Predictive distribution of the max-stable process at target positions.
Computes the predictive distribution of Y(\cdot) at a set of ungauged positions (s_1^*, ..., s_k^*), given data at gauged positions (s_1, ..., s_n), by using the output of latent.fit or
Two types of prediction are available for the HKEVP, as described in Shaby and Reich (2012). See details.
hkevp.predict(fit, targets, targets.covariates, predict.type = "kriging")
Output from the
A matrix of real values giving the spatial coordinates of the ungauged positions. Each row corresponds to an ungauged position.
A matrix of real values giving the spatial covariates of the ungauged positions. Must match with the covariates used in
Character string specifying the type of prediction. Must be one of "
The spatial prediction of Y_t(s^*) for a target site s^* and a realisation t of the process is described in Shaby and Reich (2012). This method involves a three-step procedure:
Computation of the residual dependence process θ(\cdot) at the target positions.
Computation of the conditional GEV parameters (μ^*,σ^*,ξ^*) at the target sites. See the definition of the HKEVP in Reich and Shaby (2012).
Generation of Y_t(s^*) from an independent GEV distribution with parameters (μ^*,σ^*,ξ^*).
As sketched in Shaby and Reich (2012), two types of prediction are possible: the kriging-type and the climatological-type. These two types differ when the residual dependence process θ is computed (first step of the prediction):
The kriging-type takes the actual value of A in the MCMC algorithm to compute the residual dependence process. The prediction will be the distribution of the maximum recorded at the specified targets.
The climatological-type generates A by sampling from the positive stable distribution with characteristic exponent α, where α is the actual value of the MCMC step. The prediction in climatological-type will be the distribution of what could happen in the conditions of the HKEVP dependence structure.
Posterior distribution for each realisation t of the process and each target position s^* is represented with a sample where each element corresponds to a step of the MCMC procedure.
A three-dimensional array where:
Each row corresponds to a different realisation of the process (a block).
Each column corresponds to a target position.
Each slice corresponds to a MCMC step.
Reich, B. J., & Shaby, B. A. (2012). A hierarchical max-stable spatial model for extreme precipitation. The annals of applied statistics, 6(4), 1430. <DOI:10.1214/12-AOAS591>
Shaby, B. A., & Reich, B. J. (2012). Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland. Environmetrics, 23(8), 638-648. <DOI:10.1002/env.2178>
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# Simulation of HKEVP: sites <- as.matrix(expand.grid(1:3,1:3)) targets <- as.matrix(expand.grid(1.5:2.5,1.5:2.5)) all.pos <- rbind(sites, targets) knots <- sites loc <- all.pos[,1]*10 scale <- 3 shape <- 0 alpha <- .4 tau <- 1 ysim <- hkevp.rand(10, all.pos, knots, loc, scale, shape, alpha, tau) yobs <- ysim[,1:9] # HKEVP fit (omitting first site, used as target): fit <- hkevp.fit(yobs, sites, niter = 1000) # Extrapolation: ypred <- hkevp.predict(fit, targets, predict.type = "kriging") # Plot of the density and the true value for 4 first realizations: # par(mfrow = c(2, 2)) # plot(density(ypred[1,1,]), main = "Target 1 / Year 1") # abline(v = ysim[1,10], col = 2, lwd = 2) # plot(density(ypred[2,1,]), main = "Target 1 / Year 2") # abline(v = ysim[2,10], col = 2, lwd = 2) # plot(density(ypred[1,2,]), main = "Target 2 / Year 1") # abline(v = ysim[1,11], col = 2, lwd = 2) # plot(density(ypred[2,2,]), main = "Target 2 / Year 2") # abline(v = ysim[2,11], col = 2, lwd = 2)
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