R/srexp.R

In stors: Step Optimised Rejection Sampling

#' Sampling from Exponential Distribution
#' @rdname srexp
#' @order 1
#'
#' @description
#' The \code{srexp()} function generates random samples from a Exponential Distribution using the STORS algorithm.
#' It employs an optimized proposal distribution around the mode and Inverse Transform (IT) method for the tails.
#'
#' @details
#'
#' The Exponential distribution has the probability density function (PDF):
#' \eqn{f(x | \lambda) = \lambda \exp(-\lambda x), \quad x \geq 0,}
#' where:
#' \describe{
#'   \item{\eqn{\lambda}}{is the rate parameter (\eqn{\lambda > 0}), which determines the rate of decay of the distribution.}
#' }
#' The Exponential distribution is commonly used to model the time between independent events that occur at a constant average rate.
#'
#' These two functions are for sampling using the STORS algorithm based on the proposal that has been constructed using \code{\link{srexp_optimize}}.
#'
#' By default, \code{srexp()} samples from a standard Exponential Distribution \code{rate = 1}.
#' The proposal distribution is pre-optimized at package load time using \code{srexp_optimize()} with
#' \code{steps = 4091}, creating a scalable proposal centred around the mode.
#'
#' If \code{srexp()} is called with custom \code{rate} parameter, the samples are generated
#' from the standard Exponential Distribution, then scaled accordingly.
#'
#'
#' @param n Integer, length 1. Number of samples to draw.
#' @param rate Numeric. is the rate parameter of the Exponential Distribution.
#' @param x (optional) Numeric vector of length \eqn{n}. If provided, this vector is over
#' written in place to avoid any memory allocation.
#'
#' @return
#' A numeric vector of length \code{n} containing samples from the Exponential Distribution with the specified
#' \code{rate}.
#'
#' \bold{NOTE:} When the \code{x} parameter is specified, it is updated in-place with the simulation for performance reasons.
#'
#' @seealso
#' \code{\link{srexp_optimize}} to optimize the custom or the scaled proposal.
#'
#' @examples
#' # Generate 10 samples from the standard Exponential Distribution
#' samples <- srexp(10)
#' print(samples)
#'
#' # Generate 10 samples using a pre-allocated vector
#' x <- numeric(10)
#' srexp(10, x = x)
#' print(x)
#'
#' # Generate 10 samples from a Exponential Distribution with rate = 4
#' samples <- srexp(10, rate = 4)
#' print(samples)
#'
#' @export
srexp <- function(n = 1, rate = 1, x = NULL) {
  .Call(C_srexp_scaled_check, n, c(rate), x)
}

#' Sampling from Custom Exponential Distribution
#' @rdname srexp
#' @order 2
#'
#' @export
srexp_custom <- function(n = 1, x = NULL) {
  .Call(C_srlaplace_custom_check, n, x)
}



