lmomco (version 0.88)

pwm2lmom: Probability-Weighted Moments to L-moments

Description

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.

$$\lambda_1 = \beta_0 \mbox{,}$$ $$\lambda_2 = 2\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_3 = 6\beta_2 - 6\beta_1 + \beta_0 \mbox{,}$$ $$\lambda_4 = 20\beta_3 - 30\beta_2 + 12\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_5 = 70\beta_4 - 140\beta_3 + 90\beta_2 - 20\beta_1 + \beta_0 \mbox{,}$$ $$\tau = \lambda_2/\lambda_1 \mbox{,}$$ $$\tau_3 = \lambda_3/\lambda_2 \mbox{,}$$ $$\tau_4 = \lambda_4/\lambda_2 \mbox{, and}$$ $$\tau_5 = \lambda_5/\lambda_2 \mbox{.}$$

The general expression and the expression used for computation if the argument is a vector of PWMs is

$$\lambda_{r+1} = \sum^r_{k=0} (-1)^{r-k}{r \choose k}{r+k \choose k} \beta_{k+1}$$

Usage

pwm2lmom(pwm)

Arguments

pwm

A PWM object created by pwm.ub or similar.

Value

• One of two R list are returned.
• L1
  Arithmetic mean
• L2
  L-scale---analogous to standard deviation
• LCV
  coefficient of L-variation---analogous to coe. of variation
• TAU3
  The third L-moment ratio or L-skew---analogous to skew
• TAU4
  The fourth L-moment ratio or L-kurtosis---analogous to kurtosis
• TAU5
  The fifth L-moment ratio
• L3
  The third L-moment
• L4
  The fourth L-moment
• L5
  The fifth L-moment or the following list

  lambdas{The L-moments} ratios{The L-moment ratios} source{Source of the L-moments (pwm2lmom)}

  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.

  [object Object] 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)))

  univar distribution

Details

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 $\beta_r$'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.