R/LogNormal.R

In distributions3: Probability Distributions as S3 Objects

#' Create a LogNormal distribution
#'
#' A random variable created by exponentiating a [Normal()]
#' distribution. Taking the log of LogNormal data returns in
#' [Normal()] data.
#'
#' @param log_mu The location parameter, written \eqn{\mu} in textbooks.
#'   Can be any real number. Defaults to `0`.
#' @param log_sigma The scale parameter, written \eqn{\sigma} in textbooks.
#'   Can be any positive real number. Defaults to `1`.
#'
#' @return A `LogNormal` 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 LogNormal random variable with
#'   success probability `p` = \eqn{p}.
#'
#'   **Support**: \eqn{R^+}
#'
#'   **Mean**: \eqn{\exp(\mu + \sigma^2/2)}
#'
#'   **Variance**: \eqn{[\exp(\sigma^2)-1]\exp(2\mu+\sigma^2)}
#'
#'   **Probability density function (p.d.f)**:
#'
#'   \deqn{
#'     f(x) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp \left(-\frac{(\log x - \mu)^2}{2 \sigma^2} \right)
#'   }{
#'     f(x) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp (-\frac{(\log x - \mu)^2}{2 \sigma^2})
#'   }
#'
#'   **Cumulative distribution function (c.d.f)**:
#'
#'   \deqn{F(x) = \frac{1}{2} + \frac{1}{2\sqrt{pi}}\int_{-x}^x e^{-t^2} dt}
#'
#'   **Moment generating function (m.g.f)**:
#'   Undefined.
#'
#'
#' @examples
#'
#' set.seed(27)
#'
#' X <- LogNormal(0.3, 2)
#' X
#'
#' random(X, 10)
#'
#' pdf(X, 2)
#' log_pdf(X, 2)
#'
#' cdf(X, 4)
#' quantile(X, 0.7)
LogNormal <- function(log_mu = 0, log_sigma = 1) {
  stopifnot(
    "parameter lengths do not match (only scalars are allowed to be recycled)" =
      length(log_mu) == length(log_sigma) | length(log_mu) == 1 | length(log_sigma) == 1
  )
  d <- data.frame(log_mu = log_mu, log_sigma = log_sigma)
  class(d) <- c("LogNormal", "distribution")
  d
}

#' @export
mean.LogNormal <- function(x, ...) {
  ellipsis::check_dots_used()
  mu <- x$log_mu
  sigma <- x$log_sigma
  rval <- exp(mu + sigma^2 / 2)
  setNames(rval, names(x))
}

#' @export
variance.LogNormal <- function(x, ...) {
  mu <- x$log_mu
  sigma <- x$log_sigma
  rval <- (exp(sigma^2) - 1) * exp(2 * mu + sigma^2)
  setNames(rval, names(x))
}

#' @export
skewness.LogNormal <- function(x, ...) {
  mu <- x$log_mu
  sigma <- x$log_sigma
  rval <- (exp(sigma^2) + 2) * sqrt(exp(sigma^2) - 1)
  setNames(rval, names(x))
}

#' @export
kurtosis.LogNormal <- function(x, ...) {
  mu <- x$log_mu
  sigma <- x$log_sigma
  rval <- exp(4 * sigma^2) + 2 * exp(3 * sigma^2) + 3 * exp(2 * sigma^2) - 6
  setNames(rval, names(x))
}

#' Draw a random sample from a LogNormal distribution
#'
#' @inherit LogNormal examples
#'
#' @param x A `LogNormal` object created by a call to [LogNormal()].
#' @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.
#'
#' @family LogNormal distribution
#'
#' @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.LogNormal <- function(x, n = 1L, drop = TRUE, ...) {
  n <- make_positive_integer(n)
  if (n == 0L) {
    return(numeric(0L))
  }
  FUN <- function(at, d) rlnorm(n = at, meanlog = d$log_mu, sdlog = d$log_sigma)
  apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}

#' Evaluate the probability mass function of a LogNormal distribution
#'
#' Please see the documentation of [LogNormal()] for some properties
#' of the LogNormal distribution, as well as extensive examples
#' showing to how calculate p-values and confidence intervals.
#'
#' @inherit LogNormal examples
#'
#' @param d A `LogNormal` object created by a call to [LogNormal()].
#' @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]{dlnorm}}.
#'   Unevaluated arguments will generate a warning to catch mispellings or other
#'   possible errors.
#'
#' @family LogNormal distribution
#'
#' @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.LogNormal <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  FUN <- function(at, d) dlnorm(x = at, meanlog = d$log_mu, sdlog = d$log_sigma, ...)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "density", drop = drop, elementwise = elementwise)
}

#' @rdname pdf.LogNormal
#' @export
log_pdf.LogNormal <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  FUN <- function(at, d) dlnorm(x = at, meanlog = d$log_mu, sdlog = d$log_sigma, log = TRUE)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "logLik", drop = drop, elementwise = elementwise)
}

#' Evaluate the cumulative distribution function of a LogNormal distribution
#'
#' @inherit LogNormal examples
#'
#' @param d A `LogNormal` object created by a call to [LogNormal()].
#' @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]{plnorm}}.
#'   Unevaluated arguments will generate a warning to catch mispellings or other
#'   possible errors.
#'
#' @family LogNormal distribution
#'
#' @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.LogNormal <- function(d, x, drop = TRUE, elementwise = NULL, ...) {
  FUN <- function(at, d) plnorm(q = at, meanlog = d$log_mu, sdlog = d$log_sigma, ...)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "probability", drop = drop, elementwise = elementwise)
}

#' Determine quantiles of a LogNormal distribution
#'
#' @inherit LogNormal examples
#' @inheritParams random.LogNormal
#'
#' @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]{qlnorm}}.
#'   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
#'
#' @family LogNormal distribution
#'
quantile.LogNormal <- function(x, probs, drop = TRUE, elementwise = NULL, ...) {
  FUN <- function(at, d) qlnorm(p = at, meanlog = d$log_mu, sdlog = d$log_sigma, ...)
  apply_dpqr(d = x, FUN = FUN, at = probs, type = "quantile", drop = drop, elementwise = elementwise)
}

#' Fit a Log Normal distribution to data
#'
#' @param d A `LogNormal` object created by a call to [LogNormal()].
#' @param x A vector of data.
#' @param ... Unused.
#'
#' @family LogNormal distribution
#'
#' @return A `LogNormal` object.
#' @export
#'
fit_mle.LogNormal <- function(d, x, ...) {
  ss <- suff_stat(d, x, ...)
  LogNormal(ss$mu, ss$sigma)
}

#' Compute the sufficient statistics for a Log-normal distribution from data
#'
#' @inheritParams fit_mle.LogNormal
#'
#' @return A named list of the sufficient statistics of the normal distribution:
#'
#'   - `mu`: The sample mean of the log of the data.
#'   - `sigma`: The sample standard deviation of the log of the data.
#'   - `samples`: The number of samples in the data.
#'
#' @export
#'
suff_stat.LogNormal <- function(d, x, ...) {
  valid_x <- x > 0
  if (any(!valid_x)) stop("`x` must be a vector of positive real numbers")
  log_x <- log(x)
  list(mu = mean(log_x), sigma = sd(log_x), samples = length(x))
}

#' Return the support of the LogNormal distribution
#'
#' @param d An `LogNormal` object created by a call to [LogNormal()].
#' @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.LogNormal <- 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.LogNormal <- function(d, ...) {
  ellipsis::check_dots_used()
  setNames(rep.int(FALSE, length(d)), names(d))
}

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