LOGNORM | 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 https://en.wikipedia.org/wiki/Log-normal_distribution for an introduction to the lognormal 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) = \frac{e^{ky-y^2/2}}{\alpha \sqrt{2\pi}}
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) = \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
.
\lambda_1 = \xi + \alpha(1 - e^{k^2/2})/k
\lambda_2 = \alpha/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 \ge 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
\alpha = \frac{\lambda_2 k e^{-k^2/2}}{1-2 \Phi(-k/\sqrt{2})}
\xi = \lambda_1 - \frac{\alpha}{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 (\lambda_1
, \lambda_2
, \tau_3
, \tau_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)
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