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 0.6-9 Index]