LOGNORM {nsRFA} | R Documentation |
LOGNORM
provides the link between L-moments of a sample and the three parameter
log-normal distribution.
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)
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 |
See http://en.wikipedia.org/wiki/Log-normal_distribution for an introduction to the lognormal 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) = \frac{e^{ky-y^2/2}}{α √{2π}}
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) = Φ(x)
where Φ(x)=\int_{-∞}^x φ(t)dt.
Quantile function: x(F) has no explicit analytical form.
k=0 is the Normal distribution with parameters ξ and alpha.
L-moments
L-moments are defined for all values of k.
λ_1 = ξ + α(1 - e^{k^2/2})/k
λ_2 = α/k e^{k^2/2} [1 - 2 Φ(-k/√{2})]
There are no simple expressions for the L-moment ratios τ_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 τ_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 Φ(-k/√{2})}
ξ = 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.
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, τ_3, τ_4), par.lognorm
gives the parameters (xi
, alfa
, k
), and rand.lognorm
generates random deviates.
For information on the package and the Author, and for all the references, see nsRFA
.
rnorm
, runif
, EXP
, GENLOGIS
, GENPAR
, GEV
, GUMBEL
, KAPPA
, P3
; DISTPLOTS
, GOFmontecarlo
, Lmoments
.
data(hydroSIMN) annualflows summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.lognorm(ll[1],ll[2],ll[4]) f.lognorm(1800,parameters$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)