#' Variable selection (without cross-validation)
#'
#' Perform the projection predictive variable selection for GLMs, GLMMs, GAMs,
#' and GAMMs. This variable selection consists of a *search* part and an
#' *evaluation* part. 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.
#'
#' @param 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()].
#' @param d_test A `list` of the structure outlined in section "Argument
#'   `d_test`" below, providing test data for evaluating the predictive
#'   performance of the submodels as well as of the reference model. If `NULL`,
#'   the training data is used.
#' @param 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.
#' @param 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.
#' @param 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.
#' @param 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.
#' @param 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.
#' @param 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.
#' @param 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.)
#' @param 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.
#' @param 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.
#' @param 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.
#' @param thresh Only relevant for L1 search. Convergence threshold when
#'   computing the L1 path. Usually, there is no need to change this.
#' @param 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.
#' @param 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.
#' @param verbose A single logical value indicating whether to print out
#'   additional information during the computations.
#' @param 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)`) and for drawing new group-level effects when
#'   predicting from a multilevel submodel (however, not yet in case of a GAMM).
#' @param ... 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`).
#'
#' @details
#'
#' # Argument `d_test`
#'
#' If not `NULL`, then `d_test` needs to be a `list` with the following
#' elements:
#' * `data`: a `data.frame` containing the predictor variables for the test set.
#' * `offset`: a numeric vector containing the offset values for the test set
#' (if there is no offset, use a vector of zeros).
#' * `weights`: a numeric vector containing the observation weights for the test
#' set (if there are no observation weights, use a vector of ones).
#' * `y`: a numeric vector containing the response values for the test set.
#'
#' @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 `datafit`s). 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 works, it is important to know how \pkg{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 \eqn{j \in \{1, ..., J\}}{j = 1, ..., J}. The candidate
#'   models at model size \eqn{j} are constructed from those elements from
#'   `search_terms` which yield model size \eqn{j} when combined with the
#'   `chosen` predictor terms. Note that sometimes, there may be no candidate
#'   models for model size \eqn{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")`.
#'
#' @return An object of class `vsel`. The elements of this object are not meant
#'   to be accessed directly but instead via helper functions (see the vignette
#'   or type `?projpred`).
#'
#' @seealso [cv_varsel()]
#'
#' @examples
#' 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 (here without cross-validation and 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):
#'   vs <- 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.
#' }
#'
#' @export
varsel <- function(object, ...) {
  UseMethod("varsel")
}

#' @rdname varsel
#' @export
varsel.default <- function(object, ...) {
  refmodel <- get_refmodel(object, ...)
  return(varsel(refmodel, ...))
}

#' @rdname varsel
#' @export
varsel.refmodel <- function(object, d_test = NULL, method = NULL,
                            ndraws = NULL, nclusters = 20, ndraws_pred = 400,
                            nclusters_pred = NULL,
                            refit_prj = !inherits(object, "datafit"),
                            nterms_max = NULL, verbose = TRUE,
                            lambda_min_ratio = 1e-5, nlambda = 150,
                            thresh = 1e-6, regul = 1e-4, penalty = NULL,
                            search_terms = NULL,
                            seed = sample.int(.Machine$integer.max, 1), ...) {
  # Set seed, but ensure the old RNG state is restored on exit:
  if (exists(".Random.seed", envir = .GlobalEnv)) {
    rng_state_old <- get(".Random.seed", envir = .GlobalEnv)
    on.exit(assign(".Random.seed", rng_state_old, envir = .GlobalEnv))
  }
  if (!is.na(seed)) set.seed(seed)

  refmodel <- object

  ## fetch the default arguments or replace them by the user defined values
  args <- parse_args_varsel(
    refmodel = refmodel, method = method, refit_prj = refit_prj,
    nterms_max = nterms_max, nclusters = nclusters, search_terms = search_terms
  )
  method <- args$method
  refit_prj <- args$refit_prj
  nterms_max <- args$nterms_max
  nclusters <- args$nclusters
  search_terms <- args$search_terms

  if (is.null(d_test)) {
    d_test <- list(type = "train", data = NULL, offset = refmodel$offset,
                   weights = refmodel$wobs, y = refmodel$y)
  } else {
    d_test$type <- "test"
    d_test <- d_test[nms_d_test()]
  }

  ## reference distributions for selection and prediction after selection
  p_sel <- .get_refdist(refmodel, ndraws, nclusters)
  p_pred <- .get_refdist(refmodel, ndraws_pred, nclusters_pred)

  ## perform the selection
  opt <- nlist(lambda_min_ratio, nlambda, thresh, regul)
  search_path <- select(
    method = method, p_sel = p_sel, refmodel = refmodel,
    nterms_max = nterms_max, penalty = penalty, verbose = verbose, opt = opt,
    search_terms = search_terms, ...
  )

  ## statistics for the selected submodels
  submodels <- .get_submodels(
    search_path = search_path,
    nterms = c(0, seq_along(search_path$solution_terms)),
    p_ref = p_pred, refmodel = refmodel, regul = regul, refit_prj = refit_prj,
    ...
  )
  sub <- .get_sub_summaries(submodels = submodels,
                            refmodel = refmodel,
                            test_points = NULL,
                            newdata = d_test$data,
                            offset = d_test$offset,
                            wobs = d_test$weights,
                            y = d_test$y)

