distLextreme: Extreme value stats

Description

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

Usage

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

Arguments

dat

Vector with extreme values e.g. annual discharge maxima. Ignored if dlf is given.

dlf

List as returned by distLfit. Overrides dat! DEFAULT: NULL

RPs

ReturnPeriods (in years) for which discharge is estimated. DEFAULT: c(2,5,10,20,50)

npy

Number of observations per year. 1 if you use annual block maxima. (Then leave truncate at 0). If you use a POT approach (see vignette) on e.g. daily data, use npy=365.24. DEFAULT: 1

quiet

Suppress notes and progbars? DEFAULT: FALSE

...

Further arguments passed to distLquantile (and distLfit if dlf=NULL) like truncate, selection, time, progbars

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)

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-exceedence Pn: Pn_w <- Rank /(n+1) # Weibull Pn_g <- (Rank-0.44)/(n+0.12) # Gringorton (taken from lmom:::evplot.default) Finally: RP = Returnperiod = recurrence interval = 1/P_exceedence = 1/(1-P_nonexc.), thus: RPweibull = 1/(1-Pn_w) and analogous for gringorton.

References

http://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)

# 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], decr=TRUE),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:
sort(1/(1-sapply(dlf$parameter, plmomco, x=120) )  )[1:13]
# exponential:                 every 29 years
# gev (general extreme value dist):  58,
# Weibull:                     every 72 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)

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)))

## Not run:  ## Excluded from CRAN R CMD check because of computing time
# 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")
# ## End(Not run)

ig99 <- ismev::gpd.fit(rain, dlfPOT99$threshold)
ismev::gpd.diag(ig99); title(main=paste(99, ig99$threshold))
## Not run: 
# 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))
# ## End(Not run)


# 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
## Not run: 
# ## 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("http://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(Not run) # end dontrun

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