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#' Create a Poisson distribution
#'
#' Poisson distributions are frequently used to model counts.
#'
#' @param lambda The shape parameter, which is also the mean and the
#' variance of the distribution. Can be any positive number.
#'
#' @return A `Poisson` object.
#' @export
#'
#' @family discrete distributions
#'
#' @details
#'
#' We recommend reading this documentation on
#' <https://alexpghayes.github.io/distributions3/>, where the math
#' will render with additional detail.
#'
#' In the following, let \eqn{X} be a Poisson random variable with parameter
#' `lambda` = \eqn{\lambda}.
#'
#' **Support**: \eqn{\{0, 1, 2, 3, ...\}}{{0, 1, 2, 3, ...}}
#'
#' **Mean**: \eqn{\lambda}
#'
#' **Variance**: \eqn{\lambda}
#'
#' **Probability mass function (p.m.f)**:
#'
#' \deqn{
#' P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
#' }{
#' P(X = k) = \lambda^k e^(-\lambda) / k!
#' }
#'
#' **Cumulative distribution function (c.d.f)**:
#'
#' \deqn{
#' P(X \le k) = e^{-\lambda}
#' \sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}
#' }{
#' P(X \le k) = e^(-\lambda)
#' \sum_{i = 0}^k \lambda^i / i!
#' }
#'
#' **Moment generating function (m.g.f)**:
#'
#' \deqn{
#' E(e^{tX}) = e^{\lambda (e^t - 1)}
#' }{
#' E(e^(tX)) = e^(\lambda (e^t - 1))
#' }
#'
#' @examples
#'
#' set.seed(27)
#'
#' X <- Poisson(2)
#' X
#'
#' random(X, 10)
#'
#' pdf(X, 2)
#' log_pdf(X, 2)
#'
#' cdf(X, 4)
#' quantile(X, 0.7)
#'
#' cdf(X, quantile(X, 0.7))
#' quantile(X, cdf(X, 7))
Poisson <- function(lambda) {
d <- data.frame(lambda = lambda)
class(d) <- c("Poisson", "distribution")
d
}
#' @export
mean.Poisson <- function(x, ...) {
ellipsis::check_dots_used()
rval <- x$lambda
setNames(rval, names(x))
}
#' @export
variance.Poisson <- function(x, ...) {
ellipsis::check_dots_used()
rval <- x$lambda
setNames(rval, names(x))
}
#' @export
skewness.Poisson <- function(x, ...) {
ellipsis::check_dots_used()
rval <- 1 / sqrt(x$lambda)
setNames(rval, names(x))
}
#' @export
kurtosis.Poisson <- function(x, ...) {
ellipsis::check_dots_used()
rval <- 1 / x$lambda
setNames(rval, names(x))
}
#' Draw a random sample from a Poisson distribution
#'
#' @inherit Poisson examples
#'
#' @param x A `Poisson` object created by a call to [Poisson()].
#' @param n The number of samples to draw. Defaults to `1L`.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param ... Unused. Unevaluated arguments will generate a warning to
#' catch mispellings or other possible errors.
#'
#' @return In case of a single distribution object or `n = 1`, either a numeric
#' vector of length `n` (if `drop = TRUE`, default) or a `matrix` with `n` columns
#' (if `drop = FALSE`).
#' @export
#'
random.Poisson <- function(x, n = 1L, drop = TRUE, ...) {
n <- make_positive_integer(n)
if (n == 0L) {
return(numeric(0L))
}
FUN <- function(at, d) rpois(n = at, lambda = d$lambda)
apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}
#' Evaluate the probability mass function of a Poisson distribution
#'
#' @inherit Poisson examples
#'
#' @param d A `Poisson` object created by a call to [Poisson()].
#' @param x A vector of elements whose probabilities you would like to
#' determine given the distribution `d`.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param elementwise logical. Should each distribution in \code{d} be evaluated
#' at all elements of \code{x} (\code{elementwise = FALSE}, yielding a matrix)?
#' Or, if \code{d} and \code{x} have the same length, should the evaluation be
#' done element by element (\code{elementwise = TRUE}, yielding a vector)? The
#' default of \code{NULL} means that \code{elementwise = TRUE} is used if the
#' lengths match and otherwise \code{elementwise = FALSE} is used.
#' @param ... Arguments to be passed to \code{\link[stats]{dpois}}.
#' Unevaluated arguments will generate a warning to catch mispellings or other
#' possible errors.
#'
#' @return In case of a single distribution object, either a numeric
#' vector of length `probs` (if `drop = TRUE`, default) or a `matrix` with
#' `length(x)` columns (if `drop = FALSE`). In case of a vectorized distribution
#' object, a matrix with `length(x)` columns containing all possible combinations.
