Plots the MLE of the GPD parameters against threshold
tcplot(data, tlim = NULL, nt = min(100, length(data)), p.or.n = FALSE,
alpha = 0.05, ylim.xi = NULL, ylim.sigmau = NULL,
legend.loc = "bottomright", try.thresh = quantile(data, 0.9, na.rm =
TRUE), ...)tshapeplot(data, tlim = NULL, nt = min(100, length(data)), p.or.n = FALSE,
alpha = 0.05, ylim = NULL, legend.loc = "bottomright",
try.thresh = quantile(data, 0.9, na.rm = TRUE),
main = "Shape Threshold Stability Plot", xlab = "Threshold u",
ylab = "Shape Parameter", ...)
tscaleplot(data, tlim = NULL, nt = min(100, length(data)), p.or.n = FALSE,
alpha = 0.05, ylim = NULL, legend.loc = "bottomright",
try.thresh = quantile(data, 0.9, na.rm = TRUE),
main = "Modified Scale Threshold Stability Plot", xlab = "Threshold u",
ylab = "Modified Scale Parameter", ...)
vector of sample data
vector of (lower, upper) limits of range of threshold
to plot MRL, or NULL to use default values
number of thresholds for which to evaluate MRL
logical, should tail fraction (FALSE) or number of
exceedances (TRUE) be given on upper x-axis
significance level over range (0, 1), or NULL for no CI
y-axis limits for shape parameter or NULL
y-axis limits for scale parameter or NULL
location of legend (see legend) or NULL for no legend
vector of thresholds to consider
further arguments to be passed to the plotting functions
y-axis limits or NULL
title of plot
x-axis label
y-axis label
tshapeplot and
tscaleplot produces the threshold stability plot for the
shape and scale parameter respectively. They also returns a matrix containing columns of
the threshold, number of exceedances, MLE shape/scale
and their standard devation and \(100(1 - \alpha)\%\) Wald confidence interval if requested. Where the
observed information matrix is not obtainable the standard deviation and confidence intervals
are NA. For the tscaleplot the modified scale quantities
are also provided. tcplot produces both plots on one graph and
outputs a merged dataframe of results.
The MLE of the (modified) GPD scale and shape (xi) parameters are plotted against a set of possible thresholds. If the GPD is a suitable model for a threshold \(u\) then for all higher thresholds \(v > u\) it will also be suitable, with the shape and modified scale being constant. Known as the threshold stability plots (Coles, 2001). The modified scale parameter is \(\sigma_u - u\xi\).
In practice there is sample uncertainty in the parameter estimates, which must be taken into account when choosing a threshold.
The usual asymptotic Wald confidence intervals are shown based on the observed information matrix to measure this uncertainty. The sampling density of the Wald normal approximation is shown by a greyscale image, where lighter greys indicate low density.
A pre-chosen threshold (or more than one) can be given in try.thresh.
The GPD is fitted to the excesses using maximum likelihood estimation. The
estimated parameters are shown as a horizontal line which is solid above this
threshold, for which they should be the same if the GPD is a good model (upto sample uncertainty).
The threshold should always be chosen to be as low as possible to reduce sample uncertainty.
Therefore, below the pre-chosen threshold, where the GPD should not be a good model, the line
is dashed and the parameter estimates should now deviate from the dashed line
(otherwise a lower threshold could be used).
If no threshold limits are provided tlim = NULL then the lowest threshold is set
to be just below the median data point and the maximum threshold is set to the 11th
largest datapoint. This is a slightly lower order statistic compared to that used in the MRL plot
mrlplot function to account for the fact the maximum likelihood
estimation is likely to be unreliable with 10 or fewer datapoints.
The range of permitted thresholds is just below the minimum datapoint and the second largest value. If there are less unique values of data within the threshold range than the number of threshold evalations requested, then instead of a sequence of thresholds they will be set to each unique datapoint, i.e. MLE will only be applied where there is data.
The missing (NA and NaN) and non-finite values are ignored.
The lower x-axis is the threshold and an upper axis either gives the number of
exceedances (p.or.n = FALSE) or proportion of excess (p.or.n = TRUE).
Note that unlike the gpd related functions the missing values are ignored, so
do not add to the lower tail fraction. But ignoring the missing values is consistent
with all the other mixture model functions.
Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value threshold estimation and uncertainty quantification. REVSTAT - Statistical Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf
Coles S.G. (2004). An Introduction to the Statistical Modelling of Extreme Values. Springer-Verlag: London.
# NOT RUN {
x = rnorm(1000)
tcplot(x)
tshapeplot(x, tlim = c(0, 2))
tscaleplot(x, tlim = c(0, 2), try.thresh = c(0.5, 1, 1.5))
tcplot(x, tlim = c(0, 2), try.thresh = c(0.5, 1, 1.5))
# }
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