pwm2lmom {lmomco} | R Documentation |
Converts the Probability-Weighted Moments (PWM) to the L-moments given the PWM. The conversion is linear so procedures based on PWMs and identical to those based on L-moments.
λ_1 = β_0 mbox{,}
λ_2 = 2β_1 - β_0 mbox{,}
λ_3 = 6β_2 - 6β_1 + β_0 mbox{,}
λ_4 = 20β_3 - 30β_2 + 12β_1 - β_0 mbox{,}
λ_5 = 70β_4 - 140β_3 + 90β_2 - 20β_1 + β_0 mbox{,}
tau = λ_2/λ_1 mbox{,}
tau_3 = λ_3/λ_2 mbox{,}
tau_4 = λ_4/λ_2 mbox{, and}
tau_5 = λ_5/λ_2 mbox{.}
The general expression and the expression used for computation if the argument is a vector of PWMs is
λ_{r+1} = sum^r_{k=0} (-1)^{r-k}{r choose k}{r+k choose k} β_{k+1}
pwm2lmom(pwm)
pwm |
A PWM object created by pwm.ub or similar. |
The Probability Weighted Moments (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 linearity between L-moments and PWMs means that procedures base on one are equivalent to the other.
The function can take a variety of PWM argument types in pwm
. The function checks whether the argument is a list and if so attempts to extract the β_r's from list
names such as BETA0
, BETA1
, and so on. If the extraction is successful, then a list of L-moments similar to lmom.ub
is returned. If the extraction was not successful, then a list
name betas
is checked; if betas
is found then this vector of PWMs is used to compute the L-moments. If pwm
is a list
but can not be routed in the function, a warnings
is made and NULL
returned. If the pwm
argument is a vector
, then this vector of PWMs is used to compute the L-moments are returned.
One of two R list
are returned.
Version I |
|
Version II |
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W.H. Asquith
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, vol. 15, p. 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, vol. 52, p. 105–124.
Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.
lmom.ub
, pwm.ub
, pwm
, lmom2pwm
D <- c(123,34,4,654,37,78) pwm2lmom(pwm.ub(D)) pwm2lmom(pwm(D)) pwm2lmom(pwm(rnorm(100)))