# GEV: Three parameter generalized extreme value distribution and... In nsRFA: Non-Supervised Regional Frequency Analysis

## Description

GEV provides the link between L-moments of a sample and the three parameter generalized extreme value distribution.

## Usage

 1 2 3 4 5 6 f.GEV (x, xi, alfa, k) F.GEV (x, xi, alfa, k) invF.GEV (F, xi, alfa, k) Lmom.GEV (xi, alfa, k) par.GEV (lambda1, lambda2, tau3) rand.GEV (numerosita, xi, alfa, k) 

## Arguments

 x vector of quantiles xi vector of GEV location parameters alfa vector of GEV scale parameters k vector of GEV shape parameters F vector of probabilities lambda1 vector of sample means lambda2 vector of L-variances tau3 vector of L-CA (or L-skewness) numerosita numeric value indicating the length of the vector to be generated

## Details

See http://en.wikipedia.org/wiki/Generalized_extreme_value_distribution for an introduction to the GEV distribution.

Definition

Parameters (3): ξ (location), α (scale), k (shape).

Range of x: -∞ < x ≤ ξ + α / k if k>0; -∞ < x < ∞ if k=0; ξ + α / k ≤ x < ∞ if k<0.

Probability density function:

f(x) = α^{-1} e^{-(1-k)y - e^{-y}}

where y = -k^{-1}\log\{1 - k(x - ξ)/α\} if k \ne 0, y = (x-ξ)/α if k=0.

Cumulative distribution function:

F(x) = e^{-e^{-y}}

Quantile function: x(F) = ξ + α[1-(-\log F)^k]/k if k \ne 0, x(F) = ξ - α \log(-\log F) if k=0.

k=0 is the Gumbel distribution; k=1 is the reverse exponential distribution.

L-moments

L-moments are defined for k>-1.

λ_1 = ξ + α[1 - Γ (1+k)]/k

λ_2 = α (1-2^{-k}) Γ (1+k)]/k

τ_3 = 2(1-3^{-k})/(1-2^{-k})-3

τ_4 = [5(1-4^{-k})-10(1-3^{-k})+6(1-2^{-k})]/(1-2^{-k})

Here Γ denote the gamma function

Γ (x) = \int_0^{∞} t^{x-1} e^{-t} dt

Parameters

To estimate k, no explicit solution is possible, but the following approximation has accurancy better than 9 \times 10^{-4} for -0.5 ≤ τ_3 ≤ 0.5:

k \approx 7.8590 c + 2.9554 c^2

where

c = \frac{2}{3+τ_3} - \frac{\log 2}{\log 3}

The other parameters are then given by

α = \frac{λ_2 k}{(1-2^{-k})Γ(1+k)}

ξ = λ_1 - α[1 - Γ(1+k)]/k

Lmom.GEV and par.GEV accept input as vectors of equal length. In f.GEV, F.GEV, invF.GEV and rand.GEV parameters (xi, alfa, k) must be atomic.

## Value

f.GEV gives the density f, F.GEV gives the distribution function F, invF.GEV gives the quantile function x, Lmom.GEV gives the L-moments (λ_1, λ_2, τ_3, τ_4), par.GEV gives the parameters (xi, alfa, k), and rand.GEV generates random deviates.

## Note

For information on the package and the Author, and for all the references, see nsRFA.

rnorm, runif, EXP, GENLOGIS, GENPAR, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 data(hydroSIMN) annualflows summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.GEV(ll[1],ll[2],ll[4]) f.GEV(1800,parameters$xi,parameters$alfa,parameters$k) F.GEV(1800,parameters$xi,parameters$alfa,parameters$k) invF.GEV(0.7518357,parameters$xi,parameters$alfa,parameters$k) Lmom.GEV(parameters$xi,parameters$alfa,parameters$k) rand.GEV(100,parameters$xi,parameters$alfa,parameters$k) Rll <- regionalLmoments(x,fac); Rll parameters <- par.GEV(Rll[1],Rll[2],Rll[4]) Lmom.GEV(parameters$xi,parameters$alfa,parameters\$k)