#' The zero-truncated Poisson distribution
#'
#' Density, distribution function, quantile function, and random
#' generation for the zero-truncated Poisson distribution with
#' parameter \code{lambda}.
#'
#' The Poisson distribution left-truncated at zero (or zero-truncated
#' Poisson for short) is the distribution obtained, when considering
#' a Poisson variable Y conditional on Y being greater than zero.
#'
#' All functions follow the usual conventions of d/p/q/r functions
#' in base R. In particular, all four \code{ztpois} functions for the
#' zero-truncated Poisson distribution call the corresponding \code{pois}
#' functions for the Poisson distribution from base R internally.
#'
#' @aliases dztpois pztpois qztpois rztpois
#'
#' @param x vector of (non-negative integer) quantiles.
#' @param q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of random values to return.
#' @param lambda vector of (non-negative) Poisson parameters.
#' @param log,log.p logical indicating whether probabilities p are given as log(p).
#' @param lower.tail logical indicating whether probabilities are \eqn{P[X \le x]} (lower tail) or \eqn{P[X > x]} (upper tail).
#'
#' @seealso \code{\link{ZTPoisson}}, \code{\link{dpois}}
#'
#' @keywords distribution
#'
#' @examples
#' ## theoretical probabilities for a zero-truncated Poisson distribution
#' x <- 0:8
#' p <- dztpois(x, lambda = 2.5)
#' plot(x, p, type = "h", lwd = 2)
#'
#' ## corresponding empirical frequencies from a simulated sample
#' set.seed(0)
#' y <- rztpois(500, lambda = 2.5)
#' hist(y, breaks = -1:max(y) + 0.5)
#'
#' @importFrom stats dpois ppois
#' @rdname ztpois
#' @export
dztpois <- function(x, lambda, log = FALSE) {
rval <- dpois(x, lambda, log = TRUE) - ppois(0, lambda, lower.tail = FALSE, log.p = TRUE)
rval[x < 1] <- -Inf
rval[lambda <= 0] <- -Inf
rval[(lambda <= 0) & (x == 1)] <- 0
if(log) rval else exp(rval)
}
#' @importFrom stats ppois dpois
#' @rdname ztpois
#' @export
pztpois <- function(q, lambda, lower.tail = TRUE, log.p = FALSE) {
rval <- log(ppois(q, lambda, lower.tail = lower.tail, log.p = FALSE) - dpois(0, lambda)) -
ppois(0, lambda, lower.tail = FALSE, log.p = TRUE)
rval[q < 1] <- if(lower.tail) -Inf else 0
if(log.p) rval else exp(rval)
}
#' @importFrom stats qpois ppois
#' @rdname ztpois
#' @export
qztpois <- function(p, lambda, lower.tail = TRUE, log.p = FALSE) {
p_orig <- p
p <- if(log.p) p else log(p)
p <- p + ppois(0, lambda, lower.tail = FALSE, log.p = TRUE)
p <- exp(p) + dpois(0, lambda)
rval <- qpois(p, lambda, lower.tail = lower.tail, log.p = FALSE)
if(lower.tail) rval[p_orig < dztpois(1, lambda, log = log.p)] <- 1
rval
}
#' @importFrom stats runif
#' @rdname ztpois
#' @export
rztpois <- function(n, lambda) {
qztpois(runif(n), lambda)
}
#' Create a zero-truncated Poisson distribution
#'
#' Zero-truncated Poisson distributions are frequently used to model counts
#' where zero observations cannot occur or have been excluded.
#'
#' @param lambda Parameter of the underlying untruncated Poisson distribution.
#' Can be any positive number.
#'
#' @return A `ZTPoisson` 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 zero-truncated Poisson random variable with parameter
#' `lambda` = \eqn{\lambda}.
#'
#' **Support**: \eqn{\{1, 2, 3, ...\}}{{1, 2, 3, ...}}
#'
#' **Mean**:
#' \deqn{
#' \lambda \cdot \frac{1}{1 - e^{-\lambda}}
#' }{
#' \lambda \cdot 1/(1 - e^{-\lambda})
#' }
#'
#' **Variance**: \eqn{m \cdot (\lambda + 1 - m)}, where \eqn{m} is the mean above.
