GENLOGIS {nsRFA} | R Documentation

## Three parameter generalized logistic distribution and L-moments

### Description

`GENLOGIS` provides the link between L-moments of a sample and the three parameter generalized logistic distribution.

### Usage

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

### Arguments

`x` | vector of quantiles
`xi` | vector of genlogis location parameters
`alfa` | vector of genlogis scale parameters
`k` | vector of genlogis 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/Logistic_distribution for an introduction to the Logistic 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{α^{-1} e^{-(1-k)y}}{(1+e^{-y})^2}

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) = 1/(1+e^{-y})

Quantile function: x(F) = xi + α[1-{(1-F)/F}^k]/k if k ne 0, x(F) = xi - α log{(1-F)/F} if k=0.

k=0 is the logistic distribution.

L-moments

L-moments are defined for -1<k<1.

λ_1 = xi + α[1/k - π / sin (k π)]

λ_2 = α k π / sin (k π)

tau_3 = -k

tau_4 = (1+5 k^2)/6

Parameters

k=-tau_3, α = frac{λ_2 sin (k π)}{k π}, xi = λ_1 - α (frac{1}{k} - frac{π}{sin (k π)}).

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

### Value

`f.genlogis` gives the density f, `F.genlogis` gives the distribution function F, `invF.genlogis` gives the quantile function x, `Lmom.genlogis` gives the L-moments (λ_1, λ_2, tau_3, tau_4), `par.genlogis` gives the parameters (`xi`, `alfa`, `k`), and `rand.genlogis` generates random deviates.

### Note

For information on the package and the Author, and for all the references, see `nsRFA`.

### See Also

`rnorm`, `runif`, `EXP`, `GENPAR`, `GEV`, `GUMBEL`, `KAPPA`, `LOGNORM`, `P3`; `DISTPLOTS`, `GOFmontecarlo`, `Lmoments`.

### Examples

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

    camp <- split(x,fac)$"45"
    ll <- Lmoments(camp)
    parameters <- par.genlogis(ll[1],ll[2],ll[4])
    f.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
    F.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
    invF.genlogis(0.7697433,parameters$xi,parameters$alfa,parameters$k)
    Lmom.genlogis(parameters$xi,parameters$alfa,parameters$k)
    rand.genlogis(100,parameters$xi,parameters$alfa,parameters$k)

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

[Package nsRFA version 0.6-9 Index]