rmevspec: Random samples from spectral distributions of multivariate extreme value models.

Description

Generate from $Q_{i}$ , the spectral measure of a given multivariate extreme value model based on the L1 norm.

Usage

rmevspec(
  n,
  d,
  param,
  sigma,
  model = c("log", "neglog", "bilog", "negbilog", "hr", "br", "xstud", "smith",
    "schlather", "ct", "sdir", "dirmix", "pairbeta", "pairexp", "wdirbs", "wexpbs",
    "maxlin"),
  weights = NULL,
  vario = NULL,
  coord = NULL,
  grid = FALSE,
  dist = NULL,
  ...
)

Value

an n by d exact sample from the corresponding multivariate extreme value model

Arguments

n: number of observations
d: dimension of sample
param: parameter vector for the logistic, bilogistic, negative bilogistic and extremal Dirichlet (Coles and Tawn) model. Parameter matrix for the Dirichlet mixture. Degree of freedoms for extremal student model. See Details.
sigma: covariance matrix for Brown-Resnick and extremal Student-t distributions. Symmetric matrix of squared coefficients $λ^{2}$ for the Husler-Reiss model, with zero diagonal elements.
model: for multivariate extreme value distributions, users can choose between 1-parameter logistic and negative logistic, asymmetric logistic and negative logistic, bilogistic, Husler-Reiss, extremal Dirichlet model (Coles and Tawn) or the Dirichlet mixture. Spatial models include the Brown-Resnick, Smith, Schlather and extremal Student max-stable processes. Max linear models are also supported
weights: vector of length m for the m mixture components that sum to one. For the "maxlin" model, weights should be a matrix with d columns that represent the weight of the components and whose column sum to one (if provided, this argument overrides asy).
vario: semivariogram function whose first argument must be distance. Used only if provided in conjunction with coord and if sigma is missing
coord: d by k matrix of coordinates, used as input in the variogram vario or as parameter for the Smith model. If grid is TRUE, unique entries should be supplied.
grid: Logical. TRUE if the coordinates are two-dimensional grid points (spatial models).
dist: symmetric matrix of pairwise distances. Default to NULL.
...: additional arguments for the vario function

Author

Leo Belzile

Details

The vector param differs depending on the model

log: one dimensional parameter greater than 1
neglog: one dimensional positive parameter
bilog: d-dimensional vector of parameters in $[0, 1]$
negbilog: d-dimensional vector of negative parameters
ct, dir, negdir: d-dimensional vector of positive (a)symmetry parameters. Alternatively, a $d + 1$ vector consisting of the d Dirichlet parameters and the last entry is an index of regular variation in $(0, 1]$ treated as scale
xstud: one dimensional parameter corresponding to degrees of freedom alpha
dirmix: d by m-dimensional matrix of positive (a)symmetry parameters
pairbeta, pairexp: d(d-1)/2+1 vector of parameters, containing the concentration parameter and the coefficients of the pairwise beta, in lexicographical order e.g., $β_{1, 2}, β_{1, 3}, \dots$
wdirbs, wexpbs: 2d vector of d concentration parameters followed by the d Dirichlet parameters

References

Dombry, Engelke and Oesting (2016). Exact simulation of max-stable processes, Biometrika, 103(2), 303--317.

Boldi (2009). A note on the representation of parametric models for multivariate extremes. Extremes 12, 211--218.

Examples

Run this code

set.seed(1)
rmevspec(n=100, d=3, param=2.5, model='log')
rmevspec(n=100, d=3, param=2.5, model='neglog')
rmevspec(n=100, d=4, param=c(0.2,0.1,0.9,0.5), model='bilog')
rmevspec(n=100, d=2, param=c(0.8,1.2), model='ct') #Dirichlet model
rmevspec(n=100, d=2, param=c(0.8,1.2,0.5), model='sdir') #with additional scale parameter
#Variogram gamma(h) = scale*||h||^alpha
#NEW: Variogram must take distance as argument
vario <- function(x, scale=0.5, alpha=0.8){ scale*x^alpha }
#grid specification
grid.coord <- as.matrix(expand.grid(runif(4), runif(4)))
rmevspec(n=100, vario=vario,coord=grid.coord, model='br')
## Example with Dirichlet mixture
alpha.mat <- cbind(c(2,1,1),c(1,2,1),c(1,1,2))
rmevspec(n=100, param=alpha.mat, weights=rep(1/3,3), model='dirmix')

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