# distLextreme: Extreme value stats In extremeStat: Extreme Value Statistics and Quantile Estimation

## 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

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

## 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

## 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.

## 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)

## Note

This function replaces `berryFunctions::extremeStatLmom`

## Author(s)

Berry Boessenkool, [email protected], 2012 (first draft) - 2014 & 2015 (main updates)

## References

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

`distLfit`. `distLexBoot` for confidence interval from Bootstrapping. `fevd` in the package `extRemes`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164``` ```# 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[1:17,1]),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))) ## Not run: ## 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") ## 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 ```