Conditional multivariate extreme values modelling
Description
Fit the conditional multivariate extreme value model of Heffernan and Tawn
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  ## S3 method for class 'mex'
ggplot(data = NULL, mapping, ptcol = "blue",
col = "cornflowerblue", fill = "orange", plot. = TRUE,
quantiles = seq(0.1, by = 0.2, len = 5), ..., environment)
mex(data, which, mth, mqu, dqu, margins = "laplace", constrain = TRUE,
v = 10, penalty = "gaussian", maxit = 10000, trace = 0,
verbose = FALSE, priorParameters = NULL)
mexAll(data, mqu, dqu)
## S3 method for class 'mexList'
print(x, ...)
## S3 method for class 'mex'
plot(x, quantiles = seq(0.1, by = 0.2, len = 5), col = "grey",
...)
## S3 method for class 'predict.mex'
plot(x, pch = c(1, 3, 20), col = c(2, 8, 3),
cex = c(1, 1, 1), ask = TRUE, ...)
## S3 method for class 'predict.mex'
ggplot(data = NULL, mapping, xlab, ylab, main,
ptcol = c("grey", "dark blue", "orange"), col = "dark blue",
fill = "orange", shape = 16:18, size = rep(1, 3), plot. = TRUE, ...,
environment)
## S3 method for class 'mex'
predict(object, which, pqu = 0.99, nsim = 1000, trace = 10,
smoothZdistribution = FALSE, ...)
## S3 method for class 'predict.mex'
summary(object, mth, probs = c(0.05, 0.5, 0.95), ...)

Arguments
data 
A numeric matrix or data.frame, the columns of which are to be modelled. 
col 
In 
quantiles 
A vector of quantiles taking values between 0 and 1 specifying the quantiles of the conditional distributions which will be plotted. 
... 
Further arguments to be passed to methods. 
which 
The variable on which to condition. This can be either scalar, indicating the column number of the conditioning variable, or character, giving the column name of the conditioning variable. 
mth 
Marginal thresholds. In In 
mqu 
Marginal quantiles As an alternative to specifying the marginal
GPD fitting thresholds via 
dqu 
Dependence quantile. Used to specify the quantile at which to
threshold the conditioning variable data when estimating the dependence
parameters. For example 
margins 
See documentation for 
constrain 
See documentation for 
v 
See documentation for 
penalty 
How to penalize the likelihood when estimating the marginal
generalized Pareto distributions. Defaults to “gaussian”. See the help
file for 
maxit 
The maximum number of iterations to be used by the optimizer.
defaults to 
trace 
Passed internally to 
verbose 
Whether or not to keep the user informed of progress.
Defaults to 
priorParameters 
Parameters of prior/penalty used for estimation of
the GPD parameters. This is only used if 
x, object 
Object of class 
pch, cex 
Plotting characters: colours and symbol expansion. The
observed and simulated data are plotted using different symbols, controlled
by these arguments and 
ask 
Whether or not to ask before changing the plot. Defaults to

