Plotting-Position Formula

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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. The criteria A > B > -1 must be satisfied.

Usage

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pp(x, A=NULL, B=NULL, a=0, sort=TRUE, ...)

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.

...

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.

Author(s)

W.H. Asquith

References

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

Examples

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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)

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