Four plots (selectable by which) are currently provided:
a P-P plot, a Q-Q plot, a density plot and a return level plot.
# S3 method for uvevd
plot(x, which = 1:4, main, ask = nb.fig <
length(which) && dev.interactive(), ci = TRUE, cilwd = 1,
a = 0, adjust = 1, jitter = FALSE, nplty = 2, …)An object that inherits from class "uvevd".
If a subset of the plots is required, specify a
subset of the numbers 1:4.
Title of each plot. If given, must be a character
vector with the same length as which.
Logical; if TRUE, the user is asked before
each plot.
Logical; if TRUE (the default), plot simulated
95% confidence intervals for the P-P, Q-Q and return level
plots.
Line width for confidence interval lines.
Passed through to ppoints for empirical estimation.
Larger values give less probability for extreme events.
Arguments to the density plot.
The density of the fitted model is plotted with a rug plot and
(optionally) a non-parameteric estimate. The argument
adjust controls the smoothing bandwidth for the
non-parametric estimate (see density).
jitter is logical; if TRUE, the (possibly
transformed) data are jittered to produce the rug plot.
This need only be used if the data contains repeated
values. nplty is the line type of the non-parametric
estimate. To omit the non-parametric estimate set nplty
to zero.
Other parameters to be passed through to plotting functions.
The following discussion assumes that the fitted model is stationary. For non-stationary generalized extreme value models the data are transformed to stationarity. The plot then corresponds to the distribution obtained when all covariates are zero.
The P-P plot consists of the points
$$\{(G_n(z_i), G(z_i)), i = 1,\ldots,m\}$$
where \(G_n\) is the empirical distribution function
(defined using ppoints), G is the model based
estimate of the distribution (generalized extreme value
or generalized Pareto), and \(z_1,\ldots,z_m\) are the data
used in the fitted model, sorted into ascending order.
The Q-Q plot consists of the points
$$\{(G^{-1}(p_i), z_i), i = 1,\ldots,m\}$$
where \(G^{-1}\) is the model based estimate of the quantile
function (generalized extreme value or generalized Pareto),
\(p_1,\ldots,p_m\) are plotting points defined by
ppoints, and \(z_1,\ldots,z_m\) are the data
used in the fitted model, sorted into ascending order.
The return level plot for generalized extreme value models is defined as follows.
Let \(G\) be the generalized extreme value distribution function, with location, scale and shape parameters \(a\), \(b\) and \(s\) respectively. Let \(z_t\) be defined by \(G(z_t) = 1 - 1/t\). In common terminology, \(z_t\) is the return level associated with the return period \(t\).
Let \(y_t = -1/\log(1 - 1/t)\). It follows that $$z_t = a + b(y_t^s - 1)/s.$$ When \(s = 0\), \(z_t\) is defined by continuity, so that $$z_t = a + b\log(y_t).$$ The curve within the return level plot is \(z_t\) plotted against \(y_t\) on a logarithmic scale, using maximum likelihood estimates of \((a,b,s)\). If the estimate of \(s\) is zero, the curve will be linear. For large values of \(t\), \(y_t\) is approximately equal to the return period \(t\). It is usual practice to label the x-axis as the return period.
The points on the plot are
$$\{(-1/\log(p_i), z_i), i = 1,\ldots,m\}$$
where \(p_1,\ldots,p_m\) are plotting points defined by
ppoints, and \(z_1,\ldots,z_m\) are the data
used in the fitted model, sorted into ascending order.
For a good fit the points should lie ``close'' to the curve.
The return level plot for peaks over threshold models is defined as follows.
Let \(G\) be the generalized Pareto distribution function,
with location, scale and shape parameters \(u\), \(b\)
and \(s\) respectively, where \(u\) is the model threshold.
Let \(z_m\) denote the \(m\) period return level
(see fpot and the notation therein).
It follows that
$$z_m = u + b((pmN)^s - 1)/s.$$
When \(s = 0\), \(z_m\) is defined by continuity, so that
$$z_m = u + b\log(pmN).$$
The curve within the return level plot is \(z_m\) plotted
against \(m\) on a logarithmic scale, using maximum likelihood
estimates of \((b,s,p)\). If the estimate of \(s\) is zero,
the curve will be linear.
The points on the plot are
$$\{(1/(pN(1-p_i)), z_i), i = 1,\ldots,m\}$$
where \(p_1,\ldots,p_m\) are plotting points defined by
ppoints, and \(z_1,\ldots,z_m\) are the data
used in the fitted model, sorted into ascending order.
For a good fit the points should lie ``close'' to the curve.
# NOT RUN {
uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2)
M1 <- fgev(uvdata)
# }
# NOT RUN {
par(mfrow = c(2,2))
# }
# NOT RUN {
plot(M1)
# }
# NOT RUN {
uvdata <- rgpd(100, loc = 0, scale = 1.1, shape = 0.2)
M1 <- fpot(uvdata, 1)
# }
# NOT RUN {
par(mfrow = c(2,2))
# }
# NOT RUN {
plot(M1)
# }
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