lmomco (version 2.4.14)

fpds2f: Conversion of Partial-Duration Nonexceedance Probability to Annual Nonexceedance Probability

Description

This function takes partial duration series nonexceedance probability and converts it to a an annual exceedance probability through a simple assumption that the Poisson distribution is appropriate. The relation between the cumulative distribution function \(F(x)\) for the annual series is related to the cumulative distribution function \(G(x)\) of the partial-duration series by $$F(x) = \mathrm{exp}(-\eta[1 - G(x)])\mathrm{.}$$ The core assumption is that successive events in the partial-duration series can be considered as independent. The \(\eta\) term is the arrival rate of the events. For example, suppose that 21 events have occurred in 15 years, then \(\eta = 21/15 = 1.4\) events per year.

The example documented here provides a comprehensive demonstration of the function along with a partnering function f2fpds. That function performs the opposite conversion. Lastly, the cross reference to x2xlo is made because the example contained therein provides another demonstration of partial-duration and annual series frequency analysis.

Usage

fpds2f(fpds, rate=NA)

Value

A vector of converted nonexceedance probabilities.

Arguments

fpds

A vector of partial-duration nonexceedance probabilities.

rate

The number of events per year.

Author

W.H. Asquith

References

Stedinger, J.R., Vogel, R.M., Foufoula-Georgiou, E., 1993, Frequency analysis of extreme events: in Handbook of Hydrology, ed. Maidment, D.R., McGraw-Hill, Section 18.6 Partial duration series, mixtures, and censored data, pp. 18.37--18.39.

See Also

f2fpds, x2xlo, f2flo, flo2f

Examples

if (FALSE) {
stream <- "A Stream in West Texas"
Qpds    <- c(61.8, 122, 47.3, 71.1, 211, 139, 244, 111, 233, 102)
Qann <- c(61.8, 122, 71.1, 211, 244, 0, 233)
years  <- length(Qann)  # gage has operated for about 7 years
visits <- 27  # number of visits or "events"
rate   <- visits/years
Z <- rep(0, visits-length(Qpds))
Qpds <- c(Qpds,Z) # The creation of a partial duration series
# that will contain numerous zero values.

Fs <- seq(0.001,.999, by=.005) # used to generate curves

type <- "pe3" # The Pearson type III distribution
PPpds <- pp(Qpds); Qpds <- sort(Qpds) # plotting positions (partials)
PPann <- pp(Qann); Qann <- sort(Qann) # plotting positions (annuals)
parann <- lmom2par(lmoms(Qann), type=type) # parameter estimation (annuals)
parpsd <- lmom2par(lmoms(Qpds), type=type) # parameter estimation (partials)

Fsplot    <- qnorm(Fs) # in order to produce normal probability paper
PPpdsplot <- qnorm(fpds2f(PPpds, rate=rate)) # ditto
PPannplot <- qnorm(PPann) # ditto

# There are many zero values in this particular data set that require leaving
# them out in order to achieve appropriate curvature of the Pearson type III
# distribution. Conditional probability adjustments will be used.
Qlo <- x2xlo(Qpds) # Create a left out object with an implied threshold of zero
parlo <- lmom2par(lmoms(Qlo$xin), type=type) # parameter estimation for the
# partial duration series values that are greater than the threshold, which
# defaults to zero.

plot(PPpdsplot, Qpds, type="n", ylim=c(0,400), xlim=qnorm(c(.01,.99)),
     xlab="STANDARD NORMAL VARIATE", ylab="DISCHARGE, IN CUBIC FEET PER SECOND")
mtext(stream)
points(PPannplot, Qann, col=3, cex=2, lwd=2, pch=0)
points(qnorm(fpds2f(PPpds, rate=rate)), Qpds, pch=16, cex=0.5 )
points(qnorm(fpds2f(flo2f(pp(Qlo$xin), pp=Qlo$pp), rate=rate)),
       sort(Qlo$xin), col=2, lwd=2, cex=1.5, pch=1)
points(qnorm(fpds2f(Qlo$ppout, rate=rate)),
       Qlo$xout, pch=4, col=4)

lines(qnorm(fpds2f(Fs, rate=rate)),
      qlmomco(Fs, parpsd), lwd=1, lty=2)
lines(Fsplot, qlmomco(Fs, parann), col=3, lwd=2)
lines(qnorm(fpds2f(flo2f(Fs, pp=Qlo$pp), rate=rate)),
      qlmomco(Fs, parlo), col=2, lwd=3)

# The following represents a subtle application of the probability transform
# functions. The show how one starts with annual recurrence intervals
# converts into conventional annual nonexceedance probabilities as well as
# converting these to the partial duration nonexceedance probabilities.
Tann <- c(2, 5, 10, 25, 50, 100)
Fann <- T2prob(Tann); Gpds <- f2fpds(Fann, rate=rate)
FFpds <- qlmomco(f2flo(Gpds, pp=Qlo$pp), parlo)
FFann <- qlmomco(Fann, parann)
points(qnorm(Fann), FFpds, col=2, pch=16)
points(qnorm(Fann), FFann, col=3, pch=16)

legend(-2.4,400, c("True annual series (with one zero year)",
                "Partial duration series (including 'visits' as 'events')",
                "Partial duration series (after conditional adjustment)",
                "Left-out values (<= zero) (trigger of conditional probability)",
                "PE3 partial-duration frequency curve (PE3-PDS)",
                "PE3 annual-series frequency curve (PE3-ANN)",
                "PE3 partial-duration frequency curve (zeros removed) (PE3-PDSz)",
                "PE3-ANN  T-year event: 2, 5, 10, 25, 50, 100 years",
                "PE3-PDSz T-year event: 2, 5, 10, 25, 50, 100 years"),
       bty="n", cex=.75,
       pch=c(0,  16, 1, 4, NA, NA, NA, 16, 16),
       col=c(3,  1, 2,  4,  1,  3,  2,  3, 2),
       pt.lwd=c(2,1,2,1), pt.cex=c(2, 0.5, 1.5, 1, NA, NA, NA, 1, 1),
       lwd=c(0,0,0,0,1,2,3), lty=c(0,0,0,0,2,1,1))
}