condrmaxstab {SpatialExtremes}

Conditional simulation of max-stable processes

Description

This function performs conditional simulation of various max-stable processes.

Usage

condrmaxstab(k = 1, coord, cond.coord, cond.data, cov.mod = "powexp",
..., do.sim = TRUE, thin = n.cond, burnin = 50, parts)

Arguments

k

An integer. The number of conditional simulations to be generated.

coord

A vector or matrix that gives the coordinates of each location. Each row corresponds to one location - if any.

cond.coord

A vector or matrix that gives the coordinates of each conditional location. Each row corresponds to one location - if any.

cond.data

A vector that gives the conditional values at the corresponding conditioning locations. Each row corresponds to one location - if any.

cov.mod

A character string that gives the max-stable model. This must be one of "brown" for the Brown-Resnick model, or "whitmat", "cauchy", "powexp" and "bessel" for the Schlather model with the given correlation family.

...

The parameters of the max-stable model. See rmaxstab for more details.

do.sim

A logical value. If TRUE (the default), the conditional simulations are performed; otherwise only the simulated random partitions, i.e., the hitting scenarios, are returned.

thin

A positive integer giving by which amount the generated Markov chain should be thinned. This is only useful when the number of conditioning locations is greater than 7.

burnin

A positive integer giving the duration of the burnin period of the Markov chain.

parts

A matrix giving the hitting scenarios. Each row corresponds to one hitting scenarios. If missing then a Gibbs sampler will be used to generate such hitting scenarios.

Details

The algorithm consists in three steps:

1. Draw a random partition \theta from

   \Pr\{\theta = \tau \mid Z(x) = z\}

2. Given the random partition, draw the extremal functions from

   \Pr\{\varphi^+ \in \cdot \mid Z(x) = z, \theta = \tau\}

3. Independently, draw the sub-extremal functions, i.e.,

   \max_{i \ge 1} \varphi_i 1_{\{\varphi_i(x) < z\}}

The distribution in Step 1 is usually intractable and in such cases a random scan Gibbs sampler will be used to sample from this distribution.

Value

This function returns a list whose components are

sim

The conditional simulations. Beware the first values corresponds to the conditioning values.

sub.ext.fct

The values of the sub-extremal functions.

ext.fct

The values of the extremal functions.

timings

The timings in seconds for each step of the algorithm.

Warning

This function can be extremely time consuming when the number of conditioning locations is large.

Author(s)

Mathieu Ribatet

References

Dombry, C. and Eyi-Minko, F. (2012) Regular conditional distributions of max infinitely divisible processes. Submitted.

Dombry, C., Eyi-Minko, F. and Ribatet, M. (2012) Conditional simulation of max-stable processes. To appear in Biometrika.

See Also

rmaxstab, condrgp

Examples

n.sim <- 50
n.cond <- 5

range <- 10
smooth <- 1.5

n.site <- 200
coord <- seq(-5, 5, length = n.site)
cond.coord <- seq(-4, 4, length = n.cond)
all.coord <- c(cond.coord, coord)

all.cond.data <- rmaxstab(1, all.coord, "powexp", nugget = 0, range = range,
                      smooth = smooth)
cond.data <- all.cond.data[1:n.cond]

ans <- condrmaxstab(n.sim, coord, cond.coord, cond.data, range = range,
                    smooth = smooth, cov.mod = "powexp")

idx <- order(all.coord)
matplot(coord, t(log(ans$sim)), type = "l", col = "grey", lty = 1,
        xlab = expression(x), ylab = expression(Z(x)))
lines(all.coord[idx], log(all.cond.data)[idx])
points(cond.coord, log(cond.data), pch = 15, col = 2)