R/Frechet.R

In distributions3: Probability Distributions as S3 Objects

#' Create a Frechet distribution
#'
#' The Frechet distribution is a special case of the `\link{GEV}` distribution,
#' obtained when the GEV shape parameter \eqn{\xi} is positive.
#' It may be referred to as a type II extreme value distribution.
#'
#' @param location The location (minimum) parameter \eqn{m}.
#'   `location` can be any real number.  Defaults to `0`.
#' @param scale The scale parameter \eqn{s}.
#'   `scale` can be any positive number.  Defaults to `1`.
#' @param shape The shape parameter \eqn{\alpha}.
#'   `shape` can be any positive number.  Defaults to `1`.
#'
#' @return A `Frechet` object.
#' @export
#'
#' @family continuous distributions
#'
#' @details
#'
#'   We recommend reading this documentation on
#'   <https://alexpghayes.github.io/distributions3/>, where the math
#'   will render with additional detail and much greater clarity.
#'
#'   In the following, let \eqn{X} be a Frechet random variable with location
#'   parameter  `location` = \eqn{m}, scale parameter `scale` =
#'   \eqn{s}, and shape parameter `shape` = \eqn{\alpha}.
#'   A Frechet(\eqn{m, s, \alpha}) distribution is equivalent to a
#'   `\link{GEV}`(\eqn{m + s, s / \alpha, 1 / \alpha}) distribution.
#'
#'   **Support**: \eqn{(m, \infty)}.
#'
#'   **Mean**: \eqn{m + s\Gamma(1 - 1/\alpha)}, for \eqn{\alpha > 1}; undefined
#'   otherwise.
#'
#'   **Median**: \eqn{m + s(\ln 2)^{-1/\alpha}}{m + s(\ln 2)^(-1/\alpha)}.
#'
#'   **Variance**:
#'   \eqn{s^2 [\Gamma(1 - 2 / \alpha) - \Gamma(1 - 1 / \alpha)^2]}
#'   for \eqn{\alpha > 2}; undefined otherwise.
#'
#'   **Probability density function (p.d.f)**:
#'
#'   \deqn{f(x) = \alpha s ^ {-1} [(x - m) / s] ^ {-(1 + \alpha)}%
#'         \exp\{-[(x - m) / s] ^ {-\alpha} \}}{%
#'        f(x) = (\alpha / s) [(x - m) / s] ^ [-(1 + \alpha)]%
#'         exp{-[(x - m) / s] ^ (-\alpha)}}
#'   for \eqn{x > m}.  The p.d.f. is 0 for \eqn{x \leq m}{x <= m}.
#'
#'   **Cumulative distribution function (c.d.f)**:
#'
#'   \deqn{F(x) = \exp\{-[(x - m) / s] ^ {-\alpha} \}}{%
#'        F(x) = exp{-[(x - m) / s] ^ (-\alpha)}}
#'   for \eqn{x > m}.  The c.d.f. is 0 for \eqn{x \leq m}{x <= m}.
#'
#' @examples
#'
#' set.seed(27)
#'
#' X <- Frechet(0, 2)
#' X
#'
#' random(X, 10)
#'
#' pdf(X, 0.7)
#' log_pdf(X, 0.7)
#'
#' cdf(X, 0.7)
#' quantile(X, 0.7)
#'
#' cdf(X, quantile(X, 0.7))
#' quantile(X, cdf(X, 0.7))
Frechet <- function(location = 0, scale = 1, shape = 1) {
  if (any(scale <= 0)) {
    stop("scale must be positive")
  }
  if (any(shape <= 0)) {
    stop("shape must be positive")
  }
  stopifnot(
    "parameter lengths do not match (only scalars are allowed to be recycled)" =
      length(location) == length(scale) & length(location) == length(shape) |
        sum(c(length(location) == 1, length(scale) == 1, length(shape) == 1)) >= 2 |
        length(location) == length(scale) & length(shape) == 1 |
        length(location) == length(shape) & length(scale) == 1 |
        length(scale) == length(shape) & length(location) == 1
  )

  d <- data.frame(location = location, scale = scale, shape = shape)
  class(d) <- c("Frechet", "distribution")
  d
}

#' @export
mean.Frechet <- function(x, ...) {
  ellipsis::check_dots_used()
  a <- x$shape
  m <- x$location
  s <- x$scale
  rval <- ifelse(a > 1,
    m + s * gamma(1 - 1 / a),
    Inf
  )
  setNames(rval, names(x))
}

#' @export
variance.Frechet <- function(x, ...) {
  a <- x$shape
  m <- x$location
  s <- x$scale
  rval <- ifelse(a > 2,
    s^2 * (gamma(1 - 2 / a) - gamma(1 - 1 / a)^2),
    Inf
  )
  setNames(rval, names(x))
}

#' @export
skewness.Frechet <- function(x, ...) {
  a <- x$shape
  m <- x$location
  s <- x$scale
  rval <- ifelse(a > 3,
    {
      g1 <- gamma(1 - 1 / a)
      g2 <- gamma(1 - 2 / a)
      g3 <- gamma(1 - 3 / a)
      a <- g3 - 3 * g2 * g1 + 2 * g1^3
      b <- (g2 - g1^2)^1.5
      a / b
    },
    Inf
  )
  setNames(rval, names(x))
}

