lmomco (version 2.4.14)

lmom2pwm: L-moments to Probability-Weighted Moments

Description

Converts the L-moments to the probability-weighted moments (PWMs) given the L-moments. The conversion is linear so procedures based on L-moments are identical to those based on PWMs. The expression linking PWMs to L-moments is $$ \lambda_{r+1} = \sum_{k=0}^r (-1)^{r-k} {r \choose k}{r+k \choose k}\beta_k\mbox{,} $$ where \(\lambda_{r+1}\) are the L-moments, \(\beta_r\) are the PWMs, and \(r \ge 0\).

Usage

lmom2pwm(lmom)

Value

An R

list is returned.

betas

The PWMs. Note that convention is the have a \(\beta_0\), but this is placed in the first index i=1 of the betas vector.

source

Source of the PWMs: “pwm”.

Arguments

lmom

An L-moment object created by lmoms, lmom.ub, or vec2lmom. The function also supports lmom as a vector of L-moments (\(\lambda_1\), \(\lambda_2\), \(\tau_3\), \(\tau_4\), and \(\tau_5\)).

Author

W.H. Asquith

Details

PWMs are linear combinations of the L-moments and therefore contain the same statistical information of the data as the L-moments. However, the PWMs are harder to interpret as measures of probability distributions. The PWMs are included in lmomco for theoretical completeness and are not intended for use with the majority of the other functions implementing the various probability distributions. The relations between L-moments (\(\lambda_r\)) and PWMs (\(\beta_{r-1}\)) for \(1 \le r \le 5\) order are $$\lambda_1 = \beta_0 \mbox{,}$$ $$\lambda_2 = 2\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_3 = 6\beta_2 - 6\beta_1 + \beta_0 \mbox{,}$$ $$\lambda_4 = 20\beta_3 - 30\beta_2 + 12\beta_1 - \beta_0\mbox{, and}$$ $$\lambda_5 = 70\beta_4 - 140\beta_3 + 90\beta_2 - 20\beta_1 + \beta_0\mbox{.}$$

The linearity between L-moments and PWMs means that procedures based on one are equivalent to the other. This function only accomodates the first five L-moments and PWMs. Therefore, at least five L-moments are required in the passed argument.

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.

Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments---Definition and relation to parameters of several distributions expressable in inverse form: Water Resources Research, v. 15, pp. 1,049--1,054.

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124.

See Also

lmom.ub, lmoms, pwm.ub, pwm2lmom

Examples

pwm <- lmom2pwm(lmoms(c(123,34,4,654,37,78)))
lmom2pwm(lmom.ub(rnorm(100)))
lmom2pwm(lmoms(rnorm(100)))

lmomvec1 <- c(1000,1300,0.4,0.3,0.2,0.1)
pwmvec   <- lmom2pwm(lmomvec1)
print(pwmvec)
#$betas
#[1] 1000.0000 1150.0000 1070.0000  984.5000  911.2857
#
#$source
#[1] "lmom2pwm"

lmomvec2 <- pwm2lmom(pwmvec)
print(lmomvec2)
#$lambdas
#[1] 1000 1300  520  390  260
#
#$ratios
#[1]  NA 1.3 0.4 0.3 0.2
#
#$source
#[1] "pwm2lmom"

pwm2lmom(lmom2pwm(list(L1=25, L2=20, TAU3=.45, TAU4=0.2, TAU5=0.1)))