rmaxlin {SpatialExtremes}

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, \ldots, p} f_j(x) Z_j, \qquad 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, \ldots, x_k \in R^d and an alternative but equivalent formulation is

\bf{Y} = A \odot \bf{Z},

where \mathbf{Y} = \{Y(x_1), \ldots, Y(x_k)\}, \mathbf{Z} = (Z_1, \ldots, Z_p), \odot 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 \times p matrix with non negative entries and whose i-th row is \{f_1(x_i), \ldots, f_p(x_i)\}.

Currently only the discretized Smith model is implemented for which f_j(x) = c(p) \varphi(x - u_j ; \Sigma) where \varphi(\cdot; \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))