#'
#' **Probability mass function (p.m.f.)**:
#'
#' \deqn{
#' P(X = k) = \frac{f(k; \lambda)}{1 - f(0; \lambda)}
#' }{
#' P(X = k) = f(k; \lambda)/(1 - f(0; \lambda))
#' }
#'
#' where \eqn{f(k; \lambda)} is the p.m.f. of the \code{\link{Poisson}}
#' distribution.
#'
#' **Cumulative distribution function (c.d.f.)**:
#'
#' \deqn{
#' P(X = k) = \frac{F(k; \lambda)}{1 - F(0; \lambda)}
#' }{
#' P(X = k) = F(k; \lambda)/(1 - F(0; \lambda))
#' }
#'
#' where \eqn{F(k; \lambda)} is the c.d.f. of the \code{\link{Poisson}} distribution.
#'
#' **Moment generating function (m.g.f.)**:
#'
#' \deqn{
#' E(e^{tX}) = \frac{1}{1 - e^{-\lambda}} \cdot e^{\lambda (e^t - 1)}
#' }{
#' E(e^(tX)) = 1/(1 - e^{-\lambda}) \cdot e^(\lambda (e^t - 1))
#' }
#'
#' @examples
#' ## set up a zero-truncated Poisson distribution
#' X <- ZTPoisson(lambda = 2.5)
#' X
#'
#' ## standard functions
#' pdf(X, 0:8)
#' cdf(X, 0:8)
#' quantile(X, seq(0, 1, by = 0.25))
#'
#' ## cdf() and quantile() are inverses for each other
#' quantile(X, cdf(X, 3))
#'
#' ## density visualization
#' plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)
#'
#' ## corresponding sample with histogram of empirical frequencies
#' set.seed(0)
#' x <- random(X, 500)
#' hist(x, breaks = -1:max(x) + 0.5)
ZTPoisson <- function(lambda) {
d <- data.frame(lambda = lambda)
class(d) <- c("ZTPoisson", "distribution")
return(d)
}
#' @export
mean.ZTPoisson <- function(x, ...) {
ellipsis::check_dots_used()
m <- x$lambda/ppois(0, lambda = x$lambda, lower.tail = FALSE)
m[x$lambda <= 0] <- 1
setNames(m, names(x))
}
#' @export
variance.ZTPoisson <- function(x, ...) {
ellipsis::check_dots_used()
m <- x$lambda/ppois(0, lambda = x$lambda, lower.tail = FALSE)
m[x$lambda <= 0] <- 1
setNames(m * (1 + x$lambda - m), names(x))
}
#' @export
skewness.ZTPoisson <- function(x, ...) {
ellipsis::check_dots_used()
f <- 1 / ppois(0, lambda = x$lambda, lower.tail = FALSE)
m <- x$lambda * f
s <- sqrt(m * (x$lambda + 1 - m))
rval <- (f * (x$lambda + 3 * x$lambda^2 + x$lambda^3) - 3 * m * s^2 - m^3) / s^3
rval[x$lambda <= 0] <- NaN
setNames(rval, names(x))
}
#' @export
kurtosis.ZTPoisson <- function(x, ...) {
ellipsis::check_dots_used()
f <- 1 / ppois(0, lambda = x$lambda, lower.tail = FALSE)
m <- x$lambda * f
s2 <- m * (x$lambda + 1 - m)
rval <- ( f * (x$lambda + 7 * x$lambda^2 + 6 * x$lambda^3 + x$lambda^4)
- 4 * m * f * (x$lambda + 3 * x$lambda^2 + x$lambda^3)
+ 6 * m^2 * f * (x$lambda + x$lambda^2)
- 3 * m^4 ) / s2^2 - 3
rval[x$lambda <= 0] <- NaN
setNames(rval, names(x))
}
#' Draw a random sample from a zero-truncated Poisson distribution
#'
#' @inherit ZTPoisson examples
#'
#' @param x A `ZTPoisson` object created by a call to [ZTPoisson()].
#' @param n The number of samples to draw. Defaults to `1L`.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param ... Unused. Unevaluated arguments will generate a warning to
#' catch mispellings or other possible errors.
