P3: Three parameters Pearson type III distribution and L-moments

In nsRFA: Non-Supervised Regional Frequency Analysis

Three parameters Pearson type III distribution and L-moments

Description

P3 provides the link between L-moments of a sample and the three parameter Pearson type III distribution.

Usage

f.gamma (x, xi, beta, alfa)
F.gamma (x, xi, beta, alfa)
invF.gamma (F, xi, beta, alfa)
Lmom.gamma (xi, beta, alfa)
par.gamma (lambda1, lambda2, tau3)
rand.gamma (numerosita, xi, beta, alfa)
mom2par.gamma (mu, sigma, gamm)
par2mom.gamma (alfa, beta, xi)

Arguments

x	vector of quantiles
mu	vector of gamma mean
sigma	vector of gamma standard deviation
gamm	vector of gamma third moment
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
alfa	vector of gamma shape parameters
beta	vector of gamma scale parameters
xi	vector of gamma location parameters

Details

See https://en.wikipedia.org/wiki/Pearson_distribution for an introduction to the Pearson distribution, and https://en.wikipedia.org/wiki/Gamma_distribution for an introduction to the Gamma distribution (the Pearson type III distribution is, essentially, a Gamma distribution with 3 parameters).

Definition

Parameters (3): \xi (location), \beta (scale), \alpha (shape). Moments (3): \mu (mean), \sigma (standard deviation), \gamma (skewness).

If \gamma \ne 0, let \alpha=4/\gamma^2, \beta=\frac{1}{2}\sigma |\gamma|, and \xi= \mu - 2 \sigma/\gamma. If \gamma > 0, then the range of x is \xi \le x < \infty and

f(x) = \frac{(x - \xi)^{\alpha - 1} e^{-(x-\xi)/\beta}}{\beta^{\alpha} \Gamma(\alpha)}

F(x) = G \left(\alpha, \frac{x-\xi}{\beta}\right)/ \Gamma(\alpha)

If \gamma=0, then the distribution is Normal, the range of x is -\infty < x < \infty and

f(x) = \phi \left(\frac{x-\mu}{\sigma}\right)

F(x) = \Phi \left(\frac{x-\mu}{\sigma}\right)

where \phi(x)=(2\pi)^{-1/2}\exp(-x^2/2) and \Phi(x)=\int_{-\infty}^x \phi(t)dt.

If \gamma < 0, then the range of x is -\infty < x \le \xi and

f(x) = \frac{(\xi - x)^{\alpha - 1} e^{-(\xi-x)/\beta}}{\beta^{\alpha} \Gamma(\alpha)}

F(x) = G \left(\alpha, \frac{\xi-x}{\beta}\right)/ \Gamma(\alpha)

In each case, x(F) has no explicit analytical form. Here \Gamma is the gamma function, defined as

\Gamma (x) = \int_0^{\infty} t^{x-1} e^{-t} dt

and

G(\alpha, x) = \int_0^x t^{\alpha-1} e^{-t} dt

is the incomplete gamma function.

\gamma=2 is the exponential distribution; \gamma=0 is the Normal distribution; \gamma=-2 is the reverse exponential distribution.

The parameters \mu, \sigma and \gamma are the conventional moments of the distribution.

L-moments

Assuming \gamma>0, L-moments are defined for 0<\alpha<\infty.

\lambda_1 = \xi + \alpha \beta

\lambda_2 = \pi^{-1/2} \beta \Gamma(\alpha + 1/2)/\Gamma(\alpha)

\tau_3 = 6 I_{1/3} (\alpha, 2 \alpha)-3

where I_x(p,q) is the incomplete beta function ratio

I_x(p,q) = \frac{\Gamma(p+q)}{\Gamma(p)\Gamma(q)} \int_0^x t^{p-1} (1-t)^{q-1} dt

There is no simple expression for \tau_4. Here we use the rational-funcion approximation given by Hosking and Wallis (1997, pp. 201-202).

The corresponding results for \gamma <0 are obtained by changing the signs of \lambda_1, \tau_3 and \xi wherever they occur above.

Parameters

alpha is obtained with an approximation. If 0<|\tau_3|<1/3, let z=3 \pi \tau_3^2 and use

\alpha \approx \frac{1+0.2906 z}{z + 0.1882 z^2 + 0.0442 z^3}

if 1/3<|\tau_3|<1, let z=1-|\tau_3| and use

\alpha \approx \frac{0.36067 z - 0.59567 z^2 + 0.25361 z^3}{1-2.78861 z + 2.56096 z^2 -0.77045 z^3}

Given \alpha, then \gamma=2 \alpha^{-1/2} sign(\tau_3), \sigma=\lambda_2 \pi^{1/2} \alpha^{1/2} \Gamma(\alpha)/\Gamma(\alpha+1/2), \mu=\lambda_1.

Lmom.gamma and par.gamma accept input as vectors of equal length. In f.gamma, F.gamma, invF.gamma and rand.gamma parameters (mu, sigma, gamm) must be atomic.

Value

f.gamma gives the density f, F.gamma gives the distribution function F, invFgamma gives the quantile function x, Lmom.gamma gives the L-moments (\lambda_1, \lambda_2, \tau_3, \tau_4), par.gamma gives the parameters (mu, sigma, gamm), and rand.gamma generates random deviates.

mom2par.gamma returns the parameters \alpha, \beta and \xi, given the parameters (moments) \mu, \sigma, \gamma.

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, LOGNORM; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.gamma(ll[1],ll[2],ll[4])
f.gamma(1800,parameters$xi,parameters$beta,parameters$alfa)
F.gamma(1800,parameters$xi,parameters$beta,parameters$alfa)
invF.gamma(0.7511627,parameters$xi,parameters$beta,parameters$alfa)
Lmom.gamma(parameters$xi,parameters$beta,parameters$alfa)
rand.gamma(100,parameters$xi,parameters$beta,parameters$alfa)

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

moments <- par2mom.gamma(parameters$alfa,parameters$beta,parameters$xi); moments
mom2par.gamma(moments$mu,moments$sigma,moments$gamm)

nsRFA documentation built on May 29, 2024, 1:14 a.m.