| GENPAR {nsRFA} | R Documentation |
GENPAR provides the link between L-moments of a sample and the three parameter
generalized Pareto distribution.
f.genpar (x, xi, alfa, k) F.genpar (x, xi, alfa, k) invF.genpar (F, xi, alfa, k) Lmom.genpar (xi, alfa, k) par.genpar (lambda1, lambda2, tau3) rand.genpar (numerosita, xi, alfa, k)
x |
vector of quantiles |
xi |
vector of genpar location parameters |
alfa |
vector of genpar scale parameters |
k |
vector of genpar 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 |
See http://en.wikipedia.org/wiki/Pareto_distribution for an introduction to the Pareto distribution.
Definition
Parameters (3): xi (location), α (scale), k (shape).
Range of x: xi < x <= xi + α / k if k>0; xi <= x < infty if k <= 0.
Probability density function:
f(x) = α^{-1} e^{-(1-k)y}
where y = -k^{-1}log{1 - k(x - xi)/α} if k ne 0, y = (x-xi)/α if k=0.
Cumulative distribution function:
F(x) = 1-e^{-y}
Quantile function: x(F) = xi + α[1-(1-F)^k]/k if k ne 0, x(F) = xi - α log(1-F) if k=0.
k=0 is the exponential distribution; k=1 is the uniform distribution on the interval xi < x <= xi + α.
L-moments
L-moments are defined for k>-1.
λ_1 = xi + α/(1+k)]
λ_2 = α/[(1+k)(2+k)]
tau_3 = (1-k)/(3+k)
tau_4 = (1-k)(2-k)/[(3+k)(4+k)]
The relation between tau_3 and tau_4 is given by
tau_4 = frac{tau_3 (1 + 5 tau_3)}{5+tau_3}
Parameters
If xi is known, k=(λ_1 - xi)/λ_2 - 2 and α=(1+k)(λ_1 - xi); if xi is unknown, k=(1 - 3 tau_3)/(1 + tau_3), α=(1+k)(2+k)λ_2 and xi=λ_1 - (2+k)λ_2.
f.genpar gives the density f, F.genpar gives the distribution function F, invF.genpar gives
the quantile function x, Lmom.genpar gives the L-moments (λ_1, λ_2, tau_3, tau_4), par.genpar
gives the parameters (xi, alfa, k), and rand.genpar generates random deviates.
Lmom.genpar and par.genpar accept input as vectors of equal length. In f.genpar, F.genpar, invF.genpar and rand.genpar parameters (xi, alfa, k) 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, GEV, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.
data(hydroSIMN) annualflows summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.genpar(ll[1],ll[2],ll[4]) f.genpar(1800,parameters$xi,parameters$alfa,parameters$k) F.genpar(1800,parameters$xi,parameters$alfa,parameters$k) invF.genpar(0.7161775,parameters$xi,parameters$alfa,parameters$k) Lmom.genpar(parameters$xi,parameters$alfa,parameters$k) rand.genpar(100,parameters$xi,parameters$alfa,parameters$k) Rll <- regionalLmoments(x,fac); Rll parameters <- par.genpar(Rll[1],Rll[2],Rll[4]) Lmom.genpar(parameters$xi,parameters$alfa,parameters$k)