| Type: | Package |
| Title: | The Gumbel Distribution Functions and Gradients |
| Version: | 1.0.1 |
| Date: | 2020-04-07 |
| Maintainer: | Berent Ånund Strømnes Lunde <lundeberent@gmail.com> |
| Description: | Gumbel distribution functions (De Haan L. (2007) <doi:10.1007/0-387-34471-3>) implemented with the techniques of automatic differentiation (Griewank A. (2008) <isbn:978-0-89871-659-7>). With this tool, a user should be able to quickly model extreme events for which the Gumbel distribution is the domain of attraction. The package makes available the density function, the distribution function the quantile function and a random generating function. In addition, it supports gradient functions. The package combines 'Adept' (C++ templated automatic differentiation) (Hogan R. (2017) <doi:10.5281/zenodo.1004730>) and 'Eigen' (templated matrix-vector library) for fast computations of both objective functions and exact gradients. It relies on 'RcppEigen' for easy access to 'Eigen' and bindings to R. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| URL: | https://github.com/blunde1/dgumbel |
| BugReports: | https://github.com/blunde1/dgumbel/issues |
| Encoding: | UTF-8 |
| Imports: | Rcpp (≥ 1.0.2) |
| LinkingTo: | Rcpp, RcppEigen |
| RoxygenNote: | 6.1.1 |
| NeedsCompilation: | yes |
| Packaged: | 2020-04-13 19:11:46 UTC; lunde |
| Author: | Berent Ånund Strømnes Lunde [aut, cre, cph], Robin Hogan [ctb] (Author of included Adept library), The University of Reading [cph] (Copyright holder of included Adept library) |
| Repository: | CRAN |
| Date/Publication: | 2020-04-16 21:00:03 UTC |
The Gumbel Distribution and Derivatives
Description
Density function, distribution function, quantile function and random generation, and their gradient functions for the Gumbel distribution with location and scale parameters.
Usage
dgumbel(x, location=0, scale=1, log = FALSE, grad=FALSE)
pgumbel(q, location=0, scale=1, lower.tail = TRUE, log.p = FALSE, grad=FALSE)
qgumbel(p, location=0, scale=1, lower.tail = TRUE, grad=FALSE)
rgumbel(n, location=0, scale=1)
Arguments
x, q |
Vector of quantiles. |
p |
Vector of probabilities. |
n |
Number of observations. |
location, scale |
Location and scale parameters. |
log, log.p |
Logical; if |
lower.tail |
Logical; if |
grad |
Logical; if |
Details
The Gumbel distribution function with parameters
\code{location} = a and \code{scale} = b is
G(z) = \exp\left\{-\exp\left[-\left(\frac{z-a}{b}\right)
\right]\right\}
for all real z, where b > 0.
Gradients are exact numerical derivatives implemented using automatic differentiation.
dgumbel builds on the Eigen linear algebra library, Adept for automatic differentiation and RcppEigen for bindings to R and loading Eigen.
Value
dgumbel gives the density function, pgumbel gives
the distribution function, qgumbel gives the quantile
function, and rgumbel generates random deviates.
If grad=TRUE is supplied, then the gradient is returned instead of the objective function.
Examples
dgumbel(-1:2, -1, 0.5)
pgumbel(-1:2, -1, 0.5)
qgumbel(seq(0.9, 0.6, -0.1), 2, 0.5)
rgumbel(6, -1, 0.5)
p <- (1:9)/10
pgumbel(qgumbel(p, -1, 2), -1, 2)
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## Random number generation
loc = .5
scale = 3.2
n <- 1000
x <- rgumbel(n, loc, scale)
## The density
hist(x, freq=FALSE)
xs <- sort(x)
fx <- dgumbel(xs, loc, scale)
points(xs,fx, type="l", col=2, lwd=2)
## The distribution
edf <- sapply(xs, function(x){sum(xs<=x)/n})
plot(xs, edf)
Fx <- pgumbel(xs, loc, scale)
points(xs, Fx, type="l", col=2, lwd=2)
## The quantile function
q <- qgumbel(0.6, loc, scale)
polygon(c(xs[xs <= q], q), c(Fx[xs<=q], 0), col=3)
## Negative log likelihood: Objective and gradient
nll <- function(par, data) -sum(dgumbel(data, par[1], par[2], log=TRUE))
dnll <- function(par, data) -rowSums(dgumbel(data, par[1], par[2], log=TRUE, grad=TRUE))
## Parameter estimation
par_start <- c(3,1)
opt <- nlminb(par_start, objective=nll, gradient=dnll, data=x, control = list(trace=5))
opt$convergence
opt$par