Exact unconditional sampling from max-stable random vectors
Léo Belzile, HEC Montréal
2025-09-30
Source:vignettes/mev-vignette.Rmd
mev-vignette.Rmd
The mev
package was originally introduced to implement
the exact unconditional sampling algorithms in Dombry, Engelke, and Oesting (2016). The two
algorithms therein allow one to simulate simple max-stable random
vectors. The implementation will work efficiently for moderate
dimensions.
Functions and use
There are two main functions, rmev
and
rmevspec
. rmev
samples from simple max-stable
processes, meaning it will return an
matrix of samples, where each of the column has a sample from a unit
Frechet distribution. In constrast, rmevspec
returns sample
on the unit simplex from the spectral (or angular) measure. One could
use this to test estimation based on spectral densities, or to construct
samples from Pareto processes.
The syntax is
library(mev)
#Sample of size 1000 from a 5-dimensional logistic model
x <- rmev(n=1000, d=5, param=0.5, model="log")
#Marginal parameters are all standard Frechet, meaning GEV(1,1,1)
apply(x, 2, function(col){ismev::gev.fit(col, show=FALSE)$mle})
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.9704836 0.9680742 0.9762923 0.9690277 0.9513841
## [2,] 0.9224421 0.9131969 0.9363205 0.9671969 0.8878784
## [3,] 0.9445221 0.9560178 0.9544973 0.9884361 0.9390865
#Sample from the corresponding spectral density
w <- rmevspec(n=1000, d=5, param=0.5, model="log")
#All rows sum to 1 by construction
head(rowSums(w))
## [1] 1 1 1 1 1 1
## [1] 0.20 0.21 0.19 0.20 0.20
Description of the models implemented
The different models implemented are described in Dombry, Engelke, and Oesting (2016), but some other models can be found and are described here. Throughout, we consider -variate models and let be the collection of all nonempty subsets of .
Logistic
The logistic model (log
) of Gumbel (1960) has distribution function
for
.
The spectral measure density is
Asymmetric logistic distribution
The alog
model was proposed by Tawn (1990). It shares the same parametrization
as the evd
package, merely replacing the algorithm for the
generation of logistic variates. The distribution function of the
-variate
asymmetric logistic distribution is
The parameters must be provided in a list and represent the asymmetry parameter. The sampling algorithm, from Stephenson (2003) gives some insight on the construction mechanism as a max-mixture of logistic distributions. Consider sampling from a logistic distribution of dimension (or Fréchet variates if with parameter (possibly recycled). Each marginal value corresponds to the maximum of the weighted corresponding entry. That is, for all . The max-mixture is valid provided that for As such, empirical estimates of the spectral measure will almost surely place mass on the inside of the simplex rather than on subfaces.
Negative logistic distribution
The neglog
distribution function due to Galambos (1975) is
for
(Dombry, Engelke, and Oesting 2016). The
associated spectral density is
Asymmetric negative logistic distribution
The asymmetric negative logistic (aneglog
) model is
alluded to in Joe (1990) as a
generalization of the Galambos model. It is constructed in the same way
as the asymmetric logistic distribution; see Theorem~1 in Stephenson (2003). Let
for all
and
with
for
;
the distribution function is
In particular, it does not
correspond to the ``negative logistic distribution’’ given in e.g.,
Section 4.2 of Coles and Tawn (1991) or
Section3.5.3 of Kotz and Nadarajah (2000).
The latter is not a valid distribution function in dimension
as the constraints therein on the parameters
are necessary, but not sufficient.
Joe (1990) mentions generalizations of
the distribution as given above but the constraints were not enforced
elsewhere in the literature. The proof that the distribution is valid
follows from Theorem~1 of Stephenson
(2003) as it is a max-mixture. Note that the parametrization of
the asymmetric negative logistic distribution does not match the
bivariate implementation of rbvevd
.
Multilogistic distribution
This multivariate extension of the logistic, termed multilogistic
(bilog
) proposed by M.-O. Boldi
(2009), places mass on the interior of the simplex. Let
be the solution of
where
for
and
follows a
-mixture
of Dirichlet with the
th
component being
,
so that the mixture has density function
for
.
