| GUMBEL {nsRFA} | R Documentation |
GUMBEL provides the link between L-moments of a sample and the two parameter
Gumbel distribution.
f.gumb (x, xi, alfa) F.gumb (x, xi, alfa) invF.gumb (F, xi, alfa) Lmom.gumb (xi, alfa) par.gumb (lambda1, lambda2) rand.gumb (numerosita, xi, alfa)
x |
vector of quantiles |
xi |
vector of gumb location parameters |
alfa |
vector of gumb scale parameters |
F |
vector of probabilities |
lambda1 |
vector of sample means |
lambda2 |
vector of L-variances |
numerosita |
numeric value indicating the length of the vector to be generated |
See http://en.wikipedia.org/wiki/Fisher-Tippett_distribution for an introduction to the Gumbel distribution.
Definition
Parameters (2): xi (location), α (scale).
Range of x: -infty < x < infty.
Probability density function:
f(x) = α^{-1} exp[-(x-xi)/α] exp{- exp[-(x-xi)/α]}
Cumulative distribution function:
F(x) = exp[-exp(-(x-xi)/α)]
Quantile function: x(F) = xi - α log(-log F).
L-moments
λ_1 = xi + α gamma
λ_2 = α log 2
tau_3 = 0.1699 = log(9/8)/ log 2
tau_4 = 0.1504 = (16 log 2 - 10 log 3)/ log 2
Here gamma is Euler's constant, 0.5772...
Parameters
α=λ_2 / log 2
xi = λ_1 - gamma α
f.gumb gives the density f, F.gumb gives the distribution function F, invF.gumb gives
the quantile function x, Lmom.gumb gives the L-moments (λ_1, λ_2, tau_3, tau_4)), par.gumb gives the parameters (xi, alfa), and rand.gumb generates random deviates.
Lmom.gumb and par.gumb accept input as vectors of equal length. In f.gumb, F.gumb, invF.gumb and rand.gumb parameters (xi, alfa) must be atomic.
Alberto Viglione, e-mail: alviglio@tiscali.it.
Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.
rnorm, runif, EXP, GENLOGIS, GENPAR, GEV, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.
data(hydroSIMN) annualflows[1:10,] summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.gumb(ll[1],ll[2]) f.gumb(1800,parameters$xi,parameters$alfa) F.gumb(1800,parameters$xi,parameters$alfa) invF.gumb(0.7686843,parameters$xi,parameters$alfa) Lmom.gumb(parameters$xi,parameters$alfa) rand.gumb(100,parameters$xi,parameters$alfa) Rll <- regionalLmoments(x,fac); Rll parameters <- par.gumb(Rll[1],Rll[2]) Lmom.gumb(parameters$xi,parameters$alfa)