# Predictive distribution of the max-stable process at target positions.

### Description

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 `hkevp.fit`

.

Two types of prediction are available for the HKEVP, as described in Shaby and Reich (2012). See details.

### Usage

1 | ```
hkevp.predict(fit, targets, targets.covariates, predict.type = "kriging")
``` |

### Arguments

`fit` |
Output from the |

`targets` |
A matrix of real values giving the spatial coordinates of the ungauged positions. Each row corresponds to an ungauged position. |

`targets.covariates` |
A matrix of real values giving the spatial covariates of the ungauged positions. Must match with the covariates used in |

`predict.type` |
Character string specifying the type of prediction. Must be one of " |

### Details

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.

### Value

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.

### Author(s)

Quentin Sebille

### References

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>

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
# 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)
``` |