LOGNORM {nsRFA}R Documentation

Three parameter lognormal distribution and L-moments

Description

LOGNORM provides the link between L-moments of a sample and the three parameter log-normal distribution.

Usage

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

Arguments

x vector of quantiles
xi vector of lognorm location parameters
alfa vector of lognorm scale parameters
k vector of lognorm 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/Log-normal_distribution for an introduction to the lognormal distribution.

Definition

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

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

Probability density function:

f(x) = frac{e^{ky-y^2/2}}{α sqrt{2π}}

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) = Phi(x)

where Phi(x)=int_{-infty}^x phi(t)dt.

Quantile function: x(F) has no explicit analytical form.

k=0 is the Normal distribution with parameters xi and alpha.

L-moments

L-moments are defined for all values of k.

λ_1 = xi + α(1 - e^{k^2/2})/k

λ_2 = α/k e^{k^2/2} [1 - 2 Phi(-k/sqrt{2})]

There are no simple expressions for the L-moment ratios tau_r with r >= 3. Here we use the rational-function approximation given in Hosking and Wallis (1997, p. 199).

Parameters

The shape parameter k is a function of tau_3 alone. No explicit solution is possible. Here we use the approximation given in Hosking and Wallis (1997, p. 199).

Given k, the other parameters are given by

α = frac{λ_2 k e^{-k^2/2}}{1-2 Phi(-k/sqrt{2})}

xi = λ_1 - frac{α}{k} (1 - e^{k^2/2})

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

Value

f.lognorm gives the density f, F.lognorm gives the distribution function F, invFlognorm gives the quantile function x, Lmom.lognorm gives the L-moments (λ_1, λ_2, tau_3, tau_4), par.lognorm gives the parameters (xi, alfa, k), and rand.lognorm 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, KAPPA, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.lognorm(ll[1],ll[2],ll[4])
f.lognorm(1800,parameters$xi,parameters$alfa,parameters$k)
F.lognorm(1800,parameters$xi,parameters$alfa,parameters$k)
invF.lognorm(0.7529877,parameters$xi,parameters$alfa,parameters$k)
Lmom.lognorm(parameters$xi,parameters$alfa,parameters$k)
rand.lognorm(100,parameters$xi,parameters$alfa,parameters$k)

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

[Package nsRFA version 0.6-7 Index]