refmodel
cv_varsel(
object,
method = "forward",
cv_method = if (!inherits(object, "datafit")) "LOO" else "kfold",
ndraws = NULL,
nclusters = 20,
ndraws_pred = 400,
nclusters_pred = NULL,
refit_prj = !inherits(object, "datafit"),
nterms_max = NULL,
penalty = NULL,
verbose = TRUE,
nloo = object$nobs,
K = if (!inherits(object, "datafit")) 5 else 10,
cvfits = object$cvfits,
search_control = NULL,
lambda_min_ratio = 1e-05,
nlambda = 150,
thresh = 1e-06,
validate_search = TRUE,
seed = NA,
search_terms = NULL,
search_out = NULL,
parallel = getOption("projpred.prll_cv", FALSE),
...
)An object of class refmodel (returned by get_refmodel() or
init_refmodel()) or an object that can be passed to argument object of
get_refmodel().
For cv_varsel.default(): Arguments passed to get_refmodel() as
well as to cv_varsel.refmodel(). For cv_varsel.vsel(): Arguments passed
to cv_varsel.refmodel(). For cv_varsel.refmodel(): Arguments passed to
the divergence minimizer (see argument div_minimizer of init_refmodel()
as well as section "Draw-wise divergence minimizers" of projpred-package)
when refitting the submodels for the performance evaluation (if refit_prj
is TRUE).
The CV method, either "LOO" or "kfold". In the "LOO"
case, a Pareto-smoothed importance sampling leave-one-out CV (PSIS-LOO CV)
is performed, which avoids refitting the reference model nloo times (in
contrast to a standard LOO CV). In the "kfold" case, a \(K\)-fold CV is
performed. See also section "Note" below.
Caution: Still experimental. Only relevant if cv_method = "LOO". If nloo is smaller than the number of all observations,
approximate full LOO CV using probability-proportional-to-size-sampling
(PPS) to make accurate computation only for nloo (anything from 1 to the
number of all observations) leave-one-out folds (Magnusson et al., 2019).
Smaller values lead to faster computation but higher uncertainty in the
evaluation part. If NULL, all observations are used (as by default).
Only relevant if cv_method = "kfold" and if cvfits is NULL
(which is the case for reference model objects created by
get_refmodel.stanreg() or brms::get_refmodel.brmsfit()). Number of
folds in \(K\)-fold CV.
Only relevant if cv_method = "kfold". The same as argument
cvfits of init_refmodel(), but repeated here so that output from
run_cvfun() can be inserted here straightforwardly.
A single logical value indicating whether to
cross-validate also the search part, i.e., whether to run the search
separately for each CV fold (TRUE) or not (FALSE). We strongly do not
recommend setting this to FALSE, because this is known to bias the
predictive performance estimates of the selected submodels. However,
setting this to FALSE can sometimes be useful because comparing the
results to the case where this argument is TRUE gives an idea of how
strongly the search is (over-)fitted to the data (the difference
corresponds to the search degrees of freedom or the effective number of
parameters introduced by the search).
The method for the search part. Possible options are
"forward" for forward search and "L1" for L1 search. See also section
"Details" below.
Number of posterior draws used in the search part. Ignored if
nclusters is not NULL or in case of L1 search (because L1 search always
uses a single cluster). If both (nclusters and ndraws) are NULL, the
number of posterior draws from the reference model is used for ndraws.
See also section "Details" below.
Number of clusters of posterior draws used in the search
part. Ignored in case of L1 search (because L1 search always uses a single
cluster). For the meaning of NULL, see argument ndraws. See also
section "Details" below.
Only relevant if refit_prj is TRUE. Number of
posterior draws used in the evaluation part. Ignored if nclusters_pred is
not NULL. If both (nclusters_pred and ndraws_pred) are NULL, the
number of posterior draws from the reference model is used for
ndraws_pred. See also section "Details" below.
Only relevant if refit_prj is TRUE. Number of
clusters of posterior draws used in the evaluation part. For the meaning of
NULL, see argument ndraws_pred. See also section "Details" below.
For the evaluation part, should the submodels along the
predictor ranking be fitted again (TRUE) or should their fits from the
search part be re-used (FALSE)?
Maximum submodel size (number of predictor terms) up to
which the search is continued. If NULL, then min(19, D) is used where
D is the number of terms in the reference model (or in search_terms, if
supplied). Note that nterms_max does not count the intercept, so use
nterms_max = 0 for the intercept-only model. (Correspondingly, D above
does not count the intercept.)