  ## predictive statistics of the reference model on test data. if no test data
  ## are provided,
  ## simply fetch the statistics on the train data
  if (inherits(refmodel, "datafit")) {
    ## no actual reference model, so we don't know how to predict test
    ## observations
    nobs_test <- nrow(d_test$data %||% refmodel$fetch_data())
    ref <- list(mu = rep(NA, nobs_test), lppd = rep(NA, nobs_test))
  } else {
    if (d_test$type == "train") {
      mu_test <- refmodel$mu
      if (!all(refmodel$offset == 0)) {
        mu_test <- refmodel$family$linkinv(
          refmodel$family$linkfun(mu_test) + refmodel$offset
        )
      }
    } else {
      newdata_for_ref <- d_test$data
      if (inherits(refmodel$fit, "stanreg") &&
          length(refmodel$fit$offset) > 0) {
        if ("projpred_internal_offs_stanreg" %in% names(newdata_for_ref)) {
          stop("Need to write to column `projpred_internal_offs_stanreg` of ",
               "`d_test$data`, but that column already exists. Please rename ",
               "this column in `d_test$data` and try again.")
        }
        newdata_for_ref$projpred_internal_offs_stanreg <- d_test$offset
      }
      mu_test <- refmodel$family$linkinv(
        refmodel$ref_predfun(refmodel$fit, newdata = newdata_for_ref) +
          d_test$offset
      )
    }
    ref <- .weighted_summary_means(
      y_test = d_test, family = refmodel$family, wsample = refmodel$wsample,
      mu = mu_test, dis = refmodel$dis
    )
  }

  ## store the relevant fields into the object to be returned
  vs <- nlist(
    refmodel,
    search_path,
    d_test,
    summaries = nlist(sub, ref),
    solution_terms = search_path$solution_terms,
    kl = sapply(submodels, "[[", "kl"),
    nterms_max,
    nterms_all = count_terms_in_formula(refmodel$formula),
    method = method,
    cv_method = NULL,
    validate_search = NULL,
    clust_used_search = p_sel$clust_used,
    clust_used_eval = p_pred$clust_used,
    nprjdraws_search = NCOL(p_sel$mu),
    nprjdraws_eval = NCOL(p_pred$mu)
  )
  ## suggest model size
  class(vs) <- "vsel"
  vs$suggested_size <- suggest_size(vs, warnings = FALSE)
  summary <- summary(vs)
  vs$summary <- summary$selection

  return(vs)
}

select <- function(method, p_sel, refmodel, nterms_max, penalty, verbose, opt,
                   search_terms = NULL, ...) {
  ##
  ## Auxiliary function, performs variable selection with the given method,
  ## and returns the search_path, i.e., a list with the followint entries (the
  ## last three
  ## are returned only if one cluster projection is used for selection):
  ##   solution_terms: the variable ordering
  ##   beta: coefficients along the solution path
  ##   alpha: intercepts along the solution path
  ##   p_sel: the reference distribution used in the selection (the input
  ##   argument p_sel)
  ##
  ## routine that can be used with several clusters
  if (method == "l1") {
    search_path <- search_L1(p_sel, refmodel, nterms_max - refmodel$intercept,
                             penalty, opt)
    search_path$p_sel <- p_sel
    return(search_path)
  } else if (method == "forward") {
    search_path <- search_forward(p_sel, refmodel, nterms_max, verbose, opt,
                                  search_terms = search_terms, ...)
    search_path$p_sel <- p_sel
    return(search_path)
  }
}

## Auxiliary function for parsing the input arguments for varsel.
## The arguments specified by the user (or the function calling this function)
## are treated as they are, but if some are not given, then this function
## fills them in with the default values. The purpose of this function is to
## avoid repeating the same code both in varsel and cv_varsel.
parse_args_varsel <- function(refmodel, method, refit_prj, nterms_max,
                              nclusters, search_terms) {
  search_terms_was_null <- is.null(search_terms)
  if (search_terms_was_null) {
    search_terms <- split_formula(refmodel$formula,
                                  data = refmodel$fetch_data())
  }
  search_terms <- union("1", search_terms)
  has_group_features <- formula_contains_group_terms(refmodel$formula)
  has_additive_features <- formula_contains_additive_terms(refmodel$formula)

  if (is.null(method)) {
    if (has_group_features || has_additive_features || !search_terms_was_null) {
      method <- "forward"
    } else {
      method <- "l1"
    }
  } else {
    method <- tolower(method)
    if (method == "l1") {
      if (has_group_features || has_additive_features) {
        stop("L1 search is only supported for reference models without ",
             "multilevel and without additive (\"smoothing\") terms.")
      }
      if (!search_terms_was_null) {
        warning("Argument `search_terms` only takes effect if ",
                "`method = \"forward\"`.")
      }
    }
  }

  if (!(method %in% c("l1", "forward"))) {
    stop("Unknown search method")
  }

  stopifnot(!is.null(refit_prj))
  if (refit_prj && inherits(refmodel, "datafit")) {
    warning("For an `object` of class \"datafit\", `refit_prj` is ",
            "automatically set to `FALSE`.")
    refit_prj <- FALSE
  }

  if (method == "l1") {
    nclusters <- 1
  }

  search_terms_unq <- unique(unlist(
    strsplit(search_terms, split = "+", fixed = TRUE)
  ))
  max_nv_possible <- count_terms_chosen(search_terms_unq, duplicates = TRUE)
  if (is.null(nterms_max)) {
    nterms_max <- 19
  }
  nterms_max <- min(max_nv_possible, nterms_max + refmodel$intercept)

  return(nlist(method, refit_prj, nterms_max, nclusters, search_terms))
}