#' @export
kurtosis.Frechet <- function(x, ...) {
  a <- x$shape
  m <- x$locations
  s <- x$scale
  rval <- ifelse(a > 4,
    {
      g1 <- gamma(1 - 1 / a)
      g2 <- gamma(1 - 2 / a)
      g3 <- gamma(1 - 3 / a)
      g4 <- gamma(1 - 4 / a)
      a <- 4 * g3 * g1 + 3 * g2^2
      b <- (g2 - g1^2)^2
      a / b - 6
    },
    Inf
  )
  setNames(rval, names(x))
}

#' Draw a random sample from a Frechet distribution
#'
#' @inherit Frechet examples
#'
#' @param x A `Frechet` object created by a call to [Frechet()].
#' @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.Frechet <- function(x, n = 1L, drop = TRUE, ...) {
  n <- make_positive_integer(n)
  if (n == 0L) {
    return(numeric(0L))
  }

  # Convert to the GEV parameterisation
  FUN <- function(at, d) {
    loc <- d$location + d$scale
    scale <- d$scale / d$shape
    shape <- 1 / d$shape
    revdbayes::rgev(n = at, loc = loc, scale = scale, shape = shape)
  }
  apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}

#' Evaluate the probability mass function of a Frechet distribution
#'
#' @inherit Frechet examples
#'
#' @param d A `Frechet` object created by a call to [Frechet()].
#' @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[revdbayes]{dgev}}.
#'   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.Frechet <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  # Convert to the GEV parameterisation
  FUN <- function(at, d) {
    loc <- d$location + d$scale
    scale <- d$scale / d$shape
    shape <- 1 / d$shape
    revdbayes::dgev(x = at, loc = loc, scale = scale, shape = shape, ...)
  }
  apply_dpqr(d = d, FUN = FUN, at = x, type = "density", drop = drop, elementwise = elementwise)
}

#' @rdname pdf.Frechet
#' @export
#'
log_pdf.Frechet <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  # Convert to the GEV parameterisation
  FUN <- function(at, d) {
    loc <- d$location + d$scale
    scale <- d$scale / d$shape
    shape <- 1 / d$shape
    revdbayes::dgev(x = at, loc = loc, scale = scale, shape = shape, log = TRUE)
  }
  apply_dpqr(d = d, FUN = FUN, at = x, type = "logLik", drop = drop, elementwise = elementwise)
}

#' Evaluate the cumulative distribution function of a Frechet distribution
#'
#' @inherit Frechet examples
#'
#' @param d A `Frechet` object created by a call to [Frechet()].
#' @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[revdbayes]{pgev}}.
#'   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.Frechet <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  # Convert to the GEV parameterisation
  FUN <- function(at, d) {
    loc <- d$location + d$scale
    scale <- d$scale / d$shape
    shape <- 1 / d$shape
    revdbayes::pgev(q = at, loc = loc, scale = scale, shape = shape, ...)
  }
  apply_dpqr(d = d, FUN = FUN, at = x, type = "probability", drop = drop, elementwise = elementwise)
}

#' Determine quantiles of a Frechet distribution
#'
#' `quantile()` is the inverse of `cdf()`.
#'
#' @inherit Frechet examples
#' @inheritParams random.Frechet
#'
#' @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[revdbayes]{qgev}}.
#'   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.Frechet <- function(x, probs, drop = TRUE, elementwise = NULL, ...) {
  # Convert to the GEV parameterisation
  FUN <- function(at, d) {
    loc <- x$location + x$scale
    scale <- x$scale / x$shape
    shape <- 1 / x$shape
    revdbayes::qgev(p = at, loc = loc, scale = scale, shape = shape, ...)
  }
  apply_dpqr(d = x, FUN = FUN, at = probs, type = "quantile", drop = drop, elementwise = elementwise)
}

#' Return the support of the Frechet distribution
#'
#' @param d An `Frechet` object created by a call to [Frechet()].
#' @param drop logical. Should the result be simplified to a vector if possible?
#' @param ... Currently not used.
#'
#' @return In case of a single distribution object, a numeric vector of length 2
#' with the minimum and maximum value of the support (if `drop = TRUE`, default)
#' or a `matrix` with 2 columns. In case of a vectorized distribution object, a
#' matrix with 2 columns containing all minima and maxima.
#'
#' @export
support.Frechet <- function(d, drop = TRUE, ...) {
  ellipsis::check_dots_used()
  min <- d$location
  max <- rep(Inf, length(d))
  make_support(min, max, d, drop = drop)
}

#' @exportS3Method
is_discrete.Frechet <- function(d, ...) {
  ellipsis::check_dots_used()
  setNames(rep.int(FALSE, length(d)), names(d))
}

#' @exportS3Method
is_continuous.Frechet <- function(d, ...) {
  ellipsis::check_dots_used()
  setNames(rep.int(TRUE, length(d)), names(d))
}