Only relevant for L1 search. A numeric vector determining the
relative penalties or costs for the predictors. A value of 0 means that
those predictors have no cost and will therefore be selected first, whereas
Inf means those predictors will never be selected. If NULL, then 1 is
used for each predictor.
A single logical value indicating whether to print out additional information during the computations.
A list of "control" arguments (i.e., tuning
parameters) for the search. In case of forward search, these arguments are
passed to the divergence minimizer (see argument div_minimizer of
init_refmodel() as well as section "Draw-wise divergence minimizers" of
projpred-package). In case of forward search, NULL causes ... to be
used not only for the performance evaluation, but also for the search. In
case of L1 search, possible arguments are:
lambda_min_ratio: Ratio between the smallest and largest lambda in the
L1-penalized search (default: 1e-5). This parameter essentially
determines how long the search is carried out, i.e., how large submodels
are explored. No need to change this unless the program gives a warning
about this.
nlambda: Number of values in the lambda grid for L1-penalized search
(default: 150). No need to change this unless the program gives a warning
about this.
thresh: Convergence threshold when computing the L1 path (default:
1e-6). Usually, there is no need to change this.
Deprecated (please use search_control instead).
Only relevant for L1 search. Ratio between the smallest and largest lambda
in the L1-penalized search. This parameter essentially determines how long
the search is carried out, i.e., how large submodels are explored. No need
to change this unless the program gives a warning about this.
Deprecated (please use search_control instead). Only
relevant for L1 search. Number of values in the lambda grid for
L1-penalized search. No need to change this unless the program gives a
warning about this.
Deprecated (please use search_control instead). Only relevant
for L1 search. Convergence threshold when computing the L1 path. Usually,
there is no need to change this.
Pseudorandom number generation (PRNG) seed by which the same
results can be obtained again if needed. Passed to argument seed of
set.seed(), but can also be NA to not call set.seed() at all. If not
NA, then the PRNG state is reset (to the state before calling
cv_varsel()) upon exiting cv_varsel(). Here, seed is used for
clustering the reference model's posterior draws (if !is.null(nclusters)
or !is.null(nclusters_pred)), for subsampling PSIS-LOO CV folds (if
nloo is smaller than the number of observations), for sampling the folds
in \(K\)-fold CV, and for drawing new group-level effects when predicting
from a multilevel submodel (however, not yet in case of a GAMM).
Only relevant for forward search. A custom character
vector of predictor term blocks to consider for the search. Section
"Details" below describes more precisely what "predictor term block" means.
The intercept ("1") is always included internally via union(), so
there's no difference between including it explicitly or omitting it. The
default search_terms considers all the terms in the reference model's
formula.
Intended for internal use.
A single logical value indicating whether to run costly parts
of the CV in parallel (TRUE) or not (FALSE). See also section "Note"
below.
An object of class vsel. The elements of this object are not meant
to be accessed directly but instead via helper functions (see the main
vignette and projpred-package).
Arguments ndraws, nclusters, nclusters_pred, and ndraws_pred
are automatically truncated at the number of posterior draws in the
reference model (which is 1 for datafits). Using less draws or clusters
in ndraws, nclusters, nclusters_pred, or ndraws_pred than posterior
draws in the reference model may result in slightly inaccurate projection
performance. Increasing these arguments affects the computation time
linearly.
For argument method, there are some restrictions: For a reference model
with multilevel or additive formula terms or a reference model set up for
the augmented-data projection, only the forward search is available.
Furthermore, argument search_terms requires a forward search to take
effect.
L1 search is faster than forward search, but forward search may be more accurate. Furthermore, forward search may find a sparser model with comparable performance to that found by L1 search, but it may also start overfitting when more predictors are added.
An L1 search may select an interaction term before all involved lower-order interaction terms (including main-effect terms) have been selected. In projpred versions > 2.6.0, the resulting predictor ranking is automatically modified so that the lower-order interaction terms come before this interaction term, but if this is conceptually undesired, choose the forward search instead.
The elements of the search_terms character vector don't need to be
individual predictor terms. Instead, they can be building blocks consisting
of several predictor terms connected by the + symbol. To understand how
these building blocks work, it is important to know how projpred's
forward search works: It starts with an empty vector chosen which will
later contain already selected predictor terms. Then, the search iterates
over model sizes \(j \in \{0, ..., J\}\) (with \(J\)
denoting the maximum submodel size, not counting the intercept). The
candidate models at model size \(j\) are constructed from those elements
from search_terms which yield model size \(j\) when combined with the
chosen predictor terms. Note that sometimes, there may be no candidate
models for model size \(j\). Also note that internally, search_terms is
expanded to include the intercept ("1"), so the first step of the search
(model size 0) always consists of the intercept-only model as the only
candidate.