#'
#' @return In case of a single distribution object or `n = 1`, either a numeric
#' vector of length `n` (if `drop = TRUE`, default) or a `matrix` with `n` columns
#' (if `drop = FALSE`).
#' @export
#'
random.ZTPoisson <- function(x, n = 1L, drop = TRUE, ...) {
n <- make_positive_integer(n)
if (n == 0L) return(numeric(0L))
FUN <- function(at, d) rztpois(n = at, lambda = d$lambda)
apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}
#' Evaluate the probability mass function of a zero-truncated Poisson distribution
#'
#' @inherit ZTPoisson examples
#'
#' @param d A `ZTPoisson` object created by a call to [ZTPoisson()].
#' @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{dztpois}}.
#' 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.ZTPoisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) dztpois(x = at, lambda = d$lambda, ...)
apply_dpqr(d = d, FUN = FUN, at = x, type = "density", drop = drop, elementwise = elementwise)
}
#' @rdname pdf.ZTPoisson
#' @export
#'
log_pdf.ZTPoisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) dztpois(x = at, lambda = d$lambda, log = TRUE)
apply_dpqr(d = d, FUN = FUN, at = x, type = "logLik", drop = drop, elementwise = elementwise)
}
#' Evaluate the cumulative distribution function of a zero-truncated Poisson distribution
#'
#' @inherit ZTPoisson examples
#'
#' @param d A `ZTPoisson` object created by a call to [ZTPoisson()].
#' @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{pztpois}}.
#' 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.ZTPoisson <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) pztpois(q = at, lambda = d$lambda, ...)
apply_dpqr(d = d, FUN = FUN, at = x, type = "probability", drop = drop, elementwise = elementwise)
}
#' Determine quantiles of a zero-truncated Poisson distribution
#'
#' `quantile()` is the inverse of `cdf()`.
#'
#' @inherit ZTPoisson examples
#' @inheritParams random.ZTPoisson
#'
#' @param probs A vector of probabilities.
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param elementwise logical. Should each distribution in \code{x} be evaluated
#' at all elements of \code{probs} (\code{elementwise = FALSE}, yielding a matrix)?
#' Or, if \code{x} and \code{probs} have the same length, should the evaluation be
#' done element by element (\code{elementwise = TRUE}, yielding a vector)? The
#' default of \code{NULL} means that \code{elementwise = TRUE} is used if the
#' lengths match and otherwise \code{elementwise = FALSE} is used.
#' @param ... Arguments to be passed to \code{\link{qztpois}}.
#' Unevaluated arguments will generate a warning to catch mispellings or other
#' possible errors.
#'
#' @return In case of a single distribution object, either a numeric
#' vector of length `probs` (if `drop = TRUE`, default) or a `matrix` with
#' `length(probs)` columns (if `drop = FALSE`). In case of a vectorized
#' distribution object, a matrix with `length(probs)` columns containing all
#' possible combinations.
#' @export
#'
quantile.ZTPoisson <- function(x, probs, drop = TRUE, elementwise = NULL, ...) {
FUN <- function(at, d) qztpois(p = at, lambda = d$lambda, ...)
apply_dpqr(d = x, FUN = FUN, at = probs, type = "quantile", drop = drop, elementwise = elementwise)
}
#' Return the support of the zero-truncated Poisson distribution
#'
#' @param d An `ZTPoisson` object created by a call to [ZTPoisson()].
#' @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.ZTPoisson <- function(d, drop = TRUE, ...) {
ellipsis::check_dots_used()
min <- rep(1, length(d))
max <- rep(Inf, length(d))
make_support(min, max, d, drop = drop)
}
#' @exportS3Method
is_discrete.ZTPoisson <- function(d, ...) {
ellipsis::check_dots_used()
setNames(rep.int(TRUE, length(d)), names(d))
}
#' @exportS3Method
is_continuous.ZTPoisson <- function(d, ...) {
ellipsis::check_dots_used()
setNames(rep.int(FALSE, length(d)), names(d))
}
## FIXME: currently no fit_mle.ZTPoisson and suff_stat.ZTPoisson
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.