censoredLikelihoodXS {mvPot}

Censored log-likelihood function of the extremal Student model

Description

Compute the peaks-over-threshold censored negative log-likelihood function for the extremal Student model.

Usage

censoredLikelihoodXS(
  obs,
  loc,
  corrFun,
  nu,
  u,
  p = 499L,
  vec = NULL,
  nCores = 1L,
  cl = NULL,
  likelihood = "mgp",
  ntot = NULL,
  std = FALSE,
  ...
)

Arguments

obs

List of vectors for which at least one component exceeds a high threshold.

loc

Matrix of coordinates as given by expand.grid().

corrFun

correlation function taking a vector of coordinates as input.

nu

degrees of freedom of the Student process

u

Vector of thresholds under which to censor components.

p

Number of samples used for quasi-Monte Carlo estimation. Must be a prime number.

vec

Generating vector for the quasi-Monte Carlo procedure. For a given p and dimensionality, can be computed using genVecQMC.

nCores

Number of cores used for the computation

cl

Cluster instance as created by makeCluster of the parallel package.

likelihood

vector of string specifying the contribution. Either "mgp" for multivariate generalized Pareto, "poisson" for a Poisson contribution for the observations falling below or "binom" for a binomial contribution.

ntot

integer number of observations below and above the threshold, to be used with Poisson or binomial likelihood

std

logical; if std = TRUE, consider obs/u for scalar u and exceedances over 1 rather than obs > u for potentially vector u. This affects the value of the log-likelihood function. Default to FALSE.

...

Additional arguments passed to Cpp routine.

Details

The function computes the censored log-likelihood function based on the representation developed by Ribatet (2013); see also Thibaud and Opitz (2015). Margins must have been standardized, for instance to unit Frechet.

Value

Negative censored log-likelihood function for the set of observations obs and correlation function corrFun, with attributes exponentMeasure for all of the likelihood type selected, in the order "mgp", "poisson", "binom"..

Author(s)

Leo Belzile

References

Thibaud, E. and T. Opitz (2015). Efficient inference and simulation for elliptical Pareto processes. Biometrika, 102(4), 855-870.

Ribatet, M. (2013). Spatial extremes: max-stable processes at work. JSFS, 154(2), 156-177.

Examples

#Define correlation function
corrFun <- function(h, alpha = 1, lambda = 1){
   exp(-norm(h, type = "2")^alpha/lambda)
}

#Define locations
loc <- expand.grid(1:4, 1:4)

#Compute generating vector
p <- 499L
latticeRule <- genVecQMC(p, (nrow(loc) - 1))
primeP <- latticeRule$primeP
vec <- latticeRule$genVec

#Simulate data
Sigma <- exp(-as.matrix(dist(loc))^0.8)
obs <- rExtremalStudentParetoProcess(n = 1000, nu = 5, Sigma = Sigma)
obs <- split(obs, row(obs))

#Evaluate risk functional
maxima <- sapply(obs, max)
thresh <- quantile(maxima, 0.9)

#Select exceedances
exceedances <- obs[maxima > thresh]

#Compute log-likelihood function
eval <- censoredLikelihoodXS(exceedances, loc, corrFun, nu = 5, u = thresh, primeP, vec)