#' Perform a hierarchical model fit with multiple starting values
#'
#' The purpose of this method is to check if a certain algorithm for fitting
#' nonlinear hierarchical models (also known as nonlinear mixed-effects models)
#' will reliably yield results that are sufficiently similar to each other, if
#' started with a certain range of reasonable starting parameters. It is
#' inspired by the article on practical identifiabiliy in the frame of nonlinear
#' mixed-effects models by Duchesne et al (2021).
#'
#' Currently, parallel execution of the fits is only supported using
#' [parallel::mclapply], i.e. not available on Windows.
#'
#' In case the online version of this help page contains error messages
#' in the example code and no plots, this is due to the multistart method
#' not working when called by pkgdown. Please refer to the 
#' [online vignette](https://pkgdown.jrwb.de/mkin/dev/articles/web_only/multistart.html)
#' in this case.
#'
#' @param object The fit object to work with
#' @param n How many different combinations of starting parameters should be
#' used?
#' @param cores How many fits should be run in parallel?
#' @param \dots Passed to the update function.
#' @param x The multistart object to print
#' @return A list of [saem.mmkin] objects, with class attributes
#' 'multistart.saem.mmkin' and 'multistart'.
#' @seealso [parhist], [llhist]
#'
#' @references Duchesne R, Guillemin A, Gandrillon O, Crauste F. Practical
#' identifiability in the frame of nonlinear mixed effects models: the example
#' of the in vitro erythropoiesis. BMC Bioinformatics. 2021 Oct 4;22(1):478.
#' doi: 10.1186/s12859-021-04373-4.
#' @export
#' @examples
#' \dontrun{
#' library(mkin)
#' dmta_ds <- lapply(1:7, function(i) {
#'   ds_i <- dimethenamid_2018$ds[[i]]$data
#'   ds_i[ds_i$name == "DMTAP", "name"] <-  "DMTA"
#'   ds_i$time <- ds_i$time * dimethenamid_2018$f_time_norm[i]
#'   ds_i
#' })
#' names(dmta_ds) <- sapply(dimethenamid_2018$ds, function(ds) ds$title)
#' dmta_ds[["Elliot"]] <- rbind(dmta_ds[["Elliot 1"]], dmta_ds[["Elliot 2"]])
#' dmta_ds[["Elliot 1"]] <- dmta_ds[["Elliot 2"]] <- NULL
#'
#' f_mmkin <- mmkin("DFOP", dmta_ds, error_model = "tc", cores = 7, quiet = TRUE)
#' f_saem_full <- saem(f_mmkin)
#' f_saem_full_multi <- multistart(f_saem_full, n = 16, cores = 16)
#' parhist(f_saem_full_multi, lpos = "bottomright")
#'
#' f_saem_reduced <- update(f_saem_full, covariance.model = diag(c(1, 1, 0, 1)))
#' f_saem_reduced_multi <- multistart(f_saem_reduced, n = 16, cores = 16)
#' parhist(f_saem_reduced_multi, lpos = "bottomright")
#' }
multistart <- function(object, n = 50, cores = 1, ...)
{
  UseMethod("multistart", object)
}

#' @rdname multistart
#' @export
multistart.saem.mmkin <- function(object, n = 50, cores = 1, ...) {

  mmkin_parms <- parms(object$mmkin, errparms = FALSE,
    transformed = object$transformations == "mkin")
  start_parms <- apply(
    mmkin_parms, 1,
    function(x) stats::runif(n, min(x), max(x)))
  if (n == 1) dim(start_parms) <- c(1, length(start_parms))

  res <- parallel::mclapply(1:n, function(x) {
    update(object, degparms_start = start_parms[x, ], ...)
  }, mc.cores = cores)
  attr(res, "orig") <- object
  attr(res, "start_parms") <- start_parms
  class(res) <- c("multistart.saem.mmkin", "multistart")
  return(res)
}

#' @rdname multistart
#' @export
print.multistart <- function(x, ...) {
  cat("Multistart object with", length(x), "fits of the following type:\n\n")
  print(x[[1]])
}

#' @rdname multistart
#' @export
parms.multistart <- function(object, ...) {
  t(sapply(object, parms))
}