pp {lmomco}

Plotting-Position Formula

Description

The plotting positions of a data vector (x) are returned in ascending order. The plotting-position formula is

pp_i = \frac{i-a}{n+1-2a} \mbox{,}

where pp_i is the nonexceedance probability F of the ith ascending data value. The parameter a specifies the plotting-position type, and n is the sample size (length(x)). Alternatively, the plotting positions can be computed by

pp_i = \frac{i+A}{n+B} \mbox{,}

where A and B can obviously be expressed in terms of a for B > A > -1 (Hosking and Wallis, 1997, sec. 2.8).

Usage

pp(x, A=NULL, B=NULL, a=0, sort=TRUE, ties.method="first", ...)

Arguments

x

A vector of data values. The vector is used to get sample size through length.

A

A value for the plotting-position coefficient A.

B

A value for the plotting-position coefficient B.

a

A value for the plotting-position formula from which A and B are computed, default is a=0, which returns the Weibull plotting positions.

sort

A logical whether the ranks of the data are sorted prior to F computation. It was a design mistake years ago to default this function to a sort, but it is now far too late to risk changing the logic now. The function originally lacked the sort argument for many years.

ties.method

This is the argument of the same name passed to rank.

...

Additional arguments to pass.

Value

An R vector is returned.

Note

Various plotting positions have been suggested in the literature. Stedinger and others (1992, p.18.25) comment that “all plotting positions give crude estimates of the unknown [non]exceedance probabilities associated with the largest (and smallest) events.” The various plotting positions are summarized in the follow table.

Weibull

a=0, Unbiased exceedance probability for all distributions (see discussion in pp.f).

Median

a=0.3175, Median exceedance probabilities for all distributions (if so, see pp.median).

APL

\approx 0.35, Often used with probability-weighted moments.

Blom

a=0.375, Nearly unbiased quantiles for normal distribution.

Cunnane

a=0.40, Approximately quantile unbiased.

Gringorten

a=0.44, Optimized for Gumbel distribution.

Hazen

a=0.50, A traditional choice.

The function uses the R rank function, which has specific settings to handle tied data. For implementation here, the ties.method="first" method to rank is used. The user has flexibility in changing this to their own custom purposes.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

Stedinger, J.R., Vogel, R.M., and Foufoula-Georgiou, E., 1992, Frequency analysis of extreme events, in Handbook of Hydrology, chapter 18, editor-in-chief D. A. Maidment: McGraw-Hill, New York.

See Also

nonexceeds, pwm.pp, pp.f, pp.median, headrick.sheng.lalpha

Examples

Q  <- rnorm(20)
PP <- pp(Q)
plot(PP, sort(Q))

Q <- rweibull(30, 1.4, scale=400)
WEI <- parwei(lmoms(Q))
PP <- pp(Q)
plot( PP, sort(Q))
lines(PP, quawei(PP, WEI))

# This plot looks similar, but when connecting lines are added
# the nature of the sorting is obvious.
plot( pp(Q, sort=FALSE), Q)
lines(pp(Q, sort=FALSE), Q, col=2)