rExtDepSpat {ExtremalDep}

Random generation of max-stable processes

Description

This function generates realisations from a max-stable process.

Usage

rExtDepSpat(n, coord, model="SCH", cov.mod = "whitmat", grid = FALSE, 
            control = list(), cholsky = TRUE, ...)

Arguments

n

An integer indictaing the number of observations.

coord

A vector or matrix corresponding to the coordinates of locations where the processes is simulated. Each row corresponds to a location.

model

A character string indicating the max-stable model. See details.

cov.mod

A character string indicating the correlation function function. See details.

grid

A logical value; TRUE for coordinates on a grid.

control

A named list with arguments nlines giving the number of lines of the TBM simulation, method a character string specifying the name of the simulation method and uBound the uniform upper bound. Note that method must take value 'exact', 'tbm' or 'circ'. See details.

cholsky

A logical value; if TRUE a Cholesky decomposition is performed. Used for the extremal-t and extremal skew-t models when control=list(method='exact').

...

The parameters of the max-stable model. See details.

Details

This function extends the rmaxstab function from the SpatialExtremes package in two ways:

• 1. The extremal skew-t model is included.

• 2. The function returns the hitting scenarios, i.e. the index of which 'storm' (or process) led to the maximum value for each location and observation.

The max-stable models available in this procedure and the specifics are:

• Smith model: when model='SMI', does not require cov.mod. If coord is univariate then var needs to be specified and for higher dimensions covariance parameters should be provided such as cov11, cov12, cov22, etc.

• Schlather model: when model='SCH', requires cov.mod='whitmat', 'cauchy', 'powexp' or 'bessel' depending on the correlation family. Parameters 'nugget', 'range' and 'smooth' should be specified.

• Extremal-t model: when model='ET', requires cov.mod='whitmat', 'cauchy', 'powexp' or 'bessel' depending on the correlation family. Parameters 'nugget', 'range', 'smooth' and 'DoF' should be specified.

• Extremal skew-t model: when model='EST', requires cov.mod='whitmat', 'cauchy', 'powexp' or 'bessel' depending on the correlation family. Parameters 'nugget', 'range', 'smooth', 'DoF', 'alpha' (a vector of length 3) and 'acov1' and 'acov2' (both vector of length the number of locations) should be specified. The skewness vector is defined as \alpha = \alpha_0 + \alpha_1 \textrm{acov1} + \alpha_2 \textrm{acov2} .

• Geometric gaussian model: when model='GG', requires cov.mod='whitmat', 'cauchy', 'powexp' or 'bessel' depending on the correlation family. Parameters 'sig2', 'nugget', 'range' and 'smooth' should be specified.

• Brown-Resnick model: when model='BR', does not require cov.mod. Parameters 'range' and 'smooth' should be specified.

For the argument control, details of the list components are as follows:

• methodis NULL by default, meaning that the function tries to find the most appropriate simulation technique. Current simulation techniques are a direct approach, i.e. Cholesky decomposition of the covariance matrix, the turning bands and the circular embedding methods. Note that for the extremal skew-t model it can only take value 'exact' or 'direct';

• nlines if NULL then it is set to 1000;

• uBound if NULL then it is set to reasonable values - for example 3.5 for the Schlather model.

Value

A list made of

• vals: A (n \times d) matrix containing n observations at d locations, from the specified max-stable model.

• hits: A (n \times d) matrix containing the hitting scenarios for each observations. On each row, elements with the same integer value indicate that the maxima at these two locations is coming from the same 'storm' or process.

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail] https://www.borisberanger.com;

References

Beranger, B., Stephenson, A. G. and Sisson, S.A. (2021) High-dimensional inference using the extremal skew-t process Extremes, 24, 653-685.

See Also

fExtDepSpat

Examples

# Generate some locations
set.seed(1)
lat <- lon <- seq(from=-5, to=5, length=20)
sites <- as.matrix(expand.grid(lat,lon))

# Example using the extremal-t
set.seed(2)
z <- rExtDepSpat(1, sites, model="ET", cov.mod="powexp", DoF=1, 
                 nugget=0, range=3, smooth=1.5, 
                 control=list(method="exact"))
fields::image.plot(lat, lon, matrix(z$vals,ncol=20) )

# Example using the extremal skew-t
set.seed(3)
z2 <- rExtDepSpat(1, sites, model="EST", cov.mod="powexp", DoF=5, 
                  nugget=0, range=3, smooth=1.5, alpha=c(0,5,5), 
                  acov1=sites[,1], acov2=sites[,2], 
                  control=list(method="exact"))
fields::image.plot(lat, lon, matrix(z2$vals,ncol=20) )