cv_varsel: Variable selection with cross-validation

View source: R/cv_varsel.R

cv_varselR Documentation

Variable selection with cross-validation

Description

Run the search part and the evaluation part for a projection predictive variable selection. The search part determines the 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 solution path. 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 running the evaluation part on the corresponding test set of each CV fold.

Usage

cv_varsel(object, ...)

## Default S3 method:
cv_varsel(object, ...)

## S3 method for class 'refmodel'
cv_varsel(
  object,
  method = NULL,
  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 = NULL,
  K = if (!inherits(object, "datafit")) 5 else 10,
  lambda_min_ratio = 1e-05,
  nlambda = 150,
  thresh = 1e-06,
  regul = 1e-04,
  validate_search = TRUE,
  seed = sample.int(.Machine$integer.max, 1),
  search_terms = NULL,
  ...
)

Arguments

object

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().

...

Arguments passed to get_refmodel() as well as to the divergence minimizer (during a forward search and also during the evaluation part, but the latter only if refit_prj is TRUE).

method

The method for the search part. Possible options are "L1" for L1 search and "forward" for forward search. If NULL, then internally, "L1" is used, except if the reference model has multilevel or additive terms or if !is.null(search_terms). See also section "Details" below.

cv_method

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.

ndraws

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.

nclusters

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.

ndraws_pred

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.

nclusters_pred

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.

refit_prj

A single logical value indicating whether to fit the submodels along the solution path again (TRUE) or to retrieve their fits from the search part (FALSE) before using those (re-)fits in the evaluation part.

nterms_max

Maximum number of predictor terms until 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.)

penalty

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.

verbose

A single logical value indicating whether to print out additional information during the computations.

nloo

Caution: Still experimental. Only relevant if cv_method = "LOO". Number of subsampled LOO CV folds, i.e., number of observations used for the LOO CV (anything between 1 and the original number of observations). Smaller values lead to faster computation but higher uncertainty in the evaluation part. If NULL, all observations are used, but for faster experimentation, one can set this to a smaller value.

K

Only relevant if cv_method = "kfold" and if the reference model was created with cvfits being NULL (which is the case for get_refmodel.stanreg() and brms::get_refmodel.brmsfit()). Number of folds in K-fold CV.

lambda_min_ratio

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.

nlambda

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.

thresh

Only relevant for L1 search. Convergence threshold when computing the L1 path. Usually, there is no need to change this.

regul

A number giving the amount of ridge regularization when projecting onto (i.e., fitting) submodels which are GLMs. Usually there is no need for regularization, but sometimes we need to add some regularization to avoid numerical problems.

validate_search

Only relevant if cv_method = "LOO". 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 variable selection 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).

seed

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. Here, this seed is used for clustering the reference model's posterior draws (if !is.null(nclusters) or !is.null(nclusters_pred)), for subsampling 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).

search_terms

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.

Details

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, 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 interaction terms before the corresponding main terms are selected. If this is 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 = 1, ..., J. 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 1) 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"). This search would start with y ~ 1 as the only candidate at model size 1. At model size 2, y ~ x1 would be the only candidate. At model size 3, y ~ x1 + x2 and y ~ x1 + x3 would be the two candidates. At the last model size of 4, 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").

Value

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).

Note

The case cv_method == "LOO" && !validate_search constitutes an exception where the search part is not cross-validated. In that case, the evaluation part is based on a PSIS-LOO CV also for the submodels.

For all PSIS-LOO CVs, projpred calls loo::psis() 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.

References

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.

See Also

varsel()

Examples


# Note: The code from this example is not executed when called via example().
# To execute it, you have to copy and paste it manually to the console.
if (requireNamespace("rstanarm", quietly = TRUE)) {
  # 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 = 500, refresh = 0, seed = 9876
  )

  # Variable selection with cross-validation (with small values
  # for `nterms_max`, `nclusters`, and `nclusters_pred`, but only for the
  # sake of speed in this example; this is not recommended in general):
  cvvs <- cv_varsel(fit, nterms_max = 3, nclusters = 5, nclusters_pred = 10,
                    seed = 5555)
  # Now see, for example, `?print.vsel`, `?plot.vsel`, `?suggest_size.vsel`,
  # and `?solution_terms.vsel` for possible post-processing functions.
}


projpred documentation built on Nov. 10, 2022, 5:15 p.m.