R/02_Pois.R

In joker: Probability Distributions and Parameter Estimation

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

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

setClass("Pois",
         contains = "Distribution",
         slots = c(lambda = "numeric"),
         prototype = list(lambda = 1))

#' @title Poisson Distribution
#' @name Pois
#'
#' @description
#' The Poisson distribution is a discrete probability distribution that models
#' the number of events occurring in a fixed interval of time or space, given
#' that the events occur with a constant rate \eqn{\lambda > 0} and
#' independently of the time since the last event.
#'
#' @param n number of observations. If `length(n) > 1`, the length is taken to
#' be the number required.
#' @param distr an object of class `Pois`.
#' @param x For the density function, `x` is a numeric vector of quantiles. For
#' the moments functions, `x` is an object of class `Pois`. 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 lambda numeric. The distribution parameter.
#' @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 Poisson distribution is:
#' \deqn{ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k \in
#' \mathbb{N}_0. }
#'
#' @inherit distributions return
#'
#' @seealso
#' Functions from the `stats` package: [dpois()], [ppois()], [qpois()],
#' [rpois()]
#'
#' @export
#'
#' @examples
#' # -----------------------------------------------------
#' # Pois Distribution Example
#' # -----------------------------------------------------
#'
#' # Create the distribution
#' lambda <- 5
#' D <- Pois(lambda)
#'
#' # ------------------
#' # dpqr Functions
#' # ------------------
#'
#' d(D, 0:10) # density function
#' p(D, 0:10) # 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)
#' llpois(x, lambda)
#'
#' epois(x, type = "mle")
#' epois(x, type = "me")
#'
#' mle(D, x)
#' me(D, x)
#' e(D, x, type = "mle")
#'
#' mle("pois", x) # the distr argument can be a character
#'
#' # ------------------
#' # Estimator Variance
#' # ------------------
#'
#' vpois(lambda, type = "mle")
#' vpois(lambda, type = "me")
#'
#' avar_mle(D)
#' avar_me(D)
#'
#' v(D, type = "mle")
Pois <- function(lambda = 1) {
  new("Pois", lambda = lambda)
}

setValidity("Pois", function(object) {
  if(length(object@lambda) != 1) {
    stop("lambda has to be a numeric of length 1")
  }
  if(object@lambda <= 0) {
    stop("lambda has to be positive")
  }
  TRUE
})

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

#' @rdname Pois
setMethod("d", signature = c(distr = "Pois", x = "numeric"),
          function(distr, x, log = FALSE) {
            dpois(x, lambda = distr@lambda, log = log)
          })

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

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

#' @rdname Pois
setMethod("r", signature = c(distr = "Pois", n = "numeric"),
          function(distr, n) {
            rpois(n, lambda = distr@lambda)
          })

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

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

  x@lambda

})

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

            warning("The median of a Pois(l) distribution is given by the
                    inequality: l - ln2 <= median < l + 1/3. The lower bound is
                    returned.")

            x@lambda - log(2)

          })


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

            floor(x@lambda)

          })

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

  x@lambda

})

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

  sqrt(var(x))

})

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

  1 / sqrt(x@lambda)

})

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

  1 / x@lambda

})

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

  warning("The entropy given is an approximation in the O(1 / l ^ 4) order.")
  l <- x@lambda
  0.5 * log(2 * pi  * exp(1) * l) - 1 / (12 * l) - 1 / (24 * l ^ 2) -
    19 / (360 * l ^ 3)

})

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

  1 / x@lambda

})

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

#' @rdname Pois
#' @export
llpois <- function(x, lambda) {
  ll(Pois(lambda), x)
}

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

  lam <- distr@lambda
  log(lam) * sum(x) - length(x) * lam - sum(log(factorial(x)))

})

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

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

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

  x <- check_data(x, na.rm = na.rm)
  list(lambda = mean(x))

})

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

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

})

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

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

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

  c(lambda = distr@lambda)

})

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

  avar_mle(distr)

})