grad_kweights {pGPx}

Gradient of the weights for interpolating simulations

Description

Returns a list with the gradients of the posterior mean and the gradient of the (ordinary) kriging weights for simulations points.

Usage

grad_kweights(object, simu_points, krig_points, T.mat = NULL, F.mat = NULL)

Arguments

object

km object

simu_points

simulations points, locations where the field was simulated.

krig_points

one point where the interpolation is computed.

T.mat

a matrix (n+p)x(n+p) representing the Choleski factorization of the covariance matrix for the initial design and simulation points.

F.mat

a matrix (n+p)x(fdim) representing the evaluation of the model matrix at the initial design and simulation points.

Value

A list containing the gradients of posterior mean and kriging weights for simulation points.

References

Azzimonti D. F., Bect J., Chevalier C. and Ginsbourger D. (2016). Quantifying uncertainties on excursion sets under a Gaussian random field prior. SIAM/ASA Journal on Uncertainty Quantification, 4(1):850–874.

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

Examples

######################################################################
### Compute the weights and gradient on a 2d example
if (!requireNamespace("DiceKriging", quietly = TRUE)) {
stop("DiceKriging needed for this example to work. Please install it.",
     call. = FALSE)
}
if (!requireNamespace("DiceDesign", quietly = TRUE)) {
stop("DiceDesign needed for this example to work. Please install it.",
     call. = FALSE)
}
# Define the function
g=function(x){
  return(-DiceKriging::branin(x))
}
d=2
# Fit OK km model
design<-DiceDesign::maximinESE_LHS(design = DiceDesign::lhsDesign(n=50,
                                                                  dimension = 2,
                                                                  seed=42)$design)$design
colnames(design)<-c("x1","x2")
observations<-apply(X = design,MARGIN = 1,FUN = g)
kmModel<-DiceKriging::km(formula = ~1,design = design,response = observations,
                         covtype = "matern3_2",control=list(trace=FALSE))
# Get simulation points
# Here they are not optimized, you can use optim_dist_measure to find optimized points
set.seed(1)
simu_points <- matrix(runif(100*d),ncol=d)
# obtain nsims posterior realization at simu_points
nsims <- 1
set.seed(2)
some.simu <- DiceKriging::simulate(object=kmModel,nsim=nsims,newdata=simu_points,nugget.sim=1e-6,
                         cond=TRUE,checkNames = FALSE)
nn_data<-expand.grid(seq(0,1,,50),seq(0,1,,50))
nn_data<-data.frame(nn_data)
colnames(nn_data)<-colnames(kmModel@X)
obj<-krig_weight_GPsimu(object = kmModel,simu_points = simu_points,krig_points = as.matrix(nn_data))


## Plot the approximate process realization and the gradient vector field
k_scale<-5e-4
image(matrix(obj$krig.mean.init+crossprod(obj$Lambda.end,some.simu[1,]),ncol=50),
      col=grey.colors(20))
contour(matrix(obj$krig.mean.init+crossprod(obj$Lambda.end,some.simu[1,]),ncol=50),
        nlevels = 20,add=TRUE)

for(c_ii in c(1,seq(10,2500,by = 64))){
   pp<-t(as.matrix(nn_data)[c_ii,])
   obj_deriv <- grad_kweights(object = kmModel,simu_points = simu_points,krig_points = pp)
   S_der<-obj_deriv$krig.mean.init + crossprod(obj_deriv$Lambda.end,some.simu[1,])
   points(x = pp[1],y = pp[2],pch=16)
   arrows(x0=pp[1],y0=pp[2],x1 = pp[1]+k_scale*S_der[1,1],y1=pp[2]+k_scale*S_der[2,1])
}