| Title: | Bimodal Gumbel Distribution |
|---|---|
| Description: | Bimodal Gumbel distribution. General functions for performing extreme value analysis. |
| Authors: | Pedro C. Brom [aut, cre, cph] (<https://orcid.org/0000-0002-1288-7695>, http://lattes.cnpq.br/0154064396756002), Cira E. G. Otiniano [aut, cph] (<https://orcid.org/0000-0002-5619-0478>, http://lattes.cnpq.br/0307717595727716), Roberto Vila [aut, cph] (<https://orcid.org/0000-0003-1073-0114>, http://lattes.cnpq.br/4978745622057574), Marcelo B. Pereira [aut, cph] (<https://orcid.org/0000-0002-1182-5193>, https://lattes.cnpq.br/9358366674842900) |
| Maintainer: | Pedro C. Brom <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.0.3 |
| Built: | 2025-02-26 03:29:19 UTC |
| Source: | https://github.com/pcbrom/bgumbel |
Bimodal Gumbel: Density Function
dbgumbel(x, mu, sigma, delta)dbgumbel(x, mu, sigma, delta)
x |
Domain. |
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
Vector.
dbgumbel(x = 0, mu = -2, sigma = 1, delta = -1) curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10), ylim= c(0, .4)) integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -5, upper = 0)dbgumbel(x = 0, mu = -2, sigma = 1, delta = -1) curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10), ylim= c(0, .4)) integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -5, upper = 0)
Bimodal Gumbel: Theoretical E(X)
m1bgumbel(mu, sigma, delta)m1bgumbel(mu, sigma, delta)
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
Vector.
(EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1)) # Comparison: Theoretical E(X) and empirical mean x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1) mean(x) abs(EX - mean(x))/abs(EX) # relative error # grid 1 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = 1, delta = y) ) persp(x = mu, y = delta, z = z, theta = -60, ticktype = 'detailed') # grid 2 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) sigmas <- seq(.1, 10, length.out = 20) for (sigma in sigmas) { z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = sigma, delta = y) ) persp(x = mu, y = delta, z = z, theta = -60, zlab = 'E(X)') Sys.sleep(.5) }(EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1)) # Comparison: Theoretical E(X) and empirical mean x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1) mean(x) abs(EX - mean(x))/abs(EX) # relative error # grid 1 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = 1, delta = y) ) persp(x = mu, y = delta, z = z, theta = -60, ticktype = 'detailed') # grid 2 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) sigmas <- seq(.1, 10, length.out = 20) for (sigma in sigmas) { z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m1bgumbel(mu = x, sigma = sigma, delta = y) ) persp(x = mu, y = delta, z = z, theta = -60, zlab = 'E(X)') Sys.sleep(.5) }
Bimodal Gumbel: Theoretical E(X^2)
m2bgumbel(mu, sigma, delta)m2bgumbel(mu, sigma, delta)
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
Vector.
(EX2 <- m2bgumbel(mu = -2, sigma = 1, delta = -1)) # Comparison: Theoretical E(X^2) and empirical second moment x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1) mean(x^2) abs(EX2 - mean(x))/abs(EX2) # relative error # Variance EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1) EX2 - EX^2 var(x) abs(EX2 - EX^2 - var(x))/abs(EX2 - EX^2) # relative error # grid 1 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = 1, delta = y) ) persp(x = mu, y = delta, z = z, theta = -30, ticktype = 'detailed') # grid 2 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) sigmas <- seq(.1, 10, length.out = 20) for (sigma in sigmas) { z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = sigma, delta = y) ) persp(x = mu, y = delta, z = z, theta = -45, zlab = 'E(X^2)') Sys.sleep(.5) }(EX2 <- m2bgumbel(mu = -2, sigma = 1, delta = -1)) # Comparison: Theoretical E(X^2) and empirical second moment x <- rbgumbel(100000, mu = -2, sigma = 1, delta = -1) mean(x^2) abs(EX2 - mean(x))/abs(EX2) # relative error # Variance EX <- m1bgumbel(mu = -2, sigma = 1, delta = -1) EX2 - EX^2 var(x) abs(EX2 - EX^2 - var(x))/abs(EX2 - EX^2) # relative error # grid 1 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = 1, delta = y) ) persp(x = mu, y = delta, z = z, theta = -30, ticktype = 'detailed') # grid 2 mu <- seq(-5, 5, length.out = 100) delta <- seq(-5, 5, length.out = 100) sigmas <- seq(.1, 10, length.out = 20) for (sigma in sigmas) { z <- outer( X <- mu, Y <- delta, FUN = function(x, y) m2bgumbel(mu = x, sigma = sigma, delta = y) ) persp(x = mu, y = delta, z = z, theta = -45, zlab = 'E(X^2)') Sys.sleep(.5) }
Bimodal Gumbel: Maximum Likelihood Estimation
mlebgumbel(data, theta, auto = TRUE)mlebgumbel(data, theta, auto = TRUE)
data |
A numeric vector. |
theta |
Vector. Starting parameter values for the minimization. Default: theta = c(1, 1, 1) |
auto |
Logical. Automatic search for theta initial condition. Default: TRUE |
List.
