| DISTPLOTS {nsRFA} | R Documentation |
Empirical distribution plots
Description
Sample values are plotted against their empirical distribution in graphs where points belonging to a particular distribution should lie on a straight line.
Usage
plotpos (x, a=0, orient="xF", ...)
plotposRP (x, a=0, orient="xF", ...)
loglogplot (x, a=0, orient="xF", ...)
unifplot (x, a=0, orient="xF", line=FALSE, ...)
normplot (x, a=0, orient="xF", line=FALSE, ...)
lognormplot (x, a=0, orient="xF", line=FALSE, ...)
studentplot (x, df, a=0, orient="xF", line=FALSE,...)
logisplot (x, a=0, orient="xF", line=FALSE,...)
gammaplot (x, shape, a=0, orient="xF", line=FALSE,...)
expplot (x, a=0, orient="xF", line=FALSE,...)
paretoplot (x, a=0, orient="xF", line=FALSE,...)
gumbelplot (x, a=0, orient="xF", line=FALSE, ...)
frechetplot (x, a=0, orient="xF", line=FALSE,...)
weibullplot (x, a=0, orient="xF", line=FALSE,...)
plotposRPhist (xcont, xhist=NA, infhist=NA, suphist=NA, nbans=NA, seuil=NA,
col12=c(1,1), a=0, orient="xF", ...)
pointspos (x, a=0, orient="xF", ...)
pointsposRP (x, a=0, orient="xF", ...)
loglogpoints (x, a=0, orient="xF", ...)
unifpoints (x, a=0, orient="xF", ...)
normpoints (x, a=0, orient="xF", ...)
studentpoints (x, df, a=0, orient="xF", ...)
logispoints (x, a=0, orient="xF", ...)
gammapoints (x, shape, a=0, orient="xF", ...)
exppoints (x, a=0, orient="xF", ...)
gumbelpoints (x, a=0, orient="xF", ...)
weibullpoints (x, a=0, orient="xF", ...)
regionalplotpos (x, cod, a=0, orient="xF", ...)
regionalnormplot (x, cod, a=0, orient="xF", ...)
regionallognormplot (x, cod, a=0, orient="xF", ...)
regionalexpplot (x, cod, a=0, orient="xF", ...)
regionalparetoplot (x, cod, a=0, orient="xF", ...)
regionalgumbelplot (x, cod, a=0, orient="xF", ...)
regionalfrechetplot (x, cod, a=0, orient="xF", ...)
pointsposRPhist (xcont, xhist=NA, infhist=NA, suphist=NA, nbans=NA, seuil=NA,
col12=c(1,1), a=0, orient="xF", ...)
Arguments
x |
vector representing a data-sample |
xcont |
vector of systematic data (see |
xhist |
vector of historical data (see |
infhist |
inferior limit for historical data (see |
suphist |
superior limit for historical data (see |
nbans |
period (in years) over which the threshold has not been exceeded except for the historical data (see |
seuil |
threshold non exceeded in the historical period except for the historical data (see |
df |
degrees of freedom (> 0, maybe non-integer) of the Student t distribution. 'df = Inf' is allowed. |
shape |
shape parameter of the distribution |
a |
plotting position parameter, normally between 0 and 0.5 (the default value here, corresponding to the Hazen plotting position, see details) |
orient |
if |
line |
if TRUE (default) a straight line indicating the normal, lognormal, ..., distribution with parameters estimated from |
cod |
array that defines the data subdivision among sites |
col12 |
vector of 2 elements containing the colors for the systematic and historical data respectively |
... |
graphical parameters as |
Details
A brief introduction on Probability Plots (or Quantile-Quantile plots) is available on https://en.wikipedia.org/wiki/Q-Q_plot. For plotting positions see https://en.wikipedia.org/wiki/Plotting_position.
For the quantiles of the comparison distribution typically the Weibull formula k/(n + 1) is used (default here).
Several different formulas have been used or proposed as symmetrical plotting positions.
Such formulas have the form
(k - a)/(n + 1 - 2a)
for some value of a in the range from 0 to 1/2.
The above expression k/(n+1) is one example of these, for a=0.
