distLextreme: Extreme value stats

Description

Extreme value statistics for flood risk estimation. Input: vector with annual discharge maxima (or all observations for POT approach). Output: discharge estimates for given return periods, parameters of several distributions (fit based on L-moments), quality of fits, plot with linear/logarithmic axis. (plotting positions by Weibull and Gringorton).

Usage

distLextreme(
  dat = NULL,
  dlf = NULL,
  RPs = c(2, 5, 10, 20, 50),
  npy = 1,
  truncate = 0,
  quiet = FALSE,
  ...
)

Value

invisible dlf object, see printL. The added element is returnlev, a data.frame with the return level (discharge) for all given RPs and for each distribution. Note that this differs from distLquantile (matrix output, not data.frame)

Arguments

dat: Vector with either (for Block Maxima Approach) extreme values like annual discharge maxima or (for Peak Over Threshold approach) all values in time-series. Ignored if dlf is given. DEFAULT: NULL
dlf: List as returned by distLfit. See also distLquantile. Overrides dat! DEFAULT: NULL
RPs: Return Periods (in years) for which discharge is estimated. DEFAULT: c(2,5,10,20,50)
npy: Number of observations per year. Leave npy=1 if you use annual block maxima (and leave truncate at 0). If you use a POT approach (see vignette and examples below) e.g. on daily data, use npy=365.24. DEFAULT: 1
truncate: Truncated proportion to determine POT threshold, see distLquantile. DEFAULT: 0
quiet: Suppress notes and progbars? DEFAULT: FALSE
...: Further arguments passed to distLquantile like truncate, selection, time, progbars

Author

Berry Boessenkool, berry-b@gmx.de, 2012 (first draft) - 2014 & 2015 (main updates)

Details

plotLextreme adds weibull and gringorton plotting positions to the distribution lines, which are estimated from the L-moments of the data itself.
I personally believe that if you have, say, 35 values in dat, the highest return period should be around 36 years (Weibull) and not 60 (Gringorton).
The plotting positions don't affect the distribution parameter estimation, so this dispute is not really important. But if you care, go ahead and google "weibull vs gringorton plotting positions".

Plotting positions are not used for fitting distributions, but for plotting only. The ranks of ascendingly sorted extreme values are used to compute the probability of non-exceedance Pn:
Pn_w <- Rank /(n+1) # Weibull
Pn_g <- (Rank-0.44)/(n+0.12) # Gringorton (taken from lmom:::evplot.default)
Finally: RP = Return period = recurrence interval = 1/P_exceedance = 1/(1-P_nonexc.), thus:
RPweibull = 1/(1-Pn_w) and analogous for gringorton.

References

https://RclickHandbuch.wordpress.com Chapter 15 (German)
Christoph Mudersbach: Untersuchungen zur Ermittlung von hydrologischen Bemessungsgroessen mit Verfahren der instationaeren Extremwertstatistik

Examples

Run this code


# Basic examples
# BM vs POT
# Plotting options
# weighted mean based on Goodness of fit (GOF)
# Effect of data proportion used to estimate GOF
# compare extremeStat with other packages

library(lmomco)
library(berryFunctions)

data(annMax) # annual streamflow maxima in river in Austria

# Basic examples ---------------------------------------------------------------
dlf <- distLextreme(annMax)
plotLextreme(dlf, log=TRUE)
plotLextreme(dlf, log="xy")
plotLextreme(dlf)

# Object structure:
str(dlf, max.lev=2)
printL(dlf)

# discharge levels for default return periods:
dlf$returnlev

# Estimate discharge that could occur every 80 years (at least empirically):
Q80 <- distLextreme(dlf=dlf, RPs=80)$returnlev
round(sort(Q80[1:17,1]),1)
# 99 to 143 m^3/s can make a relevant difference in engineering!
# That's why the rows weighted by GOF are helpful. Weights are given as in
plotLweights(dlf) # See also section weighted mean below
# For confidence intervals see ?distLexBoot

# Return period of a given discharge value, say 120 m^3/s:
round0(sort(1/(1-sapply(dlf$parameter, plmomco, x=120) )  ),1)
# exponential:                 every 29 years
# gev (general extreme value dist):  59,
# Weibull:                     every 73 years only


# BM vs POT --------------------------------------------------------------------
# Return levels by Block Maxima approach vs Peak Over Threshold approach:
# BM distribution theoretically converges to GEV, POT to GPD

data(rain, package="ismev")
days <- seq(as.Date("1914-01-01"), as.Date("1961-12-30"), by="days")
BM <- tapply(rain, format(days,"%Y"), max)  ;  rm(days)
dlfBM <- plotLextreme(distLextreme(BM, emp=FALSE), ylim=lim0(100), log=TRUE, nbest=10)
plotLexBoot(distLexBoot(dlfBM, quiet=TRUE), ylim=lim0(100))
plotLextreme(dlfBM, log=TRUE, ylim=lim0(100))

dlfPOT99 <- distLextreme(rain, npy=365.24, trunc=0.99, emp=FALSE)
dlfPOT99 <- plotLextreme(dlfPOT99, ylim=lim0(100), log=TRUE, nbest=10, main="POT 99")
printL(dlfPOT99)