# Let's generate some values set.seed(123) x <- rbgumbel(1000, mu = -2, sigma = 1, delta = -1) # Look for these references in the figure: hist(x, probability = TRUE) lines(density(x), col = 'blue') abline(v = c(-2.5, -.5), col = 'red') text(x = c(c(-2.5, -.5)), y = c(.05, .05), c('mu\nnear here', 'delta\nnear here')) # Time to fit! # If argument auto = FALSE fit <- mlebgumbel( data = x, # try some values near the region. Format: theta = c(mu, sigma, delta) theta = c(-3, 2, -2), auto = FALSE ) print(fit) # If argument auto = TRUE fit <- mlebgumbel( data = x, auto = TRUE ) print(fit) # Kolmogorov-Smirnov Tests mu.sigma.delta <- fit$estimate$estimate ks.test( x, y = 'pbgumbel', mu = mu.sigma.delta[[1]], sigma = mu.sigma.delta[[2]], delta = mu.sigma.delta[[3]] )# Let's generate some values set.seed(123) x <- rbgumbel(1000, mu = -2, sigma = 1, delta = -1) # Look for these references in the figure: hist(x, probability = TRUE) lines(density(x), col = 'blue') abline(v = c(-2.5, -.5), col = 'red') text(x = c(c(-2.5, -.5)), y = c(.05, .05), c('mu\nnear here', 'delta\nnear here')) # Time to fit! # If argument auto = FALSE fit <- mlebgumbel( data = x, # try some values near the region. Format: theta = c(mu, sigma, delta) theta = c(-3, 2, -2), auto = FALSE ) print(fit) # If argument auto = TRUE fit <- mlebgumbel( data = x, auto = TRUE ) print(fit) # Kolmogorov-Smirnov Tests mu.sigma.delta <- fit$estimate$estimate ks.test( x, y = 'pbgumbel', mu = mu.sigma.delta[[1]], sigma = mu.sigma.delta[[2]], delta = mu.sigma.delta[[3]] )
Bimodal Gumbel: Distribution Function
pbgumbel(q, mu, sigma, delta, lower.tail = TRUE)pbgumbel(q, mu, sigma, delta, lower.tail = TRUE)
q |
Quantile. |
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
lower.tail |
Logical; if TRUE (default), probabilities are P(X <= x) otherwise, P(X > x). |
Vector.
pbgumbel(0, mu = -2, sigma = 1, delta = -1) integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -Inf, upper = 0) pbgumbel(0, mu = -2, sigma = 1, delta = -1, lower.tail = FALSE) curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10))pbgumbel(0, mu = -2, sigma = 1, delta = -1) integrate(dbgumbel, mu = -2, sigma = 1, delta = -1, lower = -Inf, upper = 0) pbgumbel(0, mu = -2, sigma = 1, delta = -1, lower.tail = FALSE) curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 10))
Bimodal Gumbel: Quantile Function
qbgumbel(p, mu, sigma, delta, initial = -10, final = 10)qbgumbel(p, mu, sigma, delta, initial = -10, final = 10)
p |
Probability. |
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
initial |
Starting point of range in desired quantile. |
final |
Starting point of range in desired quantile. |
Vector.
# It is recommended to set up a pbgumbel # graph to see the starting and ending # range of the desired quantile. curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 5)) (value <- qbgumbel(.25, mu = -2, sigma = 1, delta = -1, initial = -4, final = -2)) pbgumbel(value, mu = -2, sigma = 1, delta = -1)# It is recommended to set up a pbgumbel # graph to see the starting and ending # range of the desired quantile. curve(pbgumbel(x, mu = -2, sigma = 1, delta = -1), xlim = c(-5, 5)) (value <- qbgumbel(.25, mu = -2, sigma = 1, delta = -1, initial = -4, final = -2)) pbgumbel(value, mu = -2, sigma = 1, delta = -1)
Bimodal Gumbel: Pseudo-Random Numbers Generator
rbgumbel(n, mu, sigma, delta)rbgumbel(n, mu, sigma, delta)
n |
Number of observations. If length(n) > 1, the length is taken to be the number required. |
mu |
First location parameter. |
sigma |
Scale parameter. |
delta |
Second location parameter. |
A matrix nx1.
x <- rbgumbel(40000, mu = -2, sigma = 1, delta = -1) hist(x, probability = TRUE) curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), add = TRUE, col = 'blue') lines(density(x), col = 'red')x <- rbgumbel(40000, mu = -2, sigma = 1, delta = -1) hist(x, probability = TRUE) curve(dbgumbel(x, mu = -2, sigma = 1, delta = -1), add = TRUE, col = 'blue') lines(density(x), col = 'red')