fExtDepSpat | R Documentation |
This function uses the Stephenson-Tawn likelihood to estimate parameters of max-stable models.
fExtDepSpat(model, z, sites, hit, jw, thresh, DoF, range, smooth, alpha, par0, acov1, acov2, parallel, ncores, args1, args2, seed=123, method = "BFGS", sandwich=TRUE, control = list(trace=1, maxit=50, REPORT=1, reltol=0.0001))
model |
A character string indicating the max-stable model, currently extremal-t ( |
z |
A (n x d) matrix containing n observations at d locations. |
sites |
A (d x 2) matrix corresponding to the coordinates of locations where the processes is simulated. Each row corresponds to a location. |
hit |
A (n x d) matrix containing the hitting scenarios for each observations. |
jw |
An integer between 2 and d, indicating the tuples considered in the composite likelihood. If |
thresh |
A positive real indicating the threshold value for pairwise distances. See |
DoF |
A positive real indicating a fixed value of the degree of freedom of the extremal-t and extremal skew-t models. |
range |
A positive real indicating a fixed value of the range parameter for the power exponential correlation function (only correlation function currently available). |
smooth |
A positive real in (0,2]) indicating a fixed value of the smoothness parameter for the power exponential correlation function (only correlation function currently available). |
alpha |
A vector of length 3) indicating fixed values of the skewness parameters α_0, α_1 and α_2 for the extremal skew-t model. If some components are |
par0 |
A vector of initial value of the parameter vector, in order the degree of freedom ν, the range r, the smoothness η and the skewness parameters α_0, α_1. Its length depends on the model and the number of fixed parameters. |
acov1, acov2 |
Vectors of length d representing covariates to model the skewness parameter of the extremal skew-t model. |
parallel |
A logical value; if |
ncores |
An integer indicating the number of cores considered in the parallel socket cluster of type |
args1, args2 |
Lists specifying details about the Monte Carlo simulation schereme to compute multivariate CDFs. See |
seed |
An integer for reproduciblity in the CDF computations. |
method |
A character string indicating the optimisation method to be used. See |
sandwich |
A logical value; if |
control |
A list of control parameter for the optimisation. See |
This routine follows the methodology developped by Beranger et al. (2021). It uses on the Stephenson-Tawn which relies on the knowledge of time occurrences of each block maxima. Rather than considering all partitions of the set {1,…,d}, the likelihood is computed using the observed partition. Let Π = (π_1, …, π_K) denote the observed partition, then the Stephenson-Tawn likelihood is given by
L(θ;z) = \exp {- V(z;θ)} x ∏^{K}_{k=1} - V_{π_k}(z;θ),
where V_π represents the partial derivative(s) of V(z;θ) with respect to π.
When jw=d
the full Stephenson-Tawn likelihood is considered whereas for values lower than d a composite likelihood approach is taken.
The argument thresh
is required when the composite likelihood is used. A tuple of size jw
, is assigned a weight of one if the maximum pairwise distance between corresponding locations is less that thresh
and a weight of zero otherwise.
Arguments args1
and args2
relate to specifications of the Monte Carlo simulation scheme to compute multivariate CDF evaluations. These should take the form of lists including the minimum and maximum number of simulations used (Nmin
and Nmax
), the absolute error (eps
) and whether the error should be controlled on the log-scale (logeps
).
A list comprising of the vector of estimated parameters (est
), the composite likelihood order (jw
), the maximised log-likelihood value (LL
). In addition, if sandwich=TRUE
the the standard errors from the sandwich information matrix are reported via stderr.sand
as well as the TIC for model selection (TIC
). Finally, if the composite likelihood is considered, a matrix with all tuples considered with a weight of 1 are reported in cmat
.
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com;
Beranger, B., Stephenson, A. G. and Sisson, S.A. (2021) High-dimensional inference using the extremal skew-t process Extremes, 24, 653-685.
fExtDepSpat
set.seed(14342) # Simulation of 20 locations Ns <- 20 sites <- matrix(runif(Ns*2)*10-5,nrow=Ns,ncol=2) for(i in 1:2) sites[,i] <- sites[,i] - mean(sites[,i]) # Simulation of 50 replicates from the Extremal-t model Ny <- 50 z <- rExtDepSpat(Ny, sites, model="ET", cov.mod="powexp", DoF=1, range=3, nugget=0, smooth=1.5, control=list(method="exact")) # Fit the extremal-t using the full Stephenson-Tawn likelihood args1 <- list(Nmax=50L, Nmin=5L, eps=0.001, logeps=FALSE) args2 <- list(Nmax=500L, Nmin=50L, eps=0.001, logeps=TRUE) if(interactive()){ fit1 <- fExtDepSpat(model="ET", z=z$vals, sites=sites, hit=z$hits, par0=c(3,1,1), parallel=TRUE, ncores=6, args1=args1, args2=args2, control = list(trace=0)) fit1$est }
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