The Filliben plotting position has a = 0.3175 and the Cunanne plotting position has a = 0.4 should be nearly quantile-unbiased for a range of distributions.
The Hazen plotting position, widely used by engineers, has a = 0.5.
The Blom's plotting position, a = 3/8, gives nearly unbiased quantiles for the normal distribution, while the Gringeton plotting position, a = 0.44, is optimized for the largest observations from a Gumbel distribution.
For the generalized Pareto, the GEV and related distributions of the Type I (Gumbel) and Weibull, a = 0.35 is suggested.
For large sample size, n, there is little difference between these various expressions.
Value
Representation of the values of x vs their empirical probability function F in a cartesian, uniform, normal, lognormal or Gumbel plot.
plotpos and unifplot are analogous except for the axis notation, unifplot has the same notation as normplot, lognormplot, ...
plotposRP is analogous to plotpos but the frequencies F are expressed as Return Periods T=1/(1-F).
With the default settings, F is defined with the Weibull plotting position F=k/(n+1).
The straight line (if line=TRUE) indicate the uniform, normal, lognormal or Gumbel distribution with parameters estimated from x.
The regional plots draw samples of a region on the same plot.
pointspos, normpoints, ... are the analogous of points, they can be used to add points or lines to plotpos, normplot, ...
normpoints can be used either on normplot or lognormplot.
exppoints can be used either on expplot or paretoplot (since the log-transformed Pareto random variable is exponentially distributed).
gumbelpoints can be used either on gumbelplot or frechetplot (since the log-transformed Frechet random variable is distributed as a Gumbel).
loglogplot plots the logarithm of sample vs the logarithm of the empirical exceedance probability. For the log-log plot, the tail probability is represented by a straight line for power-law distributions (e.g. log-pearson, log-logistic, Frechet, ..., HEAVY TAIL), but not for the other subexponential or exponential distributions (e.g. gumbel, gamma, Pearson type III, ..., MODERATE TAIL); see El Adlouni et al. (2008).
plotposRPhist is based on the method in Stedinger et al. (1993, pp. 18.41-42).
Note
For information on the package and the Author, and for all the references, see nsRFA.
See Also
These functons are analogous to qqnorm; for the distributions, see Normal, Lognormal, LOGNORM, GUMBEL.
Examples
x <- rnorm(30,10,2)
plotpos(x)
normplot(x)
normplot(x,xlab=expression(D[m]),ylab=expression(hat(F)),
main="Normal plot",cex.main=1,font.main=1)
normplot(x,line=FALSE)
x <- rlnorm(30,log(100),log(10))
normplot(x)
lognormplot(x)
x <- rand.gumb(30,1000,100)
normplot(x)
gumbelplot(x)
x <- rnorm(30,10,2)
y <- rnorm(50,10,3)
z <- c(x,y)
codz <- c(rep(1,30),rep(2,50))
regionalplotpos(z,codz)
regionalnormplot(z,codz,xlab="z")
regionallognormplot(z,codz)
regionalgumbelplot(z,codz)
plotpos(x)
pointspos(y,pch=2,col=2)
x <- rnorm(50,10,2)
F <- seq(0.01,0.99,by=0.01)
qq <- qnorm(F,10,2)
plotpos(x)
pointspos(qq,type="l")
normplot(x,line=FALSE)
normpoints(x,type="l",lty=2,col=3)
lognormplot(x)
normpoints(x,type="l",lty=2,col=3)
gumbelplot(x)
gumbelpoints(x,type="l",lty=2,col=3)
# distributions comparison in probabilistic graphs
x <- rnorm(50,10,2)
F <- seq(0.001,0.999,by=0.001)
loglikelhood <- function(param) {-sum(dgamma(x, shape=param[1],
scale=param[2], log=TRUE))}
parameters <- optim(c(1,1),loglikelhood)$par
qq <- qgamma(F,shape=parameters[1],scale=parameters[2])
plotpos(x)
pointspos(qq,type="l")
normplot(x,line=FALSE)
normpoints(qq,type="l")
lognormplot(x,line=FALSE)
normpoints(qq,type="l")