#' Optimizing Exponential Distribution proposal
#' @description
#' The \code{srexp_optimize()} function generates an optimized proposal for a targeted Exponential Distribution.
#'  The proposal can be customized and adjusted based on various options provided by the user.
#'
#'
#' @details
#'When \code{srexp_optimize()} is explicitly called:
#'\itemize{
#'  \item A proposal is created and cached. If no parameters are provided, a standard proposal is created with \code{rate = 1}.
#'  \item Providing \code{rate} creates a custom proposal, which is cached for use with \code{srexp_custom()}.
#'  \item The optimization process can be controlled via parameters such as \code{steps}, \code{proposal_range}, or
#'   \code{theta}. If no parameters are provided, the proposal is optimized via brute force based on the.
#'   \code{target_sample_size}.
#'}
#'
#' @param rate (optional) Numeric. rate parameter of the Exponential Distribution. Defaults to \code{NULL}, which implies a scalable proposal with \code{rate = 1}.
#' @param xl Numeric. Left truncation bound for the target distribution. Defaults to \code{-Inf}, representing no left truncation.
#' @param xr Numeric. Right truncation bound for the target distribution. Defaults to \code{Inf}, representing no right truncation.
#' @param steps (optional) Integer. Desired number of steps in the proposal. Defaults to \code{NULL}, which means the number of steps is determined automatically during optimization.
#' @param proposal_range (optional) Numeric vector. Specifies the range for optimizing the steps part of the proposal. Defaults to \code{NULL}, indicating automatic range selection.
#' @param theta Numeric. A parameter for proposal optimization. Defaults to 0.1.
#' @param target_sample_size (optional) Integer. Target sample size for proposal optimization. Defaults to \code{1000}.
#' @param verbose Boolean. If \code{TRUE}, detailed optimization information, including areas and steps, will be displayed. Defaults to \code{FALSE}.
#'
#'
#' @return
#' The user does not need to store the returned value, because the package internally cashes the proposal. However, we explain here the full returned proposal for advanced users.
#'
#' A list containing the optimized proposal and related parameters for the specified built-in distribution:
#' \describe{
#'   \item{\code{data}}{A data frame with detailed information about the proposal steps, including:
#'   \describe{
#'     \item{\code{x}}{The start point of each step on the x-axis.}
#'     \item{\code{s_upper}}{The height of each step on the y-axis.}
#'     \item{\code{p_a}}{Pre-acceptance probability for each step.}
#'     \item{\code{s_upper_lower}}{A vector used to scale the uniform random number when the sample is accepted.}
#'   }}
#'   \item{\code{areas}}{A numeric vector containing the areas under:
#'   \describe{
#'     \item{\code{left_tail}}{The left tail bound.}
#'     \item{\code{steps}}{The middle steps.}
#'     \item{\code{right_tail}}{The right tail bound.}
#'   }}
#'   \item{\code{steps_number}}{An integer specifying the number of steps in the proposal.}
#'   \item{\code{sampling_probabilities}}{A numeric vector with:
#'   \describe{
#'     \item{\code{left_tail}}{The probability of sampling from the left tail.}
#'     \item{\code{left_and_middle}}{The combined probability of sampling from the left tail and middle steps.}
#'   }}
#'   \item{\code{unif_scaler}}{A numeric scalar, the inverse probability of sampling from the steps part of the proposal (\eqn{\frac{1}{p(lower < x < upper)}}). Used for scaling uniform random values.}
#'   \item{\code{lt_properties}}{A numeric vector of 5 values required for Adaptive Rejection Sampling (ARS) in the left tail.}
#'   \item{\code{rt_properties}}{A numeric vector of 6 values required for ARS in the right tail.}
#'   \item{\code{alpha}}{A numeric scalar representing the uniform step area.}
#'   \item{\code{tails_method}}{A string, either \code{"ARS"} (Adaptive Rejection Sampling) or \code{"IT"} (Inverse Transform), indicating the sampling method for the tails.}
#'   \item{\code{proposal_bounds}}{A numeric vector specifying the left and right bounds of the target density.}
#'   \item{\code{cnum}}{An integer representing the cache number of the created proposal in memory.}
#'   \item{\code{symmetric}}{A numeric scalar indicating the symmetry point of the proposal, or \code{NULL} if not symmetric.}
#'   \item{\code{f_params}}{A list of parameters for the target density that the proposal is designed for.}
#'   \describe{
#'     \item{\code{rate}}{ the rate of the target distribution.}
#'   }
#'   \item{\code{is_symmetric}}{A logical value indicating whether the proposal is symmetric.}
#'   \item{\code{proposal_type}}{A string indicating the type of the generated proposal:
#'   \describe{
#'     \item{\code{"scaled"}}{The proposal is "scalable" and standardized with \code{rate = 1}. This is used when parameter \code{rate} is either \code{NULL} or not provided. Scalable proposals are compatible with \code{\link{srexp}}.}
#'     \item{\code{"custom"}}{The proposal is "custom" when \code{rate} is provided. Custom proposals are compatible with \code{\link{srexp_custom}}.}
#'   }}
#'   \item{\code{target_function_area}}{A numeric scalar estimating the area of the target distribution.}
#'   \item{\code{dens_func}}{A string containing the hardcoded density function.}
#'   \item{\code{density_name}}{A string specifying the name of the target density distribution.}
#'   \item{\code{lock}}{An identifier used for saving and loading the proposal from disk.}
#' }
#'
#' @seealso
#' \code{\link{srexp}}: Function to sample from a scalable proposal generated by \code{srexp_optimize()}.
#' \code{\link{srexp_custom}}: Function to sample from a custom proposal tailored to user specifications.
#'
#'
#' @examples
#' # Generate scalable proposal that with rate = 1, that has 4096 steps
#' scalable_proposal <- srexp_optimize(steps = 4096)
#'
#' # Generate custom proposal that with rate = 4
#' scalable_proposal <- srexp_optimize(rate = 4)
#'
#' @export
srexp_optimize <- function(rate = NULL,
                           xl = NULL,
                           xr = NULL,
                           steps = 4091,
                           proposal_range = NULL,
                           theta = 0.1,
                           target_sample_size = 1000,
                           verbose = FALSE) {

  dist_name <- "srexp"

  dendata <- built_in_proposals[[dist_name]]

  f_params <- list(rate = rate)

  if (dendata$scalable) {

    isnull <- sapply(f_params, is.null)

    if (all(isnull)) {
      cnum <- dendata$c_num
      proposal_type <- "scaled"
    } else {
      cnum <- dendata$c_num + 1
      proposal_type <- "custom"
    }

    f_params <- ifelse(isnull, dendata$std_params, f_params)

  } else {

    cnum <- dendata$c_num + 1
    proposal_type <- "custom"
  }


  modes <- 0

  symmetric <- NULL

  f <- dendata$create_f(f_params$rate)

  proposal_optimizer(dendata, dist_name, xl, xr, f, modes, f_params, steps,
                     proposal_range, theta, target_sample_size, proposal_type,
                     symmetric, cnum, verbose)

}