lmrdiscord: Compute Discordance on L-CV, L-skew, and L-kurtosis

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

Compute Discordance on L-CV, L-skew, and L-kurtosis

Description

This function computes the Hosking and Wallis discordancy of the first three L-moment ratios (L-CV, L-skew, and L-kurtosis) according to their implementation in Hosking and Wallis (1997) and earlier. Discordancy triplets of these L-moment ratios is heuristically measured by effectively locating the triplet from the mean center of the 3-dimensional cloud of values. The lmomRFA provides for discordancy embedded in the “L-moment method” of regional frequency analysis. The author of lmomco chooses to have a separate “high level” implementation for emergent ideas of his in evaluating unusual sample distributions outside of the regdata object class envisioned by Hosking in the lmomRFA package.

Let \bm{\mu_i} be a row vector of the values of \tau^{[i]}_2, \tau^{[i]}_3, \tau^{[i]}_4 and these are the L-moment ratios for the ith group or site out of n sites. Let \bm{\overline\mu} be a row vector of mean values of all the n sites. Defining a sum of squares and cross products 3\times 3 matrix as

\bm{S} = \sum_i^n (\bm{\mu} - \bm{\overline\mu})(\bm{\mu} - \bm{\overline\mu})^{T}

compute the discorancy of the ith site as

D_i = \frac{n}{3} (\bm{\mu} - \bm{\overline\mu})^T \bm{S}^{-1} (\bm{\mu} - \bm{\overline\mu}\mbox{.})

The L-moments of a sample for a location are judged to be discordance if D_i exceeds a critical value. The critical value is a function of sample size. Hosking and Wallis (1997, p. 47) provide a table for general application. By about n=14, the critical value is taken as D_c = 3, although the D_{max} increases with sample size. Specifically, the D_i has an upper limit of

D_i \le (n-1)/3\mbox{.}

However, Hosking and Wallis (1997, p. 47) recommend “that any site with D_i > 3 be regarded as discordant.” A statistical test of D_i can be constructed. Hosking and Wallis (1997, p. 47) report that the D_{critical} is

D_{critical, n, \alpha} = \frac{(n - 1)Z}{n - 4 + 3Z}\mbox{,}

where

Z = F(\alpha/n, 3, n - 4)\mbox{,}

upper-tail quantile of the F distribution with degrees of freedom 3 and n - 4. A table of critical values is preloaded into the lmrdiscord function as this mimics the table of Hosking and Wallis (1997, table 3.1) as a means for cross verification. This table corresponds to an \alpha = 0.1 significance.

Usage

lmrdiscord(site=NULL, t2=NULL, t3=NULL, t4=NULL,
           Dcrit=NULL, digits=4, lmrdigits=4, sort=TRUE,
           alpha1=0.10, alpha2=0.01, ...)

Arguments

site	An optional group or site identification; it will be sequenced from 1 to n if NULL.
t2	L-CV values; emphasis that L-scale is not used.
t3	L-skew values.
t4	L-kurtosis values.
Dcrit	An optional (user specified) critical value for discordance. This value will override the Hosking and Wallis (1997, table 3.1) critical values.
digits	The number of digits in rounding operations.
lmrdigits	The numer of digits in rounding operation for the echo of the L-moment ratios.
sort	A logical on the sort status of the returned data frame.
alpha1	A significance level that is greater (less significant, although in statistics we need to avoid assigning less or more in this context) than alpha2.
alpha2	A significance level that is less (more significant, although in statistics we need to avoid assigning less or more in this context) than alpha1.
...	Other arguments that might be used. The author added these because it was found that the function was often called by higher level functions that aggregated much of the discordance computations.

Value

An R data.frame is returned.

site	The group or site identification as used by the function.
t2	L-CV values.
t3	L-skew values.
t4	L-kurtosis.
Dmax	The maximum discordancy D_{max} = (n-1)/3.
Dalpha1	The critical value of D for \alpha_1 = 0.10 (default) significance as set by alpha1 argument.
Dalpha2	The critical value of D for \alpha_2 = 0.01 (default) significance as set by alpha1 argument.
Dcrit	The critical value of discordancy (user or tabled).
D	The discordancy of the L-moment ratios used to trigger the logical in isD.
isD	Are the L-moment ratios discordant (if starred).
signif	A hyphen, star, or double star based on the Dalpha1 and Dalpha2 values.

Author(s)

W.H. Asquith

Source

Consultation of the lmomRFA.f and regtst() function of the lmomRFA R package by J.R.M. Hosking. Thanks Jon and Jim Wallis for such a long advocation of the discordancy issue that began at least as early as the 1993 Water Resources Research Paper (-wha).

References

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

lmoms

Examples

## Not run: 
# This is the canonical test of lmrdiscord().
library(lmomRFA) # Import lmomRFA, needs lmom package too
data(Cascades)   # Extract Hosking's data use in his examples
data <- as.regdata(Cascades) # A "regional" data structure
Dhosking <- sort(regtst(data)$D, decreasing=TRUE) # Discordancy

Dlmomco <- lmrdiscord(site=data$name, t2=data$t, t3=data$t_3, t4=data$t_4)

Dasquith <- Dlmomco$D
# Now show the site id, and the two discordancy computations
print(data.frame(NAME=data$name, Dhosking=Dhosking,
                                 Dasquith=Dasquith))
# The Dhosking and Dasquith columns had better match!

set.seed(3) # This seed produces a "*" and "**", but users
# are strongly encouraged to repeat the folowing code block
# over and over with an unspecified seed and look at the table.
n <- 30 # simulation sample size
par1 <- lmom2par(vec2lmom(c(1, .23, .2, .1)), type="kap")
par2 <- lmom2par(vec2lmom(c(1, .5, -.1)),      type="gev")
name <- t2 <- t3 <- t4 <- vector(mode="numeric")
for(i in 1:20) {
  X <- rlmomco(n, par1); lmr <- lmoms(X)
  t2[i] <- lmr$ratios[2]
  t3[i] <- lmr$ratios[3]
  t4[i] <- lmr$ratios[4]
  name[i] <- "kappa"
}
j <- length(t2)
for(i in 1:3) {
  X <- rlmomco(n, par2); lmr <- lmoms(X)
  t2[j + i] <- lmr$ratios[2]
  t3[j + i] <- lmr$ratios[3]
  t4[j + i] <- lmr$ratios[4]
  name[j + i] <- "gev"
}
D <- lmrdiscord(site=name, t2=t2, t3=t3, t4=t4)
print(D)

plotlmrdia(lmrdia(), xlim=c(-.2,.6), ylim=c(-.1, .4),
           autolegend=TRUE, xleg=0.1, yleg=.4)
points(D$t3,D$t4)
text(D$t3,D$t4,D$site, cex=0.75, pos=3)
text(D$t3,D$t4,D$D, cex=0.75, pos=1) #
## End(Not run)

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.