#' @export
#'
pdf.Poisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) dpois(x = at, lambda = d$lambda, ...)
apply_dpqr(d = d, FUN = FUN, at = x, type = "density", drop = drop, elementwise = elementwise)
}
#' @rdname pdf.Poisson
#' @export
#'
log_pdf.Poisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) dpois(x = at, lambda = d$lambda, log = TRUE)
apply_dpqr(d = d, FUN = FUN, at = x, type = "logLik", drop = drop, elementwise = elementwise)
}
#' Evaluate the cumulative distribution function of a Poisson distribution
#'
#' @inherit Poisson examples
#'
#' @param d A `Poisson` object created by a call to [Poisson()].
#' @param x A vector of elements whose cumulative probabilities you would
#' like to determine given the distribution `d`.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param elementwise logical. Should each distribution in \code{d} be evaluated
#' at all elements of \code{x} (\code{elementwise = FALSE}, yielding a matrix)?
#' Or, if \code{d} and \code{x} have the same length, should the evaluation be
#' done element by element (\code{elementwise = TRUE}, yielding a vector)? The
#' default of \code{NULL} means that \code{elementwise = TRUE} is used if the
#' lengths match and otherwise \code{elementwise = FALSE} is used.
#' @param ... Arguments to be passed to \code{\link[stats]{ppois}}.
#' Unevaluated arguments will generate a warning to catch mispellings or other
#' possible errors.
#'
#' @return In case of a single distribution object, either a numeric
#' vector of length `probs` (if `drop = TRUE`, default) or a `matrix` with
#' `length(x)` columns (if `drop = FALSE`). In case of a vectorized distribution
#' object, a matrix with `length(x)` columns containing all possible combinations.
#' @export
#'
cdf.Poisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) ppois(q = at, lambda = d$lambda, ...)
apply_dpqr(d = d, FUN = FUN, at = x, type = "probability", drop = drop, elementwise = elementwise)
}
#' Determine quantiles of a Poisson distribution
#'
#' `quantile()` is the inverse of `cdf()`.
#'
#' @inherit Poisson examples
#' @inheritParams random.Poisson
#'
#' @param probs A vector of probabilities.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param elementwise logical. Should each distribution in \code{x} be evaluated
#' at all elements of \code{probs} (\code{elementwise = FALSE}, yielding a matrix)?
#' Or, if \code{x} and \code{probs} have the same length, should the evaluation be
#' done element by element (\code{elementwise = TRUE}, yielding a vector)? The
#' default of \code{NULL} means that \code{elementwise = TRUE} is used if the
#' lengths match and otherwise \code{elementwise = FALSE} is used.
#' @param ... Arguments to be passed to \code{\link[stats]{qpois}}.
#' Unevaluated arguments will generate a warning to catch mispellings or other
#' possible errors.
#'
#' @return In case of a single distribution object, either a numeric
#' vector of length `probs` (if `drop = TRUE`, default) or a `matrix` with
#' `length(probs)` columns (if `drop = FALSE`). In case of a vectorized
#' distribution object, a matrix with `length(probs)` columns containing all
#' possible combinations.
#' @export
#'
quantile.Poisson <- function(x, probs, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) qpois(p = at, lambda = d$lambda, ...)
apply_dpqr(d = x, FUN = FUN, at = probs, type = "quantile", drop = drop, elementwise = elementwise)
}
#' Fit an Poisson distribution to data
#'
#' @param d An `Poisson` object created by a call to [Poisson()].
#' @param x A vector of data.
#' @param ... Unused.
#'
#' @family Poisson distribution
#'
#' @return An `Poisson` object.
#' @export
fit_mle.Poisson <- function(d, x, ...) {
ss <- suff_stat(d, x, ...)
Poisson(ss$sum / ss$samples)
}
#' Compute the sufficient statistics of an Poisson distribution from data
#'
#' @inheritParams fit_mle.Poisson
#'
#' @return A named list of the sufficient statistics of the Poisson
#' distribution:
#'
#' - `sum`: The sum of the data.
#' - `samples`: The number of samples in the data.
#'
#' @export
suff_stat.Poisson <- function(d, x, ...) {
valid_x <- (x >= 0) & (x %% 1 == 0)
if (any(!valid_x)) stop("`x` must only contain positive integers")
list(sum = sum(x), samples = length(x))
}
#' Return the support of the Poisson distribution
#'
#' @param d An `Poisson` object created by a call to [Poisson()].
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param ... Currently not used.
#'
#' @return A vector of length 2 with the minimum and maximum value of the support.
#'
#' @export
support.Poisson <- function(d, drop = TRUE, ...) {
ellipsis::check_dots_used()
min <- rep(0, length(d))
max <- rep(Inf, length(d))
make_support(min, max, d, drop = drop)
}
#' @exportS3Method
is_discrete.Poisson <- function(d, ...) {
ellipsis::check_dots_used()
setNames(rep.int(TRUE, length(d)), names(d))
}
#' @exportS3Method
is_continuous.Poisson <- function(d, ...) {
ellipsis::check_dots_used()
setNames(rep.int(FALSE, length(d)), names(d))
}
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