The spectral density of the multilogistic distribution is thus
for
.
Coles and Tawn Dirichlet distribution
The Dirichlet (ct
) model of Coles
and Tawn (1991) has spectral density
for
Scaled extremal Dirichlet
The angular density of the scaled extremal Dirichlet
(sdir
) model with parameters
and
is given, for all
,
by
where
is the
-vector
with entries
for
and
denotes the inner product between two vectors.
Huesler–Reiss
The Huesler–Reiss model (hr
), due to Hüsler and Reiss (1989), is a special case of
the Brown–Resnick process. While Engelke et al.
(2015) state that H"usler–Reiss variates can be sampled following
the same scheme, the spatial analog is conditioned on a particular site
(),
which complicates the comparisons with the other methods.
Let and be entries of a strictly conditionally negative definite matrix , for which . Then, following Nikoloulopoulos, Joe, and Li (2009) (Remark~2.5) and Huser and Davison (2013), we can write the distribution function as where the partial correlation matrix has elements and for all so that the diagonal entries . Engelke et al. (2015) uses the covariance matrix with entries are , so the resulting expression is evaluated at instead. We recover the same expression by standardizing, since this amounts to division by the standard deviations
The package implementation has a bivariate implementation of the H"usler–Reiss distribution with dependence parameter , with or for for the Brown–Resnick model. In this setting, it is particularly easy since the only requirement is non-negativity of the parameter. For inference in dimension , one needs to impose the constraint (cf. Engelke et al. (2015), p.3), where denotes the set of symmetric conditionally negative definite matrices with zero diagonal entries. An avenue to automatically satisfy these requirements is to optimize over a symmetric positive definite matrix parameter , where is an upper triangular matrix whose diagonal element are on the log-scale to ensure uniqueness of the Cholesky factorization; see Pinheiro and Bates (1996). By taking one can perform unconstrained optimization for the non-zero elements of which are in one-to-one correspondence with those of .
It easily follows that generating from a dimensional log-Gaussian distribution with covariance for with mean vector gives the finite dimensional analog of the Brown–Resnick process in the mixture representation of Dombry, Engelke, and Oesting (2016).
The function checks conditional negative definiteness of the matrix. The easiest way to do so negative definiteness of with real entries is to form , where is an matrix with ones on the diagonal, on the entries for and zeros elsewhere. If the matrix , then the eigenvalues of the leading submatrix of will all be negative.
For a set of locations, one can supply the variogram matrix as valid input to the method.
Brown–Resnick process
The Brown–Resnick process (br
) is the functional
extension of the H"usler–Reiss distribution, and is a max-stable process
associated with the log-Gaussian distribution. It is often in the
spatial setting conditioned on a location (typically the origin). Users
can provide a variogram function that takes distance as argument and is
vectorized. If vario
is provided, the model will simulate
from an intrinsically stationary Gaussian process. The user can
alternatively provide a covariance matrix sigma
obtained by
conditioning on a site, in which case simulations are from a stationary
Gaussian process. See Engelke et al.
(2015) or Dombry, Engelke, and Oesting
(2016) for more information.
Extremal Student
The extremal Student (extstud
) model of Nikoloulopoulos, Joe, and Li (2009), eq. 2.8,
with unit Fréchet margins is
where
is the distribution function of the $d-1 $ dimensional
Student-
distribution and the partial correlation matrix
has diagonal entry
for
.
The user must provide a valid correlation matrix (the function checks for diagonal elements), which can be obtained from a variogram.
Dirichlet mixture
The Dirichlet mixture (dirmix
) proposed by M.-O. Boldi and Davison (2007), see Dombry, Engelke, and Oesting (2016) for details
on the mixture. The spectral density of the model is
The argument param
is thus a
matrix of coefficients, while the argument for the
-vector
weights
gives the relative contribution of each Dirichlet
mixture component.
Smith model
The Smith model (smith
) is from the unpublished report
of Smith (1990). It corresponds to a
moving maximum process on a domain
.
The de Haan representation of the process is
where
is a Poisson point process on
with intensity measure
and
is the density of the multivariate Gaussian distribution. Other
could be used in principle, but are not implemented.