shape, size, mapping, ptcol, fill, plot., environment, xlab, ylab, main 
Further arguments to plot and ggplot methods. 
pqu 
Prediction quantile. Argument to 
nsim 
Argument to 
smoothZdistribution 
In 
probs 
In 
Details
The function mex
works as follows. First, Generalized Pareto
distributions (GPD) are fitted to the upper tails of each of the marginal
distributions of the data: the GPD parameters are estimated for each column
of the data in turn, independently of all other columns. Then, the
conditional multivariate approach of Heffernan and Tawn is used to model the
dependence between variables. The returned object is of class "mex".
This function is a wrapper for calls to migpd
and
mexDependence
, which estimate parameters of the marginal and
dependence components of the Heffernan and Tawn model respectively. See
documentation of these functions for details of modelling issues including
the use of penalties / priors, threshold choice and checking for convergence
of parameter estimates.
The plot
method produces diagnostic plots for the fitted dependence
model described by Heffernan and Tawn, 2004. The plots are best viewed by
using the plotting area split by par(mfcol=c(.,.))
rather than
mfrow
, see examples below. Three diagnostic plots are produced for
each dependent variable:
1) Scatterplots of the residuals Z from the fitted model of Heffernan and
Tawn (2004) are plotted against the quantile of the conditioning variable,
with a lowess curve showing the local mean of these points. 2) The absolute
value of Zmean(Z)
is also plotted, again with the lowess curve
showing the local mean of these points. Any trend in the location or
scatter of these variables with the conditioning variable indicates a
violation of the model assumption that the residuals Z are indepenendent of
the conditioning variable. This can be indicative of the dependence
threshold used being too low. 3) The final plots show the original data (on
the original scale) and the fitted quantiles (specified by quantiles
)
of the conditional distribution of each dependent variable given the
conditioning variable. A model that fits well will have good agreement
between the distribution of the raw data (shown by the scatter plot) and the
fitted quantiles. Note that the raw data are a sample from the joint
distribution, whereas the quantiles are those of the estimated conditional
distribution given the value of the conditioning variable, and while these
two distributions should move into the same part of the sample space as the
conditioning variable becomes more extreme, they are not the same thing!
The predict
method for mex
works as follows. The returned
object has class "predict.mex". Simulated values of the dependent variables
are created, given that the conditioning variable is above its 100pqu
quantile. If predict.mex
is passed an object object
of class
"mex"
then the simulated values are based only on the point estimate
of the dependence model parameters, and the original data. If
predict.mex
is passed an object object
of class
"bootmex"
then the returned value additionally contains simulated
replicate data sets corresponding to the bootstrap model parameter
estimates. In both cases, the simulated values based on the original data
and point estimates appear in component object$data$simulated
. The
simulated data from the bootstrap estimates appear in
object$replicates
.
The plot
method for class "predict.mex"
displays both the
original data and the simulated data generated above the threshold for
prediction; it shows the threshold for prediction (vertical line) and also
the curve joining equal quantiles of the marginal distributions – this is
for reference: variables that are perfectly dependent will lie exactly on
this curve. Original data are shown with one plotting character and
simulated data with another; colours of simulated point distinguish those
points which have the conditioning variable as the largest (on a quantile
scale) or not the largest.
The function mexAll
fits a collection of GPD and conditional
dependence models, the same fitted GPD being used for all of the dependence
model fits. This can be used in turn to generate Monte Carlo samples from
the entire sample space usign the collected dependence models.
Value
A call to mex
returns an list of class mex
containing
the following three items:
margins 
An object of class

dependence 
An object of class

call 
This matches the original function call. 
There are plot
, summary
, coef
and predict
methods for this class.
A call to predict.mex
does the importance sampling for prediction,
and returns a list of class "predict.mex"
for which there are print
and plot methods available. The summary method for this class of object is
intended to be used following a call to the predict method, to estimate
quantiles or probabilities of threshold excesses for the fitted conditional
distributions given the conditioning variable above the threshold for
prediction. See examples below.
There are print
, summary
and plot
methods available for
the class "predict.mex".
Note
The package texmex
is equipped to fit GPD models to the upper
marginal tails only, not the lower tails. This is appropriate for
extrapolating into the tails of any dependent variable when dependence
between this variable and the conditioning variable is positive. In the
case of negative dependence between the conditioning variable and any
dependent variable, estimation of the conditional distribution of the
dependent variable for extreme values of the conditioning variable would
naturally visit the lower tail of the dependent variable. Extrapolation
beyond the range of the observed lower tail is not supported in the current
version of texmex
. In cases where negative dependence is observed and
extrapolation is required into the lower tail of the dependent variable, the
situation is trivially resolved by working with a reflection of the
dependent variable (Y becomes Y and so the upper and lower tails are
swapped). Results can be calculated for the reflected variable then
reflected back to the correct scale. This is satisfactory when only the
pair of variables (the conditioning and single dependent variable) are of
interest, but when genuine multivariate (as opposed to simply bivariate)
structure is of interest, this approach will destroy the dependence
structure between the reflected dependent variable and the remaining
dependent variables.
Author(s)
Harry Southworth, Janet E. Heffernan
References
J. E. Heffernan and J. A. Tawn, A conditional approach for multivariate extreme values, Journal of the Royal Statistical Society B, 66, 497  546, 2004
See Also
migpd
, mexDependence
,
bootmex
, mexMonteCarlo
Examples
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