Estimate the dependence parameters in a conditional multivariate extreme values model using the approach of Heffernan and Tawn, 2004.
mexDependence(x, which, dqu, margins="laplace",
constrain=TRUE, v = 10, maxit=1000000, start=c(.01, .01),
marTransform="mixture", referenceMargin = NULL, nOptim = 1,
PlotLikDo=FALSE, PlotLikRange=list(a=c(-1,1),b=c(-3,1)),
PlotLikTitle=NULL)An object of class mex which is a list containing
the following three objects:
An object of class
migpd.
An object of class
mexDependence.
This matches the original function call.
An object of class "migpd" as returned by
migpd.
The name of the variable on which to condition. This
is the name of a column of the data that was passed into
migpd.
See documentation for this argument in
mex.
The form of margins to which the data are transformed for carrying out dependence estimation. Defaults to "laplace", with the alternative option being "gumbel". The choice of margins has an impact on the interpretation of the fitted dependence parameters. Under Gumbel margins, the estimated parameters a and b describe only positive dependence, while c and d describe negative dependence in this case. For Laplace margins, only parameters a and b are estimated as these capture both positive and negative dependence.
Logical value. Defaults to constrain=TRUE
although this will subsequently be changed to FALSE if
margins="gumbel" for which constrained estimation is not
implemented. If margins="laplace" and
constrain=TRUE then the dependence parameter space is
constrained to allow only combinations of parameters which give
the correct stochastic ordering between (positively and
negatively) asymptotically dependent variables and variables
which are asymptotically independent.
Scalar. Tuning parameter used to carry out constrained
estimation of dependence structure under
constrain=TRUE. Takes positive values greater than 1;
values between 2 and 10 are recommended.
The maximum number of iterations to be used by the
optimizer. Defaults to maxit = 1000000.
Optional starting value for dependence estimation.
This can be: a vector of length two, with values corresponding
to dependence parameters a and b respectively, and in which
case start is used as a starting value for numerical
estimation of each of the dependence models to be estimated; a
matrix with two rows corresponding to dependence parameters a
and b respectively and number of columns equal to the number of
dependence models to be estimated (the ordering of the columns
will be as in the original data matrix); or a previously
estimated object of class "mex" whose dependence parameter
estimates are used as a starting point for estimation. Note
that under constrain=TRUE, if supplied, start
must lie within the permitted area of the parameter space.
Optional form of transformation to be used for
probability integral transform of data from original to Gumbel
or Laplace margins. Takes values marTransform="mixture"
(the default) or marTransform="empirical". When
marTransform="mixture", the rank transform is used below
the corresponding GPD fitting threshold used in x, and
the fitted gpd tail model is used above this threshold. When
marTransform="empirical" the rank transform is used for
the entire range of each marginal distribution.
Optional set of reference marginal
distributions to use for marginal transformation if the data's
own marginal distribution is not appropriate (for instance if
only data for which one variable is large is available, the
marginal distributions of the other variables will not be
represented by the available data). This object can be created
from a combination of datasets and fitted GPDs using the
function makeReferenceMarginalDistribution.
Number of times to run optimiser when estimating
dependence model parameters. Defaults to 1. In the case of
nOptim > 1 the first call to the optimiser uses the
value start as a starting point, while subsequent calls
to the optimiser are started at the parameter value to which
the previous call converged.
Logical value: whether or not to plot the profile likelihood surface for dependence model parameters under constrained estimation.
This is used to specify a region of the
parameter space over which to plot the profile log-likelihood
surface. List of length 2; each item being a vector of length
two corresponding to the plotting ranges for dependence
parameters a and b respectively. If this argument is not
missing, then PlotLikDo is set equal to TRUE.
Used only if PlotLikDo=TRUE. Character
string. Optional title added to the profile log-likelihood
surface plot.
Harry Southworth, Janet E. Heffernan
Estimates the extremal dependence structure of the data in x. The
precise nature of the estimation depends on the value of margins. If
margins="laplace" (the default) then dependence parameters a and b
are estimated after transformation of the data to Laplace marginal
distributions. These parameters can describe both positive and negative
dependence. If margins="gumbel" then the parameters a, b, c and d in
the dependence structure described by Heffernan and Tawn (2004) are
estimated in the following two steps: first, a and b are estimated; then, if
a=0 and b is negative, parameters c and d are estimated (this is the case of
negative dependence). Otherwise c and d will be fixed at zero (this is the
case of positive dependence).
If margins="laplace" then the option of constrained parameter
estimation is available by setting argument constrain=TRUE. The
default is to constrain the values of the parameters
(constrain=TRUE). This constrained estimation ensures validity of
the estimated model, and enforces the consistency of the fitted dependence
model with the strength of extremal dependence exhibited by the data. More
details are given in Keef et al. (2013). The effect of this constraint is
to limit the shape of the dependence parameter space so that its boundary is
curved rather than following the original box constraints suggested by
Heffernan and Tawn (2004). The constraint brings with it some performance
issues for the optimiser used to estimate the dependence parameters, in
particular sensitivity to choice of starting value which we describe now.
The dependence parameter estimates returned by this function can be
particularly sensitive to the choice of starting value used for the
optimisation. This is especially true when margins="laplace" and
constrain=TRUE, in which case the maximum of the objective function
can lie on the edge of the (possibly curved) constrained parameter space.
It is therefore up to the user to check that the reported parameter
estimates really do correspond to the maximum of the profile lilkelihood
surface. This is easily carried out by using the visual diagnostics invoked
by setting PlotLikDo=TRUE and adjusting the plotting area by using
the argument PlotLikRange to focus on the region containing the
surface maximum. See an example below which illustrates the use of this
diagnostic.
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.
C. Keef, I. Papastathopoulos and J. A. Tawn. Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model, Journal of Multivariate Analysis, 115, 396 -- 404, 2013
migpd, bootmex,
predict.mex, plot.mex
data(winter)
mygpd <- migpd(winter , mqu=.7, penalty="none")
mexDependence(mygpd , which = "NO", dqu=.7)
# focus on 2-d example with parameter estimates on boundary of constrained parameter space:
NO.NO2 <- migpd(winter[,2:3] , mqu=.7, penalty="none")
# starting value gives estimate far from true max:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.01,0.01),
PlotLikDo=TRUE,PlotLikTitle=c("NO2 | NO"))
# zoom in on plotting region containing maximum:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.01,0.01),
PlotLikDo=TRUE,PlotLikTitle=c("NO2 | NO"),
PlotLikRange = list(a=c(0,0.8),b=c(-0.2,0.6)))
# try different starting value:
mexDependence(NO.NO2, which = "NO",dqu=0.7,start=c(0.1,0.1),
PlotLikDo=TRUE,PlotLikTitle=c("NO2 | NO"),
PlotLikRange = list(a=c(0,0.8),b=c(-0.2,0.6)))
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