KAPPA {nsRFA}R Documentation

Four parameter kappa distribution and L-moments

Description

KAPPA provides the link between L-moments of a sample and the four parameter kappa distribution.

Usage

f.kappa (x, xi, alfa, k, h)
F.kappa (x, xi, alfa, k, h)
invF.kappa (F, xi, alfa, k, h)
Lmom.kappa (xi, alfa, k, h)
par.kappa (lambda1, lambda2, tau3, tau4)
rand.kappa (numerosita, xi, alfa, k, h)

Arguments

x vector of quantiles
xi vector of kappa location parameters
alfa vector of kappa scale parameters
k vector of kappa third parameters
h vector of kappa fourth parameters
F vector of probabilities
lambda1 vector of sample means
lambda2 vector of L-variances
tau3 vector of L-CA (or L-skewness)
tau4 vector of L-kurtosis
numerosita numeric value indicating the length of the vector to be generated

Details

Definition

Parameters (4): xi (location), α (scale), k, h.

Range of x: upper bound is xi + α/k if k>0, inf if k <= 0; lower bound is xi + α(1-h^(-k))/k if h>0, xi + α/k if h <= 0 and k<0 and -inf if h <= 0 and k >= 0

Probability density function:

f(x) = α^(-1) [1 - k (x - xi)/α]^(1/k-1) [F(x)]^(1-h)

Cumulative distribution function:

F(x) = {1 - h[1 - k(x - xi)/α]^(1/k)}^(1/h)

Quantile function:

x(F) = xi + α/k [1 - ((1-F^h)/h)^k]

h=-1 is the generalized logistic distribution; h=0 is the generalized eztreme value distribution; h=1 is the generalized Pareto distribution.

L-moments

L-moments are defined for h >= 0 and k>-1, or if h<0 and -1<k<-1/h.

λ1 = xi + α(1 - g1)/k

λ2 = α(g1 - g2)/k

tau3 = (-g1 + 3g2 - 2g3)/(g1 - g2)

tau4 = (-g1 + 6g2 - 10g3 + 5g4)/(g1 - g2)

where gr = (r Γ(1+k)Γ(r/h))/(h^(1+k) Γ(1+k+r/h)) if h>0; gr = (rΓ(1+k)Γ(-k-r/h))/((-h)^(1+k)Γ(1-r/h)) if h<0;

Here Γ denote the gamma function

Γ(x) = integral[from 0 to inf] t^(x-1) e^(-t) dt

Parameters

There are no simple expressions for the parameters in terms of the L-moments. However they can be obtained with a numerical algorithm considering the formulations of tau3 and tau4 in terms of k and h. Here we use the function optim to minimize (t3-tau3)^2 + (t4-tau4)^2 where t3 and t4 are the sample L-moment ratios.

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

Value

f.kappa gives the density f, F.kappa gives the distribution function F, invFkappa gives the quantile function x, Lmom.kappa gives the L-moments (λ_1, λ_2, tau_3, tau_4), par.kappa gives the parameters (xi, alfa, k, h), and rand.kappa 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, GEV, GUMBEL, LOGNORM, P3; optim, DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

data(hydroSIMN)
annualflows
summary(annualflows)
x <- annualflows["dato"][,]
fac <- factor(annualflows["cod"][,])
split(x,fac)

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.kappa(ll[1],ll[2],ll[4],ll[5])
f.kappa(1800,parameters$xi,parameters$alfa,parameters$k,parameters$h)
F.kappa(1800,parameters$xi,parameters$alfa,parameters$k,parameters$h)
invF.kappa(0.771088,parameters$xi,parameters$alfa,parameters$k,parameters$h)
Lmom.kappa(parameters$xi,parameters$alfa,parameters$k,parameters$h)
rand.kappa(100,parameters$xi,parameters$alfa,parameters$k,parameters$h)

Rll <- regionalLmoments(x,fac); Rll
parameters <- par.kappa(Rll[1],Rll[2],Rll[4],Rll[5])
Lmom.kappa(parameters$xi,parameters$alfa,parameters$k,parameters$h)

[Package nsRFA version 0.6-7 Index]