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| #' The log-normal distribution | ||
| #' | ||
| #' \lifecycle{stable} | ||
| #' | ||
| #' The log-normal distribution is a commonly used transformation of the Normal | ||
| #' distribution. If \eqn{X} follows a log-normal distribution, then \eqn{\ln{X}} | ||
| #' would be characteristed by a Normal distribution. | ||
| #' | ||
| #' @param mu The mean (location parameter) of the distribution, which is the | ||
| #' mean of the associated Normal distribution. Can be any real number. | ||
| #' @param sigma The standard deviation (scale parameter) of the distribution. | ||
| #' Can be any positive number. | ||
| #' | ||
| #' @details | ||
| #' | ||
| #' We recommend reading this documentation on | ||
| #' <https://pkg.mitchelloharawild.com/distributional/>, where the math | ||
| #' will render nicely. | ||
| #' | ||
| #' In the following, let \eqn{Y} be a Normal random variable with mean | ||
| #' `mu` = \eqn{\mu} and standard deviation `sigma` = \eqn{\sigma}. The | ||
| #' log-normal distribution \eqn{X = exp(Y)} is characterised by: | ||
| #' | ||
| #' **Support**: \eqn{R+}, the set of all real numbers greater than or equal to 0. | ||
| #' | ||
| #' **Mean**: \eqn{e^(\mu + \sigma^2/2} | ||
| #' | ||
| #' **Variance**: \eqn{(e^(\sigma^2)-1) e^(2\mu + \sigma^2} | ||
| #' | ||
| #' **Probability density function (p.d.f)**: | ||
| #' | ||
| #' \deqn{ | ||
| #' f(x) = \frac{1}{x\sqrt{2 \pi \sigma^2}} e^{-(\ln{x} - \mu)^2 / 2 \sigma^2} | ||
| #' }{ | ||
| #' f(x) = 1 / (x * sqrt(2 \pi \sigma^2)) exp(-(log(x) - \mu)^2 / (2 \sigma^2)) | ||
| #' } | ||
| #' | ||
| #' **Cumulative distribution function (c.d.f)**: | ||
| #' | ||
| #' The cumulative distribution function has the form | ||
| #' | ||
| #' \deqn{ | ||
| #' F(x) = \Phi((\ln{x} - \mu)/\sigma) | ||
| #' }{ | ||
| #' F(x) = Phi((log(x) - \mu)/\sigma) | ||
| #' } | ||
| #' | ||
| #' Where \eqn{Phi}{Phi} is the CDF of a standard Normal distribution, N(0,1). | ||
| #' | ||
| #' @seealso [stats::Lognormal] | ||
| #' | ||
| #' @examples | ||
| #' dist <- dist_lognormal(mu = 1:5, sigma = 0.1) | ||
| #' | ||
| #' dist | ||
| #' mean(dist) | ||
| #' variance(dist) | ||
| #' skewness(dist) | ||
| #' kurtosis(dist) | ||
| #' | ||
| #' generate(dist, 10) | ||
| #' | ||
| #' density(dist, 2) | ||
| #' density(dist, 2, log = TRUE) | ||
| #' | ||
| #' cdf(dist, 4) | ||
| #' | ||
| #' quantile(dist, 0.7) | ||
| #' | ||
| #' # A log-normal distribution X is exp(Y), where Y is a Normal distribution of | ||
| #' # the same parameters. So log(X) will produce the Normal distribution Y. | ||
| #' log(dist) | ||
| #' @export | ||
| dist_lognormal <- function(mu = 0, sigma = 1){ | ||
| mu <- vec_cast(mu, double()) | ||
| sigma <- vec_cast(sigma, double()) | ||
| if(any(sigma[!is.na(sigma)] < 0)){ | ||
| abort("Standard deviation of a log-normal distribution must be non-negative") | ||
| } | ||
| new_dist(mu = mu, sigma = sigma, class = "dist_lognormal") | ||
| } | ||
|
|
||
| #' @export | ||
| format.dist_lognormal <- function(x, digits = 2, ...){ | ||
| sprintf( | ||
| "lN(%s, %s)", | ||
| format(x[["mu"]], digits = digits, ...), | ||
| format(x[["sigma"]]^2, digits = digits, ...) | ||
| ) | ||
| } | ||
|
|
||
| #' @export | ||
| density.dist_lognormal <- function(x, at, ...){ | ||
| stats::dlnorm(at, x[["mu"]], x[["sigma"]]) | ||
| } | ||
|
|
||
| #' @export | ||
| log_density.dist_lognormal <- function(x, at, ...){ | ||
| stats::dlnorm(at, x[["mu"]], x[["sigma"]], log = TRUE) | ||
| } | ||
|
|
||
| #' @export | ||
| quantile.dist_lognormal <- function(x, p, ...){ | ||
| stats::qlnorm(p, x[["mu"]], x[["sigma"]], ...) | ||
| } | ||
| #' @export | ||
| log_quantile.dist_lognormal <- function(x, p, ...){ | ||
| stats::qlnorm(p, x[["mu"]], x[["sigma"]], ..., log.p = TRUE) | ||
| } | ||
|
|
||
| #' @export | ||
| cdf.dist_lognormal <- function(x, q, ...){ | ||
| stats::plnorm(q, x[["mu"]], x[["sigma"]], ...) | ||
| } | ||
| #' @export | ||
| log_cdf.dist_lognormal <- function(x, q, ...){ | ||
| stats::plnorm(q, x[["mu"]], x[["sigma"]], ..., log.p = TRUE) | ||
| } | ||
|
|
||
| #' @export | ||
| generate.dist_lognormal <- function(x, times, ...){ | ||
| stats::rlnorm(times, x[["mu"]], x[["sigma"]]) | ||
| } | ||
|
|
||
| #' @export | ||
| mean.dist_lognormal <- function(x, ...){ | ||
| exp(x[["mu"]] + x[["sigma"]]^2/2) | ||
| } | ||
|
|
||
| #' @export | ||
| covariance.dist_lognormal <- function(x, ...){ | ||
| s2 <- x[["sigma"]]^2 | ||
| (exp(s2)-1)*exp(2*x[["mu"]] + s2) | ||
| } | ||
|
|
||
| #' @export | ||
| skewness.dist_lognormal <- function(x, ...) { | ||
| es2 <- exp(x[["sigma"]]^2) | ||
| (es2+2)*sqrt(es2-1) | ||
| } | ||
|
|
||
| #' @export | ||
| kurtosis.dist_lognormal <- function(x, ...) { | ||
| s2 <- x[["sigma"]]^2 | ||
| exp(4*s2) + 2*exp(3*s2) + 3*exp(2*s2) - 6 | ||
| } | ||
|
|
||
| #' @method Math dist_lognormal | ||
| #' @export | ||
| Math.dist_lognormal <- function(x, ...) { | ||
| # Shortcut to get Normal distribution from log-normal. | ||
| if(.Generic == "log") return(vec_data(dist_normal(x[["mu"]], x[["sigma"]]))[[1]]) | ||
| NextMethod() | ||
| } |
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