Description Usage Arguments Details Value Note Author(s) References See Also Examples
Fit the conditional multivariate extreme value model of Heffernan and Tawn
1 2 3 4 5 6 7 8 9 10 11  mex(data, which, mth, mqu, dqu, margins = "laplace", constrain = TRUE, v = 10,
penalty = "gaussian", maxit = 10000, trace = 0, verbose = FALSE, priorParameters = NULL)
## S3 method for class 'mex'
plot(x, quantiles = seq(0.1, by = 0.2, len = 5), col = "grey", ...)
## S3 method for class 'mex'
predict(object, which, pqu=0.99, nsim=1000, trace=10, ...)
## S3 method for class 'predict.mex'
summary(object, mth, probs=c(0.05, 0.5, 0.95), ...)
## S3 method for class 'predict.mex'
plot(x, pch=c(1, 3), col=c(2, 8), cex=c(1, 1), ask=TRUE, ...)

data 
A numeric matrix or data.frame, the columns of which are to be modelled. 
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 
In In 
mqu 
As an alternative to specifying the marginal GPD fitting thresholds via mth, you can specify the quantile (a probability) above which to fit generalized Pareto distributions. If this is a vector of length 1, the same quantile will be used for each variable. Otherwise, it should be a vector whose length is equal to the number of columns in data. 
dqu 
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 internall 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 
quantiles 
A vector of quantiles taking values between 0 and 1 specifying the quantiles of the conditional distributions which will be plotted. 
col 
In 
x, object 
Object of class 
pqu 
Argument to 
nsim 
Argument to 
probs 
In 
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

... 
Further arguments to be passed to methods. 
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 "oredict.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.
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'.
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.
Harry Southworth, Janet E. Heffernan
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
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