GEV: Three parameter generalized extreme value distribution and...

In nsRFA: Non-Supervised Regional Frequency Analysis

Three parameter generalized extreme value distribution and L-moments

Description

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

Usage

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

Definition

Parameters (3): \xi (location), \alpha (scale), k (shape).

Range of x: -\infty < x \le \xi + \alpha / k if k>0; -\infty < x < \infty if k=0; \xi + \alpha / k \le x < \infty if k<0.

Probability density function:

f(x) = \alpha^{-1} e^{-(1-k)y - e^{-y}}

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

Cumulative distribution function:

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

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

\lambda_1 = \xi + \alpha[1 - \Gamma (1+k)]/k

\lambda_2 = \alpha (1-2^{-k}) \Gamma (1+k)]/k

\tau_3 = 2(1-3^{-k})/(1-2^{-k})-3

\tau_4 = [5(1-4^{-k})-10(1-3^{-k})+6(1-2^{-k})]/(1-2^{-k})

Here \Gamma denote the gamma function

\Gamma (x) = \int_0^{\infty} 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 \le \tau_3 \le 0.5:

k \approx 7.8590 c + 2.9554 c^2

where

c = \frac{2}{3+\tau_3} - \frac{\log 2}{\log 3}

The other parameters are then given by

\alpha = \frac{\lambda_2 k}{(1-2^{-k})\Gamma(1+k)}

\xi = \lambda_1 - \alpha[1 - \Gamma(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 (\lambda_1, \lambda_2, \tau_3, \tau_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.

See Also

rnorm, runif, EXP, GENLOGIS, GENPAR, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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)

nsRFA documentation built on May 29, 2024, 1:14 a.m.