R: Conditional simulation of max-linear random fields

condrmaxlin {SpatialExtremes} | R Documentation

## Conditional simulation of max-linear random fields

### Description

This function generates (approximate) conditional simulation of unit Frechet max-linear random fields. It can be used to get approximate conditional simulation for max-stable processes.

### Usage

    condrmaxlin(n, coord, data.coord, data, cov.mod = "gauss", ..., grid =
    FALSE, p = 10000)

### Arguments

`n` | Integer. The number of conditional simulations.
`coord` | A numeric vector or matrix specifying the coordinates where the process has to be generated. If `coord` is a matrix, each row specifies one locations.
`data.coord` | A numeric vector or matrix specifying the coordinates where the process is conditioned.
`data` | A numeric vector giving the conditioning observations.
`cov.mod` | A character string specifying the max-stable model. See section Details.
`...` | The parameters of the max-stable model. See section Details.
`grid` | Logical. Does `coord` defines a grid?
`p` | An integer. The number of unit Frechet random variables used in the max-linear approximation.

### Details

Any unit Frechet max-stable processes Z(x) can be approximated by a unit Frechet max-linear process, i.e.,

Z(x) ~ max_{j=1, …, p} f_j(x) Z_j,

where f_j are non-negative deterministic functions, p is a sufficiently large integer and Z_j are independent unit Frechet random variables. Note that to ensure unit Frechet margins, the following condition has to be satisfied

∑_{j=1, …, p} f_j(x) = 1,

for all x.

Currently only the discretized Smith model is implemented for which f_j(x) = c(p) φ(x - u_j ; Σ) where φ( . ; Σ) is the zero mean (multivariate) normal density with covariance matrix Σ, u_j is a sequence of deterministic points appropriately chosen and c(p) is a constant ensuring unit Frechet margins.

### Value

A matrix containing observations from the required max-stable model. Each column represents one stations. If `grid = TRUE`, the function returns an array of dimension nrow(coord) x nrow(coord) x n.

### Warnings

It may happen that some conditional observations are not honored because the approximation of a max-stable process by a max-linear one isn't accurate enough! Sometimes taking a larger `p` solves the issue.

### Author(s)

Mathieu Ribatet

### References

Wang, Y. and Stoev, S. A. (2011) Conditional Sampling for Max-Stable Random Fields. Advances in Applied Probability.

### See Also

`rmaxstab`, `condrmaxlin`

### Examples

    ## One dimensional conditional simulations
    n.cond.site <- 10
    cond.coord <- runif(n.cond.site, -10, 10)
    data <- rmaxlin(1, cond.coord, var = 3, p = 10000)

    x <- seq(-10, 10, length = 250)
    cond.sim <- condrmaxlin(5, x, cond.coord, data, var = 3)

    matplot(x, t(log(cond.sim)), type = "l", lty = 1, pch = 1)
    points(cond.coord, log(data))

    ## Two dimensional conditional simulation
    cond.coord <- matrix(runif(2 * n.cond.site, -10, 10), ncol = 2)
    data <- rmaxstab(1, cond.coord, "gauss", cov11 = 4, cov12 = 0, cov22 = 4)

    x <- y <- seq(-10, 10, length = 75)
    cond.sim <- condrmaxlin(4, cbind(x, y), cond.coord, data, cov11 = 4,
    cov12 = 0, cov22 = 4, grid = TRUE, p = 2000)
    ## Note p is set to 2000 for CPU reasons but is likely to be too small

    op <- par(mfrow = c(2, 2), mar = rep(1, 4))
    for (i in 1:4){
    image(x, y, log(cond.sim[,,i]), col = heat.colors(64), xaxt = "n", yaxt
    = "n", bty = "n")
    contour(x, y, log(cond.sim[,,i]), add = TRUE)
    text(cond.coord[,1], cond.coord[,2], round(log(data), 2), col = 3)
    }
    par(op)

[Package SpatialExtremes version 2.0-5 Index]