As a search_terms example, consider a reference model with formula y ~ x1 + x2 + x3. Then, to ensure that x1 is always included in the
candidate models, specify search_terms = c("x1", "x1 + x2", "x1 + x3", "x1 + x2 + x3") (or, in a simpler way that leads to the same results,
search_terms = c("x1", "x1 + x2", "x1 + x3"), for which helper function
force_search_terms() exists). This search would start with y ~ 1 as the
only candidate at model size 0. At model size 1, y ~ x1 would be the only
candidate. At model size 2, y ~ x1 + x2 and y ~ x1 + x3 would be the
two candidates. At the last model size of 3, y ~ x1 + x2 + x3 would be
the only candidate. As another example, to exclude x1 from the search,
specify search_terms = c("x2", "x3", "x2 + x3") (or, in a simpler way
that leads to the same results, search_terms = c("x2", "x3")).
If validate_search is FALSE, the search is not included in the CV
so that only a single full-data search is run.
For PSIS-LOO CV, projpred calls loo::psis() (or, exceptionally,
loo::sis(), see below) with r_eff = NA. This is only a problem if there
was extreme autocorrelation between the MCMC iterations when the reference
model was built. In those cases however, the reference model should not
have been used anyway, so we don't expect projpred's r_eff = NA to
be a problem.
PSIS cannot be used if the draws have different (i.e., nonconstant) weights or if the number of draws is too small. In such cases, projpred resorts to standard importance sampling (SIS) and throws a warning about this. Throughout the documentation, the term "PSIS" is used even though in fact, projpred resorts to SIS in these special cases.
With parallel = TRUE, costly parts of projpred's CV are run in
parallel. Costly parts are the fold-wise searches and performance
evaluations in case of validate_search = TRUE. (Note that in case of
\(K\)-fold CV, the \(K\) reference model refits are not affected by
argument parallel; only projpred's CV is affected.) The
parallelization is powered by the foreach package. Thus, any parallel
(or sequential) backend compatible with foreach can be used, e.g.,
the backends from packages doParallel, doMPI, or
doFuture. For GLMs, this CV parallelization should work reliably, but
for other models (such as GLMMs), it may lead to excessive memory usage
which in turn may crash the R session (on Unix systems, setting an
appropriate memory limit via unix::rlimit_as() may avoid crashing the
whole machine). However, the problem of excessive memory usage is less
pronounced for the CV parallelization than for the projection
parallelization described in projpred-package. In that regard, the CV
parallelization is recommended over the projection parallelization.
Magnusson, Måns, Michael Andersen, Johan Jonasson, and Aki Vehtari. 2019. "Bayesian Leave-One-Out Cross-Validation for Large Data." In Proceedings of the 36th International Conference on Machine Learning, edited by Kamalika Chaudhuri and Ruslan Salakhutdinov, 97:4244--53. Proceedings of Machine Learning Research. PMLR. https://proceedings.mlr.press/v97/magnusson19a.html.
Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. "Practical Bayesian Model Evaluation Using Leave-One-Out Cross-Validation and WAIC." Statistics and Computing 27 (5): 1413--32. doi:10.1007/s11222-016-9696-4 .
Vehtari, Aki, Daniel Simpson, Andrew Gelman, Yuling Yao, and Jonah Gabry. 2022. "Pareto Smoothed Importance Sampling." arXiv. doi:10.48550/arXiv.1507.02646 .
# Data:
dat_gauss <- data.frame(y = df_gaussian$y, df_gaussian$x)
# The `stanreg` fit which will be used as the reference model (with small
# values for `chains` and `iter`, but only for technical reasons in this
# example; this is not recommended in general):
fit <- rstanarm::stan_glm(
y ~ X1 + X2 + X3 + X4 + X5, family = gaussian(), data = dat_gauss,
QR = TRUE, chains = 2, iter = 1000, refresh = 0, seed = 9876
)
# Run cv_varsel() (with L1 search and small values for `K`, `nterms_max`, and
# `nclusters_pred`, but only for the sake of speed in this example; this is
# not recommended in general):
cvvs <- cv_varsel(fit, method = "L1", cv_method = "kfold", K = 2,
nterms_max = 3, nclusters_pred = 10, seed = 5555,
verbose = FALSE)
#> Fitting model 1 out of 2
#> Fitting model 2 out of 2
# Now see, for example, `?print.vsel`, `?plot.vsel`, `?suggest_size.vsel`,
# and `?ranking` for possible post-processing functions.