retlev is a generic function used to show return level plot.
The function invokes particular methods
which depend on the class of the first argument.
So the function makes a return level plot for POT models.
retlev(object, ...)# S3 method for uvpot
retlev(object, npy, main, xlab, ylab, xlimsup,
ci = TRUE, points = TRUE, ...)
# S3 method for mcpot
retlev(object, opy, exi, main, xlab, ylab, xlimsup,
...)
A graphical window. In addition, it returns invisibly the return level function.
A fitted object. When using the POT package, an object
of class 'uvpot' or 'mcpot'. Most often, the
return of fitgpd or fitmcgpd functions.
The mean Number of events Per Year (or more generally per block).if missing, setting it to 1.
The title of the graphic. If missing, the title is set to
"Return Level Plot".
The labels for the x and y axis. If missing, they are
set to "Return Period (Years)" and "Return Level" respectively.
Numeric. The right limit for the x-axis. If missing, a suited value is computed.
Logical. Should the 95% pointwise confidence interval be plotted?
Logical. Should observations be plotted?
Other arguments to be passed to the plot
function.
The number of Observations Per Year (or more generally per block). If missing, it is set it to 365 i.e. daily values with a warning.
Numeric. The extremal index. If missing, an estimate is
given using the fitexi function.
For class "mcpot", though this is computationally expensive, we recommend to give the
extremal index estimate using the dexi function. Indeed,
there is a severe bias when using the Ferro and Segers (2003)
estimator - as it is estimated using observation and not the Markov
chain model.
Mathieu Ribatet
For class "uvpot", the return level plot consists of plotting the theoretical quantiles
in function of the return period (with a logarithmic scale for the
x-axis). For the definition of the return period see the
prob2rp function. Thus, the return level plot consists
of plotting the points defined by:
$$(T(p), F^{-1}(p))$$ where \(T(p)\) is the return period related to the non exceedance probability \(p\), \(F^{-1}\) is the fitted quantile function.
If points = TRUE, the probabilities \(p_j\) related to
each observation are computed using the following plotting position
estimator proposed by Hosking (1995):
$$p_j = \frac{j - 0.35}{n}$$ where \(n\) is the total number of observations.
If the theoretical model is correct, the points should be ``close'' to the ``return level'' curve.
For class "mcpot", let \(X_1, \ldots,X_n\) be the first \(n\)
observations from a stationary sequence with marginal distribution
function \(F\). Thus, we can use the following (asymptotic)
approximation:
$$\Pr\left[\max\left\{X_1,\ldots,X_n\right\} \leq x \right] = \left[ F(x) \right]^{n \theta}$$
where \(\theta\) is the extremal index.
Thus, to obtain the T-year return level, we equate this equation to \(1 - 1/T\) and solve for \(x\).
Hosking, J. R. M. and Wallis, J. R. (1995). A comparison of unbiased and plotting-position estimators of L moments. Water Resources Research. 31(8): 2019--2025.
Ferro, C. and Segers, J. (2003). Inference for clusters of extreme values. Journal of the Royal Statistical Society B. 65: 545--556.
prob2rp, fitexi.
#for uvpot class
x <- rgpd(75, 1, 2, 0.1)
pwmu <- fitgpd(x, 1, "pwmu")
rl.fun <- retlev(pwmu)
rl.fun(100)
#for mcpot class
data(ardieres)
Mcalog <- fitmcgpd(ardieres[,"obs"], 5, "alog")
retlev(Mcalog, opy = 990)
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