# Conversion of Partial-Duration Nonexceedance Probability to Annual Nonexceedance Probability 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 [REMOVE_ME]$$ F(x) = \mathrm{exp}(-\eta[1 - G(x)])\mathrm{.} [REMOVE_ME_2]$$ 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. ## 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. ```r fpds2f(fpds, rate=NA) ``` ## Arguments - `fpds`: A vector of partial-duration nonexceedance probabilities. - `rate`: The number of events per year. ## Returns A vector of converted nonexceedance probabilities. ## 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. ## Author(s) W.H. Asquith ## See Also `f2fpds`, `x2xlo`, `f2flo`, `flo2f` ## Examples ```r ## Not run: 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)) ## End(Not run) ```

fpds2f function