# using only nonzero values (normally yields better fits, but not here)
rainnz <- rain[rain>0]
dlfPOT99nz <- distLextreme(rainnz, npy=length(rainnz)/48, trunc=0.99, emp=FALSE)
dlfPOT99nz <- plotLextreme(dlfPOT99nz, ylim=lim0(100), log=TRUE, nbest=10,
                           main=paste("POT 99 x>0, npy =", round(dlfPOT99nz$npy,2)))

if (FALSE)  ## Excluded from CRAN R CMD check because of computing time

dlfPOT99boot <- distLexBoot(dlfPOT99, prop=0.4)
printL(dlfPOT99boot)
plotLexBoot(dlfPOT99boot)


dlfPOT90 <- distLextreme(rain, npy=365.24, trunc=0.90, emp=FALSE)
dlfPOT90 <- plotLextreme(dlfPOT90, ylim=lim0(100), log=TRUE, nbest=10, main="POT 90")

dlfPOT50 <- distLextreme(rain, npy=365.24, trunc=0.50, emp=FALSE)
dlfPOT50 <- plotLextreme(dlfPOT50, ylim=lim0(100), log=TRUE, nbest=10, main="POT 50")


ig99 <- ismev::gpd.fit(rain, dlfPOT99$threshold)
ismev::gpd.diag(ig99); title(main=paste(99, ig99$threshold))
if (FALSE) {
ig90 <- ismev::gpd.fit(rain, dlfPOT90$threshold)
ismev::gpd.diag(ig90); title(main=paste(90, ig90$threshold))
ig50 <- ismev::gpd.fit(rain, dlfPOT50$threshold)
ismev::gpd.diag(ig50); title(main=paste(50, ig50$threshold))
}


# Plotting options -------------------------------------------------------------
plotLextreme(dlf=dlf)
# Line colors / select distributions to be plotted:
plotLextreme(dlf, nbest=17, distcols=heat.colors(17), lty=1:5) # lty is recycled
plotLextreme(dlf, selection=c("gev", "gam", "gum"), distcols=4:6, PPcol=3, lty=3:2)
plotLextreme(dlf, selection=c("gpa","glo","wei","exp"), pch=c(NA,NA,6,8),
                 order=TRUE, cex=c(1,0.6, 1,1), log=TRUE, PPpch=c(16,NA), n_pch=20)
# use n_pch to say how many points are drawn per line (important for linear axis)

plotLextreme(dlf, legarg=list(cex=0.5, x="bottom", box.col="red", col=3))
# col in legarg list is (correctly) ignored
if (FALSE) {
## Excluded from package R CMD check because it's time consuming

plotLextreme(dlf, PPpch=c(1,NA)) # only Weibull plotting positions
# add different dataset to existing plot:
distLextreme(Nile/15, add=TRUE, PPpch=NA, distcols=1, selection="wak", legend=FALSE)

# Logarithmic axis
plotLextreme(distLextreme(Nile), log=TRUE, nbest=8)



# weighted mean based on Goodness of fit (GOF) ---------------------------------
# Add discharge weighted average estimate continuously:
plotLextreme(dlf, nbest=17, legend=FALSE)
abline(h=115.6, v=50)
RP <- seq(1, 70, len=100)
DischargeEstimate <- distLextreme(dlf=dlf, RPs=RP, plot=FALSE)$returnlev
lines(RP, DischargeEstimate["weighted2",], lwd=3, col="orange")

# Or, on log scale:
plotLextreme(dlf, nbest=17, legend=FALSE, log=TRUE)
abline(h=115.9, v=50)
RP <- unique(round(logSpaced(min=1, max=70, n=200, plot=FALSE),2))
DischargeEstimate <- distLextreme(dlf=dlf, RPs=RP)$returnlev
lines(RP, DischargeEstimate["weighted2",], lwd=5)


# Minima -----------------------------------------------------------------------

browseURL("https://nrfa.ceh.ac.uk/data/station/meanflow/39072")
qfile <- system.file("extdata/discharge39072.csv", package="berryFunctions")
Q <- read.table(qfile, skip=19, header=TRUE, sep=",", fill=TRUE)[,1:2]
rm(qfile)
colnames(Q) <- c("date","discharge")
Q$date <- as.Date(Q$date)
plot(Q, type="l")
Qmax <- tapply(Q$discharge, format(Q$date,"%Y"), max)
plotLextreme(distLextreme(Qmax, quiet=TRUE))
Qmin <- tapply(Q$discharge, format(Q$date,"%Y"), min)
dlf <- distLextreme(-Qmin, quiet=TRUE, RPs=c(2,5,10,20,50,100,200,500))
plotLextreme(dlf, ylim=c(0,-31), yaxs="i", yaxt="n", ylab="Q annual minimum", nbest=14)
axis(2, -(0:3*10), 0:3*10, las=1)
-dlf$returnlev[c(1:14,21), ]
# Some distribution functions are an obvious bad choice for this, so I use
# weighted 3: Values weighted by GOF of dist only for the best half.
# For the Thames in Windsor, we will likely always have > 9 m^3/s streamflow


# compare extremeStat with other packages: ---------------------------------------
library(extRemes)
plot(fevd(annMax))
par(mfrow=c(1,1))
return.level(fevd(annMax, type="GEV")) # "GP", "PP", "Gumbel", "Exponential"
distLextreme(dlf=dlf, RPs=c(2,20,100))$returnlev["gev",]
# differences are small, but noticeable...
# if you have time for a more thorough control, please pass me the results!


# yet another dataset for testing purposes:
Dresden_AnnualMax <- c(403, 468, 497, 539, 542, 634, 662, 765, 834, 847, 851, 873,
885, 983, 996, 1020, 1028, 1090, 1096, 1110, 1173, 1180, 1180,
1220, 1270, 1285, 1329, 1360, 1360, 1387, 1401, 1410, 1410, 1456,
1556, 1580, 1610, 1630, 1680, 1734, 1740, 1748, 1780, 1800, 1820,
1896, 1962, 2000, 2010, 2238, 2270, 2860, 4500)
plotLextreme(distLextreme(Dresden_AnnualMax))
} # end dontrun

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