R: Simulation from max-linear models

rmaxlin {SpatialExtremes} | R Documentation

## Simulation from max-linear models

### Description

This function generates realisations from a max-linear model.

### Usage

    rmaxlin(n, coord, cov.mod = "gauss", dsgn.mat, grid = FALSE, p = 500,
    ...)

### Arguments

`n` | Integer. The number of observations.
`coord` | A vector or matrix that gives the coordinates of each location. Each row corresponds to one location - if any. May be missing if `dsgn.mat` is specified.
`cov.mod` | A character string that specifies the max-linear model. Currently only the discretized Smith model is implemented, i.e., `cov.mod = "gauss"`.
`dsgn.mat` | The design matrix of the max-linear model — see Section Details. May be missing if `coord` and `cov.mod` are given.
`grid` | Logical. Does `coord` defines a grid?
`p` | An integer corresponding to the number of unit Frechet random variable used in the max-linear model — see Section Details.
`...` | The parameters of the max-stable model — see Section Details.

### Details

A max-linear process {Y(x)} is defined by

Y(x) = max_{j=1, …, p} f_j(x) Z_j, x in R^d,

where p is a positive integer, f_j are non negative functions and Z_j are independent unit Frechet random variables. Most often, the max-linear process will be generated at locations x_1, …, x_k in R^d and an alternative but equivalent formulation is

Y = A * Z,

where Y = {Y(x_1), …, Y(x_k)}, Z = (Z_1, …, Z_p), * is the max-linear operator, see the first equation, and A is the design matrix of the max-linear model. The design matrix A is a k x p matrix with non negative entries and whose i-th row is {f_1(x_i), …, f_i(x_p)}.

Currently only the discretized Smith model is implemented for which f_j(x) = c(p) phi(x - u_j ; Sigma) where phi( . ; Sigma) is the zero mean (multivariate) normal density with covariance matrix Sigma, u_j is a sequence of deterministic points appropriately chosen and c(p) is a constant ensuring unit Frechet margins. Hence if this max-linear model is used, users must specify `var` for one dimensional processes, and `cov11`, `cov12`, `cov22` for two dimensional processes.

### Value

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

### 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

`condrmaxlin`, `rmaxstab`

### Examples

    ## A one dimensional simulation from a design matrix. This design matrix
    ## corresponds to a max-moving average process MMA(alpha)
    n.site <- 250
    x <- seq(-10, 10, length = n.site)

    ## Build the design matrix
    alpha <- 0.8
    dsgn.mat <- matrix(0, n.site, n.site)
    dsgn.mat[1,1] <- 1

    for (i in 2:n.site){
    dsgn.mat[i,1:(i-1)] <- alpha * dsgn.mat[i-1,1:(i-1)]
    dsgn.mat[i,i] <- 1 - alpha
    }

    data <- rmaxlin(3, dsgn.mat = dsgn.mat)
    matplot(x, t(log(data)), pch = 1, type = "l", lty = 1, ylab =
    expression(log(Y(x))))

    ## One realisation from the discretized Smith model (2d sim)
    x <- y <- seq(-10, 10, length = 100)
    data <- rmaxlin(1, cbind(x, y), cov11 = 3, cov12 = 1, cov22 = 4, grid =
    TRUE, p = 2000)
    image(x, y, log(data), col = heat.colors(64))

[Package SpatialExtremes version 2.0-5 Index]