R/02_Geom.R

In joker: Probability Distributions and Parameter Estimation

# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Geom Distribution                                                         ----
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Distribution           ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

setClass("Geom",
         contains = "Distribution",
         slots = c(prob = "numeric"),
         prototype = list(prob = 0.5))

#' @title Geometric Distribution
#' @name Geom
#'
#' @description
#' The Geometric distribution is a discrete probability distribution that models
#' the number of failures before the first success in a sequence of independent
#' Bernoulli trials, each with success probability \eqn{0 < p \leq 1}.
#'
#' @param n number of observations. If `length(n) > 1`, the length is taken to
#' be the number required.
#' @param distr an object of class `Geom`.
#' @param x For the density function, `x` is a numeric vector of quantiles. For
#' the moments functions, `x` is an object of class `Geom`. For the
#' log-likelihood and the estimation functions, `x` is the sample of
#' observations.
#' @param p numeric. Vector of probabilities.
#' @param q numeric. Vector of quantiles.
#' @param prob numeric. Probability of success.
#' @param type character, case ignored. The estimator type (mle or me).
#' @param log,log.p logical. Should the logarithm of the probability be
#' returned?
#' @param lower.tail logical. If TRUE (default), probabilities are
#' \eqn{P(X \leq x)}, otherwise \eqn{P(X > x)}.
#' @param na.rm logical. Should the `NA` values be removed?
#' @param ... extra arguments.
#'
#' @details
#' The probability mass function (PMF) of the Geometric distribution is:
#' \deqn{ P(X = k) = (1 - p)^k p, \quad k \in \mathbb{N}_0.}
#'
#' @inherit distributions return
#'
#' @seealso
#' Functions from the `stats` package: [dgeom()], [pgeom()], [qgeom()],
#' [rgeom()]
#'
#' @export
#'
#' @examples
#' # -----------------------------------------------------
#' # Geom Distribution Example
#' # -----------------------------------------------------
#'
#' # Create the distribution
#' p <- 0.4
#' D <- Geom(p)
#'
#' # ------------------
#' # dpqr Functions
#' # ------------------
#'
#' d(D, 0:4) # density function
#' p(D, 0:4) # distribution function
#' qn(D, c(0.4, 0.8)) # inverse distribution function
#' x <- r(D, 100) # random generator function
#'
#' # alternative way to use the function
#' df <- d(D) ; df(x) # df is a function itself
#'
#' # ------------------
#' # Moments
#' # ------------------
#'
#' mean(D) # Expectation
#' median(D) # Median
#' mode(D) # Mode
#' var(D) # Variance
#' sd(D) # Standard Deviation
#' skew(D) # Skewness
#' kurt(D) # Excess Kurtosis
#' entro(D) # Entropy
#' finf(D) # Fisher Information Matrix
#'
#' # List of all available moments
#' mom <- moments(D)
#' mom$mean # expectation
#'
#' # ------------------
#' # Point Estimation
#' # ------------------
#'
#' ll(D, x)
#' llgeom(x, p)
#'
#' egeom(x, type = "mle")
#' egeom(x, type = "me")
#'
#' mle(D, x)
#' me(D, x)
#' e(D, x, type = "mle")
#'
#' mle("geom", x) # the distr argument can be a character
#'
#' # ------------------
#' # Estimator Variance
#' # ------------------
#'
#' vgeom(p, type = "mle")
#' vgeom(p, type = "me")
#'
#' avar_mle(D)
#' avar_me(D)
#'
#' v(D, type = "mle")
Geom <- function(prob = 0.5) {
  new("Geom", prob = prob)
}

setValidity("Geom", function(object) {
  if(length(object@prob) != 1) {
    stop("prob has to be a numeric of length 1")
  }
  if(object@prob <= 0 || object@prob >= 1) {
    stop("prob has to be between 0 and 1")
  }
  TRUE
})

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## d, p, q, r             ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

#' @rdname Geom
setMethod("d", signature = c(distr = "Geom", x = "numeric"),
          function(distr, x, log = FALSE) {
            dgeom(x, prob = distr@prob, log = log)
          })

