Gradient score function

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Description

Compute the peaks-over-threhold gradient score function for the Brown–Resnick model.

Usage

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scoreEstimation(obs, loc, vario, weigthFun, dWeigthFun, nCores = 1,
  cl = NULL, ...)

Arguments

obs

List of vectors exceeding an R-threshold, see Fondeville and Davison (2016) for more details.

loc

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

vario

Semi-variogram function taking a vector of coordinates as input.

weigthFun

Function of weigths.

dWeigthFun

Partial derivative function of weigthFun.

nCores

Number of cores used for the computation

cl

Cluster instance as created by makeCluster of the parallel package.

...

Parameters for weigthFun and dWeigthFun.

Details

The function computes the gradient score based on the representation developped by Wadsworth et al. (2013). Margins must have been standardized. The weighting function must differentiable and verify some properties for consitency, see Fondeville and Davison (2016) for more details.

Value

Evaluation of the gradient score function for the set of observations obs and semi-variogram vario.

References

Fondeville, R. de and Davison A. (2016). High-dimensional Peaks-over-threshold Inference for Brown-Resnick Processes. Submitted.

Examples

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#Define variogram function
vario <- function(h){
   1 / 2 * norm(h,type = "2")^1.5
}

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

#Simulate data
obs <- simulPareto(1000, loc, vario)

#Evaluate risk functional
sums <- sapply(obs, sum)

#Define weighting function
weigthFun <- function(x, u){
x * (1 - exp(-(sum(x) / u - 1)))
}

#Define partial derivative of weighting function
dWeigthFun <- function(x, u){
(1 - exp(-(sum(x) / u - 1))) + (x / u) * exp( - (sum(x) / u - 1))
}

#Select exceedances
threshold <- quantile(sums, 0.9)
exceedances <- obs[sums > threshold]

#Evaluate gradient score function
scoreEstimation(exceedances, loc, vario, weigthFun, dWeigthFun, u = threshold)