| % File src/library/stats/man/Logistic.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Logistic} |
| \alias{Logistic} |
| \alias{dlogis} |
| \alias{plogis} |
| \alias{qlogis} |
| \alias{rlogis} |
| \title{The Logistic Distribution} |
| \concept{logit} |
| \concept{sigmoid} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the logistic distribution with parameters |
| \code{location} and \code{scale}. |
| } |
| \usage{ |
| dlogis(x, location = 0, scale = 1, log = FALSE) |
| plogis(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) |
| qlogis(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) |
| rlogis(n, location = 0, scale = 1) |
| } |
| \arguments{ |
| \item{x, q}{vector of quantiles.} |
| \item{p}{vector of probabilities.} |
| \item{n}{number of observations. If \code{length(n) > 1}, the length |
| is taken to be the number required.} |
| \item{location, scale}{location and scale parameters.} |
| \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).} |
| \item{lower.tail}{logical; if TRUE (default), probabilities are |
| \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.} |
| } |
| |
| \value{ |
| \code{dlogis} gives the density, |
| \code{plogis} gives the distribution function, |
| \code{qlogis} gives the quantile function, and |
| \code{rlogis} generates random deviates. |
| |
| The length of the result is determined by \code{n} for |
| \code{rlogis}, and is the maximum of the lengths of the |
| numerical arguments for the other functions. |
| |
| The numerical arguments other than \code{n} are recycled to the |
| length of the result. Only the first elements of the logical |
| arguments are used. |
| } |
| \details{ |
| If \code{location} or \code{scale} are omitted, they assume the |
| default values of \code{0} and \code{1} respectively. |
| |
| The Logistic distribution with \code{location} \eqn{= \mu}{= m} and |
| \code{scale} \eqn{= \sigma}{= s} has distribution function |
| \deqn{ |
| F(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}% |
| }{F(x) = 1 / (1 + exp(-(x-m)/s))} and density |
| \deqn{ |
| f(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}% |
| }{f(x) = 1/s exp((x-m)/s) (1 + exp((x-m)/s))^-2.} |
| |
| It is a long-tailed distribution with mean \eqn{\mu}{m} and variance |
| \eqn{\pi^2/3 \sigma^2}{\pi^2 /3 s^2}. |
| } |
| \note{ |
| \code{qlogis(p)} is the same as the well known \sQuote{\emph{logit}} |
| function, \eqn{logit(p) = \log p/(1-p)}{logit(p) = log(p/(1-p))}, |
| and \code{plogis(x)} has consequently been called the \sQuote{inverse logit}. |
| |
| The distribution function is a rescaled hyperbolic tangent, |
| \code{plogis(x) == (1+ \link{tanh}(x/2))/2}, and it is called a |
| \emph{sigmoid function} in contexts such as neural networks. |
| } |
| \source{ |
| \code{[dpq]logis} are calculated directly from the definitions. |
| |
| \code{rlogis} uses inversion. |
| } |
| \references{ |
| Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) |
| \emph{The New S Language}. |
| Wadsworth & Brooks/Cole. |
| |
| Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) |
| \emph{Continuous Univariate Distributions}, volume 2, chapter 23. |
| Wiley, New York. |
| } |
| \seealso{ |
| \link{Distributions} for other standard distributions. |
| } |
| \examples{ |
| var(rlogis(4000, 0, scale = 5)) # approximately (+/- 3) |
| pi^2/3 * 5^2 |
| } |
| \keyword{distribution} |