#' @rdname Geom
setMethod("p", signature = c(distr = "Geom", q = "numeric"),
          function(distr, q, lower.tail = TRUE, log.p = FALSE) {
            pgeom(q, prob = distr@prob,
                  lower.tail = lower.tail, log.p = log.p)
          })

#' @rdname Geom
setMethod("qn", signature = c(distr = "Geom", p = "numeric"),
          function(distr, p, lower.tail = TRUE, log.p = FALSE) {
            qgeom(p, prob = distr@prob,
                  lower.tail = lower.tail, log.p = log.p)
          })

#' @rdname Geom
setMethod("r", signature = c(distr = "Geom", n = "numeric"),
          function(distr, n) {
            rgeom(n, prob = distr@prob)
          })

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Moments                ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

#' @rdname Geom
setMethod("mean",
          signature  = c(x = "Geom"),
          definition = function(x) {

  1 / x@prob - 1

})

#' @rdname Geom
setMethod("median",
          signature  = c(x = "Geom"),
          definition = function(x) {

  y <- - 1 / log(1 - x@prob, base = 2)
  if (is_whole(y) & x@prob != 0.5) {
    warning("The median of the Geom distribution is not uniquely defined in this
            case.")
  }

  ceiling(y) - 1

})

#' @rdname Geom
setMethod("mode",
          signature  = c(x = "Geom"),
          definition = function(x) {

  0

})

#' @rdname Geom
setMethod("var",
          signature  = c(x = "Geom"),
          definition = function(x) {

  (1 - x@prob) / x@prob ^ 2

})

#' @rdname Geom
setMethod("sd",
          signature  = c(x = "Geom"),
          definition = function(x) {

  sqrt(var(x))

})

#' @rdname Geom
setMethod("skew",
          signature  = c(x = "Geom"),
          definition = function(x) {

  (2 - x@prob) / sqrt(1 - x@prob)

})

#' @rdname Geom
setMethod("kurt",
          signature  = c(x = "Geom"),
          definition = function(x) {

  6 + x@prob ^ 2 / (1 - x@prob)

})

#' @rdname Geom
setMethod("entro",
          signature  = c(x = "Geom"),
          definition = function(x) {

  p <- x@prob
  (- (1 - p) * log(1 - p) - p * log(p)) / p

})

#' @rdname Geom
setMethod("finf",
          signature  = c(x = "Geom"),
          definition = function(x) {

  1 / (x@prob ^ 2 * (1 - x@prob))

})

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Likelihood             ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

#' @rdname Geom
#' @export
llgeom <- function(x, prob) {
  ll(Geom(prob), x)
}

#' @rdname Geom
setMethod("ll",
          signature  = c(distr = "Geom", x = "numeric"),
          definition = function(distr, x) {

  p <- distr@prob
  log(1 - p) * sum(x) + log(p) * length(x)

})

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimation             ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

#' @rdname Geom
#' @export
egeom <- function(x, type = "mle", ...) {
  type <- match.arg(tolower(type), choices = c("mle", "me"))
  distr <- Geom()
  do.call(type, list(distr = distr, x = x, ...))
}

#' @rdname Geom
setMethod("mle",
          signature  = c(distr = "Geom", x = "numeric"),
          definition = function(distr, x, na.rm = FALSE) {

  x <- check_data(x, na.rm = na.rm)
  list(prob = 1 / (1 + mean(x)))

})

#' @rdname Geom
setMethod("me",
          signature  = c(distr = "Geom", x = "numeric"),
          definition = function(distr, x, na.rm = FALSE) {

  mle(distr, x, na.rm = na.rm)

})

## ~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Variance               ----
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~

#' @rdname Geom
#' @export
vgeom <- function(prob, type = "mle") {
  type <- match.arg(tolower(type), choices = c("mle", "me"))
  distr <- Geom(prob)
  do.call(paste0("avar_", type), list(distr = distr))
}

#' @rdname Geom
setMethod("avar_mle",
          signature  = c(distr = "Geom"),
          definition = function(distr) {

  prob <- distr@prob
  c(prob = prob ^ 2 * (1 - prob))

})

#' @rdname Geom
setMethod("avar_me",
          signature  = c(distr = "Geom"),
          definition = function(distr) {

  avar_mle(distr)

})